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Chapter 5 Geometric correction
Chapter 5 G e o m e t r i c Correction Multiple geometric objects that axe supposedly interrelated by a constraint may not satisfy it if each object is separately observed in the presence of noise. This chapter presents a statistically optimal way to correct the positions of geometric objects so that they satisfy a required constraint. The fundamental principle is the minimization of the Mahalanobis distance defined in terms of the covariance matrices of the objects. First, a general theory is formulated in abstract terms. Then, it is applied to typical geometric problems in two and three dimensions-optimally imposing coincidence and incidence on image points, image lines, conics, space points, space lines, and space planes. For each problem, explicit expressions for the correction and the a posteriori covariance matrices are derived. Optimal correction for orthogonality constraints is also studied.
5.1 5.1.1
General Theory Basic formulation
Consider N geometric objects in two or t h r e e dimensions, the c~th object being r e p r e s e n t e d by an n ~ - v e c t o r u~, c~ - 1, ..., N . Let n a be t h e dimension of vector u~. T h e N objects are a s s u m e d to be statistically i n d e p e n d e n t , b u t the c o m p o n e n t s of each u ~ m a y be correlated. We a s s u m e t h a t each u a is c o n s t r a i n e d to be in an n~~ -dimensional manifold b/~ C 7Zn~ which we call t h e data space of u~. Let ft~ be the t r u e value we should observe in the absence of noise, a n d write u ~ = ~2~ + A u ~ . T h e error A u a is, to a first a p p r o x i m a t i o n , constrained to be in the tangent space Tu~ (b/a) to the manifold Ha at ~ . Let V[u~] be the covariance m a t r i x of the error A u ~ . W e a s s u m e t h a t no constraint exists on A u a other t h a n A u a E T~t~ (Ha) a n d hence the range of V[ua] coincides with T~to (L/a). It follows t h a t
where P~tu" is the n ~ - d i m e n s i o n a l projection m a t r i x onto Tu~ (b/a). Suppose L s m o o t h functions F ( k ) ( . , . . . , 9)" 7~nl • the t r u e values Ul, .... , u N are k n o w n to satisfy
F(k)(a,,..., ~ ) --0, 131
~ - N _~ 7~ exist and
k = 1, ..., L.
(5.2)
132
C h a p t e r 5.
S(1) '
Geometric Correction
S
S
(~)
(b)
Fig. 5.1. (a) Nonsingular constraint imposed by three equations that are not independent. (b) Singular constraint imposed by two independent equations. We call eq. (5.2) simply the constraint, whereas we refer to the constraint u~ E L/a as the inherent constraint on u a . We now consider the problem of optimally correcting u l , .... , u g so t h a t these L equations are satisfied. Since each u~ is constrained to be in the d a t a space Ha C 7~ n~ , the direct sum ~[~N__1 u~ is constrained to be in its d a t a space
U_(DN ~=1/4~
C ~=~n~.
We say t h a t the constraint imposed by eq. (5.2) is
nonsingular if each of the L equations defines a manifold S (k) of codimension i i n / 4 and the L manifolds S (k), k = 1, ..., L, meet each other transversally in b/ (Fig. 5.1a; see Section 3.2.1); otherwise, the constraint is said to be singular I (Fig. 5.1b). In this chapter, we assume t h a t the constraint (5.2) is nonsingular, although the following theory can also be applied to singular constraints if appropriately modified 2. T h e L equations (5.2) m a y not necessarily be i n d e p e n d e n t (Fig. 5.1a). We call the n u m b e r r of i n d e p e n d e n t equations the rank of the constraint. It L S(k) of codifollows t h a t the constraint (5.2) defines a manifold S - Nk=l mension r in H; we call S the (geometric) modal of th~ constraint (5.2). F r o m the above definition, we see t h a t the rank r equals the dimension of the linear subspace
N
N
-
~N
a=l
(5.a)
ot=l
where V u / ~ ( k ) is the abbreviation of ~7u,~F(k)('Y_tl,...,ftN). Note t h a t the dimension of this subspace m a y not be equal to the dimension of the linear subspace N
v-
us { G P,"oV, oF
c~=l
N
~ 9 ...j
p u s V, o F(L) Us
c
N- ~n~
(5.4)
ct=l
1For example, if the L equations (5.2) are expressed as one equation, say, in the form ~N_ 1F(k)(fil, ..., fiN) 2 = 0, the constraint is singular. 2-'~Vewill see an example of a singular constraint in the motion analysis in Chapter 11.
5.1.
General Theory
133
for u~ # ~t~, where V u F (k) is the abbreviation of VuF(k)(Ul,..., UN). We say that the constraint (5.2) is degenerate 3 if the dimension of )2 is larger that t h e dimension of the subspace 12. Substituting u~ - ~ + Au~ i n t o F(k)(Ul,...,UL) and taking a linear approximation, we can replace eq. (5.2) to a first approximation by _
N
F(a) - E (Vu~/~(a)' A u a ) ,
k - 1, ..., L,
(5.5)
or--1
where F (k) is the abbreviation of F(k)(ul,..., UL). This linearized constraint is assumed to be satisfiable, i.e., there exists at least one set of solutions A u ~ E T~t. (b/a), c~ = 1, ..., N, that satisfies eq. (5.5). If A u ~ is a solution of eq. (5.5), the correction takes the f o r m / t ~ = u~ Au~ to a first approximation. However, infinitely many solutions may exist for Au~, c~ = 1, ..., N. From among them, we choose the one which minimizes the square sum of the Mahalanobis distance ]]Au~[l~,[u~] (see eq. (3.66)), i.e., N
J-
E ( A u ~ , IT[u~l-Au~)--4 min
(5.6)
c~--1
under the inherent constraint A u ~ E T~t. (t4~), c~ = 1, ..., N. N Geometrically, we are projecting the direct sum ( ~ = 1 u~ onto the "closest point" in the model $ determined by eq. (5.2), where the "closeness" is measured in the Mahalanobis distance with respect to the total covariance N matrix (~)~=1 17[u~] (Fig. 5.2). This criterion can be justified as maximum likelihood estimation for Gaussian noise. Namely, if the errors AUl, ..., AUN have the probability density
~=1
(5.7) maximizing the likelihood is equivalent to minimizing the function J given in (5.6) (see eqs. (3.46), ( 3 . 8 1 ) a n d (3.82)).
5.1.2
Optimal solution
1,..., n ~ - n ~ , ' be an orthonormal basis of T~t. (t4~) • Let ~"-( V j(a) } , j inherent constraint A u ~ E T u . (L/a) can be written as 9v j , Au (:'(~)
~
) -
0,
j -
1,
...,
3A more rigorous argument will be given in Chapter 14.
n ~ - n' ~ .
The
(5.8)
134
Chapter 5.
Geometric Correction
N ~ t t
a
a=l
Fig. 5.2. Projecting ~[~=1 N us onto the closest point in the model ,.q measured in the Mahalanobis distance. Introducing Lagrange multipliers Ai and pl ~), differentiating L
N
N
J-2EAkE(Vu k=l
/~ ( k ) , A u ~ ) - 2 E
a=l
a=l
n~-n,~
! 9
E
(~) ,-(~)
t~j (vj
,Au~)
(5.9)
j=l
with respect to each Au~, and setting the result zero, we obtain L
n ~ - - n IoL
-
v[~ol-~uo - Z ~ v ~ o p(~) + ~ k=l
=(~)
.~(o)~
(5.10)
j=l
Multiplying this by V[u~] on both sides and noting that =(~) u~ ~ T~to(U~) • we obtain L
)~kV[u~lVu~T'(k)'
PuU~ Au~ - E
(5.11)
k=l
where eqs. (5.1) have been used. Since Au~ E T~t(b/~), the solution is given by L
~
- ~ ~ [ ~ o l V ~ o ~ (~).
(5.12)
k=l
Substitution of this into eq. (5.5) yields
(v~oF (k), ~?[u~lVuoP (~)) ~ - F(k) /=1
(5.13)
c~=1
Since eq. (5.5) is assumed to be satisfiable, this equation is solvable (see Section 2.3.2); the solution is given in the following form: L
~ - Z w(~')r('). /=1
(5.14)
5.1.
General Theory
135
Here, l~ (kl) is the (kl) element of the (LL)-matrix I ~ - (I~ (kl)) defined by 12V - V - , where V - (~(kl)) is the (LL)-matrix defined by
(~?(~')) - ( L~=1(vuo F(k)' r
P('))) "
(5.15)
In the following, we use the following abbreviation to denote the (LL)-matrix
W-
(w(k~)):
(~v(kl))--(~(~u' ) ~-'a(k)'~/r[ua]vu' = l ~(l))
(5.~6)
It can be shown 4 that the rank of the matrix V (hence of I?V) equals the rank r of the constraint (5.2). It follows that the optimal correction is given in the following form (see eqs. (2.140)and (3.85)): L
(5.17)
k,l--1
This equation has the following geometric interpretation. If the noise is Gaussian, the equiprobability surface for ua has the form
(5.~8)
(u a - u a, V[u ~ ](ua - ~ a ) ) - constant.
As discussed in Section 4.5, this equation defines a nonsingular quadric in the tangent space T~t~ (L/a). Let Sa be the restriction of the model S to L/a obtained by fixing u~ - ~tZ for/3 ~ a. We now show that the optimal correction A u a given by eq. (5.17) is in the conjugate direction of the tangent space T~t~ ($a) to the model Sa at ~a (see Section 4.5.2). In T u , (L/a), the tangent hyperplane to the quadric defined by eq. (5.18) at ua + A u a has the following surface normal (see eq. (4.105)): L
no o~ e [ ~ o ] - ~
- Z
w<~')F(~)P,~~
k,l=l
p<~).
(5.~0)
Let v be an arbitrary tangent vector to the manifold Sa at ua (Fig. 5.3). Since the orthogonal complement of T~t~ ($a) with respect to T~t~ (L/a) is generated by P~tu~ Vu/~(k), k - 1, ..., L, eq. (5.19)implies L
(',"o) ~ Z w(~')F(~)(', Pu~V~o r(*)) - 0.
k,l-=l
Thus, Au~ is in the conjugate direction of T~t~ ($a). 4The proof will be given in C h a p t e r 14 in a more general framework.
(5.20)
136
Chapter 5.
r
Geometric Correction
.2v
Fig. 5.3. Geometric interpretation of optimal correction.
5.1.3
Practical considerations
Eq. (5.17) is merely a theoretical expression, because the right-hand side involves the covariance matrices V[ua] evaluated at the true values ~ta, a 1, ..., N, which we want to compute. It appears that they can be approximated by the covariance matrices V[ua] evaluated at the observed values u a , a - 1, ..., N. However, if the matrix V - (~(k0) defined by eq. (5.15) is approximated by V - (V (kl)) in the form
(5.21)
(v(k'))--(~(Vu"F(k)'V[ualVu"F(O)) ' a 1
matrices V and V may have different ranks" the rank of V is larger than that of Y if the constraint (5.2) is degenerate (see eqs. ( 5 . 3 ) a n d (5.4)). Hence, even if V is a good approximation to V, its generalized inverse W - V may be very different from W - V . A practical solution to this difficulty is to compute the rank-constrained generalized inverse (see eq. (2.82)). Namely, if the rank of the constraint (5.2) is r, eq. (5.17) is approximated by L
(5.22)
A u a - V[ua] E W
where
W - (W (kO) is an (LL)
matrix defined by W - (V)~-, which we write
as
a=l
r
The use of V[ua] instead of V[ua] has the following geometric interpretation. The quadric defined by eq. (5.18) is centered at the true value u a in the tangent space Tu~ (bG) at ua; the correction A u given by eq. (5.17) is an element of Tuo (bG) (Fig. 5.4). Using V[ua] instead of V[ua] means replacing eq. (5.18) by (ua - Ua, V[ua](~ta - u a ) ) - constant. (5.24)
5.1.
General Theory
137
a
Fig. 5.4. Theoretical analysis is done in the tangent space Tu~ (b/~) at fi~.
"
Js~
Fig. 5.5. Actual correction is done in the tangent space Tu~ (U~) at u~. If we regard ~ts as a variable, this equation defines a quadric centered at the data value u s in the tangent space Tu~ (Sts) at u s . Hence, the correction A u given by eq. (5.22) is an element of Tu~(Ua) (Fig. 5.5). This means that the data value u~ is corrected within Tu~ (Lts) in such a way that the Mahalanobis distance IIAusllv[u~] is minimized. This observation implies that as long as A u s E Tu~ (b/s), the inherent c o n s t r a i n t / t s E L/s on the corrected v a l u e / t s - u s - A u s is satisfied to a first approximation but may be violated if higher order terms are considered (Fig. 5.5). It follows that if we want to impose the inherent c o n s t r a i n t / t s E U~ exactly, we need a higher order correction, which we denote by C[. ] (see eqs. (4.113)): = C[uo -
(5.25)
This higher order correction can be made rather arbitrarily, since the correction is optimal in the first order. Because the correction given by eq. (5.22) is based on the linear approximation (5.5), the values {/t~} corrected by eq. (5.25) may not exactly satisfy the constraint (5.2) (Fig. 5.5). In order to impose it exactly, the computation is iterated by replacing the original values { u s } by the corrected values {/ts }. This process is essentially the Newton iterations, so the convergence is quadratic; usually two or three iterations are sufficient. In these iterations, the covariance matrix V[us] must also be updated, be-
138
Chapter 5.
Geometric Correction
U(~
Ua
Fig. 5.6. The error of optimal correction. cause the range of V[ua] at u~ is Tu~ (Ha) and is generally different from the range at /ta, which should be T/t ~ (Ha) (Fig. 5.5). If the covariance matrix V[ua] is given as a function of u~, it only needs to be re-evaluated at the updated value/~a. In many practical problems, however, the covariance matrix V[ua] is given only at the initial value ua. In such a case, a practical compromise is to "project" the covariance matrix V[ua] onto the tangent space T/t ~ (Ha) a t / t a in the form V[ua] - P i ~
V[ua]PU"its'
(5.26)
where P/t~ is the projection matrix onto the the tangent space T/t = (Ha) at
Ua.
5.1.4
A posteriori covariance matrices
Even if the constraint (5.2) is exactly imposed on the corrected values {/ta }, they are random variables because they are computed from the data {ua }. Let ua - ua + Aria, a - 1, ..., N, be the observed values, where Aria is the actual error in ua. After the correction A u a given by eq. (5.17) is subtracted, the data value ua is modified into
L Ua -- ('Ua + /k~ta)- ~r[Ua] E ~Tv
= ~ +
(
~-
k,l-1
r
~ k,l=l
w(k~)~(v~.~(k),~)v~o~(~) ~=1
)
(5.27)
to a first approximation. Let A/ta = /ta - u a be the error in the corrected v a l u e / t a (Fig. 5.6). The covariance matrix V[/ta,/t~] - E[A/taA/t~-] of the corrected values {/ta } is computed as follows:
~[~,~] - z[~oa~]
5.1.
General Theory
--
E
139
l~r(mn)(~/'[us
g[A~t
-T
6=1 m,n=l N L ~,=1 k,l--1 N L 3,,6=1 k,l,m,n=l
(Vu /~(k)) r E[A ~ A ~ - ] Vu~ F(,n)(s
Vu, F(n) ) T
= ~r['l/,o~](~c~ -- ( L ~v(mn)(~'r['uj3]Vu/~F(n))(V,cto,~-~(m))T~/r['llto~]) m~n----1 L
- ~
r162
k,/=l L
+
}~
r162
(~
k,l,m,n=l N
(Vu /~(k), ~/[u~lVuf.(m))(f/[u~]Vuof.(n))T
3,=1 L
-
~
s
l;V(m')(f/[u~]Vu,F('))(f/[u~]Vu~F(m))-r
re,n--1 L
k,l=l L + E ~v(kl)~v(mn)(~'r[U~176 k,l,m,n=l L
= ~[~1~-
~
"/(~~163
~.
(5.2s)
k,l=l Here, we have invoked the assumption that each us is independent and hence E[Aft~A~Z] - f/[u~]6~/3. We have also used the identity ITVVW
= ww-w-
w (s~e eqs. (2.123)).
Letting a - fl in eq. (5.28), we obtain L
(~[~1- ~ [ ~ ] - ~ -~(~)((~[~]v~oP(~))(r~[~o]v~o~(~ ~ k,/=l
(5.29)
140
Chapter 5.
Geometric Correction
/ T
J
ft--
N
---
a
,/
Fig. 5.7. The a priori and a posteriori standard confidence regions. For a ~ fl, we obtain L =
_
(5.30)
k,/----1
Thus, the corrected values {ira } are correlated even though the original values {u~} are independent. It can be confirmed 5 that the ranges of V[/ta] and Y[/ta, $tZ] coincide with the tangent space T/t" (b/a) to the manifold U~ at ~ta.
Eqs. (5.29) and (5.30) can be given the following geometric interpretation. The a priori covariance matrix ~[~N (x--1 Vinci defines the a priori standard confidence region in the tangent space T~=I ~t. (/4) to the manifold/4 = ~aN__1 Ua (Fig. 5.7). The a posteriori covariance matrix given by eq. (5.28) defines the a posteriori standard confidence region in the tangent space TeN=I ~t. (S) to the model $ defined by the constraint (5.2). This confidence region is the "projection" of the a priori standard confidence region onto Tey=l u . ($) along the "conjugate direction". It will be shown in Chapter 14 that eq. (5.28) coincides with the Cramer-Rao lower bound on the covariance matrix of the corrected values, meaning that the correction given by eq. (5.17) is indeed theoretically optimal. Eqs. (5.29) and (5.30) are mere theoretical expressions, since they involve values evaluated at ~ , a = 1, ..., N. Furthermore, eqs. (5.29) and (5.30) define the covariance matrices evaluated at ~a, a - 1, ..., N. In actual computation, a consistent approximation is identifying ~ w i t h / t ~ , i.e., we compute L T
k,l=l
5The proof will be given in Chapter 14 in a more general framework.
5.1.
General Theory
141 L
V[/ta,/tfi] = -
l~(kO(f/'[ua]Vu.~'(k))(f/[u~]Vu~ "f'(O)-r"
E
(5.32)
k,l=l
Here, IY[ua] is the matrix defined by eq. (5.26), and Vu/#(k) is the abbreviation of Vu F(k)(/tl,...,/tN). The matrix I ~ = (l~ (kl)) is defined by
(5.aa) a:l
r
The ranges of V[/ta] and V[~ia,/till thus defined coincide with the tangent space T~t ~ (b/a).
5.1.5
Hypothesis testing and noise level estimation
So far, the constraint (5.2) has been assumed given. However, it can be byand the above correction procedure can provide a means of testing this hypothesis. Let us hypothesize that the true values ua, a - 1, ..., N, satisfy eq. (5.2). Let ua = ua + A u a , a = 1, ..., N, be the observed values, and regard each A u a as an independent Gaussian random variable with mean 0 and covariance matrix I?[ua]. Since Vu/~(k) are deterministic values, eq. (5.5) implies that to a first approximation F (k) is a Gaussian random variable of mean 0. Noting that each A u a is independent, we can compute the covariance of F (k) and F (l) in the following form:
pothetical,
N
N
a=l
fl=l
N
=
(~ a,fl=l N
= E(Vu
/~(k) ' lF[ua]Vu /~(t)) _ ~(kt).
(5.34)
a--1
The matrix I~r - (W (kt)) defined by eq. (5.16) has rank r ( - the rank of eq. (5.2)), so the quadratic form L
d -
W(kOf(k)V(O
(5.35)
k,l=l
is a X2 variable with r degrees of freedom (see eq. (3.61)). It is easily confirmed that the right-hand side of eq. (5.35) coincides with the value obtained by substituting eq. (5.17) into eq. (5.6) (see eqs. (2.141) and
142
Chapter 5.
Geometric Correction
(3.87)). In other words, j is the residual of the optimization (5.6). If this value is much larger than can be accounted for by the statistical behavior of the noise, the hypothesis (5.2) should be rejected. It follows that the hypothesis (5.2) can be tested by the standard X2 test" the hypothesis is rejected with significance level a% if J > x ,o, where X,.,a 2 is the a% significance value of X2 with r degrees of freedom (see Section 3.3.4). Intuitively, the hypothesis that eq. (5.2) holds is rejected if the Mahalanobis distance over which the data {ua} must be displaced for imposing eq. (5.2)is too large (see Fig. 5.2). Eq. (5.35) is merely a theoretical expression, since it involves 1~r = (l~ (kl)). A simple approximation is using W - (W (kl)) defined by eq. (5.23), but now that the optimal estimate/t~ has been obtained, we can alternatively use ITr = (1~ (kt)) defined by eq. (5.33). However, the use of W rather than W makes only a second order difference. In many application problems, the geometric characteristics of noise (e.g., the degree of homogeneity/inhomogeneity and isotropy//anisotropy) can be relatively easily predicted but the absolute magnitude of noise is very difficult to estimate a priori. In such a case, we can write the covariance matrix Vinci in the form = = ..., N, (5.a7) where V0[u,] has a known form while e is unknown. Let us call V0[u~] the normalized covariance matrix of u~, and e the noise level. It is easily seen from eqs. (5.22) and (5.23) that the optimal correction is not affected by multiplication of Vinci by an arbitrary positive constant. Hence, V[u,] can be replaced by the normalized covariance matrix V0[u~]. In other words, we need not know the absolute noise level .for the optimal correction. Once the optimal solution is computed, the noise level e can be estimated a posteriori as follows. If the normalized covariance matrix V0[u~] is used for V[u,] in eq. (5.35), the resulting residual J0 equals e2J. Since 2 is a X2 variable with r degrees of freedom, its expectation and variance are r and 2r, respectively (see eqs. (3.59)). Hence, an unbiased estimator ~2 of e2 is obtained in the form
J0.
(5.as)
r
Its expectation and variance are respectively given by
(5.39) r
In geometric terms, we are estimating the noise level from the Mahalanobis distance with respect to V0[u,] over which the data ( u , } must be displaced for imposing eq. (5.2) (see Fig. 5.2). It follows that the hypothesis test (5.a6)
5.1.
General Theory
143
can be interpreted as comparing the a priori value e with the a posteriori estimate ~ computed on the assumption that the hypothesis is true. In fact, eq. (5.36) is equivalently rewritten as 6 ~2
2
~r,a
-7 e > --'r
5.1.6
(5.40)
Linear constraint
In many problems, the constraint is linear in the form N
EAters=b,
(5.41)
s=l
where A s is an L n s - m a t r i x and b is an L-vector. The rank r of this constraint equals the rank of the L ( ~ N s= 1 n s)-matrix
A-
(A1P~tU11,..., ANPuU~),
(5.42)
in terms of which eq. (5.41) can be written as A (~)N s = l ~ ' s - - b. Suppose the observed values u s , c~ = i, ..., N, do not satisfy the constraint (5.41). If we write uo = ~o + A , ~ , eq. (5.41) can be written as N
N
s 88
-b.
(5.43)
s=l
The correction A u s is determined by the optimization (5.6) under the inherent constraint A u s e Tu~ (L/s). The solution given by eq. (5.17) reduces to N
Aus - V [ u s l A ~ W ( E
Azu~ - b),
(5.44)
~=1 where the (LL)-matrix W is given as follows (see eq. (5.23)):
w
=
,545, s--1
r
Since the constraint (5.41) is linear, no approximation has been made to obtain eq. (5.43). Hence, no iterations are necessary. The a posteriori covariance matrices given by eqs. (5.31) and (5.32) reduce to V [ / t ~ ] - i Y [ u ~ ] - V[u~]A~WA~/[u~], (5.46) 6 T h i s is a c o n s e q u e n c e of t h e fact t h a t ~2/e2 is a modified X 2 variable with r degrees of freedom if the h y p o t h e s i s is t r u e (see eq. (3.72)).
144
Chapter 5. V[/~,/~fl]
-
Geometric Correction
V[u,~]A~WAzV[uz].
(5.47)
The residual of the optimization (5.6) can be written in the form N J
-
(E
N
A~/G
-
b, W ( E
c~=l
AZ/~Z
-
b)),
(5.48)
fl--1
which is a X2 variable with r degrees of freedom if the noise is Gaussian.
5.2
Correction of Image Points and Image Lines
5.2.1 A. Image
Optimal correction for coincidence points
Let xl and x2 be two image points, and V[Xl] and V[x2] their respective a priori covariance matrices. Suppose X l and x2 are two different estimates of the same image point. Consider the problem of estimating the true position. Let 51 and 52 be the true values of xl and x2, respectively. The constraint to be imposed is 51 52, (5.49) -
-
which has rank 2 because both sides are orthogonal to k = (0, 0, 1) T. If the two image points are statistically independent, the optimal estimate ~ = Xl - AXl = x2 -- Ax2 is obtained by finding AXl and Ax2 such that 7 J -- ( A X l , J~:[Xl]- A X l ) --[- ( A x 2 , ~ ' r [ x 2 ] - A x 2 ) ~ m i n
(5.50)
under the linearized constraint
e
/~X2 -- /~Xl -- X2 -- Xl,
(5.51)
The first order solution is given by 8 /XXlAx2
-
V[Xl]W(Xl-X2), -
V[x2]W(x2
-
-
X 1),
(5.52)
where W is a (33)-matrix defined by (5.53) 7We adopt the convention that V[. ] denotes the value of the covariance matrix V[. ] evaluated at the true value of the variable. SWe mean by "first order solution" the approximation expressed in terms of the data and the covariance matrices evaluated at the data values (see eq. (5.22)).
5.2.
Correction of Image Points and Image Lines
145
The a posteriori covariance matrix of the estimate 5~ is
V[xllWV[xl]- V[xl]WV[x2] v[~]- v[x:]wv[x:].
Fix] - V [ X l ] =
(5.54)
The residual of J can be written as J -- ( X 2 -
Xl, W(x2
(5.55)
-- X l ) ) ,
which is a X2 variable with two degrees of freedom 9. This fact provides a coincidence test for image points" the hypothesis that image points X l and x2 coincide with each other is rejected with significance level 6% if J ~" ~2,a.
(5.56)
E x a m p l e 5.1 If each coordinate is perturbed independently by Gaussian noise of mean 0 and variance e 2, the covariance matrices of AXl and Ax2 are V[xl] - V[x2] - e2Pk (see Example 4.1). The optimal estimate 5~ is 1
(5.57)
-- Xl -- A X l -- X2 -- A X 2 -- ~ ( X l "~" X2).
The a posteriori covariance matrix of ~ is 62
v[~]-
-~Pk.
(5.58)
The residual (5.55) can be written as 1
J - ~-j2 IIx2 - xl[I 2
(5.59)
Hence, an unbiased estimator of the variance e2 is obtained in the form ~2 _ ~11~ 1 - ~, I12 9
(5.60)
The value ~ thus estimated equals the half-distance between the two image points. If the value e is given a priori, the coincidence test takes the form ~2
X 2,a
--~ > - - ~ .
(5.61)
9We assume Gaussian noise and do first order analysis whenever we refer to X2 distributions and X2 tests.
146
Chapter 5.
Geometric Correction
B. Image lines
The same analysis can be done for two image 0. Let Y [ n l ] and V[n2] be their respective a signs of n l and n2 are chosen in such a way and (n2, x ) , = 0 be the true image lines. The
lines ( n l , x) - 0 and (n2, x) priori covariance matrices. The that n l ~ n2. Let (n l, x) - 0 constraint to be imposed is (5.62)
~1 -- n2,
which has rank 2 because both sides are unit vectors 1~ If the two image lines are statistically independent, the first order solution of the optimization g - ( A n i , ~r[l"~l]-- A1"~1) ~- (A1"~2, ~-7[n2]-A?'$2) --+ min
(5.63)
under the linearized constraint An2 - An1 - n2 - n l, A n I e {~t,1}L~,
n~O,2 e {~2}L~,
(5.64)
is given by A1"~1- V[nl]W(lo,1 - n2), A~~2 -- V[?'~2]W(n2 - 1"~1),
(5.65)
where W is a (33)-matrix defined by 11 W - (V[ni]-~- V[~o,21):.
(5.66)
A realistic form of the correction is i~, - N [ n i - A n 1 ] -
- An2].
i[n2
(5.67)
The a posteriori covariance matrix of the estimate/~ is V[~] -- ? [ r t i ] -
?[19,11WV[rg1 ] - V[l"~llWg[TI,2]
(5.68)
=
Here, i = 1, 2,
V[ni] - P f t V [ n i ] P n ,
(5.69)
where Pr is the projection matrix along/~. The matrix I ~ is obtained by replacing V[ni] by V[ni], i - 1, 2, in eq. (5.66). The residual of J can be written as J
-
-
-
(5.70)
a~ constraint is degenerate. 11The rank-constrained generalized inverse (.)2 is used because the ranges of V[nl] and V[n2] are different from the ranges of V[nl] and V[n2]. Consequently, although V[nl] + V[n2] is a singular matrix of rank 2, the matrix V[nl] + V[n2] is generally nonsingular.
5.2.
Correction of Image Points and Image Lines
which is a
X2
147
variable with two degrees of freedom.
This fact provides a
coincidence test for image lines: the hypothesis that image lines (n l, x) - 0 and (n2, x) - 0 coincide with each other is rejected with significance level a ~ if
j > ,~2,a.
5.2.2
(5.71)
Optimal correction for incidence
A. S i m u l t a n e o u s correction Suppose image point x and image line (n, x) - 0 are, respectively, estimates of an image point p and an image line 1 that should be incident to each other in the absence of noise. Consider the problem of optimally correcting them so as to make them incident. In other words, we want to find A x and A n such that 9 - x - A x and ~ - n - A n satisfy
(~, ~) - 0.
(5.72)
The rank of this constraint is 1. Let V[x] and V[n] be the a priori covariance matrices of x and n, respectively. If the image point and the image line are statistically independent, the problem can be written as the optimization J - (Ax, V [ x ] - A x ) + (An, V [ n ] - A n ) -+ min
(5.73)
under the linearized constraint
(g. ~ )
+ (~. ~ n ) - (n. ~).
(5.74)
k • The first order solution is given by A~ -
(n, ~ ) v [ ~ ] ~
v[~]n) + (~. v[.~]~) '
(~. A,~ -
(n, ~ ) v [ n ] ~
(.. v[~]~) + (~. v [ ~ ] ~ )
(5.75)
A realistic form of the correction is - ~-
A~,
a-
N[n-
An].
(5.76)
The a posteriori covariance matrices of the corrected values & and ~ are V[~] - V[x] -
(V[~']'~)(V[~I'~)T (~, v[~]~) + (~, ~ [ ~ ] ~ ) '
v[,~] - ?In]
(r (~, v[~],~) + (~, ~ [ n ] ~ ) '
-
148
Chapter 5.
( r l ~(~r 1 T ~,v~x~,,_~n~,,
v[~,h]--
(~. v[~]~) + (~. ~[.]~)
Geometric Correction
= V[fi, &]T,
(5.77)
where V [ n ] - P n V[n]Pn.
(5.78)
The residual of J can be written as J -
(5.79)
(n'x)2
(g, V[~lg) + (~, ~[.]~)' which is a X2 variable with one degree of freedom. This fact provides an incidence test for an image point and an image line: the hypothesis that image point x and image line (n, x) - 0 are incident to each other is rejected with significance level a ~ if J > X2,a 9 (5.80)
B. Image point correction If the image line (n, x) - 0 is fixed, the linearized constraint is
(-,~)
• A ~ e {k)L.
- (-,~),
(5.81)
The first-order correction of the image point x is obtained by letting V[n] O in eqs. (5.75)"
~ = (~,~)v[~]~
(~, v [ ~ ] ~ ) "
(5.82)
The a posteriori covariance matrix of the corrected value ~ is
v[~]-
v[~]-
(v[~]')(v[~]~)~
(~. v[~].)
(5.83)
Since V[hc]n = 0, the rank of V[~] is 1; its null space is {k, n } L , which is orthogonal to the orientation m - N[k x n] of the space line (n, x) = 0. The residual
j_
(n, ~)~
(n, V[x]n)
(5.84)
is a X2 variable with one degree of freedom. Hence, and the incidence test given by eq. (5.80) can be applied. E x a m p l e 5.2 If each coordinate is perturbed independently by Gaussian noise of mean 0 and variance e2, the covariance matrix of A x is V[x] e2Pk . The optimal correction (5.82) reduces to
~
- - (~' ~ ) P k ~ 1 - (k, ~ ) ~
(5.s5)
5.2.
Correction of Image Points and Image Lines
149
X
^ x
m
Fig. 5.8. Optimal incidence correction for an image point. Consequently, x is displaced onto the image line ( n , x ) - 0 perpendicularly (Fig. 5.8). The a posteriori covariance matrix of the corrected value 5~ can be written as V[~] = e2....~T. (5.86) where m = X[k x n I is the orientation of the image line (n, x) = 0. The residual (5.84) can be written as
af -
~
(n'~):
-'T E 1 -(k,
n) 2"
(5.87)
Hence, an unbiased estimator of the variance e2 is obtained in the form ~2 __--
( n , X) 2
- (k,.)~"
(5.8s)
The value @thus estimated equals the distance between the image point and the image line (see eq. (4.11)). If the value e is given a priori, the incidence test takes the form ~2 E---~-
> ~1~,o.
(5.s9)
C. Image line correction
If the image point x = 0 is fixed, the linearized constraint is ( A n , x) - (n, x),
A n 6 {fi}LI .
(5.90)
The first-order correction of the image line (n, x) = 0 is obtained by letting V[x] = O in eqs. (5.75):
A,~ - (n, ~ ) v [ n ] ~ (~, v[n]~) "
(5.91)
The a posteriori covariance matrix of the corrected value ~ is
V[n]- ~tT[n]- (~/"[fl']X)(Y[f't]x)T
(5.92)
150
Chapter 5.
Geometric Correction
The rank of V[~] is 1; its null space is {~, X}L. The residual J -- (x, l}'[n]x)
(5.93)
is a X2 variable with one degree of freedom. Hence, the incidence test given by eq. (5.80) can be applied.
5.3
5.3.1
Correction of Space Points and Space Lines
Optimal correction for coincidence
A. Space points Two space points ?'1 and ?'2 that are supposed to coincide can be optimally corrected in the same way as in the case of image points. Let V[?'l] and V[?'2] be their respective a priori covariance matrices. Let r l and r2 be the true positions of ?'1 and ?'2, respectively. The constraint to be imposed is
r, - r2,
(5.94)
which has rank 3. If the two space points are statistically independent, the problem is finding A?'I and A?'2 such that J - (A?'I, V[?'1]-1A?'1) -~-(A?'2, V[?'2]-1A?'2) --+ min
(5.95)
under the linearized constraint
A?'2 -- A?'I -- ?'2 -- ?'1.
(5.96)
The first order solution is given by
A?'I-
V[?'1]W(?'1-?'2),
A?'2 -- V[?'2]W(?'2 - ?'1),
(5.97)
where W is a (33)-matrix defined by
W-
(V[?'l]-~- V[?'2]) -1 9
(5.98)
The a posteriori covariance matrix of the estimate ~ is
V [ ' r ' ] - V [ ? ' l ] - V[?'I]WV[?'I] -- V[?'I]WV[?'2] =
(5.99)
The residual of J can be written as
J-
(?'2-?'1,W(?'2-?'1)),
(5.100)
5.3.
Correction of Space Points and Space Lines
151
which is a X2 variable with three degrees of freedom. This fact provides a coincidence test for space points: the hypothesis that space points r l and ?'2 coincide with each other is rejected with significance level a% if
J >
(5.101)
E x a m p l e 5.3 If each coordinate is perturbed independently by Gaussian noise of mean 0 and variance e 2, the covariance matrices of A r l and At2 are V[rl] = V[r2] = e2I. The optimal estimate ~ is 1
~" -- ~'1 -- A / ' I -- /'2 -- A/'2 -- ~(7"1 ~t- ~'2).
(5.102)
The a posteriori covariance matrix of ~ is s
V[e]-
~-I.
(5.103)
The residual (5.100) can be written as
1
12
J -- ~E2 11~2 -- ~'1[ 9
(5.104)
Hence, an unbiased estimator of the variance e 2 is obtained in the form ~2_ 1 ~llr2 - rill 2.
(5.105)
The value ~ thus estimated equals 1/vf3 times the half-distance between the two space points. If the value e is given a priori, the coincidence test takes the form ~:2
X 2 ,a
J > -i-"
(5.106)
B. Space lines
The same analysis can be done n2. Let V[p 1 | n l] and V[p 2 | signs of {Pl, n l } and {P2, n2} /' X P l - - ~'1 and r x P2 - n2 imposed is
for two space lines r x Pl - n l and r x P2 = n2] be their a priori covariance matrices. The are chosen so that Pl ~ P2 and ~t 1 ~,~ ~t 2. Let be the true space lines. The constraint to be
P l -- P2,
~1 -- fi2,
(5.107)
which has rank 4 because the representations {Pl, fil } and {102, fi2 } have four degrees of freedom 12 (see Section 4.2.2). If the two space lines are statistically independent, the problem is finding Apl , An1, A p 2, and An2 such that J -
( / k p l 9 ATtl, ~r[p 1 9 n l l - ( / k p
I 9 ATt,1))
_
+ ( A p 2 | An2, V[p 2 | n : ] - ( A p 2 | An2)) ~ min 12This constraint is degenerate.
(5.108)
152
Chapter 5.
Geometric Correction
under the linearized constraint Ap 2 | An2 - Ap 1 | An1 - P2 | n2 - Pl | n l , A p l ~ A n l e {Pl 0 ~/,I, ~,1 O P l } f , 2_
(5.109)
Ap2 | An2 E {P2 @ fi2,fi2 @ P2}L" The first order solution is
~XV~ 9 ~X,~ - V[p~ ~ ,~]W(p~ 9 ,~ - p, 9 nl),
(5.110)
where W is a (66)-matrix defined by 13 W-(V[p
1
@rtl] "4-V[p2 O n 2 ] )
(5.111)
4 9
A realistic form of the correction is (lb)
_N•
PI--Apl nl --/knl
h
where the operation N•
N• ( a ) b
rt2 -- Art2
'
(5.112)
] is defined by
{ N[a | PN[a] b] ]-
p2--Ap2 )]
)I-N•
N[PN[b]a | b]
if"a">--"bl"
(5.113)
otherwise.
The a posteriori covariance matrix of the estimate lb @ h is W[i~ @ ~] - l$[pl @ rtl] - V[Pl @ n l]l~rV[Pl 9 n l]
(5.114) Here,
V[Pi | nil = Px~,.~ V[pi | n~]P~r
i = 1, 2,
(5.115)
where PX~.~ is the six-dimensional projection matrix onto N ' p.1_. n
(see
eq. (4.44:)). The matrix I ~ is obtained by replacing V[p i @ hi] by l~[p i @ hi], i - 1, 2, in eq. (15.111). The residual of J can be written as
(5.116) which is a X2 variable with four degrees of freedom. This fact provides a coincidence test for space lines" the hypothesis that space lines r x Pl = n l and r x P2 - n2 coincide with each other is rejected with significance level a% if J > X~,a" (5.117) 13The ranges of V[pl • nl] and V[p2 9 n2] are different from the ranges of l/[pl (~ nl] and 17[p2 @ n2]. Consequently, although IY[pl @ nl] + l?[p2 @ n2] is a singular matrix of rank 4, the matrix V[pl | nl] + V[p2 | he] is generally nonsingular.
5.3.
5.3.2
Correction of Space Points and Space Lines
153
Optimal correction for incidence
A. Simultaneous correction As in two dimensions, a space point r and a space line r • p - n can be optimally corrected so as to make them incident. Let V[r] and V[p | n] be their a priori covariance matrices. The problem is finding A t , Ap, and A n such that ~ = r - A t , p = p - Ap, and ~ = n - A n satisfy
e xp--ff,.
(5.118)
The rank of this constraint is 2 because the three component equations are algebraically dependent 14. If the space point and the space line are statistically independent, the problem can be written as the optimization J - (At,
~r[~,]--I A/')
-~- (Ap | An, fT[p | n ] - ( A p | A n ) ) -+ min
(5.119)
under the linearized constraint Ar
x
p+f"
x
Ap--An=r
x
p_n,
Ap | A n E {p | n, fi @ lb}~.
(5.120)
The first order solution is given by
=-(vH
x r,)w(,, x p - n),
Ap = (V[p] x r -
V[p, n l ) W ( r x p -
A n = ( V t n , p] x r -
V[n])W(r x p-
n), (5.121)
n),
where W is a (33)-matrix defined by 15 W-
(p x V[v] x p + v x V[p] x r -
)-
2S[v x V[p,n]l + V[n] 2 "
(5.122)
The symbol S[. ] denotes the symmetrization operator (see eqs. (2.205)). A realistic form of the correction is f--r-At,
h
n--An
]'
(5.123)
where the operator N• ]is defined by eq. (5.113). The a posteriori covariance matrices of the corrected values ?,/~, and h are
U p ] - V[,']- (V[r] x/,)W(p x V[,']), 14This constraint is degenerate. 15The rank-constrained generalized inverse ( . ) 2 is used because ( . ) is generally nonsingular if evaluated at the data values; it should be a singular matrix of rank 2 if evaluated at the true values.
154
Chapter 5.
Geometric Correction
v[v] - ?[v]- (?Iv] • e - ?[v,,~])W(e • ?Iv]- ?In, v]), V[Ib, ~t] - V[p, n] - (V[p] • ? - lY[p, n])I~(/" • V[p, n] - I7[n]) - V[/t, lb] 7-,
v[,~]- ? [ n l - (?[n,v] x e - ?[n])Cc(e • ?[v, n ] - ?[n]), v[~,/,]- (v[r] x/,)g,(~ x
?[v]- ~[n, v ] ) -
v[/,, ~]T,
V[O, ~t] = (V[r] x p)I~(O x V[p,n]- I Y [ n ] ) - V[~t, ?]T.
(5.124)
The matrices IY[p], ~Z[p, n], and V[n] are obtained as submatrices of
V[p | n] - P~'~e~V[p @ nlPx~e~.
(s.~2s)
The matrix I&r is obtained by replacing r, p, V[p], V[p, n], and V[n] b y / ' , lb, V[p], V[p, n], and V[n], respectively, in eq. (5.122). The residual of J can be written as j - (r • p - n, W ( r x p - n)), (5.126) which is a ~:2 variable with two degrees of freedom. This fact provides an incidence test for a space point and a space line: the hypothesis that space point r and space line r • p - n are incident to each other is rejected with significance level a ~ if j > ~2,a. (5.127) B. Space point correction
If the space line r • p - n is fixed, the linearized constraint is
Arxp=rxp-n.
(5.128)
The first order correction of the space point r is
~
= -(v[~] • v)w(~
• v-
n),
(s.129)
where W is a (33)-matrix given by
The a posteriori covm'iance matrix V[~] of the corrected value ~ is given in the form shown in eqs. (5.124), where lb and ~ are replaced by p and n, respectively. Matrix V[~] has rank 1; its null space is {n, rH}L in the {m, rH}-representation. The residual J is given in the form of eq. (5.126) and is a X2 variable with two degrees of freedom. Hence, the incidence test given by eq. (5.127) can be applied.
5.3.
Correction of Space Points and Space Lines
155
Ar r
O Fig. 5.9. Optimal incidence correction for a space point. E x a m p l e 5.4 If each coordinate is p e r t u r b e d independently by Gaussian noise of mean 0 and variance e2, the covariance m a t r i x of A r is V[r] - e2I. The optimal correction (5.129) reduces to
p
At-
x
(r x p - n ) ilPl12
-- P r o m -
?"H.
(5.131)
Consequently, r is displaced onto the space line r x p - n perpendicularly (Fig. 5.9). The a posteriori covariance matrix of the corrected value ~ has the form V [ ~ ] - e 2 m m -T-, (5.132) where m - N[p] is the orientation of the space line r x p - n. The residual (5.126) can be written as
J-
1
--~
I1~ x p - n i l ~ 1 ilpl12 = -~llPmr-
r.II 2.
(5.133)
Hence, an unbiased estimator of the variance e e is obtained in the form
~2
I1',' x
p-.,112
211pll2
12
1
= ~llPmr-
rill 9
(5.134)
The value ~ thus estimated equals 1 / v ~ times the distance between the space point and the space line (see eq. 4.49)). If the value e is given a priori, the incidence test takes the form ~2 X2,~ e-Y > ---~--.
(5.135)
C. Space line correction If the space point r is fixed, the linearized constraint is r x Ap-
An - v x p-
n,
156
Chapter 5. AV+n
Geometric Correction (5.136)
~ {p+ n,n +p}~.
The optimal correction of the space line r x p - n is Ap~.
V[p, n l ) W ( r
(V[p] x r -
- ((~ • v [ v , . ] ) ~
x p-
n), (5.137)
-),
- V[.l)W(~ • v-
where W is a (33)-matrix given by w
-
(\ ~ • v [ v ] • ~ - ~ • v [ v , . ] -
(~ • v [ v , . ] ) ~
+ vM
.
(5.138)
The a posteriori covariance matrices of the corrected values/5 and fi are given in the form shown in eqs. (5.124), where ~ is replaced by r. The residual is (5.139)
J - (," • v - - , w ( , - • v - -)).
This is a X2 variable with two degrees of freedom. Hence, the incidence test given by eq. (5.127) can be applied. 5.4
5.~.1
Correction
of Space
Planes
Optimal correction for coincidence
Two space planes (v,, p) = 0 and (v2, p) = 0 that are supposed to coincide can also be optimally corrected. Let Y[vl] and Y[v2] be their respective a priori covariance matrices. The signs of the 4-vectors Vl and v2 are chosen so that vl ~-- v2. Let (Pl, P) - 0 and (P2, P) - 0 be the true space planes. The constraint to be imposed is (5.140)
Vl -- V2,
which has rank 3 because both sides are unit vectors 16. If the two space planes are statistically independent, the problem is finding AVl and Av2 such that J - (AVl, ~r[vl]-lAvl)
-[- ( A v 2 , ~ r [ / 2 2 ] - l n v 2 )
--=+m i n
(5.141)
under the linearized constraint A v 2 -- A V l -- /22 -- V l ,
Av2 e {P2}L~. The first order solution is given by ~1 16This constraint is degenerate.
- V[~l]W(~l - ~),
(5.142)
5.4.
Correction of Space Planes
157
AI/2 -- V[I22]W(v 2 - Vl) ,
(5.143)
where W is a (44)-matrix defined by a7 (5.144) A realistic form of the correction is ~' = N[L,1 - A I / 1 ] - N[L,2 - A!21].
(5.145)
The a posteriori covariance matrix of the estimate s is Villi -
?[Vl]W?[Vl]-
?[Vl]W?[v2]
(5.146)
Here, -
i - 1, 2,
(5.147)
where P~,, is the four-dimensional projection matrix along u i. The matrix IrV is obtained by replacing V[vi] by Y[~'i], i - 1, 2, in eq. (5.144). The residual of J can be written as ^
J - - (V2 - V l , W ( v 2 - V l ) ) ,
(5.148)
which is a X2 variable with three degrees of freedom. This fact provides a coincidence test for space planes: the hypothesis that two space planes (Vl, p) = 0 and (L'2, p) = 0 coincide with each other is rejected with significance level a% if J > ~2,a. (5.149)
5.~.2
Optimal incidence with space points
A. S i m u l t a n e o u s c o r r e c t i o n
A space point p and a space plane (v, p) - 0 can be optimally corrected so as to make them incident. Let V[p] and V[v] be their respective a priori covariance matrices. The problem is finding A p and A v such that ~ = p - A p and P - L , - A v satisfy (P, h) - 0. (5.150) The rank of this constraint is 1. If the space point and the space plane are statistically independent, the problem can be written as the optimization J - (Ap, V [ p ] - A p ) + (Av, V [ v ] - A v ) --+ min
(5.151)
aTThe the ranges of V[ua] and V[u2] are different from the ranges of l)'[pa] and V[v2]. Consequently, although ~'[Vl]+ V[u2] is a singular matrix of rank 3, the matrix VIal]+ V[v2] is generally nonsingular.
158
Chapter 5.
Geometric Correction
under the linearized constraint
(~,,/,) + (~, ~p) = (~,, p), tg /
(5.152)
where ~ = (0, 0, 0, 1) T. The first order solution is given by
(~,, p)V[plv ~t, - (~, v[p]v) + (p, v[~,]p)' (~,, p)v[~,]p (v, V[plv) + (p, v[~,lp)
Alp
(5.153)
A realistic form of the correction is /5 - p -
Ap,
~, = N [ ~ , -
(5.154)
A~,].
The a posteriori covariance matrices of the corrected values/~ and 1) are
v[/,]- v [ d -
(v[p]~)(v[p]~) ~ (~,, v[p]~,) + (/,, ?[~,]/,)'
(?[,,,]/,)(?[,,].a) T v[~] = ~ [ ~ ] v[/,, ~,1 -
(~,, v[p]~,) + (/,, ?[~,]/,)'
(v[p]~,) (?[~,]/,)T (~,, v[p]~,) + (h, ~'[~,]h)
-
= v[~,,/,]T
.
(5.155)
Here,
r
p~v[~]p~,
(5.156)
where P/, is the four-dimensional projection matrix along ~,. The residual of J can be written as
.]
-
(u' P)2
(5.157)
(~', V[d~') + (h, ?[~']/')' which is a X2 variable with one degree of freedom. This fact provides an incidence test for a space point and a space plane: the hypothesis that space point p and space plane (u, p) - 0 are incident to each other is rejected with significance level a~0 if j > X2,a .
(5.158)
5.4.
Correction of Space Plmles
159
O Fig. 5.10. Optimal incidence correction for a space point. B. Space point correction
If the space plane (v, p) = O is fixed, the linearized constraint is
(~, Ap)
(~, p),
-
Ap
e {~}~.
(5.159)
The optimal correction of p is
Ap
(v,p)V[p]u
(5.160)
(~, v[p]~) "
-
The a posteriori covariance matrix of the corrected value/5 is V[~b]-
V[p]- (Y[P]v)(Y[P]V)-r (•, V[plv)
"
(5.161)
Since Viably = O, the rank of V[/~] is 2; its null space is {~, V}L. The residual j
_
(v, p)2
(~, v[p]~)
(5.162)
is a X2 variable with one degree of freedom. Hence, the incidence test given by eq. (5.158) can be applied. E x a m p l e 5.5 If each coordinate is perturbed independently by Gaussia~ noise of mean 0 rand variance e2, the covariance matrix of p is V[p] = e2P~ (=
~ I ~ 0). In the (n, d}-represent~tion, the optim~ correctio~ (5.160) reduces to
zx~ = ((n, ~) - d)n.
(5.~63)
Consequently, r is displaced onto the space plane (n, r) = d perpendicularly (Fig. 5.10). The a posteriori covariance matrix of the corrected value ? is V[/'] =
e2pn.
(5.164)
The residual (5.162) can be written as 1 j - j ( ( n , r) - d)2.
(5.165)
160
Chapter 5.
Geometric Correction
Hence, an unbiased estimator of the variance e2 is obtained in the form -
-
d)
(5.166)
The value ~ thus estimated equals the distance between the space plane and the space point (see eq. (4.68)). If the value e is given a priori, the incidence test takes the form ~2 e-~ > X~,a" (5.167) C. Space plane correction
If the space point p is fixed, the linearized constraint is (Av, p ) -
(v,p),
Av e {P}~.
(5.168)
The optimal correction of space plane (v, p) - 0 is (p, V[v]p) "
(5.169)
The a posteriori covariance matrix the corrected value D is V[~]- ?[~l-
(V[~IP)(?[~]P)~ (p, ~[~]p) .
(5.~70)
The rank of V[9] is 2; its null space is {/~, P}L. The residual
J -
P):
(5. 71)
is a X2 variable with one degree of freedom. Hence, the incidence test given by eq. (5.158) can be applied.
5.4.3
Optimal incidence with space lines
A. Simultaneous correction
A space line (r - rH) • m = 0 and a space plane (n, r) = d can be optimally corrected so as to make them incident. Let V[m (DrH] and V[n @d] be their respective a priori covariance matrices, and ( r - OH) • rh -- 0 and (fi, r) d their true equations. The constraint is (~, ,~) - 0,
(~, ~ , ) - d,
(5.172)
which has rank 2. If the space line and the space plane are statistically independent, the problem is finding A m , Art/, An, and Ad such that _
J - (Am + ArH, Vim + rHI-(Am + Art/)) + ( A n | Ad, V[n 9 d]- ( A n | Ad)) -+ min _
(5.173)
5.49
Correction of Space Planes
161
under the linearized constraint
(An,~,)+(~,Am)=(n,m),
(~.. e.) + (~. ~ . ) - ~d = (....) - d. • A m | ArH e { m | 0, OH | vh }L,
An e {n}f.
(5.174)
The first order solution is given by
vimlo
Am
V[rH,m]Tn
Ad
--
(m, V[n, d])
v[...l.
(.. ...) - d
(rH, V[n, d]) - V[d]
W
'
(n, rH) - d ' (5.175)
where W is a (22)-matrix defined by
(.. vim].) + (m. v[.]m) w =
(.. vim. ~.1.) + (~.. v [ . ] m ) (n. V[~?l.. 7".]n) -}- (771..V[n]~'.) - (~11..V[n. d]) ~ -1 (n,
V[rH]n) +
(vii, V[nIrH ) -- (rH, V[n, d])
)
9 (5.176)
A realistic form of the correction is
vh = N [ m -
Am],
if, : N [ n -
An],
d = d - Ad.
(5.177)
The a posteriori covariance matrices of the corrected values rh,/'H, fi, and d
are
v[.~l
v[.~. #~]
#[m]
#[m. ~1
v[+~,m] v[+~])= (~[~,m] ~[~] ) ~[m..'~/].~ ) v[r.].~
viii
(v[d,,~]
v[~. d] vial]
~[.] #[n. d] ~[d. nl V[dl ^ ) _ ( ~[.].~ ~[.]~ - ~[.,d] (.h. e[.. dl) (~.. e[-. dl/- ~[d] ) W ( ?[-l.h e[n]~H-'~[-.dl (.h. e[.. dl) (+.. r d]) - V[~I )
)=(
162
Chapter 5.
v[.~. ,~] ( v[e..a]
v[.~. d] v[~. , d] ) - - (
?[m]a
?[rH, m] .~
i~r ( ?[n]~q~
Geometric Correction
?[m.~.]a
?[rH]/~ )
?[nigH -- ?[n, d]
(rh, ?In, d]) (/~g,?[n, d]) - ?[4 )
, (5.178)
where V[m], V[m, rH], etc. are computed as submatrices of V[m | r , ]
- Px~e~ V[m |
lY[n | d] - ( P n | 1)V[n | Here, P x ~ e ~
rg]Px~e~,
d](Pr~ |
1).
(5.179) • N,i~**. (se~ m, rH, n, V[m],
is the six-dimensional projection matrix onto
eq. (4.40)). The matrix I ~ is obtained by replacing Vim, rH], etc. by rh, rH,/~, ?[m], ? I r a , rH], etc., respectively, in eq. (5.176). The residual of J can be written as J -- ~TV "(11) (1"~,m ) 2 +212V(12) (n, m)((n,
rH)-d)+IV (22)((n, rH)--d) 2,
(5.180)
which is a X2 variable with two degrees of freedom. This fact provides an incidence test for a space line and a space plane: the hypothesis that space line ( r - r H ) • m -- 0 and space plane are incident to each other is rejected with significance level a% if J > X~,a. (5.181) B. Space line correction
If the space plane (n, r) = d is fixed, the linearized constraint is (n, A m ) = (n, m),
(n, Art/) = (n,
rH) -- d,
A m G A~'H e {?~ O 0, ~H O m } f .
(5.182)
The optimal correction of the space line (r - r H ) X m = 0 is
~.
v[~.. m] ~ .
v[~.l.
(~. ~.) - d
'
where W is a (22)-matrix defined by
w-
(n,V[m]n) ((,vim, ~.1~)
(n, V[m, rH]n) )-1 (~. v[~.].) 9
(5.~84)
The a posteriori covariance matrices of the corrected values ~h and rH are given in the form shown in eqs.^(5.178), where/~ and d are replaced by n and d, respectively. The residual J is given in the form of eq. (5.178) and is a X2 variable with two degrees of freedom. Hence, the incidence test given by eq. (5.181) can be applied.
5.5.
Orthogonality Correction
163
C. Space plane correction
If the space line (r - r H ) • m - 0 is fixed, the linearized constraint is (Art, m ) = (n, m),
(An, r g ) -- Ad = (n, rH) -- d, _k A n e {n}L.
(5.185)
The optimal correction of the space plane (n, r) = d is
Ad
=
(m, V[n, d])
(r, V[n, dl) - V[d]
W
(n, rH) - d
' (5.186)
where W is a (22)-matrix defined by
W -
(m, V[n]m) ( ~ , Via]m)
(m, v[,~]rH) - (m, v [ . , d]) ( ~ , v[n]~H) - (~, v[n, d])
-i
)
(5.187)
The a posteriori covariance matrices of the corrected values fi and d are given in the form shown in eqs. (5.178), where ~'H and ~h are replaced by r H and m , respectively. The residual J is given in the form of eq. (5.178) and is a X2 variable with two degrees of freedom. Hence, the incidence test given by eq. (5.181) can be applied.
5.5
Orthogonality Correction
5.5.1
Correction of two orientations
A. Simultaneous correction
Let m l and m2 be unit 3-vectors that indicate orientations supposedly orthogonal. Let V[ml] and Vim2] be their respective a priori covariance matrices. In the presence of noise, m l and m2 are not exactly orthogonal. Consider the problem of optimally correcting them so as to make them orthogonal (Fig. 5.11). In other words, we want to find Am1 and Am2 such that m l = m l -- A m l a n d m 2 - m 2 - A m 2 satisfy (rhl, m2) - 0.
(5.188)
The rank of this constraint is 1. If the two orientations are statistically independent, the problem can be written as the optimization
J
-
(Am1,
V [ m l ] - A m 1) +
(Am2,
V[m2]-Am2) .-+ r a i n
under the linearized constraint
(~ml, m2)+ (rex, ~m2) - (ml, m2),
(5.189)
164
Chapter 5.
Geometric Correction
Am I
ml
m I
Am2 0
n-t2
Fig. 5.11. Orthogonality correction for two orientations.
A m l E {ml}L~,
Am2 E {?~2}~.
(5.19o)
The first order solution is given by
(ml,m2)V[ml]m2 A m i -- (m2,V[mi]~T/,2) ~- (ml, V[m2]mi)'
(mi,m2)V[m2]ml Am2 = (m2, V [ m i ] m 2 ) + (m~, V[m2]mi)"
(5.191)
A realistic form of the correction is
ml
-
-
N[ml
-
-
Am:],
vh2 - N[m2 - Am2].
(5.192)
The a posteriori covariance matrices of the corrected values vhl and ~h2 are
V[mi] -- ? [ m i ] - (T~2,V[ml]m2) 4- ( m l , V [ m 2 ] m l ) ' V[zh2] - ?[m2] - (zh2, V[ml]~~2) -~- (rex, V[/2]~/~l)' ( ? [ m l ]m2 ) ( ? [ m 2 l m i ) T
V[T~I, m21 -- --(m2, V[mi]m2) + (ziz~, V[m2]ziZl)"
(5.193)
Here, V[mi] - P v h V [ m i ] P v i z , ,
i-
1, 2,
(5.194)
where Pviz~ is the projection matrix along zizi. The residual of J can be written as
,] --
(mi, m2) 2 (~TI'2,~/[ml]m2) -}- (#/,1, IY[m2]ziZl)'
(5.195)
which is a )/2 variable with one degree of freedom. This fact provides an orthogonality test for two orientations" the hypothesis that the two orientations m l and m2 are orthogonal to each other is rejected with significance level a% if
J > x i,o.
(5.196)
5.5.
Orthogonality Correction
165
Am3 in3 ~ m3 Am, Am 2 Fig. 5.12. Orthogonality correction for three orientations. B. Correction of one orientation If m2 is fixed, the linearized constraint is (Am1, m2) - ( m l , m2),
A m l e {Vhl }L I.
(5.197)
The optimal correction of m l is =
(m2, U[ml]m2) .
(5.198)
The a posteriori covariance matrix of the corrected value vhl is
V[?~I]-- Y[ml]- (?[ml]m2)(?[ml]m2)T (m2, ? [ m i ] m 2 )
"
(5.199)
Since V[Vhl]m2 -- O, the rank of V[vh]] is 1; its null space is {Vhl,m2}L. The residual
j --
(ml, m2) 2 (m2, V[ml]m2)
(5.200)
is a X2 variable with one degree of freedom. Hence, the orthogonality test given by eq. (5.196) can be applied.
5.5.2
C o r r e c t i o n of three o r i e n t a t i o n s
A. Simultaneous correction The same procedure can be applied to three orientations. Let m l, m2, and m3 be unit 3-vectors that indicate three orientations supposedly orthogonal (Fig. 5.12). The problem is finding A m i such that m i = m i - A m i satisfies
(~'~,i, ~'j) "-- (~ij,
i, j = 1, 2, 3.
(5.201)
The rank of this constraint is 3. Let V[mi] be the a priori covariance matrix of mi. If the three orientations are statistically independent, the problem can be written as the optimization
3 J-
~-~(Ami, < d [ m i ] - A m i ) ~ min
i=1
(5.202)
166
Chapter 5.
Geometric Correction
under the linearized constraint (m3, Am2) + (m2, Am3) = (m2, m3), (rhl, Am3)+ (rn3, Am1) -- ( m 3 , m l ) , (rh2, Am1)+ (ml,Am2) = ( m l , m 2 ) , mi e {rhi}Ll,
i = 1, 2, 3.
(5.203)
The first order solution is given by
( mx) ( o Vmlm3VEmxlm) 2 (,m2,m3,) Am2
--
Am3
V[m2]m3
0
V[m2]m I
V[m3lm2
W[m3lml
0
W
(m3,ml)
,
(ml,m2) (5.204)
where W is a (33)-matrix defined by W
/(o m3m2) ( viral] v[m~] m3 m2
0 ml
ml 0
) v[~]
(
0 m3 m2
m3 0 ml
m2 ml 0
))
(5.205)
A realistic form of the correction is i = 1,2,3.
~'1,i - N [ m i - Ami],
(5.206)
The a posteriori covariance matrices of the corrected values rhi are
V[ml] V['r~2, 'r~,l] V["~3, ~'Y/'I]
V[~,I, vh2] V[rhl, m3] ) V ['rll,2] V["~3, '/~2]
V['rh2, 'r?t,3] V['/~3]
o --
( V[ml]
f'[m~]ma ?[ml]~t2 )
~'[,ml)
V[m2l'rn3
0 V[m2]~'~t1 0 l}'[m3lrh2 ~Z[m3]'m 1 0
V[ml]~,3
?[m~],~ o ~[m~],~ ~[,m],~ where
f'[m~]
-
f~[m~]-Pm,V[,,~,lP,u,,
?[ml]~'h2 ) V[m2]~,l
,
(5.207)
0
i = 1,2,3.
(5.208)
5.5.
Orthogonality Correction
167
V[mi]
The matrix I ~ is obtained by replacing m~ and by ~h~ and IY[mi], respectively, in eq. (5.207). The residual of J can be written as - (
(m3, m l )
,W
(ml,m )
(m3, m l )
),
(5.209)
(ml,m )
which is a ~2 variable with three degrees of freedom. This fact provides an for three orientations: the hypothesis that the three orientations m i , i - 1, 2, 3, are orthogonal to each other is rejected with significance level a% if
orthogonalitytest
J > X ,a. B.
Correction
of one
(5.e10)
orientation
If m l and m2 are fixed in such a way that ( m l, m2) - 0, the rank of the constraint decreases to 2, and the linearized constraint is (ml,Am3)-
( m l , m3),
(m2, A m 3 ) -
(m2,m3),
(5.211)
e The optimal correction of m3 is
Am3-V[m3](ml'm2)W((me'm3))(m2,m3),
(5.212)
where W is a (22)-matrix defined by
W_ ((ml,V[m3]ml) (ml,V[m3]m2))-1 (m2, V [ m 3 ] m l )
(m2,V[m3]m2)
.
(5.213)
It is evident from the underlying geometry that ~4~3 - -~-ml x m2
(5.214)
is the exact solution if the sign is appropriately chosen. Hence, its covariance is V[,~I] = O. (5.215) The residual can be written as ,]
--
w ( l l ) ( m l , m3)2 +
2I~(12)(ml,m3)(m2,m3)+W(22)(m2,m3)2, (5.2 6)
which is a X2 variable with two degrees of freedom. Here, the matrix I~r = (l~ (kl)) is obtained by replacing m l , m2, and Vim3] by ~hl, ~h2, and Vim3], respectively, in eq. (5.213). The orthogonality test takes the form
j >
(5.217)
168
Chapter 5.
Geometric Correction
C. C o r r e c t i o n o f t w o o r i e n t a t i o n s
If m3 is fixed, the rank of the constraint is 3, and the linearized constraint is
(m3, Aml)--(m3, ml),
(m3, Am2) -- (m3, m2),
( A m l , m 2 ) + ( m l , A m 2 ) -- ( m l , m 2 ) ,
AT/%I E {Thl}t ,
Am2 6 {~2}L~.
(5.218)
The optimal correction of m l and m2 is given by
( iml ) _ ( V[~l,1]m3 ~m:
o
O u[m:].%
V[ml]m2 ) W
((m3,~D,1))
V[m~]m~
(m~,m~)
'
(5.2~9)
where W is a (33)-matrix defined by
W = ( ( m 3 , V[ml]m3)
\
(m3,V[m2]m3)
(ml,V[m2lm3) (m3,V[ml]m2) ) -1 (lYt,3,V[m2]m l ) (m2, V[ml]m2) + (ml, V[m2]ml)
(m2, V[~I]TYt3)
The a posteriori covariance matrices of the corrected values
(5.220)
m l and vh2 are
V[~rrt,1]-- Vii1] - w(ll)(i~,r[ml]m3)(~,r[ml]m3)T,
u[,~] VITal,m2]--
-
?Imp] w(~)(?[.~].~)(?[.~]m~) ~, -
-w(i2)(V[ml]m3)(?[m2]m3) T -
U[m2,~l] T,
(5.221)
where the matrix W is obtained by replacing m i and V[mi] by vhi and ? [ m i ] , respectively, in eq. (5.220). The residual can be written as J - Iiv(ll) (m3, m i ) 2 +
21~(12)(m3, mi)(m3,m2)+ Wi22)(m3,m1) 2,
(5.222) which is a ~(2 variable with two degrees of freedom. Hence, the orthogonality test given by eq. (5.217) can be applied.
5.6
Conic Incidence C o r r e c t i o n
Consider a conic (x, Qx) - 0 (see eq. (4.80)). Let x be an image point not on conic (x, Qx) = O. We consider the problem of optimally correcting x so as to make it incident to the conic (x, Qx) = O. In other words, we want to find A x such that 5~ = x - Ax satisfies (~, Q $ ) = 0.
(5.223)
5.6.
Conic Incidence Correction
169
The rank of this constraint is 1. Let V[x] be the a priori covariance matrix of x. The problem can be written as the optimization J-
(Ax, V [ x ] - A x ) ~ min
(5.224)
under the linearized constraint 1
( ~ , Q~) - 5(~, Q~),
(5.225)
The first order solution is given by ~,~
-
(x Q x ) V [ x ] Q x ' 2(x, QV[x]Qx) "
(5.226)
If we put n - N[Qx], eq. (5.226) can be written as
~_
(~,~)v[~]~ 2(., v[~]~)
(5.227)
This problem can be viewed as imposing the incidence constraint on the image point x and its polar (n, x) = 0 with respect to the conic (x, Q x ) = 0 (see eq. (4.82)). The difference in the factor 2 (see eq. (5.82)) is due to the fact that as the image point x approaches, its polar (n, x) - 0 also approaches its pole x by the same distance. The a posteriori covariance matrix of the corrected position :b is
v i i i - v[~]- (v[~]a)(v[~la)~ (a, via]a) '
(5.22s)
i-~ = N[Q~,].
(5.229)
where Eq. (5.228) has the same form as eq. (5.83). Hence, the rank of V[5~] is 1; its null space is {k, n } i , which is orthogonal to the orientation vh - N[k • nl of the polar (~, x) - 0. The residual of J can be written as
j -
(~, Q~)~ 4(~,,QV[x]Q~,)'
(5.230)
which is a X2 variable with one degree of freedom. This fact provides a conic incidence test: the hypothesis that image point x is on conic (x, Q x ) = 0 is rejected with significance level a ~ if
j > ~2,a.
(5.231)
170
Chapter 5.
Geometric Correction
X
Fig. 5.13. Optimal incidence correction for an image point. E x a m p l e 5.6 If each coordinate is perturbed independently by Gaussian noise of mean 0 and variance e 2, the covariance matrix of x is V[x] = e2Pk . The optimal correction (5.226) reduces to
A x - (x, Q X ) P k Q x . 2[[PkQx[[2
(5.232)
Consequently, x is displaced onto the conic perpendicularly (Fig. 5.13). The a posteriori covariance matrix of the corrected value 5~ is
V[5~]- E2ff~vhT,
(5.233)
where rh - N[k x r is the orientation of the polar (r x) = 0. The residual (5.230) can be written as
j-
1 (~, Q~)~ 4e 2 [[pkQS~[[2.
(5.234)
Hence, an unbiased estimator of the variance e2 is obtained in the form
~ = 1 (~, Q:~):
411PkQSc[I 2"
(5.235)
The value ~ thus estimated equals half the distance between the image point and its polar with respect to the conic. If the value e is given a priori, the conic incidence test takes the form
~:2 E--~ > ~2,a.
(5.236)
5.1 5.1.1
General Theory Basic formulation
Consider N geometric objects in two or t h r e e dimensions, the c~th object being r e p r e s e n t e d by an n ~ - v e c t o r u~, c~ - 1, ..., N . Let n a be t h e dimension of vector u~. T h e N objects are a s s u m e d to be statistically i n d e p e n d e n t , b u t the c o m p o n e n t s of each u ~ m a y be correlated. We a s s u m e t h a t each u a is c o n s t r a i n e d to be in an n~~ -dimensional manifold b/~ C 7Zn~ which we call t h e data space of u~. Let ft~ be the t r u e value we should observe in the absence of noise, a n d write u ~ = ~2~ + A u ~ . T h e error A u a is, to a first a p p r o x i m a t i o n , constrained to be in the tangent space Tu~ (b/a) to the manifold Ha at ~ . Let V[u~] be the covariance m a t r i x of the error A u ~ . W e a s s u m e t h a t no constraint exists on A u a other t h a n A u a E T~t~ (Ha) a n d hence the range of V[ua] coincides with T~to (L/a). It follows t h a t
where P~tu" is the n ~ - d i m e n s i o n a l projection m a t r i x onto Tu~ (b/a). Suppose L s m o o t h functions F ( k ) ( . , . . . , 9)" 7~nl • the t r u e values Ul, .... , u N are k n o w n to satisfy
F(k)(a,,..., ~ ) --0, 131
~ - N _~ 7~ exist and
k = 1, ..., L.
(5.2)
132
C h a p t e r 5.
S(1) '
Geometric Correction
S
S
(~)
(b)
Fig. 5.1. (a) Nonsingular constraint imposed by three equations that are not independent. (b) Singular constraint imposed by two independent equations. We call eq. (5.2) simply the constraint, whereas we refer to the constraint u~ E L/a as the inherent constraint on u a . We now consider the problem of optimally correcting u l , .... , u g so t h a t these L equations are satisfied. Since each u~ is constrained to be in the d a t a space Ha C 7~ n~ , the direct sum ~[~N__1 u~ is constrained to be in its d a t a space
U_(DN ~=1/4~
C ~=~n~.
We say t h a t the constraint imposed by eq. (5.2) is
nonsingular if each of the L equations defines a manifold S (k) of codimension i i n / 4 and the L manifolds S (k), k = 1, ..., L, meet each other transversally in b/ (Fig. 5.1a; see Section 3.2.1); otherwise, the constraint is said to be singular I (Fig. 5.1b). In this chapter, we assume t h a t the constraint (5.2) is nonsingular, although the following theory can also be applied to singular constraints if appropriately modified 2. T h e L equations (5.2) m a y not necessarily be i n d e p e n d e n t (Fig. 5.1a). We call the n u m b e r r of i n d e p e n d e n t equations the rank of the constraint. It L S(k) of codifollows t h a t the constraint (5.2) defines a manifold S - Nk=l mension r in H; we call S the (geometric) modal of th~ constraint (5.2). F r o m the above definition, we see t h a t the rank r equals the dimension of the linear subspace
N
N
-
~N
a=l
(5.a)
ot=l
where V u / ~ ( k ) is the abbreviation of ~7u,~F(k)('Y_tl,...,ftN). Note t h a t the dimension of this subspace m a y not be equal to the dimension of the linear subspace N
v-
us { G P,"oV, oF
c~=l
N
~ 9 ...j
p u s V, o F(L) Us
c
N- ~n~
(5.4)
ct=l
1For example, if the L equations (5.2) are expressed as one equation, say, in the form ~N_ 1F(k)(fil, ..., fiN) 2 = 0, the constraint is singular. 2-'~Vewill see an example of a singular constraint in the motion analysis in Chapter 11.
5.1.
General Theory
133
for u~ # ~t~, where V u F (k) is the abbreviation of VuF(k)(Ul,..., UN). We say that the constraint (5.2) is degenerate 3 if the dimension of )2 is larger that t h e dimension of the subspace 12. Substituting u~ - ~ + Au~ i n t o F(k)(Ul,...,UL) and taking a linear approximation, we can replace eq. (5.2) to a first approximation by _
N
F(a) - E (Vu~/~(a)' A u a ) ,
k - 1, ..., L,
(5.5)
or--1
where F (k) is the abbreviation of F(k)(ul,..., UL). This linearized constraint is assumed to be satisfiable, i.e., there exists at least one set of solutions A u ~ E T~t. (b/a), c~ = 1, ..., N, that satisfies eq. (5.5). If A u ~ is a solution of eq. (5.5), the correction takes the f o r m / t ~ = u~ Au~ to a first approximation. However, infinitely many solutions may exist for Au~, c~ = 1, ..., N. From among them, we choose the one which minimizes the square sum of the Mahalanobis distance ]]Au~[l~,[u~] (see eq. (3.66)), i.e., N
J-
E ( A u ~ , IT[u~l-Au~)--4 min
(5.6)
c~--1
under the inherent constraint A u ~ E T~t. (t4~), c~ = 1, ..., N. N Geometrically, we are projecting the direct sum ( ~ = 1 u~ onto the "closest point" in the model $ determined by eq. (5.2), where the "closeness" is measured in the Mahalanobis distance with respect to the total covariance N matrix (~)~=1 17[u~] (Fig. 5.2). This criterion can be justified as maximum likelihood estimation for Gaussian noise. Namely, if the errors AUl, ..., AUN have the probability density
~=1
(5.7) maximizing the likelihood is equivalent to minimizing the function J given in (5.6) (see eqs. (3.46), ( 3 . 8 1 ) a n d (3.82)).
5.1.2
Optimal solution
1,..., n ~ - n ~ , ' be an orthonormal basis of T~t. (t4~) • Let ~"-( V j(a) } , j inherent constraint A u ~ E T u . (L/a) can be written as 9v j , Au (:'(~)
~
) -
0,
j -
1,
...,
3A more rigorous argument will be given in Chapter 14.
n ~ - n' ~ .
The
(5.8)
134
Chapter 5.
Geometric Correction
N ~ t t
a
a=l
Fig. 5.2. Projecting ~[~=1 N us onto the closest point in the model ,.q measured in the Mahalanobis distance. Introducing Lagrange multipliers Ai and pl ~), differentiating L
N
N
J-2EAkE(Vu k=l
/~ ( k ) , A u ~ ) - 2 E
a=l
a=l
n~-n,~
! 9
E
(~) ,-(~)
t~j (vj
,Au~)
(5.9)
j=l
with respect to each Au~, and setting the result zero, we obtain L
n ~ - - n IoL
-
v[~ol-~uo - Z ~ v ~ o p(~) + ~ k=l
=(~)
.~(o)~
(5.10)
j=l
Multiplying this by V[u~] on both sides and noting that =(~) u~ ~ T~to(U~) • we obtain L
)~kV[u~lVu~T'(k)'
PuU~ Au~ - E
(5.11)
k=l
where eqs. (5.1) have been used. Since Au~ E T~t(b/~), the solution is given by L
~
- ~ ~ [ ~ o l V ~ o ~ (~).
(5.12)
k=l
Substitution of this into eq. (5.5) yields
(v~oF (k), ~?[u~lVuoP (~)) ~ - F(k) /=1
(5.13)
c~=1
Since eq. (5.5) is assumed to be satisfiable, this equation is solvable (see Section 2.3.2); the solution is given in the following form: L
~ - Z w(~')r('). /=1
(5.14)
5.1.
General Theory
135
Here, l~ (kl) is the (kl) element of the (LL)-matrix I ~ - (I~ (kl)) defined by 12V - V - , where V - (~(kl)) is the (LL)-matrix defined by
(~?(~')) - ( L~=1(vuo F(k)' r
P('))) "
(5.15)
In the following, we use the following abbreviation to denote the (LL)-matrix
W-
(w(k~)):
(~v(kl))--(~(~u' ) ~-'a(k)'~/r[ua]vu' = l ~(l))
(5.~6)
It can be shown 4 that the rank of the matrix V (hence of I?V) equals the rank r of the constraint (5.2). It follows that the optimal correction is given in the following form (see eqs. (2.140)and (3.85)): L
(5.17)
k,l--1
This equation has the following geometric interpretation. If the noise is Gaussian, the equiprobability surface for ua has the form
(5.~8)
(u a - u a, V[u ~ ](ua - ~ a ) ) - constant.
As discussed in Section 4.5, this equation defines a nonsingular quadric in the tangent space T~t~ (L/a). Let Sa be the restriction of the model S to L/a obtained by fixing u~ - ~tZ for/3 ~ a. We now show that the optimal correction A u a given by eq. (5.17) is in the conjugate direction of the tangent space T~t~ ($a) to the model Sa at ~a (see Section 4.5.2). In T u , (L/a), the tangent hyperplane to the quadric defined by eq. (5.18) at ua + A u a has the following surface normal (see eq. (4.105)): L
no o~ e [ ~ o ] - ~
- Z
w<~')F(~)P,~~
k,l=l
p<~).
(5.~0)
Let v be an arbitrary tangent vector to the manifold Sa at ua (Fig. 5.3). Since the orthogonal complement of T~t~ ($a) with respect to T~t~ (L/a) is generated by P~tu~ Vu/~(k), k - 1, ..., L, eq. (5.19)implies L
(',"o) ~ Z w(~')F(~)(', Pu~V~o r(*)) - 0.
k,l-=l
Thus, Au~ is in the conjugate direction of T~t~ ($a). 4The proof will be given in C h a p t e r 14 in a more general framework.
(5.20)
136
Chapter 5.
r
Geometric Correction
.2v
Fig. 5.3. Geometric interpretation of optimal correction.
5.1.3
Practical considerations
Eq. (5.17) is merely a theoretical expression, because the right-hand side involves the covariance matrices V[ua] evaluated at the true values ~ta, a 1, ..., N, which we want to compute. It appears that they can be approximated by the covariance matrices V[ua] evaluated at the observed values u a , a - 1, ..., N. However, if the matrix V - (~(k0) defined by eq. (5.15) is approximated by V - (V (kl)) in the form
(5.21)
(v(k'))--(~(Vu"F(k)'V[ualVu"F(O)) ' a 1
matrices V and V may have different ranks" the rank of V is larger than that of Y if the constraint (5.2) is degenerate (see eqs. ( 5 . 3 ) a n d (5.4)). Hence, even if V is a good approximation to V, its generalized inverse W - V may be very different from W - V . A practical solution to this difficulty is to compute the rank-constrained generalized inverse (see eq. (2.82)). Namely, if the rank of the constraint (5.2) is r, eq. (5.17) is approximated by L
(5.22)
A u a - V[ua] E W
where
W - (W (kO) is an (LL)
matrix defined by W - (V)~-, which we write
as
a=l
r
The use of V[ua] instead of V[ua] has the following geometric interpretation. The quadric defined by eq. (5.18) is centered at the true value u a in the tangent space Tu~ (bG) at ua; the correction A u given by eq. (5.17) is an element of Tuo (bG) (Fig. 5.4). Using V[ua] instead of V[ua] means replacing eq. (5.18) by (ua - Ua, V[ua](~ta - u a ) ) - constant. (5.24)
5.1.
General Theory
137
a
Fig. 5.4. Theoretical analysis is done in the tangent space Tu~ (b/~) at fi~.
"
Js~
Fig. 5.5. Actual correction is done in the tangent space Tu~ (U~) at u~. If we regard ~ts as a variable, this equation defines a quadric centered at the data value u s in the tangent space Tu~ (Sts) at u s . Hence, the correction A u given by eq. (5.22) is an element of Tu~(Ua) (Fig. 5.5). This means that the data value u~ is corrected within Tu~ (Lts) in such a way that the Mahalanobis distance IIAusllv[u~] is minimized. This observation implies that as long as A u s E Tu~ (b/s), the inherent c o n s t r a i n t / t s E L/s on the corrected v a l u e / t s - u s - A u s is satisfied to a first approximation but may be violated if higher order terms are considered (Fig. 5.5). It follows that if we want to impose the inherent c o n s t r a i n t / t s E U~ exactly, we need a higher order correction, which we denote by C[. ] (see eqs. (4.113)): = C[uo -
(5.25)
This higher order correction can be made rather arbitrarily, since the correction is optimal in the first order. Because the correction given by eq. (5.22) is based on the linear approximation (5.5), the values {/t~} corrected by eq. (5.25) may not exactly satisfy the constraint (5.2) (Fig. 5.5). In order to impose it exactly, the computation is iterated by replacing the original values { u s } by the corrected values {/ts }. This process is essentially the Newton iterations, so the convergence is quadratic; usually two or three iterations are sufficient. In these iterations, the covariance matrix V[us] must also be updated, be-
138
Chapter 5.
Geometric Correction
U(~
Ua
Fig. 5.6. The error of optimal correction. cause the range of V[ua] at u~ is Tu~ (Ha) and is generally different from the range at /ta, which should be T/t ~ (Ha) (Fig. 5.5). If the covariance matrix V[ua] is given as a function of u~, it only needs to be re-evaluated at the updated value/~a. In many practical problems, however, the covariance matrix V[ua] is given only at the initial value ua. In such a case, a practical compromise is to "project" the covariance matrix V[ua] onto the tangent space T/t ~ (Ha) a t / t a in the form V[ua] - P i ~
V[ua]PU"its'
(5.26)
where P/t~ is the projection matrix onto the the tangent space T/t = (Ha) at
Ua.
5.1.4
A posteriori covariance matrices
Even if the constraint (5.2) is exactly imposed on the corrected values {/ta }, they are random variables because they are computed from the data {ua }. Let ua - ua + Aria, a - 1, ..., N, be the observed values, where Aria is the actual error in ua. After the correction A u a given by eq. (5.17) is subtracted, the data value ua is modified into
L Ua -- ('Ua + /k~ta)- ~r[Ua] E ~Tv
= ~ +
(
~-
k,l-1
r
~ k,l=l
w(k~)~(v~.~(k),~)v~o~(~) ~=1
)
(5.27)
to a first approximation. Let A/ta = /ta - u a be the error in the corrected v a l u e / t a (Fig. 5.6). The covariance matrix V[/ta,/t~] - E[A/taA/t~-] of the corrected values {/ta } is computed as follows:
~[~,~] - z[~oa~]
5.1.
General Theory
--
E
139
l~r(mn)(~/'[us
g[A~t
-T
6=1 m,n=l N L ~,=1 k,l--1 N L 3,,6=1 k,l,m,n=l
(Vu /~(k)) r E[A ~ A ~ - ] Vu~ F(,n)(s
Vu, F(n) ) T
= ~r['l/,o~](~c~ -- ( L ~v(mn)(~'r['uj3]Vu/~F(n))(V,cto,~-~(m))T~/r['llto~]) m~n----1 L
- ~
r162
k,/=l L
+
}~
r162
(~
k,l,m,n=l N
(Vu /~(k), ~/[u~lVuf.(m))(f/[u~]Vuof.(n))T
3,=1 L
-
~
s
l;V(m')(f/[u~]Vu,F('))(f/[u~]Vu~F(m))-r
re,n--1 L
k,l=l L + E ~v(kl)~v(mn)(~'r[U~176 k,l,m,n=l L
= ~[~1~-
~
"/(~~163
~.
(5.2s)
k,l=l Here, we have invoked the assumption that each us is independent and hence E[Aft~A~Z] - f/[u~]6~/3. We have also used the identity ITVVW
= ww-w-
w (s~e eqs. (2.123)).
Letting a - fl in eq. (5.28), we obtain L
(~[~1- ~ [ ~ ] - ~ -~(~)((~[~]v~oP(~))(r~[~o]v~o~(~ ~ k,/=l
(5.29)
140
Chapter 5.
Geometric Correction
/ T
J
ft--
N
---
a
,/
Fig. 5.7. The a priori and a posteriori standard confidence regions. For a ~ fl, we obtain L =
_
(5.30)
k,/----1
Thus, the corrected values {ira } are correlated even though the original values {u~} are independent. It can be confirmed 5 that the ranges of V[/ta] and Y[/ta, $tZ] coincide with the tangent space T/t" (b/a) to the manifold U~ at ~ta.
Eqs. (5.29) and (5.30) can be given the following geometric interpretation. The a priori covariance matrix ~[~N (x--1 Vinci defines the a priori standard confidence region in the tangent space T~=I ~t. (/4) to the manifold/4 = ~aN__1 Ua (Fig. 5.7). The a posteriori covariance matrix given by eq. (5.28) defines the a posteriori standard confidence region in the tangent space TeN=I ~t. (S) to the model $ defined by the constraint (5.2). This confidence region is the "projection" of the a priori standard confidence region onto Tey=l u . ($) along the "conjugate direction". It will be shown in Chapter 14 that eq. (5.28) coincides with the Cramer-Rao lower bound on the covariance matrix of the corrected values, meaning that the correction given by eq. (5.17) is indeed theoretically optimal. Eqs. (5.29) and (5.30) are mere theoretical expressions, since they involve values evaluated at ~ , a = 1, ..., N. Furthermore, eqs. (5.29) and (5.30) define the covariance matrices evaluated at ~a, a - 1, ..., N. In actual computation, a consistent approximation is identifying ~ w i t h / t ~ , i.e., we compute L T
k,l=l
5The proof will be given in Chapter 14 in a more general framework.
5.1.
General Theory
141 L
V[/ta,/tfi] = -
l~(kO(f/'[ua]Vu.~'(k))(f/[u~]Vu~ "f'(O)-r"
E
(5.32)
k,l=l
Here, IY[ua] is the matrix defined by eq. (5.26), and Vu/#(k) is the abbreviation of Vu F(k)(/tl,...,/tN). The matrix I ~ = (l~ (kl)) is defined by
(5.aa) a:l
r
The ranges of V[/ta] and V[~ia,/till thus defined coincide with the tangent space T~t ~ (b/a).
5.1.5
Hypothesis testing and noise level estimation
So far, the constraint (5.2) has been assumed given. However, it can be byand the above correction procedure can provide a means of testing this hypothesis. Let us hypothesize that the true values ua, a - 1, ..., N, satisfy eq. (5.2). Let ua = ua + A u a , a = 1, ..., N, be the observed values, and regard each A u a as an independent Gaussian random variable with mean 0 and covariance matrix I?[ua]. Since Vu/~(k) are deterministic values, eq. (5.5) implies that to a first approximation F (k) is a Gaussian random variable of mean 0. Noting that each A u a is independent, we can compute the covariance of F (k) and F (l) in the following form:
pothetical,
N
N
a=l
fl=l
N
=
(~ a,fl=l N
= E(Vu
/~(k) ' lF[ua]Vu /~(t)) _ ~(kt).
(5.34)
a--1
The matrix I~r - (W (kt)) defined by eq. (5.16) has rank r ( - the rank of eq. (5.2)), so the quadratic form L
d -
W(kOf(k)V(O
(5.35)
k,l=l
is a X2 variable with r degrees of freedom (see eq. (3.61)). It is easily confirmed that the right-hand side of eq. (5.35) coincides with the value obtained by substituting eq. (5.17) into eq. (5.6) (see eqs. (2.141) and
142
Chapter 5.
Geometric Correction
(3.87)). In other words, j is the residual of the optimization (5.6). If this value is much larger than can be accounted for by the statistical behavior of the noise, the hypothesis (5.2) should be rejected. It follows that the hypothesis (5.2) can be tested by the standard X2 test" the hypothesis is rejected with significance level a% if J > x ,o, where X,.,a 2 is the a% significance value of X2 with r degrees of freedom (see Section 3.3.4). Intuitively, the hypothesis that eq. (5.2) holds is rejected if the Mahalanobis distance over which the data {ua} must be displaced for imposing eq. (5.2)is too large (see Fig. 5.2). Eq. (5.35) is merely a theoretical expression, since it involves 1~r = (l~ (kl)). A simple approximation is using W - (W (kl)) defined by eq. (5.23), but now that the optimal estimate/t~ has been obtained, we can alternatively use ITr = (1~ (kt)) defined by eq. (5.33). However, the use of W rather than W makes only a second order difference. In many application problems, the geometric characteristics of noise (e.g., the degree of homogeneity/inhomogeneity and isotropy//anisotropy) can be relatively easily predicted but the absolute magnitude of noise is very difficult to estimate a priori. In such a case, we can write the covariance matrix Vinci in the form = = ..., N, (5.a7) where V0[u,] has a known form while e is unknown. Let us call V0[u~] the normalized covariance matrix of u~, and e the noise level. It is easily seen from eqs. (5.22) and (5.23) that the optimal correction is not affected by multiplication of Vinci by an arbitrary positive constant. Hence, V[u,] can be replaced by the normalized covariance matrix V0[u~]. In other words, we need not know the absolute noise level .for the optimal correction. Once the optimal solution is computed, the noise level e can be estimated a posteriori as follows. If the normalized covariance matrix V0[u~] is used for V[u,] in eq. (5.35), the resulting residual J0 equals e2J. Since 2 is a X2 variable with r degrees of freedom, its expectation and variance are r and 2r, respectively (see eqs. (3.59)). Hence, an unbiased estimator ~2 of e2 is obtained in the form
J0.
(5.as)
r
Its expectation and variance are respectively given by
(5.39) r
In geometric terms, we are estimating the noise level from the Mahalanobis distance with respect to V0[u,] over which the data ( u , } must be displaced for imposing eq. (5.2) (see Fig. 5.2). It follows that the hypothesis test (5.a6)
5.1.
General Theory
143
can be interpreted as comparing the a priori value e with the a posteriori estimate ~ computed on the assumption that the hypothesis is true. In fact, eq. (5.36) is equivalently rewritten as 6 ~2
2
~r,a
-7 e > --'r
5.1.6
(5.40)
Linear constraint
In many problems, the constraint is linear in the form N
EAters=b,
(5.41)
s=l
where A s is an L n s - m a t r i x and b is an L-vector. The rank r of this constraint equals the rank of the L ( ~ N s= 1 n s)-matrix
A-
(A1P~tU11,..., ANPuU~),
(5.42)
in terms of which eq. (5.41) can be written as A (~)N s = l ~ ' s - - b. Suppose the observed values u s , c~ = i, ..., N, do not satisfy the constraint (5.41). If we write uo = ~o + A , ~ , eq. (5.41) can be written as N
N
s 88
-b.
(5.43)
s=l
The correction A u s is determined by the optimization (5.6) under the inherent constraint A u s e Tu~ (L/s). The solution given by eq. (5.17) reduces to N
Aus - V [ u s l A ~ W ( E
Azu~ - b),
(5.44)
~=1 where the (LL)-matrix W is given as follows (see eq. (5.23)):
w
=
,545, s--1
r
Since the constraint (5.41) is linear, no approximation has been made to obtain eq. (5.43). Hence, no iterations are necessary. The a posteriori covariance matrices given by eqs. (5.31) and (5.32) reduce to V [ / t ~ ] - i Y [ u ~ ] - V[u~]A~WA~/[u~], (5.46) 6 T h i s is a c o n s e q u e n c e of t h e fact t h a t ~2/e2 is a modified X 2 variable with r degrees of freedom if the h y p o t h e s i s is t r u e (see eq. (3.72)).
144
Chapter 5. V[/~,/~fl]
-
Geometric Correction
V[u,~]A~WAzV[uz].
(5.47)
The residual of the optimization (5.6) can be written in the form N J
-
(E
N
A~/G
-
b, W ( E
c~=l
AZ/~Z
-
b)),
(5.48)
fl--1
which is a X2 variable with r degrees of freedom if the noise is Gaussian.
5.2
Correction of Image Points and Image Lines
5.2.1 A. Image
Optimal correction for coincidence points
Let xl and x2 be two image points, and V[Xl] and V[x2] their respective a priori covariance matrices. Suppose X l and x2 are two different estimates of the same image point. Consider the problem of estimating the true position. Let 51 and 52 be the true values of xl and x2, respectively. The constraint to be imposed is 51 52, (5.49) -
-
which has rank 2 because both sides are orthogonal to k = (0, 0, 1) T. If the two image points are statistically independent, the optimal estimate ~ = Xl - AXl = x2 -- Ax2 is obtained by finding AXl and Ax2 such that 7 J -- ( A X l , J~:[Xl]- A X l ) --[- ( A x 2 , ~ ' r [ x 2 ] - A x 2 ) ~ m i n
(5.50)
under the linearized constraint
e
/~X2 -- /~Xl -- X2 -- Xl,
(5.51)
The first order solution is given by 8 /XXlAx2
-
V[Xl]W(Xl-X2), -
V[x2]W(x2
-
-
X 1),
(5.52)
where W is a (33)-matrix defined by (5.53) 7We adopt the convention that V[. ] denotes the value of the covariance matrix V[. ] evaluated at the true value of the variable. SWe mean by "first order solution" the approximation expressed in terms of the data and the covariance matrices evaluated at the data values (see eq. (5.22)).
5.2.
Correction of Image Points and Image Lines
145
The a posteriori covariance matrix of the estimate 5~ is
V[xllWV[xl]- V[xl]WV[x2] v[~]- v[x:]wv[x:].
Fix] - V [ X l ] =
(5.54)
The residual of J can be written as J -- ( X 2 -
Xl, W(x2
(5.55)
-- X l ) ) ,
which is a X2 variable with two degrees of freedom 9. This fact provides a coincidence test for image points" the hypothesis that image points X l and x2 coincide with each other is rejected with significance level 6% if J ~" ~2,a.
(5.56)
E x a m p l e 5.1 If each coordinate is perturbed independently by Gaussian noise of mean 0 and variance e 2, the covariance matrices of AXl and Ax2 are V[xl] - V[x2] - e2Pk (see Example 4.1). The optimal estimate 5~ is 1
(5.57)
-- Xl -- A X l -- X2 -- A X 2 -- ~ ( X l "~" X2).
The a posteriori covariance matrix of ~ is 62
v[~]-
-~Pk.
(5.58)
The residual (5.55) can be written as 1
J - ~-j2 IIx2 - xl[I 2
(5.59)
Hence, an unbiased estimator of the variance e2 is obtained in the form ~2 _ ~11~ 1 - ~, I12 9
(5.60)
The value ~ thus estimated equals the half-distance between the two image points. If the value e is given a priori, the coincidence test takes the form ~2
X 2,a
--~ > - - ~ .
(5.61)
9We assume Gaussian noise and do first order analysis whenever we refer to X2 distributions and X2 tests.
146
Chapter 5.
Geometric Correction
B. Image lines
The same analysis can be done for two image 0. Let Y [ n l ] and V[n2] be their respective a signs of n l and n2 are chosen in such a way and (n2, x ) , = 0 be the true image lines. The
lines ( n l , x) - 0 and (n2, x) priori covariance matrices. The that n l ~ n2. Let (n l, x) - 0 constraint to be imposed is (5.62)
~1 -- n2,
which has rank 2 because both sides are unit vectors 1~ If the two image lines are statistically independent, the first order solution of the optimization g - ( A n i , ~r[l"~l]-- A1"~1) ~- (A1"~2, ~-7[n2]-A?'$2) --+ min
(5.63)
under the linearized constraint An2 - An1 - n2 - n l, A n I e {~t,1}L~,
n~O,2 e {~2}L~,
(5.64)
is given by A1"~1- V[nl]W(lo,1 - n2), A~~2 -- V[?'~2]W(n2 - 1"~1),
(5.65)
where W is a (33)-matrix defined by 11 W - (V[ni]-~- V[~o,21):.
(5.66)
A realistic form of the correction is i~, - N [ n i - A n 1 ] -
- An2].
i[n2
(5.67)
The a posteriori covariance matrix of the estimate/~ is V[~] -- ? [ r t i ] -
?[19,11WV[rg1 ] - V[l"~llWg[TI,2]
(5.68)
=
Here, i = 1, 2,
V[ni] - P f t V [ n i ] P n ,
(5.69)
where Pr is the projection matrix along/~. The matrix I ~ is obtained by replacing V[ni] by V[ni], i - 1, 2, in eq. (5.66). The residual of J can be written as J
-
-
-
(5.70)
a~ constraint is degenerate. 11The rank-constrained generalized inverse (.)2 is used because the ranges of V[nl] and V[n2] are different from the ranges of V[nl] and V[n2]. Consequently, although V[nl] + V[n2] is a singular matrix of rank 2, the matrix V[nl] + V[n2] is generally nonsingular.
5.2.
Correction of Image Points and Image Lines
which is a
X2
147
variable with two degrees of freedom.
This fact provides a
coincidence test for image lines: the hypothesis that image lines (n l, x) - 0 and (n2, x) - 0 coincide with each other is rejected with significance level a ~ if
j > ,~2,a.
5.2.2
(5.71)
Optimal correction for incidence
A. S i m u l t a n e o u s correction Suppose image point x and image line (n, x) - 0 are, respectively, estimates of an image point p and an image line 1 that should be incident to each other in the absence of noise. Consider the problem of optimally correcting them so as to make them incident. In other words, we want to find A x and A n such that 9 - x - A x and ~ - n - A n satisfy
(~, ~) - 0.
(5.72)
The rank of this constraint is 1. Let V[x] and V[n] be the a priori covariance matrices of x and n, respectively. If the image point and the image line are statistically independent, the problem can be written as the optimization J - (Ax, V [ x ] - A x ) + (An, V [ n ] - A n ) -+ min
(5.73)
under the linearized constraint
(g. ~ )
+ (~. ~ n ) - (n. ~).
(5.74)
k • The first order solution is given by A~ -
(n, ~ ) v [ ~ ] ~
v[~]n) + (~. v[.~]~) '
(~. A,~ -
(n, ~ ) v [ n ] ~
(.. v[~]~) + (~. v [ ~ ] ~ )
(5.75)
A realistic form of the correction is - ~-
A~,
a-
N[n-
An].
(5.76)
The a posteriori covariance matrices of the corrected values & and ~ are V[~] - V[x] -
(V[~']'~)(V[~I'~)T (~, v[~]~) + (~, ~ [ ~ ] ~ ) '
v[,~] - ?In]
(r (~, v[~],~) + (~, ~ [ n ] ~ ) '
-
148
Chapter 5.
( r l ~(~r 1 T ~,v~x~,,_~n~,,
v[~,h]--
(~. v[~]~) + (~. ~[.]~)
Geometric Correction
= V[fi, &]T,
(5.77)
where V [ n ] - P n V[n]Pn.
(5.78)
The residual of J can be written as J -
(5.79)
(n'x)2
(g, V[~lg) + (~, ~[.]~)' which is a X2 variable with one degree of freedom. This fact provides an incidence test for an image point and an image line: the hypothesis that image point x and image line (n, x) - 0 are incident to each other is rejected with significance level a ~ if J > X2,a 9 (5.80)
B. Image point correction If the image line (n, x) - 0 is fixed, the linearized constraint is
(-,~)
• A ~ e {k)L.
- (-,~),
(5.81)
The first-order correction of the image point x is obtained by letting V[n] O in eqs. (5.75)"
~ = (~,~)v[~]~
(~, v [ ~ ] ~ ) "
(5.82)
The a posteriori covariance matrix of the corrected value ~ is
v[~]-
v[~]-
(v[~]')(v[~]~)~
(~. v[~].)
(5.83)
Since V[hc]n = 0, the rank of V[~] is 1; its null space is {k, n } L , which is orthogonal to the orientation m - N[k x n] of the space line (n, x) = 0. The residual
j_
(n, ~)~
(n, V[x]n)
(5.84)
is a X2 variable with one degree of freedom. Hence, and the incidence test given by eq. (5.80) can be applied. E x a m p l e 5.2 If each coordinate is perturbed independently by Gaussian noise of mean 0 and variance e2, the covariance matrix of A x is V[x] e2Pk . The optimal correction (5.82) reduces to
~
- - (~' ~ ) P k ~ 1 - (k, ~ ) ~
(5.s5)
5.2.
Correction of Image Points and Image Lines
149
X
^ x
m
Fig. 5.8. Optimal incidence correction for an image point. Consequently, x is displaced onto the image line ( n , x ) - 0 perpendicularly (Fig. 5.8). The a posteriori covariance matrix of the corrected value 5~ can be written as V[~] = e2....~T. (5.86) where m = X[k x n I is the orientation of the image line (n, x) = 0. The residual (5.84) can be written as
af -
~
(n'~):
-'T E 1 -(k,
n) 2"
(5.87)
Hence, an unbiased estimator of the variance e2 is obtained in the form ~2 __--
( n , X) 2
- (k,.)~"
(5.8s)
The value @thus estimated equals the distance between the image point and the image line (see eq. (4.11)). If the value e is given a priori, the incidence test takes the form ~2 E---~-
> ~1~,o.
(5.s9)
C. Image line correction
If the image point x = 0 is fixed, the linearized constraint is ( A n , x) - (n, x),
A n 6 {fi}LI .
(5.90)
The first-order correction of the image line (n, x) = 0 is obtained by letting V[x] = O in eqs. (5.75):
A,~ - (n, ~ ) v [ n ] ~ (~, v[n]~) "
(5.91)
The a posteriori covariance matrix of the corrected value ~ is
V[n]- ~tT[n]- (~/"[fl']X)(Y[f't]x)T
(5.92)
150
Chapter 5.
Geometric Correction
The rank of V[~] is 1; its null space is {~, X}L. The residual J -- (x, l}'[n]x)
(5.93)
is a X2 variable with one degree of freedom. Hence, the incidence test given by eq. (5.80) can be applied.
5.3
5.3.1
Correction of Space Points and Space Lines
Optimal correction for coincidence
A. Space points Two space points ?'1 and ?'2 that are supposed to coincide can be optimally corrected in the same way as in the case of image points. Let V[?'l] and V[?'2] be their respective a priori covariance matrices. Let r l and r2 be the true positions of ?'1 and ?'2, respectively. The constraint to be imposed is
r, - r2,
(5.94)
which has rank 3. If the two space points are statistically independent, the problem is finding A?'I and A?'2 such that J - (A?'I, V[?'1]-1A?'1) -~-(A?'2, V[?'2]-1A?'2) --+ min
(5.95)
under the linearized constraint
A?'2 -- A?'I -- ?'2 -- ?'1.
(5.96)
The first order solution is given by
A?'I-
V[?'1]W(?'1-?'2),
A?'2 -- V[?'2]W(?'2 - ?'1),
(5.97)
where W is a (33)-matrix defined by
W-
(V[?'l]-~- V[?'2]) -1 9
(5.98)
The a posteriori covariance matrix of the estimate ~ is
V [ ' r ' ] - V [ ? ' l ] - V[?'I]WV[?'I] -- V[?'I]WV[?'2] =
(5.99)
The residual of J can be written as
J-
(?'2-?'1,W(?'2-?'1)),
(5.100)
5.3.
Correction of Space Points and Space Lines
151
which is a X2 variable with three degrees of freedom. This fact provides a coincidence test for space points: the hypothesis that space points r l and ?'2 coincide with each other is rejected with significance level a% if
J >
(5.101)
E x a m p l e 5.3 If each coordinate is perturbed independently by Gaussian noise of mean 0 and variance e 2, the covariance matrices of A r l and At2 are V[rl] = V[r2] = e2I. The optimal estimate ~ is 1
~" -- ~'1 -- A / ' I -- /'2 -- A/'2 -- ~(7"1 ~t- ~'2).
(5.102)
The a posteriori covariance matrix of ~ is s
V[e]-
~-I.
(5.103)
The residual (5.100) can be written as
1
12
J -- ~E2 11~2 -- ~'1[ 9
(5.104)
Hence, an unbiased estimator of the variance e 2 is obtained in the form ~2_ 1 ~llr2 - rill 2.
(5.105)
The value ~ thus estimated equals 1/vf3 times the half-distance between the two space points. If the value e is given a priori, the coincidence test takes the form ~:2
X 2 ,a
J > -i-"
(5.106)
B. Space lines
The same analysis can be done n2. Let V[p 1 | n l] and V[p 2 | signs of {Pl, n l } and {P2, n2} /' X P l - - ~'1 and r x P2 - n2 imposed is
for two space lines r x Pl - n l and r x P2 = n2] be their a priori covariance matrices. The are chosen so that Pl ~ P2 and ~t 1 ~,~ ~t 2. Let be the true space lines. The constraint to be
P l -- P2,
~1 -- fi2,
(5.107)
which has rank 4 because the representations {Pl, fil } and {102, fi2 } have four degrees of freedom 12 (see Section 4.2.2). If the two space lines are statistically independent, the problem is finding Apl , An1, A p 2, and An2 such that J -
( / k p l 9 ATtl, ~r[p 1 9 n l l - ( / k p
I 9 ATt,1))
_
+ ( A p 2 | An2, V[p 2 | n : ] - ( A p 2 | An2)) ~ min 12This constraint is degenerate.
(5.108)
152
Chapter 5.
Geometric Correction
under the linearized constraint Ap 2 | An2 - Ap 1 | An1 - P2 | n2 - Pl | n l , A p l ~ A n l e {Pl 0 ~/,I, ~,1 O P l } f , 2_
(5.109)
Ap2 | An2 E {P2 @ fi2,fi2 @ P2}L" The first order solution is
~XV~ 9 ~X,~ - V[p~ ~ ,~]W(p~ 9 ,~ - p, 9 nl),
(5.110)
where W is a (66)-matrix defined by 13 W-(V[p
1
@rtl] "4-V[p2 O n 2 ] )
(5.111)
4 9
A realistic form of the correction is (lb)
_N•
PI--Apl nl --/knl
h
where the operation N•
N• ( a ) b
rt2 -- Art2
'
(5.112)
] is defined by
{ N[a | PN[a] b] ]-
p2--Ap2 )]
)I-N•
N[PN[b]a | b]
if"a">--"bl"
(5.113)
otherwise.
The a posteriori covariance matrix of the estimate lb @ h is W[i~ @ ~] - l$[pl @ rtl] - V[Pl @ n l]l~rV[Pl 9 n l]
(5.114) Here,
V[Pi | nil = Px~,.~ V[pi | n~]P~r
i = 1, 2,
(5.115)
where PX~.~ is the six-dimensional projection matrix onto N ' p.1_. n
(see
eq. (4.44:)). The matrix I ~ is obtained by replacing V[p i @ hi] by l~[p i @ hi], i - 1, 2, in eq. (15.111). The residual of J can be written as
(5.116) which is a X2 variable with four degrees of freedom. This fact provides a coincidence test for space lines" the hypothesis that space lines r x Pl = n l and r x P2 - n2 coincide with each other is rejected with significance level a% if J > X~,a" (5.117) 13The ranges of V[pl • nl] and V[p2 9 n2] are different from the ranges of l/[pl (~ nl] and 17[p2 @ n2]. Consequently, although IY[pl @ nl] + l?[p2 @ n2] is a singular matrix of rank 4, the matrix V[pl | nl] + V[p2 | he] is generally nonsingular.
5.3.
5.3.2
Correction of Space Points and Space Lines
153
Optimal correction for incidence
A. Simultaneous correction As in two dimensions, a space point r and a space line r • p - n can be optimally corrected so as to make them incident. Let V[r] and V[p | n] be their a priori covariance matrices. The problem is finding A t , Ap, and A n such that ~ = r - A t , p = p - Ap, and ~ = n - A n satisfy
e xp--ff,.
(5.118)
The rank of this constraint is 2 because the three component equations are algebraically dependent 14. If the space point and the space line are statistically independent, the problem can be written as the optimization J - (At,
~r[~,]--I A/')
-~- (Ap | An, fT[p | n ] - ( A p | A n ) ) -+ min
(5.119)
under the linearized constraint Ar
x
p+f"
x
Ap--An=r
x
p_n,
Ap | A n E {p | n, fi @ lb}~.
(5.120)
The first order solution is given by
=-(vH
x r,)w(,, x p - n),
Ap = (V[p] x r -
V[p, n l ) W ( r x p -
A n = ( V t n , p] x r -
V[n])W(r x p-
n), (5.121)
n),
where W is a (33)-matrix defined by 15 W-
(p x V[v] x p + v x V[p] x r -
)-
2S[v x V[p,n]l + V[n] 2 "
(5.122)
The symbol S[. ] denotes the symmetrization operator (see eqs. (2.205)). A realistic form of the correction is f--r-At,
h
n--An
]'
(5.123)
where the operator N• ]is defined by eq. (5.113). The a posteriori covariance matrices of the corrected values ?,/~, and h are
U p ] - V[,']- (V[r] x/,)W(p x V[,']), 14This constraint is degenerate. 15The rank-constrained generalized inverse ( . ) 2 is used because ( . ) is generally nonsingular if evaluated at the data values; it should be a singular matrix of rank 2 if evaluated at the true values.
154
Chapter 5.
Geometric Correction
v[v] - ?[v]- (?Iv] • e - ?[v,,~])W(e • ?Iv]- ?In, v]), V[Ib, ~t] - V[p, n] - (V[p] • ? - lY[p, n])I~(/" • V[p, n] - I7[n]) - V[/t, lb] 7-,
v[,~]- ? [ n l - (?[n,v] x e - ?[n])Cc(e • ?[v, n ] - ?[n]), v[~,/,]- (v[r] x/,)g,(~ x
?[v]- ~[n, v ] ) -
v[/,, ~]T,
V[O, ~t] = (V[r] x p)I~(O x V[p,n]- I Y [ n ] ) - V[~t, ?]T.
(5.124)
The matrices IY[p], ~Z[p, n], and V[n] are obtained as submatrices of
V[p | n] - P~'~e~V[p @ nlPx~e~.
(s.~2s)
The matrix I&r is obtained by replacing r, p, V[p], V[p, n], and V[n] b y / ' , lb, V[p], V[p, n], and V[n], respectively, in eq. (5.122). The residual of J can be written as j - (r • p - n, W ( r x p - n)), (5.126) which is a ~:2 variable with two degrees of freedom. This fact provides an incidence test for a space point and a space line: the hypothesis that space point r and space line r • p - n are incident to each other is rejected with significance level a ~ if j > ~2,a. (5.127) B. Space point correction
If the space line r • p - n is fixed, the linearized constraint is
Arxp=rxp-n.
(5.128)
The first order correction of the space point r is
~
= -(v[~] • v)w(~
• v-
n),
(s.129)
where W is a (33)-matrix given by
The a posteriori covm'iance matrix V[~] of the corrected value ~ is given in the form shown in eqs. (5.124), where lb and ~ are replaced by p and n, respectively. Matrix V[~] has rank 1; its null space is {n, rH}L in the {m, rH}-representation. The residual J is given in the form of eq. (5.126) and is a X2 variable with two degrees of freedom. Hence, the incidence test given by eq. (5.127) can be applied.
5.3.
Correction of Space Points and Space Lines
155
Ar r
O Fig. 5.9. Optimal incidence correction for a space point. E x a m p l e 5.4 If each coordinate is p e r t u r b e d independently by Gaussian noise of mean 0 and variance e2, the covariance m a t r i x of A r is V[r] - e2I. The optimal correction (5.129) reduces to
p
At-
x
(r x p - n ) ilPl12
-- P r o m -
?"H.
(5.131)
Consequently, r is displaced onto the space line r x p - n perpendicularly (Fig. 5.9). The a posteriori covariance matrix of the corrected value ~ has the form V [ ~ ] - e 2 m m -T-, (5.132) where m - N[p] is the orientation of the space line r x p - n. The residual (5.126) can be written as
J-
1
--~
I1~ x p - n i l ~ 1 ilpl12 = -~llPmr-
r.II 2.
(5.133)
Hence, an unbiased estimator of the variance e e is obtained in the form
~2
I1',' x
p-.,112
211pll2
12
1
= ~llPmr-
rill 9
(5.134)
The value ~ thus estimated equals 1 / v ~ times the distance between the space point and the space line (see eq. 4.49)). If the value e is given a priori, the incidence test takes the form ~2 X2,~ e-Y > ---~--.
(5.135)
C. Space line correction If the space point r is fixed, the linearized constraint is r x Ap-
An - v x p-
n,
156
Chapter 5. AV+n
Geometric Correction (5.136)
~ {p+ n,n +p}~.
The optimal correction of the space line r x p - n is Ap~.
V[p, n l ) W ( r
(V[p] x r -
- ((~ • v [ v , . ] ) ~
x p-
n), (5.137)
-),
- V[.l)W(~ • v-
where W is a (33)-matrix given by w
-
(\ ~ • v [ v ] • ~ - ~ • v [ v , . ] -
(~ • v [ v , . ] ) ~
+ vM
.
(5.138)
The a posteriori covariance matrices of the corrected values/5 and fi are given in the form shown in eqs. (5.124), where ~ is replaced by r. The residual is (5.139)
J - (," • v - - , w ( , - • v - -)).
This is a X2 variable with two degrees of freedom. Hence, the incidence test given by eq. (5.127) can be applied. 5.4
5.~.1
Correction
of Space
Planes
Optimal correction for coincidence
Two space planes (v,, p) = 0 and (v2, p) = 0 that are supposed to coincide can also be optimally corrected. Let Y[vl] and Y[v2] be their respective a priori covariance matrices. The signs of the 4-vectors Vl and v2 are chosen so that vl ~-- v2. Let (Pl, P) - 0 and (P2, P) - 0 be the true space planes. The constraint to be imposed is (5.140)
Vl -- V2,
which has rank 3 because both sides are unit vectors 16. If the two space planes are statistically independent, the problem is finding AVl and Av2 such that J - (AVl, ~r[vl]-lAvl)
-[- ( A v 2 , ~ r [ / 2 2 ] - l n v 2 )
--=+m i n
(5.141)
under the linearized constraint A v 2 -- A V l -- /22 -- V l ,
Av2 e {P2}L~. The first order solution is given by ~1 16This constraint is degenerate.
- V[~l]W(~l - ~),
(5.142)
5.4.
Correction of Space Planes
157
AI/2 -- V[I22]W(v 2 - Vl) ,
(5.143)
where W is a (44)-matrix defined by a7 (5.144) A realistic form of the correction is ~' = N[L,1 - A I / 1 ] - N[L,2 - A!21].
(5.145)
The a posteriori covariance matrix of the estimate s is Villi -
?[Vl]W?[Vl]-
?[Vl]W?[v2]
(5.146)
Here, -
i - 1, 2,
(5.147)
where P~,, is the four-dimensional projection matrix along u i. The matrix IrV is obtained by replacing V[vi] by Y[~'i], i - 1, 2, in eq. (5.144). The residual of J can be written as ^
J - - (V2 - V l , W ( v 2 - V l ) ) ,
(5.148)
which is a X2 variable with three degrees of freedom. This fact provides a coincidence test for space planes: the hypothesis that two space planes (Vl, p) = 0 and (L'2, p) = 0 coincide with each other is rejected with significance level a% if J > ~2,a. (5.149)
5.~.2
Optimal incidence with space points
A. S i m u l t a n e o u s c o r r e c t i o n
A space point p and a space plane (v, p) - 0 can be optimally corrected so as to make them incident. Let V[p] and V[v] be their respective a priori covariance matrices. The problem is finding A p and A v such that ~ = p - A p and P - L , - A v satisfy (P, h) - 0. (5.150) The rank of this constraint is 1. If the space point and the space plane are statistically independent, the problem can be written as the optimization J - (Ap, V [ p ] - A p ) + (Av, V [ v ] - A v ) --+ min
(5.151)
aTThe the ranges of V[ua] and V[u2] are different from the ranges of l)'[pa] and V[v2]. Consequently, although ~'[Vl]+ V[u2] is a singular matrix of rank 3, the matrix VIal]+ V[v2] is generally nonsingular.
158
Chapter 5.
Geometric Correction
under the linearized constraint
(~,,/,) + (~, ~p) = (~,, p), tg /
(5.152)
where ~ = (0, 0, 0, 1) T. The first order solution is given by
(~,, p)V[plv ~t, - (~, v[p]v) + (p, v[~,]p)' (~,, p)v[~,]p (v, V[plv) + (p, v[~,lp)
Alp
(5.153)
A realistic form of the correction is /5 - p -
Ap,
~, = N [ ~ , -
(5.154)
A~,].
The a posteriori covariance matrices of the corrected values/~ and 1) are
v[/,]- v [ d -
(v[p]~)(v[p]~) ~ (~,, v[p]~,) + (/,, ?[~,]/,)'
(?[,,,]/,)(?[,,].a) T v[~] = ~ [ ~ ] v[/,, ~,1 -
(~,, v[p]~,) + (/,, ?[~,]/,)'
(v[p]~,) (?[~,]/,)T (~,, v[p]~,) + (h, ~'[~,]h)
-
= v[~,,/,]T
.
(5.155)
Here,
r
p~v[~]p~,
(5.156)
where P/, is the four-dimensional projection matrix along ~,. The residual of J can be written as
.]
-
(u' P)2
(5.157)
(~', V[d~') + (h, ?[~']/')' which is a X2 variable with one degree of freedom. This fact provides an incidence test for a space point and a space plane: the hypothesis that space point p and space plane (u, p) - 0 are incident to each other is rejected with significance level a~0 if j > X2,a .
(5.158)
5.4.
Correction of Space Plmles
159
O Fig. 5.10. Optimal incidence correction for a space point. B. Space point correction
If the space plane (v, p) = O is fixed, the linearized constraint is
(~, Ap)
(~, p),
-
Ap
e {~}~.
(5.159)
The optimal correction of p is
Ap
(v,p)V[p]u
(5.160)
(~, v[p]~) "
-
The a posteriori covariance matrix of the corrected value/5 is V[~b]-
V[p]- (Y[P]v)(Y[P]V)-r (•, V[plv)
"
(5.161)
Since Viably = O, the rank of V[/~] is 2; its null space is {~, V}L. The residual j
_
(v, p)2
(~, v[p]~)
(5.162)
is a X2 variable with one degree of freedom. Hence, the incidence test given by eq. (5.158) can be applied. E x a m p l e 5.5 If each coordinate is perturbed independently by Gaussia~ noise of mean 0 rand variance e2, the covariance matrix of p is V[p] = e2P~ (=
~ I ~ 0). In the (n, d}-represent~tion, the optim~ correctio~ (5.160) reduces to
zx~ = ((n, ~) - d)n.
(5.~63)
Consequently, r is displaced onto the space plane (n, r) = d perpendicularly (Fig. 5.10). The a posteriori covariance matrix of the corrected value ? is V[/'] =
e2pn.
(5.164)
The residual (5.162) can be written as 1 j - j ( ( n , r) - d)2.
(5.165)
160
Chapter 5.
Geometric Correction
Hence, an unbiased estimator of the variance e2 is obtained in the form -
-
d)
(5.166)
The value ~ thus estimated equals the distance between the space plane and the space point (see eq. (4.68)). If the value e is given a priori, the incidence test takes the form ~2 e-~ > X~,a" (5.167) C. Space plane correction
If the space point p is fixed, the linearized constraint is (Av, p ) -
(v,p),
Av e {P}~.
(5.168)
The optimal correction of space plane (v, p) - 0 is (p, V[v]p) "
(5.169)
The a posteriori covariance matrix the corrected value D is V[~]- ?[~l-
(V[~IP)(?[~]P)~ (p, ~[~]p) .
(5.~70)
The rank of V[9] is 2; its null space is {/~, P}L. The residual
J -
P):
(5. 71)
is a X2 variable with one degree of freedom. Hence, the incidence test given by eq. (5.158) can be applied.
5.4.3
Optimal incidence with space lines
A. Simultaneous correction
A space line (r - rH) • m = 0 and a space plane (n, r) = d can be optimally corrected so as to make them incident. Let V[m (DrH] and V[n @d] be their respective a priori covariance matrices, and ( r - OH) • rh -- 0 and (fi, r) d their true equations. The constraint is (~, ,~) - 0,
(~, ~ , ) - d,
(5.172)
which has rank 2. If the space line and the space plane are statistically independent, the problem is finding A m , Art/, An, and Ad such that _
J - (Am + ArH, Vim + rHI-(Am + Art/)) + ( A n | Ad, V[n 9 d]- ( A n | Ad)) -+ min _
(5.173)
5.49
Correction of Space Planes
161
under the linearized constraint
(An,~,)+(~,Am)=(n,m),
(~.. e.) + (~. ~ . ) - ~d = (....) - d. • A m | ArH e { m | 0, OH | vh }L,
An e {n}f.
(5.174)
The first order solution is given by
vimlo
Am
V[rH,m]Tn
Ad
--
(m, V[n, d])
v[...l.
(.. ...) - d
(rH, V[n, d]) - V[d]
W
'
(n, rH) - d ' (5.175)
where W is a (22)-matrix defined by
(.. vim].) + (m. v[.]m) w =
(.. vim. ~.1.) + (~.. v [ . ] m ) (n. V[~?l.. 7".]n) -}- (771..V[n]~'.) - (~11..V[n. d]) ~ -1 (n,
V[rH]n) +
(vii, V[nIrH ) -- (rH, V[n, d])
)
9 (5.176)
A realistic form of the correction is
vh = N [ m -
Am],
if, : N [ n -
An],
d = d - Ad.
(5.177)
The a posteriori covariance matrices of the corrected values rh,/'H, fi, and d
are
v[.~l
v[.~. #~]
#[m]
#[m. ~1
v[+~,m] v[+~])= (~[~,m] ~[~] ) ~[m..'~/].~ ) v[r.].~
viii
(v[d,,~]
v[~. d] vial]
~[.] #[n. d] ~[d. nl V[dl ^ ) _ ( ~[.].~ ~[.]~ - ~[.,d] (.h. e[.. dl) (~.. e[-. dl/- ~[d] ) W ( ?[-l.h e[n]~H-'~[-.dl (.h. e[.. dl) (+.. r d]) - V[~I )
)=(
162
Chapter 5.
v[.~. ,~] ( v[e..a]
v[.~. d] v[~. , d] ) - - (
?[m]a
?[rH, m] .~
i~r ( ?[n]~q~
Geometric Correction
?[m.~.]a
?[rH]/~ )
?[nigH -- ?[n, d]
(rh, ?In, d]) (/~g,?[n, d]) - ?[4 )
, (5.178)
where V[m], V[m, rH], etc. are computed as submatrices of V[m | r , ]
- Px~e~ V[m |
lY[n | d] - ( P n | 1)V[n | Here, P x ~ e ~
rg]Px~e~,
d](Pr~ |
1).
(5.179) • N,i~**. (se~ m, rH, n, V[m],
is the six-dimensional projection matrix onto
eq. (4.40)). The matrix I ~ is obtained by replacing Vim, rH], etc. by rh, rH,/~, ?[m], ? I r a , rH], etc., respectively, in eq. (5.176). The residual of J can be written as J -- ~TV "(11) (1"~,m ) 2 +212V(12) (n, m)((n,
rH)-d)+IV (22)((n, rH)--d) 2,
(5.180)
which is a X2 variable with two degrees of freedom. This fact provides an incidence test for a space line and a space plane: the hypothesis that space line ( r - r H ) • m -- 0 and space plane are incident to each other is rejected with significance level a% if J > X~,a. (5.181) B. Space line correction
If the space plane (n, r) = d is fixed, the linearized constraint is (n, A m ) = (n, m),
(n, Art/) = (n,
rH) -- d,
A m G A~'H e {?~ O 0, ~H O m } f .
(5.182)
The optimal correction of the space line (r - r H ) X m = 0 is
~.
v[~.. m] ~ .
v[~.l.
(~. ~.) - d
'
where W is a (22)-matrix defined by
w-
(n,V[m]n) ((,vim, ~.1~)
(n, V[m, rH]n) )-1 (~. v[~.].) 9
(5.~84)
The a posteriori covariance matrices of the corrected values ~h and rH are given in the form shown in eqs.^(5.178), where/~ and d are replaced by n and d, respectively. The residual J is given in the form of eq. (5.178) and is a X2 variable with two degrees of freedom. Hence, the incidence test given by eq. (5.181) can be applied.
5.5.
Orthogonality Correction
163
C. Space plane correction
If the space line (r - r H ) • m - 0 is fixed, the linearized constraint is (Art, m ) = (n, m),
(An, r g ) -- Ad = (n, rH) -- d, _k A n e {n}L.
(5.185)
The optimal correction of the space plane (n, r) = d is
Ad
=
(m, V[n, d])
(r, V[n, dl) - V[d]
W
(n, rH) - d
' (5.186)
where W is a (22)-matrix defined by
W -
(m, V[n]m) ( ~ , Via]m)
(m, v[,~]rH) - (m, v [ . , d]) ( ~ , v[n]~H) - (~, v[n, d])
-i
)
(5.187)
The a posteriori covariance matrices of the corrected values fi and d are given in the form shown in eqs. (5.178), where ~'H and ~h are replaced by r H and m , respectively. The residual J is given in the form of eq. (5.178) and is a X2 variable with two degrees of freedom. Hence, the incidence test given by eq. (5.181) can be applied.
5.5
Orthogonality Correction
5.5.1
Correction of two orientations
A. Simultaneous correction
Let m l and m2 be unit 3-vectors that indicate orientations supposedly orthogonal. Let V[ml] and Vim2] be their respective a priori covariance matrices. In the presence of noise, m l and m2 are not exactly orthogonal. Consider the problem of optimally correcting them so as to make them orthogonal (Fig. 5.11). In other words, we want to find Am1 and Am2 such that m l = m l -- A m l a n d m 2 - m 2 - A m 2 satisfy (rhl, m2) - 0.
(5.188)
The rank of this constraint is 1. If the two orientations are statistically independent, the problem can be written as the optimization
J
-
(Am1,
V [ m l ] - A m 1) +
(Am2,
V[m2]-Am2) .-+ r a i n
under the linearized constraint
(~ml, m2)+ (rex, ~m2) - (ml, m2),
(5.189)
164
Chapter 5.
Geometric Correction
Am I
ml
m I
Am2 0
n-t2
Fig. 5.11. Orthogonality correction for two orientations.
A m l E {ml}L~,
Am2 E {?~2}~.
(5.19o)
The first order solution is given by
(ml,m2)V[ml]m2 A m i -- (m2,V[mi]~T/,2) ~- (ml, V[m2]mi)'
(mi,m2)V[m2]ml Am2 = (m2, V [ m i ] m 2 ) + (m~, V[m2]mi)"
(5.191)
A realistic form of the correction is
ml
-
-
N[ml
-
-
Am:],
vh2 - N[m2 - Am2].
(5.192)
The a posteriori covariance matrices of the corrected values vhl and ~h2 are
V[mi] -- ? [ m i ] - (T~2,V[ml]m2) 4- ( m l , V [ m 2 ] m l ) ' V[zh2] - ?[m2] - (zh2, V[ml]~~2) -~- (rex, V[/2]~/~l)' ( ? [ m l ]m2 ) ( ? [ m 2 l m i ) T
V[T~I, m21 -- --(m2, V[mi]m2) + (ziz~, V[m2]ziZl)"
(5.193)
Here, V[mi] - P v h V [ m i ] P v i z , ,
i-
1, 2,
(5.194)
where Pviz~ is the projection matrix along zizi. The residual of J can be written as
,] --
(mi, m2) 2 (~TI'2,~/[ml]m2) -}- (#/,1, IY[m2]ziZl)'
(5.195)
which is a )/2 variable with one degree of freedom. This fact provides an orthogonality test for two orientations" the hypothesis that the two orientations m l and m2 are orthogonal to each other is rejected with significance level a% if
J > x i,o.
(5.196)
5.5.
Orthogonality Correction
165
Am3 in3 ~ m3 Am, Am 2 Fig. 5.12. Orthogonality correction for three orientations. B. Correction of one orientation If m2 is fixed, the linearized constraint is (Am1, m2) - ( m l , m2),
A m l e {Vhl }L I.
(5.197)
The optimal correction of m l is =
(m2, U[ml]m2) .
(5.198)
The a posteriori covariance matrix of the corrected value vhl is
V[?~I]-- Y[ml]- (?[ml]m2)(?[ml]m2)T (m2, ? [ m i ] m 2 )
"
(5.199)
Since V[Vhl]m2 -- O, the rank of V[vh]] is 1; its null space is {Vhl,m2}L. The residual
j --
(ml, m2) 2 (m2, V[ml]m2)
(5.200)
is a X2 variable with one degree of freedom. Hence, the orthogonality test given by eq. (5.196) can be applied.
5.5.2
C o r r e c t i o n of three o r i e n t a t i o n s
A. Simultaneous correction The same procedure can be applied to three orientations. Let m l, m2, and m3 be unit 3-vectors that indicate three orientations supposedly orthogonal (Fig. 5.12). The problem is finding A m i such that m i = m i - A m i satisfies
(~'~,i, ~'j) "-- (~ij,
i, j = 1, 2, 3.
(5.201)
The rank of this constraint is 3. Let V[mi] be the a priori covariance matrix of mi. If the three orientations are statistically independent, the problem can be written as the optimization
3 J-
~-~(Ami, < d [ m i ] - A m i ) ~ min
i=1
(5.202)
166
Chapter 5.
Geometric Correction
under the linearized constraint (m3, Am2) + (m2, Am3) = (m2, m3), (rhl, Am3)+ (rn3, Am1) -- ( m 3 , m l ) , (rh2, Am1)+ (ml,Am2) = ( m l , m 2 ) , mi e {rhi}Ll,
i = 1, 2, 3.
(5.203)
The first order solution is given by
( mx) ( o Vmlm3VEmxlm) 2 (,m2,m3,) Am2
--
Am3
V[m2]m3
0
V[m2]m I
V[m3lm2
W[m3lml
0
W
(m3,ml)
,
(ml,m2) (5.204)
where W is a (33)-matrix defined by W
/(o m3m2) ( viral] v[m~] m3 m2
0 ml
ml 0
) v[~]
(
0 m3 m2
m3 0 ml
m2 ml 0
))
(5.205)
A realistic form of the correction is i = 1,2,3.
~'1,i - N [ m i - Ami],
(5.206)
The a posteriori covariance matrices of the corrected values rhi are
V[ml] V['r~2, 'r~,l] V["~3, ~'Y/'I]
V[~,I, vh2] V[rhl, m3] ) V ['rll,2] V["~3, '/~2]
V['rh2, 'r?t,3] V['/~3]
o --
( V[ml]
f'[m~]ma ?[ml]~t2 )
~'[,ml)
V[m2l'rn3
0 V[m2]~'~t1 0 l}'[m3lrh2 ~Z[m3]'m 1 0
V[ml]~,3
?[m~],~ o ~[m~],~ ~[,m],~ where
f'[m~]
-
f~[m~]-Pm,V[,,~,lP,u,,
?[ml]~'h2 ) V[m2]~,l
,
(5.207)
0
i = 1,2,3.
(5.208)
5.5.
Orthogonality Correction
167
V[mi]
The matrix I ~ is obtained by replacing m~ and by ~h~ and IY[mi], respectively, in eq. (5.207). The residual of J can be written as - (
(m3, m l )
,W
(ml,m )
(m3, m l )
),
(5.209)
(ml,m )
which is a ~2 variable with three degrees of freedom. This fact provides an for three orientations: the hypothesis that the three orientations m i , i - 1, 2, 3, are orthogonal to each other is rejected with significance level a% if
orthogonalitytest
J > X ,a. B.
Correction
of one
(5.e10)
orientation
If m l and m2 are fixed in such a way that ( m l, m2) - 0, the rank of the constraint decreases to 2, and the linearized constraint is (ml,Am3)-
( m l , m3),
(m2, A m 3 ) -
(m2,m3),
(5.211)
e The optimal correction of m3 is
Am3-V[m3](ml'm2)W((me'm3))(m2,m3),
(5.212)
where W is a (22)-matrix defined by
W_ ((ml,V[m3]ml) (ml,V[m3]m2))-1 (m2, V [ m 3 ] m l )
(m2,V[m3]m2)
.
(5.213)
It is evident from the underlying geometry that ~4~3 - -~-ml x m2
(5.214)
is the exact solution if the sign is appropriately chosen. Hence, its covariance is V[,~I] = O. (5.215) The residual can be written as ,]
--
w ( l l ) ( m l , m3)2 +
2I~(12)(ml,m3)(m2,m3)+W(22)(m2,m3)2, (5.2 6)
which is a X2 variable with two degrees of freedom. Here, the matrix I~r = (l~ (kl)) is obtained by replacing m l , m2, and Vim3] by ~hl, ~h2, and Vim3], respectively, in eq. (5.213). The orthogonality test takes the form
j >
(5.217)
168
Chapter 5.
Geometric Correction
C. C o r r e c t i o n o f t w o o r i e n t a t i o n s
If m3 is fixed, the rank of the constraint is 3, and the linearized constraint is
(m3, Aml)--(m3, ml),
(m3, Am2) -- (m3, m2),
( A m l , m 2 ) + ( m l , A m 2 ) -- ( m l , m 2 ) ,
AT/%I E {Thl}t ,
Am2 6 {~2}L~.
(5.218)
The optimal correction of m l and m2 is given by
( iml ) _ ( V[~l,1]m3 ~m:
o
O u[m:].%
V[ml]m2 ) W
((m3,~D,1))
V[m~]m~
(m~,m~)
'
(5.2~9)
where W is a (33)-matrix defined by
W = ( ( m 3 , V[ml]m3)
\
(m3,V[m2]m3)
(ml,V[m2lm3) (m3,V[ml]m2) ) -1 (lYt,3,V[m2]m l ) (m2, V[ml]m2) + (ml, V[m2]ml)
(m2, V[~I]TYt3)
The a posteriori covariance matrices of the corrected values
(5.220)
m l and vh2 are
V[~rrt,1]-- Vii1] - w(ll)(i~,r[ml]m3)(~,r[ml]m3)T,
u[,~] VITal,m2]--
-
?Imp] w(~)(?[.~].~)(?[.~]m~) ~, -
-w(i2)(V[ml]m3)(?[m2]m3) T -
U[m2,~l] T,
(5.221)
where the matrix W is obtained by replacing m i and V[mi] by vhi and ? [ m i ] , respectively, in eq. (5.220). The residual can be written as J - Iiv(ll) (m3, m i ) 2 +
21~(12)(m3, mi)(m3,m2)+ Wi22)(m3,m1) 2,
(5.222) which is a ~(2 variable with two degrees of freedom. Hence, the orthogonality test given by eq. (5.217) can be applied.
5.6
Conic Incidence C o r r e c t i o n
Consider a conic (x, Qx) - 0 (see eq. (4.80)). Let x be an image point not on conic (x, Qx) = O. We consider the problem of optimally correcting x so as to make it incident to the conic (x, Qx) = O. In other words, we want to find A x such that 5~ = x - Ax satisfies (~, Q $ ) = 0.
(5.223)
5.6.
Conic Incidence Correction
169
The rank of this constraint is 1. Let V[x] be the a priori covariance matrix of x. The problem can be written as the optimization J-
(Ax, V [ x ] - A x ) ~ min
(5.224)
under the linearized constraint 1
( ~ , Q~) - 5(~, Q~),
(5.225)
The first order solution is given by ~,~
-
(x Q x ) V [ x ] Q x ' 2(x, QV[x]Qx) "
(5.226)
If we put n - N[Qx], eq. (5.226) can be written as
~_
(~,~)v[~]~ 2(., v[~]~)
(5.227)
This problem can be viewed as imposing the incidence constraint on the image point x and its polar (n, x) = 0 with respect to the conic (x, Q x ) = 0 (see eq. (4.82)). The difference in the factor 2 (see eq. (5.82)) is due to the fact that as the image point x approaches, its polar (n, x) - 0 also approaches its pole x by the same distance. The a posteriori covariance matrix of the corrected position :b is
v i i i - v[~]- (v[~]a)(v[~la)~ (a, via]a) '
(5.22s)
i-~ = N[Q~,].
(5.229)
where Eq. (5.228) has the same form as eq. (5.83). Hence, the rank of V[5~] is 1; its null space is {k, n } i , which is orthogonal to the orientation vh - N[k • nl of the polar (~, x) - 0. The residual of J can be written as
j -
(~, Q~)~ 4(~,,QV[x]Q~,)'
(5.230)
which is a X2 variable with one degree of freedom. This fact provides a conic incidence test: the hypothesis that image point x is on conic (x, Q x ) = 0 is rejected with significance level a ~ if
j > ~2,a.
(5.231)
170
Chapter 5.
Geometric Correction
X
Fig. 5.13. Optimal incidence correction for an image point. E x a m p l e 5.6 If each coordinate is perturbed independently by Gaussian noise of mean 0 and variance e 2, the covariance matrix of x is V[x] = e2Pk . The optimal correction (5.226) reduces to
A x - (x, Q X ) P k Q x . 2[[PkQx[[2
(5.232)
Consequently, x is displaced onto the conic perpendicularly (Fig. 5.13). The a posteriori covariance matrix of the corrected value 5~ is
V[5~]- E2ff~vhT,
(5.233)
where rh - N[k x r is the orientation of the polar (r x) = 0. The residual (5.230) can be written as
j-
1 (~, Q~)~ 4e 2 [[pkQS~[[2.
(5.234)
Hence, an unbiased estimator of the variance e2 is obtained in the form
~ = 1 (~, Q:~):
411PkQSc[I 2"
(5.235)
The value ~ thus estimated equals half the distance between the image point and its polar with respect to the conic. If the value e is given a priori, the conic incidence test takes the form
~:2 E--~ > ~2,a.
(5.236)