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A perspective theorem and rectification
Photogrammetria - Elsevier Publishing Company, Amsterdam - Printed in The Netherlands
A PERSPECTIVE THEOREM AND RECTIFICATION E. H. THOMPSON Department oJ Photogrammetry and Surveying, University College, London (Great Britain)
(Received July 8, 1965) SUMMARY The paper considers the general theory underlying the setting of an optical rectifier from ground control data. It is shown that the problem amounts to studying if, and when, arbitrary quadrangles can be brought into perspective. The conclusion reached is that all arbitrary quadrangles can be put into perspective, save (but with certain exceptions, however) those related by affine transformations.
THE PROBLEM OF INDIRECT RECTIFICATION Two problems arise in the optical rectification of air photographs: the direct and the indirect. In the former the elements of rectification are known a priori as, for example, when the tilts have been determined by rigorous photogrammetric methods, or when we wish to transform the peripheral pictures of a multMens camera. In the indirect problem the elements are determined by setting up a correspondence between a suitable number of points whose "true" and photographed positions are known. The former problem presents no real difficulty and we do not deal with it here; the latter has, however, been inadequately treated in the literature and the case that arises in practice has been almost completely neglected. My attention has been drawn to a paper by KR.~TKY (1960) which does at least take account of the conditions that arise with photographs of the ground as it really is. The object of Krfitk~'s paper is however different from mine and it does not deal with the matter so fundamentally. The essence of the problem is that when we set up a correspondence in a rectifier between two quadrangles i which is the usual procedure, we are putting two figures into perspective that have not been in perspective before. The reason for this is simply that the topographical surface of the earth, to the accuracy we now expect in photogrammetry, is not a plane, let alone a horizontal plane. The photograph is thus, in general, a central projection of four non-coplanar points, which is not a perspective; while the points we plot on the 1 Four-point figures, no three of whose points are collinear. Photogrammetria, 20 (1965) 143-161
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E.H. THOMPSON
positive plane of the rectifier are orthogonal projections (at a reduced scale) of the ground points. The two quadrangles we bring together in the rectifier are thus arbitrary and it is surely of some importance that we should establish rigorously if and when this can be done, particularly as some exceptions have been known for a long time. It has been ~-trggested that such procedures can hardly be called photogrammetry, let alone rectification; and it is certainly true to say that when a rectifier is used in this way the transformed image will not in general be a map nor will it indeed be a projection to some predetermined plane such as the horizontal1. My excuse for dealing with the matter is that, whether we call it photogrammetry or not, it is what everyone does in practice and if there are to be theories of rectification they should include the one that deals with the common problem. In some private correspondence on this paper it was suggested that the problem is an old one and that it was solved by VON GRUBER (1932, pp. ll--13). A glance at his treatment should convince the reader that he does not even consider the problem, let alone solve it. It is clear from his figures 4, 5 and 6 that he is, in common with almost everyone who has treated the problem, considering figures in distinct planes that are already in perspective. In practice the situation is quite otherwise: the images that appear on the negative are not a perspective projection of the points that are plotted on the table of the rectifier; and what we have to show is that, in spite of this, they can, in general, be put into perspective with them. The discussion in this paper draws attention to the relation between perspective and general projective transformation; a relation inadequately explained, if at all, in photogrammetric text books. Although it is known that figures related by certain affine transformations cannot be put into perspective, it is an outcome of this paper that we arrive at the precise algebraic conditions under which perspective is or is not possible. Since not all affine transformations exclude a perspective relationship between the figures, an exact statement of the conditions is necessary for any complete theory.
PLANE PERSPECTIVE
In photogrammetry it is usual to treat figures in perspective as lying in distinct planes, as is the case in an optical rectifier. For the present purpose it is, however, slightly more convenient to take the related figures as being in the same plane. DESARGUES' (1936) theorem states that, if figures are in perspective, then corresponding lines meet on a line (the axis of perspective). By "figures in perspective" is meant simply plane figures, the joins of corresponding points of which are concurrent in a point (the vertex). The converse of this theorem says that, if corresponding lines 1 But it will be a correct perspective projection to some plane.
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A P E R S P E C T I V E T H E O R E M AND R E C T I F I C A T I O N
meet in collinear points, then plane figures are in perspective. From the theorem and its converse we thus see that if we have two figures in perspective lying in distinct planes we may rotate the planes relative to each other about the axis of perspective until they coincide and the figures, now lying in one plane, will still be in perspective. Conversely, figures in perspective in one plane may be put into distinct planes by reversing the process and they will still be in perspective.
I S O M E T R I C LINES
The argument in this paper is based on consideration of what we call isometric lines and a preliminary word on this may be helpful.
Fig. 1 shows figures in distinct planes ~ and ~' in perspective from a vertex S. Corresponding lines p, q and p', q' meet in corresponding points A and A', respectively. The lines meet by pairs on the axis of perspective. In general, corresponding points, such as A and A', are distinct; as are corresponding lines such as p and p'. However, points lying on the axis of perspective, such as P and P', correspond to themselves; and the axis of perspective corresponds to itself and can be regarded as a line d in ~ coinciding with a line d' in ~'. If now we separate the two planes as in Fig.2 the axis will break up into two distinct lines d and d' having the property that the distances between pairs of their corresponding points such as P, Q and P', Q' must be equal. We call lines such as d and d' isometric. The value of isometric lines is in the reverse process. Suppose, Fig.3, that we have two figures in a plane such that (straight) lines correspond to (straight) lines. Let d and d' be corresponding lines having the
S
A
Q~Q' Fig. l. Photogrammetria, 20 (1965) 143-161
146
E.H. THOMPSON
/ / p'j /
q
//"
_.1/ d
P
Q
p"
/
S'
//
d"
Q'
Fig.2.
I
---____Q
d'
Fig.3. isometric property, then we may rotate and translate one of the figures so that, not only will d coincide with d' (this is possible with any pair of lines), but so that every point on d will coincide with its corresponding point on d'. Moreover this will be possible only with isometric lines. Consider any two corresponding lines l, l', distinct from d and d' respectively, and intersecting the latter in L and L'. When we have superimposed d' and d we will have superimposed L' and L and Fig.4 will result. We will then have arrived at the situation that corresponding lines such as l and l' intersect on a line (did'). By the converse of DESARGUES' theorem the figures will now be in perspective. It follows that if plane figures, in which lines correspond to lines, can be put into perspective at least one pair of isometric lines must exist, one member of the pair being in each figure. The problem of deciding whether figures can be put into perspective is thus reduced simply to looking for isometric pairs. Once a pair has been found a rotation and translation in a plane suffice to achieve the perspective. It is shown below that there are, in general, two pairs of isometric lines in projectively related figures and thus two figures in perspective must have a pair in addition to those coinciding in the axis. Fig.5 shows this.
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A PERSPECTIVE THEOREM AND RECTIFICATION
Fig.5 is the principal plane of a perspective projection from 3 to 3' from a vertex S. The vanishing points in the principal plane are respectively V and V'. Consider a point A in ~ such that V A = D V . If A' corresponds to A then the triangles S A ' V ' and A S V are similar, but when V A = D V ---- V ' S , the triangles are also congruent, so that A ' S = S A . It is then clear that the corresponding lines through A and A' parallel to the axis of perspective are isometric. The result is interesting in that it shows that if two figures are in perspective we may put them into perspective in an alternative and distinct manner by separating the planes and then making the second pair of isometric lines coincide, as in Fig.6. It will be noted that it has been necessary to rotate zt through 180 ° in its own plane to achieve coincidence, since corresponding points run in opposite directions along the isometric lines through A and A' in Fig.5.
//
t/,t '
Fig.4.
A
Fig.5.
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E.H. THOMPSON 17
D'
Fig.6.
PROJECTIVE TRANSFORMATIONAND PERSPECTIVE Consider points in a plane referred to a fixed rectangular coordinate system; and suppose these points transformed according to the scheme: a l l X + a12Y + a13 X, _
azlX + a32Y + az3
(1) a21X + a22Y + am yP
~
.
.
.
.
.
.
.
.
.
azlX ÷ a32Y + a33
where the 3 X 3 matrix of the nine elements a~ is non-singular but otherwise arbitrary. Such a transformation is called pro]ective. It is a fundamental theorem of projective geometry (see SEMELE and KNEEBONE, 1960, p. 399) that the eight rations of the nine elements of this transformation are uniquely determinable if we know the coordinates, before and after transformation, of the points of a quadrangle. In other words, a correspondence between two arbitrary quadrangles determines a projective transformation uniquely and there is no exception to this: any two quadrangles will determine some one projective transformation. This is by no means a new idea to photogrammetry and is, in fact, the theoretical basis of the cross ratio method of graphical rectification, although it will be noted that the assumption, usually made, that the quadrangles have previously been in perspective, is not necessary. The difficulty arises only when we subsequently attempt to put the quadrangles into perspective which we do when we set up a rectifier, for there is nothing obvious in eq.1 that says that figures related in this way are in, or can be Photogrammetria, 20 (1965) 143-161
A PERSPECTIVE THEOREM AND RECTIFICATION
149
put into perspective. Before we can understand the problem we must be clear about the distinction between perspectively and projectively related figtrres. Perspectively related figures can be derived one from the other by a single projection, as in a rectifier. Algebraically their coordinates are related by expressions similar to eq.1, but with the difference that the a~j no longer have the sole restriction that the determinant does not vanish but are further dependent on each other. In other words the 3 )< 3 matrix of the a~ is not the same kind of matrix when it represents a perspective but is more specialised. The technical difference, which does not concern us here, is given in Appendix, Note 1. Descriptively we say that perspectively related figures are obtainable one from the other by a single projection while, in general, projectively related figures (i.e., figures connected by eq.1) can be shown to require up to five successive projections (e.g., ARCHBOLD, 1948, p.205). The question we ask is this, given two projectively related figures in a plane, can they be put into perspective by a relative rotation 1 of the figures? In other words, we ask ourselves if a general projectivity can be factorised into a rotation and a perspective. The answer, which we derive in the next paragraph, is that it is possible provided the projectivity it not a general 2 affine transformation. It then follows that two arbitrary quadrangles, not related by the exceptional affine transformations, can always be brought into perspective correspondence in a unique manner in the sense that the projection of any fifth point is uniquely determined. We argue as follows: the two quadrangles define a unique projectivity which may (with certain exceptions) be reduced to a perspectivity by a relative rotation of the figures (a movement that forms part of the setting procedure). The resulting perspectivity is still a projectivity defined by two quadrangles and is therefore unique.
A PERSPECTIVE THEOREM We express the result indicated towards the end of the last paragraph as a formal theorem: Every plane projectivity, that is not a general affine transformation, when
combined with a suitably chosen rotation, is a perspective. As we have indicated earlier (in the section on isometric lines), our task in proving this amounts simply to looking for isometric lines in figures related by eq.1. We consider two cases: first, when the transformation is not affine and, second, when it is.
1 A rotation and translation in a plane is equivalent to a rotation about some point. 2 We say "general" because not all affine transformations are exceptions. The matter is fully considered in the text.
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E . H . THOMPSON
I
f
Fig.7.
Non-affine transformations An affine transformation is characterised by the fact that points at infinity remain points at infinity 1. This can be the case only if the denominators of eq.1 contain neither X nor Y, i.e., if aal = a32 ---- 0. The non-affine transformations are therefore characterised by the circumstance that not both of aal and a:~2 can be zero; and in this case the equation of the vanishing line v is given by:
a31X -[- a,a2Y + aaa -----0 and it has a definite gradient. Suppose at least one isometric pair exists and let the equation of one member of the pair be:
aX q-bY-kc:O We show that it is necessarily parallel to the vanishing line v and that its equation is of the form:
a31X + a32Y + c = 0 Suppose (Fig.7) that the isometric line l intersects the vanishing line v in A. Then, since A lies on the vanishing line, it will become a point at infinity after transformation. On the other hand B, not on v, will project to a point at a finite distance. Thus the distance A B cannot be invariant, and an isometric line, if one exists, must be parallel to v, z and, since v has a definite gradient and 1 has the s a m e gradient, the result follows. Let X~ and Y~ be the coordinates of any point on l. For such a point:
a31X~ q- a82Y~ ÷ c = 0
1 Photogrammetrists may not like this definition, which is, hoewever, that given in geometrical text-books. The matter is considered in the Appendix, Note 2. '2 We have seen (Fig.5) how the isometric lines in a perspective are obviously parallel to the vanishing lines.
Photogrammetria, 20
(1965) 143-161
A PERSPECTIVE
THEOREM
AND
RECTIFICATION
l 5 1
and, again for points on l, eq.1 m a y be written:
a11Xi q- a12Yi + aaa Xi ~ a33 -
C
a,.X, + a22Y, + a2a Yi* = 333 -- C
C o n s i d e r two points X1, Y~ a n d X2, Y2 on I. W e have: a a l ( X 1 - X2) + 332(Y1- Y2) = 0
XI' - X',2 . Y I ' - Y2' =
6'11(X1- X2) + a 1 2 ( Y 1 - Y2)
.
.
.
.
.
aaa -- c
.
a e l ( X l - X2) + a z z ( Y 1 - Y2) aaa - c
Then:
(Y1 - Y2)
(Y1 - Y'-')
aal(X( - X2') = (al2aal - a11a32)
-a33 -
A2 3 -
C
aaa -- C
(Y1 - Yg) a a f f Y l ' - Y2') ---- (a22a,31- amaa,2)
a33 - C
(Y1 - Y,_,) -
-
ala
a33 -- C
W h e r e A13 and A2a are the cofactors of ala and a2a in the matrix: 11 21
a12 a22
alq 32~ I
31
a32
aaa..l
I!
If the line is isometric: a,.l 2 [ ( X 1 - X2) 2 + ( Y 1 - Y2) 2 ] = a312 [ ( X I ' - X2t) 2 ~- ( Y I ' - Y 2 ' ) 2 ] and, hence: (aal = + aaz 2) (Y1 - Y2) 2 = (Aaa 2 + A 232)(Y1 ( a a a- UY')= C)y so that:
c
=
~ + A,,.~'~~/= aaa _+ 12.2,2 + aa._,"J = 0
T h e r e are thus two p a i r s of i s o m e t r i c lines the first m e m b e r s of which are:
pla2 + A2311/2 aalX -4- aa2Y + a,3a ___ [.a31~ -+- ~ _ ]
= 0
(2)
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E. H. THOMPSON
Evidently they are parallel to and equally spaced on either side of the vanishing line (see Fig.5). The term under the root cannot be zero or the isometric lines would coincide with the vanishing line which is impossible: As a check we show in the Appendix, Note 3 that, if not both a31 and a3.~ are zero, then not both A,3 and A23 can be zero. The second members of the isometric pairs are easily obtained by transforming eq.2 to accented letters but the result is not of interest to us in our present enquiry. A f[ine trans[ormations
We next consider affine transformations which we take (Appendix, Note 2) as: X" : a~lX ~- a12Y + a13
(3) y,
= aelX
Af_ a2eY + a23
with the sole restriction that: a12
el,, 021
~
0
a22
If figures related by this transformation can be put into prespective either the axis of perspective will be the line at infinity or it will not. Suppose, first, that the figures are in perspective with the line at infinity as axis, then, by Desargues' theorem, corresponding lines will meet on this line, i.e., all corresponding lines will be parallel and the figures will be similar. By the converse of Desargues' theorem similar figures can always be put into perspective by a suitable rotation about any point, for corresponding lines can then be brought parallel, meeting on the line at infinity which will be the axis. Thus figures related by forms of eq.3 that are similarity transformations can be put into perspective in an infinite number of ways and these are the only perspectives in which the axis is the line at infinity. Eq.3 represents a similarity transformation if, and only if, a l l = a 2 2 and a,2 = - az, (but see Appendix, Note 4). We are thus left to consider the case where figures related by eq.3 can be put into perspective with an accessible axis, i.e., an axis not at infinity. When we have done this all cases will have been covered. Since a rotation is a special form of eq.3, under it the line at infinity always remains the line at infinity. If therefore a rotation is going to enable us to superimpose isometric lines to give us an accessible axis, the isometric lines nmst be accessible before rotation. We therefore look for real isometric lines in the accessible part of the plane. Photogrammetria, 20 (1965) 143-161
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A PERSPECTIVE THEOREM AND RECTIFICATION
From eq.3 we have, for any two distinct points with coordinates X~, Y1 and -,t"2, Y2: Sl'-S2' YI'-
= a l l ( X 1 - S2) @ al2(Y1- Y2)
Y2" = a m ( X 1 - X2) + a22(Ya- Y_~)
If these points lie on an isometric line: ( X I ' - X 2 ' ) 2 + ( Y I ' - Y 2 ' ) 2 = ( X i - X2) 2 ~[- ( Y 1 - Y2) 2 = (all2 _~ a212) (X 1 _ X2)2 _t_ 2(alia12 -t- a21a22) ( X I - X2) (Y1 - Y2) -~- (a122 q- a222) (Y1 - Y2) 2 If the slope of the line is ~ then: (a122 -~- a222- 1)Q2 + 2(alla12 ~- a21a22)o + (all 2 @ a212- 1) = 0
and the condition that the roots of this equation are real, i.e., that a real isometric line exists, is: (alia12 -4- a~ia22) 2 >~ (a122 + a222 _ 1) (all 2 + a212 - 1)
or, after some manipulation: all a12[ 2 all 2 -- a12z + a2(' + az,~2 ~> 1 +
(4) a21 a22
When this inequality is satisfied the real values of ~ are given by: -(a~1a12 + a21a22) -I- (all 2 -~- a122 -~ a,212 -]- /7222- 1 -
IAI2)'/,
(a122 ~- a22 z - 1)
where: A =
all a12I a21 a221
Since every line with a gradient Q is isometric there are in general two families of parallel isometric lines; but if strict equality holds in eq.4 there is only one. In all these cases the vertex is a point at infinity (see Appendix, Note 2) and when there is only one family of parallel isometric lines it is not difficult to see that the vertex is in a direction perpendicular to the axis of perspective. As an example of the use of the inequality eq.4 see Appendix, Note 5 where a well-known example (ScnWIDEFSKY, 1958, p.179) is considered. The general affine case is considered descriptively and elegantly in KLEIN'S famous book (1939, p.78). In the Appendix, Note 6, the inequality eq.4 is shown to agree with Klein's criterion. Photogrammetria, 20 (1965) 143-161
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E.H.
THOMPSON
It is easily shown that figures related by the general affine transformation, when combined with a similarity transformation (i.e., a rotation and scale change), can always be put into perspective. We have in fact to show that there is always a scalar a' such that the inequality eq.4 is satisfied when the transformation is a A. It is sufficient, and easier, to show that a can be found so as to give strict equality in eq.4. We have to show, then, that we can find a such that: ( a l l 2 + a122 q- a212 -]- a222) a 2 =
1
+
( a l i a 2 2 - a~2a~) 2 a 4
or: (alla22 - a12a21) 2 a 4 - ( a l l 2 -~- aafi +
a212
-~ azfi) a z + 1 = 0
The condition that there shall be a real root f o r a is, in this case (see Appendix, Note 6), that: (axl = + a122 DF" a212 ~- a222)2 ~>.4(aHaz2-alza,.,1) ~ or:
(all 2 + a12e + a~l 2 + a222 + 2allaee - 2av.,a20 (all 2 + aa22 + a212 + + a222 - 2alla2e + 2a12a21) ~> 0 that is: [(al, + ae2) 2 + (a12- a20 ~] [ ( a , 1 - a22)2 + (ale -~- a~l) ~] ~> 0 which is obviously always the case.
CONCLUSIONS
We can summarise the results as follows: (1) The transformation: a11X + a l 2 Y + a13 Xt
~
_
_
az~lX + a.~2Y + a3.~ a,.,1X + a z z Y + ae.~ a31X + a.~zY + a~:3
when combined with a suitable rotation, is a perspective if the transformation is not affine, i.e., if not both a:~a and a82 are zero. When combined with a suitable similarity transformation, it is always a perspective. (2) The proper similarity transformation: X' = aX - bY+
a13
Y ' = - b X + a Y + a~3
when combined with a suitable rotation about any point, is a perspective. Photogrammetria,
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A PERSPECTIVE THEOREM AND RECTIFICATION
155
(3) The general affine transformation: X ' = a l l X + a12Y + a13 Y' = a21X -~- a2zY + a2a
excluding the transformation in (2) is, when combined with a suitable rotation, a perspective provided: a11" A- a122 d- a212 ~- az2z ~> 1 + aal a21
a1212
I
a,2,2
Since a perspective with an accessible axis which is also affine requires the vertex to be a point at infinity, it must be accepted that figures so related cannot be brought into correspondence in a conventional rectifier even though the inequality is satisfied. Our conclusion then is this: correspondence between two quadrangles defines a unique projectivity. When this projectivity it not a certain well-defined type of affine transformation it may be reduced, by a rotation of one of the figures, to a perspective which, being itself a projectivity determined from two quadrangles, is unique. Hence the correspondence of two arbitrary quadrangles defines, with certain exceptions, a unique perspective. By unique we mean that the transformation of any fifth point is thereby uniquely determined.
ACKNOWLEDGMENTS
I am grateful to Mr. J. W. Archbold, Dr. P. Du Val and Mr. H. Kestelman, all of University College London, for having allowed me to discuss this problem with them and for having made a number of helpful suggestions. A draft of this paper was seen by several colleagues and as a result of their comments the presentation is, I hope, clearer. In this connexion I am particularly grateful to Professor W. Schermerhorn, Professor Bertil Hallert, Professor Van der Weele and Mr. J. Visser. The manuscript was read by Mr. Naji Tawfik and Mr. M. W. Grist of University College London and I am grateful to them for having corrected a number of mistakes. I do not hold them responsible for having overlooked any that may remain.
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E . H . THOMPSON
REFERENCES ARCHBOLD, J. W., 1948. The Algebraic Geometry o] a Plane. Arnold, London, 300 pp. DESARGUES, G., 1936. Mdthode Universelle de Mettre en Perspective les Objets Donn&. Appendix, 1: Proposition G6ometrique. Paris. (The original text is not extant. The theorem is given in Bosse, A., 1648. ManiOre Universelle de Mr. Desargues pour Pratiquer la Perspective par Petit-Pied. Imprimerie de Pierre Des-Hayes, Paris, 343 pp.) KLEIN, F., 1939. Elementary Mathematics / t o m an Advanced Standpoint. 2. Geometry. Constable, London, 213 pp. (Translation). KR~,XKC,', V., 1960. Affine rectification of vertical air photographs of non-flat ground. Intern. Arch. Photogrammetry, 13(4). SCnWIOEFSKV, K., 1959. A n Outline of Photogrammetry. Pitman, London. 326 pp. (Translation). SEMELE, J. G. and KNEEBONE, G. T., 1960. Algebraic Projective Geometry. Oxford Univ. Press, London, 404 pp. VON GRUBER, O., 1932. Collected Lectures and Essays. C h a p m a n and Hall, London, 454 pp. (Translation).
APPENDIX Note 1 Let A be the matrix of the nine elements. In the general projectivity A is arbitrary with the sole proviso that it is non-singular, i.e., [ A [ ~- O. The roots of its characteristic equation:
rA-~I] =0 are all three distinct and each distinct root gives rise to a single united point, i.e., a point that corresponds to itself. If, however, A represents a perspectivity then two of the roots of the characteristic equation are the same and the repeated root (say 21) results in a matrix A - 21 I which has rank 1. This, in its turn, gives rise to a "line" of united points (the axis of perspective) while the third root gives rise to a single united point (the vertex).
Note 2 A / f i n e trans[ormations In the general projective transformation: X" = aa~X + aa2Y + a13 aalX + aazY + aaa
(1) Y ' = a,2lX -k a22Y -Jr-a23 aalX + aa2Y -k aaa there are accessible points that transform to points at infinity, vlz., any point lying on the (vanishing) line: aalX + aa2Y + aaa = 0 In the special case however when an1 := aa2 = 0 the transformation reduces, effectively, to: X ' = aaaX ÷ a j 2 Y + ala
(2)
y , = a21X + a22Y + a2a Photogrammetria, 20 (1965) 14:3-16 !
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A PERSPECTIVE THEOREM AND RECTIFICATION for %3 cannot then be zero in eq.1 or the matrix of the transformation, which is now:
11 a12 a13 21 [~ 0a22 aa233 would be singular. W e m a y thus divide out by a33. It is clear that points at infinity are now the only points that t r a n s f o r m to points at infinity. T h e feature that distinguishes affine t r a n s f o r m a t i o n s for p h o t o g r a m m e t r i s t s appears to be their p r o p e r t y that scale is constant in any given direction, but, in general, varies as the direction is changed. This is easily deduced f r o m eq.2. Consider a displacement f r o m X1, Y1 to X 2, Y2 and put X 2 - X x = A X, Y 2 - Y1 AY so that A X and AY are the c o m p o n e n t s of the displacement vector. It follows at once that:
IX7 alq [~] Yj = pll R121a2d Thus the components of the transformed vector are homogeneous linear functions of the original components and do not depend upon the position of the displacement in the plane. The scale is thus a function of direction only and, for the same reason, parallel lines transf o r m to parallel lines. The last result is useful in the present context for it shows that, if a perspective with an accessible axis is affine, the vertex must be a point at infinity or parallelism could not be preserved.
Note 3
r h e t r a n s f o r m a t i o n in h o m o g e n e o u s coordinates is:
I~'i1, pll 1031a12 a32 alI la2l
a22
a2 a a3
Ii 1
and the inverse t r a n s f o r m a t i o n is:
Ii I ~ 11 12A21 A22A31~ A32I F;il 12,
.*,3
L w'__l
where A~j is the cofactor o f % . This t r a n s f o r m a t i o n must also be non-affine and therefore not both A13 and A2a can be zero.
Note 4
W h e n a l l -- a22 and a12 -- - a 2 1 the t r a n s f o r m a t i o n takes the form: X'--
aX - bY+
ala
Y ' = b X ~- a Y + a23
and:
a -b ]
l
b
a2 + b 2
a P h o t o g r a m m e t r i a , 20 (1965) 143-161
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E . H . THOMPSON
which is positive. This transformation is sometimes called in contrast to the improper transformation:
proper similarity
transformation
X ' = - a X + bY + alz Y'= bX + aY + am where:
I - a=b-l ( a 2 + b 2 ) b which is negative. Figures obtained by this last transformation are indeed similar to the original figures but are, in addition, mirror reflexions of them. Corresponding lines in figures related by such a transformation cannot therefore he brought simultaneously parallel by a rotation. If such figures can be brought into perspective, the axis cannot then be the line at infinity and we may use the criterion (eq.4) to study the problem.
[allb12[ 2=(a2+b2)21721 ~a, and: ala 2 + aao2 + a212 + a222- 1 : 2(a 2 + b 2) - 1 If a 2 + b 2 = 1 we have equality in eq.4 and a perspective is possible. This transformation is a rotation and reflexion without change of scale. F o r values of a and b such that a 2 + b 2~- 1 we must thus have: 2 (a 2 + b 2) > 1 + (a 2 + b2)2 or: (a 2 + b 2 - 1)2 < 0 But: (a 2 + b2 _ 1)2 is positive when a2 + b 3 ~ 1 and hence the condition cannot be satisfied. We conclude that figures related by an improper similarity transformation can be put into perspective only in the special case when the transformation is a rotation and a reflexion.
Note 5 A picture (principal distance ]) is taken parallel to and a distance D from a ground plane inclined to the horizontal at an angle ~ (@ 0) (Fig.8). The orthogonal projections of ground points are plotted at a scale m on the positive plane of a rectifier having its vertex at infinity (otherwise rectification is clearly impossible). We first construct the general projective transformation corresponding to this case. Apart from translation the relationship between the photocoordinates and the orthogonal projection when the figures are arbitrarily placed in a common plane is:
where a and b are the elements of an orthogonal matrix so that a 2 + b 2 = 1. Multiplying out the square matrices: EX--mD
Ec°st9-bcos
Photogrammetria, 20
(1965) 143-161
A PERSPECTIVE THEOREM AND RECTIFICATION
/
159
\o
Fig.8. The determinant of the transformation is: m2O 2 [: - cos and the inequality (eq.4) gives:
m20'-' -/.~
(1 + cos :~v~)
~
(m2V2
1+ \
a9
cos
as the required condition. This analysis is given as a simple example of the use of the inequality, and it is not suggested that the problem could not be, nor has not been, solved in a more direct way.
Note 6 Very briefly, KLEIN (1939) argues as follows. In the present context an affine transformation can be considered as transforming a circle with centre at the origin into an ellipse also with its centre at the origin. This ellipse will either intersect the circle in four (real) points or will lie wholly inside or outside the circle (apart, that is to say, from borderline cases). In the first case we can take the line joining a pair of opposite points of intersection as the axis of perspective but in the last two cases we cannot do this. Klein's criterion is then simply whether the transformed circle intersects the original circle in four real points or not. Klein's argument seems to fail in the case of a similarity transformation, for the circle now projects into a circle of a different size. We take our transformation to be:
Where: A ~r
all
a12]
La x Photogrammetria, 20 (1965) 143-16l
160
E . H . THOMPSON
Let the circle be:
(XY) E X - q =a2 Then the ellipse is: (XY) A T - 1 A - I [ - y X
3
=a 2
That is: (a212 + a222)X2- 2(alia21 +
a~2a22)XY + (aaa 2 + a122)y2 = a 2 1 A 12
To obtain the points of intersection of the circle and ellipse we substitute for Y from the equation of the circle, putting: y2 = a 2 _ X 2 and: y
= -+- (a 2_X2)%
in the equation of the ellipse. The resulting equation, after the surd has been removed, is: (p2 + 4Q2)X4 +
2a2(pR_ 202)X 2 + a4R2 = 0
Where: P = (a212 + a222 - a a l 2 -a122) Q = (alia21 + a12a22) R = (a112 + a122 - I A
12)
If we write the quartic:
pX 4 + qX 2 + r = 0 we have:
X = ± ~ q +- (q2-4pr)~/~-~ and the conditions that all four values of X shall be real are:
q2 ~ 4pr q ~ r
0
(i.e., q is zero or negative)
~> 0
(i.e., r is zero or positive)
when we have taken p positive. In the present case the first condition gives:
(PR 202) 2 ~> R2(P 2 + 4Q 2) -
That is:
Q2 ~ R(R + P) If this is satisfied then:
Q2 ~ RP
(a fortiori)
and: 202 ~
RP
so that the second condition is also satisfied. Moreover since r = (a2R) 2 the third condition
Photogrammetria, 20 (1965) 143-161
A PERSPECTIVE THEOREM AND RECTIFICATION
161
is always satisfied. When expressed in terms of the elements of the transformation matrix it is easily verified that:
Q2 ~ R(R ÷ P) is our inequality (4), viz: all 2 -4- a122 d,-a212 Jr- a222 ~
1 ~+- I A 12
Photogrammetria, 20 (1965) 143-161
A PERSPECTIVE THEOREM AND RECTIFICATION E. H. THOMPSON Department oJ Photogrammetry and Surveying, University College, London (Great Britain)
(Received July 8, 1965) SUMMARY The paper considers the general theory underlying the setting of an optical rectifier from ground control data. It is shown that the problem amounts to studying if, and when, arbitrary quadrangles can be brought into perspective. The conclusion reached is that all arbitrary quadrangles can be put into perspective, save (but with certain exceptions, however) those related by affine transformations.
THE PROBLEM OF INDIRECT RECTIFICATION Two problems arise in the optical rectification of air photographs: the direct and the indirect. In the former the elements of rectification are known a priori as, for example, when the tilts have been determined by rigorous photogrammetric methods, or when we wish to transform the peripheral pictures of a multMens camera. In the indirect problem the elements are determined by setting up a correspondence between a suitable number of points whose "true" and photographed positions are known. The former problem presents no real difficulty and we do not deal with it here; the latter has, however, been inadequately treated in the literature and the case that arises in practice has been almost completely neglected. My attention has been drawn to a paper by KR.~TKY (1960) which does at least take account of the conditions that arise with photographs of the ground as it really is. The object of Krfitk~'s paper is however different from mine and it does not deal with the matter so fundamentally. The essence of the problem is that when we set up a correspondence in a rectifier between two quadrangles i which is the usual procedure, we are putting two figures into perspective that have not been in perspective before. The reason for this is simply that the topographical surface of the earth, to the accuracy we now expect in photogrammetry, is not a plane, let alone a horizontal plane. The photograph is thus, in general, a central projection of four non-coplanar points, which is not a perspective; while the points we plot on the 1 Four-point figures, no three of whose points are collinear. Photogrammetria, 20 (1965) 143-161
144
E.H. THOMPSON
positive plane of the rectifier are orthogonal projections (at a reduced scale) of the ground points. The two quadrangles we bring together in the rectifier are thus arbitrary and it is surely of some importance that we should establish rigorously if and when this can be done, particularly as some exceptions have been known for a long time. It has been ~-trggested that such procedures can hardly be called photogrammetry, let alone rectification; and it is certainly true to say that when a rectifier is used in this way the transformed image will not in general be a map nor will it indeed be a projection to some predetermined plane such as the horizontal1. My excuse for dealing with the matter is that, whether we call it photogrammetry or not, it is what everyone does in practice and if there are to be theories of rectification they should include the one that deals with the common problem. In some private correspondence on this paper it was suggested that the problem is an old one and that it was solved by VON GRUBER (1932, pp. ll--13). A glance at his treatment should convince the reader that he does not even consider the problem, let alone solve it. It is clear from his figures 4, 5 and 6 that he is, in common with almost everyone who has treated the problem, considering figures in distinct planes that are already in perspective. In practice the situation is quite otherwise: the images that appear on the negative are not a perspective projection of the points that are plotted on the table of the rectifier; and what we have to show is that, in spite of this, they can, in general, be put into perspective with them. The discussion in this paper draws attention to the relation between perspective and general projective transformation; a relation inadequately explained, if at all, in photogrammetric text books. Although it is known that figures related by certain affine transformations cannot be put into perspective, it is an outcome of this paper that we arrive at the precise algebraic conditions under which perspective is or is not possible. Since not all affine transformations exclude a perspective relationship between the figures, an exact statement of the conditions is necessary for any complete theory.
PLANE PERSPECTIVE
In photogrammetry it is usual to treat figures in perspective as lying in distinct planes, as is the case in an optical rectifier. For the present purpose it is, however, slightly more convenient to take the related figures as being in the same plane. DESARGUES' (1936) theorem states that, if figures are in perspective, then corresponding lines meet on a line (the axis of perspective). By "figures in perspective" is meant simply plane figures, the joins of corresponding points of which are concurrent in a point (the vertex). The converse of this theorem says that, if corresponding lines 1 But it will be a correct perspective projection to some plane.
Photogrammetria,
20 (1965) 143 161
145
A P E R S P E C T I V E T H E O R E M AND R E C T I F I C A T I O N
meet in collinear points, then plane figures are in perspective. From the theorem and its converse we thus see that if we have two figures in perspective lying in distinct planes we may rotate the planes relative to each other about the axis of perspective until they coincide and the figures, now lying in one plane, will still be in perspective. Conversely, figures in perspective in one plane may be put into distinct planes by reversing the process and they will still be in perspective.
I S O M E T R I C LINES
The argument in this paper is based on consideration of what we call isometric lines and a preliminary word on this may be helpful.
Fig. 1 shows figures in distinct planes ~ and ~' in perspective from a vertex S. Corresponding lines p, q and p', q' meet in corresponding points A and A', respectively. The lines meet by pairs on the axis of perspective. In general, corresponding points, such as A and A', are distinct; as are corresponding lines such as p and p'. However, points lying on the axis of perspective, such as P and P', correspond to themselves; and the axis of perspective corresponds to itself and can be regarded as a line d in ~ coinciding with a line d' in ~'. If now we separate the two planes as in Fig.2 the axis will break up into two distinct lines d and d' having the property that the distances between pairs of their corresponding points such as P, Q and P', Q' must be equal. We call lines such as d and d' isometric. The value of isometric lines is in the reverse process. Suppose, Fig.3, that we have two figures in a plane such that (straight) lines correspond to (straight) lines. Let d and d' be corresponding lines having the
S
A
Q~Q' Fig. l. Photogrammetria, 20 (1965) 143-161
146
E.H. THOMPSON
/ / p'j /
q
//"
_.1/ d
P
Q
p"
/
S'
//
d"
Q'
Fig.2.
I
---____Q
d'
Fig.3. isometric property, then we may rotate and translate one of the figures so that, not only will d coincide with d' (this is possible with any pair of lines), but so that every point on d will coincide with its corresponding point on d'. Moreover this will be possible only with isometric lines. Consider any two corresponding lines l, l', distinct from d and d' respectively, and intersecting the latter in L and L'. When we have superimposed d' and d we will have superimposed L' and L and Fig.4 will result. We will then have arrived at the situation that corresponding lines such as l and l' intersect on a line (did'). By the converse of DESARGUES' theorem the figures will now be in perspective. It follows that if plane figures, in which lines correspond to lines, can be put into perspective at least one pair of isometric lines must exist, one member of the pair being in each figure. The problem of deciding whether figures can be put into perspective is thus reduced simply to looking for isometric pairs. Once a pair has been found a rotation and translation in a plane suffice to achieve the perspective. It is shown below that there are, in general, two pairs of isometric lines in projectively related figures and thus two figures in perspective must have a pair in addition to those coinciding in the axis. Fig.5 shows this.
Photogrammetria, 20 (1965) 143-161
147
A PERSPECTIVE THEOREM AND RECTIFICATION
Fig.5 is the principal plane of a perspective projection from 3 to 3' from a vertex S. The vanishing points in the principal plane are respectively V and V'. Consider a point A in ~ such that V A = D V . If A' corresponds to A then the triangles S A ' V ' and A S V are similar, but when V A = D V ---- V ' S , the triangles are also congruent, so that A ' S = S A . It is then clear that the corresponding lines through A and A' parallel to the axis of perspective are isometric. The result is interesting in that it shows that if two figures are in perspective we may put them into perspective in an alternative and distinct manner by separating the planes and then making the second pair of isometric lines coincide, as in Fig.6. It will be noted that it has been necessary to rotate zt through 180 ° in its own plane to achieve coincidence, since corresponding points run in opposite directions along the isometric lines through A and A' in Fig.5.
//
t/,t '
Fig.4.
A
Fig.5.
Photogrammetria, 20 (1965) 143-161
148
E.H. THOMPSON 17
D'
Fig.6.
PROJECTIVE TRANSFORMATIONAND PERSPECTIVE Consider points in a plane referred to a fixed rectangular coordinate system; and suppose these points transformed according to the scheme: a l l X + a12Y + a13 X, _
azlX + a32Y + az3
(1) a21X + a22Y + am yP
~
.
.
.
.
.
.
.
.
.
azlX ÷ a32Y + a33
where the 3 X 3 matrix of the nine elements a~ is non-singular but otherwise arbitrary. Such a transformation is called pro]ective. It is a fundamental theorem of projective geometry (see SEMELE and KNEEBONE, 1960, p. 399) that the eight rations of the nine elements of this transformation are uniquely determinable if we know the coordinates, before and after transformation, of the points of a quadrangle. In other words, a correspondence between two arbitrary quadrangles determines a projective transformation uniquely and there is no exception to this: any two quadrangles will determine some one projective transformation. This is by no means a new idea to photogrammetry and is, in fact, the theoretical basis of the cross ratio method of graphical rectification, although it will be noted that the assumption, usually made, that the quadrangles have previously been in perspective, is not necessary. The difficulty arises only when we subsequently attempt to put the quadrangles into perspective which we do when we set up a rectifier, for there is nothing obvious in eq.1 that says that figures related in this way are in, or can be Photogrammetria, 20 (1965) 143-161
A PERSPECTIVE THEOREM AND RECTIFICATION
149
put into perspective. Before we can understand the problem we must be clear about the distinction between perspectively and projectively related figtrres. Perspectively related figures can be derived one from the other by a single projection, as in a rectifier. Algebraically their coordinates are related by expressions similar to eq.1, but with the difference that the a~j no longer have the sole restriction that the determinant does not vanish but are further dependent on each other. In other words the 3 )< 3 matrix of the a~ is not the same kind of matrix when it represents a perspective but is more specialised. The technical difference, which does not concern us here, is given in Appendix, Note 1. Descriptively we say that perspectively related figures are obtainable one from the other by a single projection while, in general, projectively related figures (i.e., figures connected by eq.1) can be shown to require up to five successive projections (e.g., ARCHBOLD, 1948, p.205). The question we ask is this, given two projectively related figures in a plane, can they be put into perspective by a relative rotation 1 of the figures? In other words, we ask ourselves if a general projectivity can be factorised into a rotation and a perspective. The answer, which we derive in the next paragraph, is that it is possible provided the projectivity it not a general 2 affine transformation. It then follows that two arbitrary quadrangles, not related by the exceptional affine transformations, can always be brought into perspective correspondence in a unique manner in the sense that the projection of any fifth point is uniquely determined. We argue as follows: the two quadrangles define a unique projectivity which may (with certain exceptions) be reduced to a perspectivity by a relative rotation of the figures (a movement that forms part of the setting procedure). The resulting perspectivity is still a projectivity defined by two quadrangles and is therefore unique.
A PERSPECTIVE THEOREM We express the result indicated towards the end of the last paragraph as a formal theorem: Every plane projectivity, that is not a general affine transformation, when
combined with a suitably chosen rotation, is a perspective. As we have indicated earlier (in the section on isometric lines), our task in proving this amounts simply to looking for isometric lines in figures related by eq.1. We consider two cases: first, when the transformation is not affine and, second, when it is.
1 A rotation and translation in a plane is equivalent to a rotation about some point. 2 We say "general" because not all affine transformations are exceptions. The matter is fully considered in the text.
Photogrammetria, 20 (1965) 143-161
150
E . H . THOMPSON
I
f
Fig.7.
Non-affine transformations An affine transformation is characterised by the fact that points at infinity remain points at infinity 1. This can be the case only if the denominators of eq.1 contain neither X nor Y, i.e., if aal = a32 ---- 0. The non-affine transformations are therefore characterised by the circumstance that not both of aal and a:~2 can be zero; and in this case the equation of the vanishing line v is given by:
a31X -[- a,a2Y + aaa -----0 and it has a definite gradient. Suppose at least one isometric pair exists and let the equation of one member of the pair be:
aX q-bY-kc:O We show that it is necessarily parallel to the vanishing line v and that its equation is of the form:
a31X + a32Y + c = 0 Suppose (Fig.7) that the isometric line l intersects the vanishing line v in A. Then, since A lies on the vanishing line, it will become a point at infinity after transformation. On the other hand B, not on v, will project to a point at a finite distance. Thus the distance A B cannot be invariant, and an isometric line, if one exists, must be parallel to v, z and, since v has a definite gradient and 1 has the s a m e gradient, the result follows. Let X~ and Y~ be the coordinates of any point on l. For such a point:
a31X~ q- a82Y~ ÷ c = 0
1 Photogrammetrists may not like this definition, which is, hoewever, that given in geometrical text-books. The matter is considered in the Appendix, Note 2. '2 We have seen (Fig.5) how the isometric lines in a perspective are obviously parallel to the vanishing lines.
Photogrammetria, 20
(1965) 143-161
A PERSPECTIVE
THEOREM
AND
RECTIFICATION
l 5 1
and, again for points on l, eq.1 m a y be written:
a11Xi q- a12Yi + aaa Xi ~ a33 -
C
a,.X, + a22Y, + a2a Yi* = 333 -- C
C o n s i d e r two points X1, Y~ a n d X2, Y2 on I. W e have: a a l ( X 1 - X2) + 332(Y1- Y2) = 0
XI' - X',2 . Y I ' - Y2' =
6'11(X1- X2) + a 1 2 ( Y 1 - Y2)
.
.
.
.
.
aaa -- c
.
a e l ( X l - X2) + a z z ( Y 1 - Y2) aaa - c
Then:
(Y1 - Y2)
(Y1 - Y'-')
aal(X( - X2') = (al2aal - a11a32)
-a33 -
A2 3 -
C
aaa -- C
(Y1 - Yg) a a f f Y l ' - Y2') ---- (a22a,31- amaa,2)
a33 - C
(Y1 - Y,_,) -
-
ala
a33 -- C
W h e r e A13 and A2a are the cofactors of ala and a2a in the matrix: 11 21
a12 a22
alq 32~ I
31
a32
aaa..l
I!
If the line is isometric: a,.l 2 [ ( X 1 - X2) 2 + ( Y 1 - Y2) 2 ] = a312 [ ( X I ' - X2t) 2 ~- ( Y I ' - Y 2 ' ) 2 ] and, hence: (aal = + aaz 2) (Y1 - Y2) 2 = (Aaa 2 + A 232)(Y1 ( a a a- UY')= C)y so that:
c
=
~ + A,,.~'~~/= aaa _+ 12.2,2 + aa._,"J = 0
T h e r e are thus two p a i r s of i s o m e t r i c lines the first m e m b e r s of which are:
pla2 + A2311/2 aalX -4- aa2Y + a,3a ___ [.a31~ -+- ~ _ ]
= 0
(2)
Photogrammetria, 20 (1965) 143-161
152
E. H. THOMPSON
Evidently they are parallel to and equally spaced on either side of the vanishing line (see Fig.5). The term under the root cannot be zero or the isometric lines would coincide with the vanishing line which is impossible: As a check we show in the Appendix, Note 3 that, if not both a31 and a3.~ are zero, then not both A,3 and A23 can be zero. The second members of the isometric pairs are easily obtained by transforming eq.2 to accented letters but the result is not of interest to us in our present enquiry. A f[ine trans[ormations
We next consider affine transformations which we take (Appendix, Note 2) as: X" : a~lX ~- a12Y + a13
(3) y,
= aelX
Af_ a2eY + a23
with the sole restriction that: a12
el,, 021
~
0
a22
If figures related by this transformation can be put into prespective either the axis of perspective will be the line at infinity or it will not. Suppose, first, that the figures are in perspective with the line at infinity as axis, then, by Desargues' theorem, corresponding lines will meet on this line, i.e., all corresponding lines will be parallel and the figures will be similar. By the converse of Desargues' theorem similar figures can always be put into perspective by a suitable rotation about any point, for corresponding lines can then be brought parallel, meeting on the line at infinity which will be the axis. Thus figures related by forms of eq.3 that are similarity transformations can be put into perspective in an infinite number of ways and these are the only perspectives in which the axis is the line at infinity. Eq.3 represents a similarity transformation if, and only if, a l l = a 2 2 and a,2 = - az, (but see Appendix, Note 4). We are thus left to consider the case where figures related by eq.3 can be put into perspective with an accessible axis, i.e., an axis not at infinity. When we have done this all cases will have been covered. Since a rotation is a special form of eq.3, under it the line at infinity always remains the line at infinity. If therefore a rotation is going to enable us to superimpose isometric lines to give us an accessible axis, the isometric lines nmst be accessible before rotation. We therefore look for real isometric lines in the accessible part of the plane. Photogrammetria, 20 (1965) 143-161
153
A PERSPECTIVE THEOREM AND RECTIFICATION
From eq.3 we have, for any two distinct points with coordinates X~, Y1 and -,t"2, Y2: Sl'-S2' YI'-
= a l l ( X 1 - S2) @ al2(Y1- Y2)
Y2" = a m ( X 1 - X2) + a22(Ya- Y_~)
If these points lie on an isometric line: ( X I ' - X 2 ' ) 2 + ( Y I ' - Y 2 ' ) 2 = ( X i - X2) 2 ~[- ( Y 1 - Y2) 2 = (all2 _~ a212) (X 1 _ X2)2 _t_ 2(alia12 -t- a21a22) ( X I - X2) (Y1 - Y2) -~- (a122 q- a222) (Y1 - Y2) 2 If the slope of the line is ~ then: (a122 -~- a222- 1)Q2 + 2(alla12 ~- a21a22)o + (all 2 @ a212- 1) = 0
and the condition that the roots of this equation are real, i.e., that a real isometric line exists, is: (alia12 -4- a~ia22) 2 >~ (a122 + a222 _ 1) (all 2 + a212 - 1)
or, after some manipulation: all a12[ 2 all 2 -- a12z + a2(' + az,~2 ~> 1 +
(4) a21 a22
When this inequality is satisfied the real values of ~ are given by: -(a~1a12 + a21a22) -I- (all 2 -~- a122 -~ a,212 -]- /7222- 1 -
IAI2)'/,
(a122 ~- a22 z - 1)
where: A =
all a12I a21 a221
Since every line with a gradient Q is isometric there are in general two families of parallel isometric lines; but if strict equality holds in eq.4 there is only one. In all these cases the vertex is a point at infinity (see Appendix, Note 2) and when there is only one family of parallel isometric lines it is not difficult to see that the vertex is in a direction perpendicular to the axis of perspective. As an example of the use of the inequality eq.4 see Appendix, Note 5 where a well-known example (ScnWIDEFSKY, 1958, p.179) is considered. The general affine case is considered descriptively and elegantly in KLEIN'S famous book (1939, p.78). In the Appendix, Note 6, the inequality eq.4 is shown to agree with Klein's criterion. Photogrammetria, 20 (1965) 143-161
154
E.H.
THOMPSON
It is easily shown that figures related by the general affine transformation, when combined with a similarity transformation (i.e., a rotation and scale change), can always be put into perspective. We have in fact to show that there is always a scalar a' such that the inequality eq.4 is satisfied when the transformation is a A. It is sufficient, and easier, to show that a can be found so as to give strict equality in eq.4. We have to show, then, that we can find a such that: ( a l l 2 + a122 q- a212 -]- a222) a 2 =
1
+
( a l i a 2 2 - a~2a~) 2 a 4
or: (alla22 - a12a21) 2 a 4 - ( a l l 2 -~- aafi +
a212
-~ azfi) a z + 1 = 0
The condition that there shall be a real root f o r a is, in this case (see Appendix, Note 6), that: (axl = + a122 DF" a212 ~- a222)2 ~>.4(aHaz2-alza,.,1) ~ or:
(all 2 + a12e + a~l 2 + a222 + 2allaee - 2av.,a20 (all 2 + aa22 + a212 + + a222 - 2alla2e + 2a12a21) ~> 0 that is: [(al, + ae2) 2 + (a12- a20 ~] [ ( a , 1 - a22)2 + (ale -~- a~l) ~] ~> 0 which is obviously always the case.
CONCLUSIONS
We can summarise the results as follows: (1) The transformation: a11X + a l 2 Y + a13 Xt
~
_
_
az~lX + a.~2Y + a3.~ a,.,1X + a z z Y + ae.~ a31X + a.~zY + a~:3
when combined with a suitable rotation, is a perspective if the transformation is not affine, i.e., if not both a:~a and a82 are zero. When combined with a suitable similarity transformation, it is always a perspective. (2) The proper similarity transformation: X' = aX - bY+
a13
Y ' = - b X + a Y + a~3
when combined with a suitable rotation about any point, is a perspective. Photogrammetria,
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A PERSPECTIVE THEOREM AND RECTIFICATION
155
(3) The general affine transformation: X ' = a l l X + a12Y + a13 Y' = a21X -~- a2zY + a2a
excluding the transformation in (2) is, when combined with a suitable rotation, a perspective provided: a11" A- a122 d- a212 ~- az2z ~> 1 + aal a21
a1212
I
a,2,2
Since a perspective with an accessible axis which is also affine requires the vertex to be a point at infinity, it must be accepted that figures so related cannot be brought into correspondence in a conventional rectifier even though the inequality is satisfied. Our conclusion then is this: correspondence between two quadrangles defines a unique projectivity. When this projectivity it not a certain well-defined type of affine transformation it may be reduced, by a rotation of one of the figures, to a perspective which, being itself a projectivity determined from two quadrangles, is unique. Hence the correspondence of two arbitrary quadrangles defines, with certain exceptions, a unique perspective. By unique we mean that the transformation of any fifth point is thereby uniquely determined.
ACKNOWLEDGMENTS
I am grateful to Mr. J. W. Archbold, Dr. P. Du Val and Mr. H. Kestelman, all of University College London, for having allowed me to discuss this problem with them and for having made a number of helpful suggestions. A draft of this paper was seen by several colleagues and as a result of their comments the presentation is, I hope, clearer. In this connexion I am particularly grateful to Professor W. Schermerhorn, Professor Bertil Hallert, Professor Van der Weele and Mr. J. Visser. The manuscript was read by Mr. Naji Tawfik and Mr. M. W. Grist of University College London and I am grateful to them for having corrected a number of mistakes. I do not hold them responsible for having overlooked any that may remain.
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E . H . THOMPSON
REFERENCES ARCHBOLD, J. W., 1948. The Algebraic Geometry o] a Plane. Arnold, London, 300 pp. DESARGUES, G., 1936. Mdthode Universelle de Mettre en Perspective les Objets Donn&. Appendix, 1: Proposition G6ometrique. Paris. (The original text is not extant. The theorem is given in Bosse, A., 1648. ManiOre Universelle de Mr. Desargues pour Pratiquer la Perspective par Petit-Pied. Imprimerie de Pierre Des-Hayes, Paris, 343 pp.) KLEIN, F., 1939. Elementary Mathematics / t o m an Advanced Standpoint. 2. Geometry. Constable, London, 213 pp. (Translation). KR~,XKC,', V., 1960. Affine rectification of vertical air photographs of non-flat ground. Intern. Arch. Photogrammetry, 13(4). SCnWIOEFSKV, K., 1959. A n Outline of Photogrammetry. Pitman, London. 326 pp. (Translation). SEMELE, J. G. and KNEEBONE, G. T., 1960. Algebraic Projective Geometry. Oxford Univ. Press, London, 404 pp. VON GRUBER, O., 1932. Collected Lectures and Essays. C h a p m a n and Hall, London, 454 pp. (Translation).
APPENDIX Note 1 Let A be the matrix of the nine elements. In the general projectivity A is arbitrary with the sole proviso that it is non-singular, i.e., [ A [ ~- O. The roots of its characteristic equation:
rA-~I] =0 are all three distinct and each distinct root gives rise to a single united point, i.e., a point that corresponds to itself. If, however, A represents a perspectivity then two of the roots of the characteristic equation are the same and the repeated root (say 21) results in a matrix A - 21 I which has rank 1. This, in its turn, gives rise to a "line" of united points (the axis of perspective) while the third root gives rise to a single united point (the vertex).
Note 2 A / f i n e trans[ormations In the general projective transformation: X" = aa~X + aa2Y + a13 aalX + aazY + aaa
(1) Y ' = a,2lX -k a22Y -Jr-a23 aalX + aa2Y -k aaa there are accessible points that transform to points at infinity, vlz., any point lying on the (vanishing) line: aalX + aa2Y + aaa = 0 In the special case however when an1 := aa2 = 0 the transformation reduces, effectively, to: X ' = aaaX ÷ a j 2 Y + ala
(2)
y , = a21X + a22Y + a2a Photogrammetria, 20 (1965) 14:3-16 !
157
A PERSPECTIVE THEOREM AND RECTIFICATION for %3 cannot then be zero in eq.1 or the matrix of the transformation, which is now:
11 a12 a13 21 [~ 0a22 aa233 would be singular. W e m a y thus divide out by a33. It is clear that points at infinity are now the only points that t r a n s f o r m to points at infinity. T h e feature that distinguishes affine t r a n s f o r m a t i o n s for p h o t o g r a m m e t r i s t s appears to be their p r o p e r t y that scale is constant in any given direction, but, in general, varies as the direction is changed. This is easily deduced f r o m eq.2. Consider a displacement f r o m X1, Y1 to X 2, Y2 and put X 2 - X x = A X, Y 2 - Y1 AY so that A X and AY are the c o m p o n e n t s of the displacement vector. It follows at once that:
IX7 alq [~] Yj = pll R121a2d Thus the components of the transformed vector are homogeneous linear functions of the original components and do not depend upon the position of the displacement in the plane. The scale is thus a function of direction only and, for the same reason, parallel lines transf o r m to parallel lines. The last result is useful in the present context for it shows that, if a perspective with an accessible axis is affine, the vertex must be a point at infinity or parallelism could not be preserved.
Note 3
r h e t r a n s f o r m a t i o n in h o m o g e n e o u s coordinates is:
I~'i1, pll 1031a12 a32 alI la2l
a22
a2 a a3
Ii 1
and the inverse t r a n s f o r m a t i o n is:
Ii I ~ 11 12A21 A22A31~ A32I F;il 12,
.*,3
L w'__l
where A~j is the cofactor o f % . This t r a n s f o r m a t i o n must also be non-affine and therefore not both A13 and A2a can be zero.
Note 4
W h e n a l l -- a22 and a12 -- - a 2 1 the t r a n s f o r m a t i o n takes the form: X'--
aX - bY+
ala
Y ' = b X ~- a Y + a23
and:
a -b ]
l
b
a2 + b 2
a P h o t o g r a m m e t r i a , 20 (1965) 143-161
158
E . H . THOMPSON
which is positive. This transformation is sometimes called in contrast to the improper transformation:
proper similarity
transformation
X ' = - a X + bY + alz Y'= bX + aY + am where:
I - a=b-l ( a 2 + b 2 ) b which is negative. Figures obtained by this last transformation are indeed similar to the original figures but are, in addition, mirror reflexions of them. Corresponding lines in figures related by such a transformation cannot therefore he brought simultaneously parallel by a rotation. If such figures can be brought into perspective, the axis cannot then be the line at infinity and we may use the criterion (eq.4) to study the problem.
[allb12[ 2=(a2+b2)21721 ~a, and: ala 2 + aao2 + a212 + a222- 1 : 2(a 2 + b 2) - 1 If a 2 + b 2 = 1 we have equality in eq.4 and a perspective is possible. This transformation is a rotation and reflexion without change of scale. F o r values of a and b such that a 2 + b 2~- 1 we must thus have: 2 (a 2 + b 2) > 1 + (a 2 + b2)2 or: (a 2 + b 2 - 1)2 < 0 But: (a 2 + b2 _ 1)2 is positive when a2 + b 3 ~ 1 and hence the condition cannot be satisfied. We conclude that figures related by an improper similarity transformation can be put into perspective only in the special case when the transformation is a rotation and a reflexion.
Note 5 A picture (principal distance ]) is taken parallel to and a distance D from a ground plane inclined to the horizontal at an angle ~ (@ 0) (Fig.8). The orthogonal projections of ground points are plotted at a scale m on the positive plane of a rectifier having its vertex at infinity (otherwise rectification is clearly impossible). We first construct the general projective transformation corresponding to this case. Apart from translation the relationship between the photocoordinates and the orthogonal projection when the figures are arbitrarily placed in a common plane is:
where a and b are the elements of an orthogonal matrix so that a 2 + b 2 = 1. Multiplying out the square matrices: EX--mD
Ec°st9-bcos
Photogrammetria, 20
(1965) 143-161
A PERSPECTIVE THEOREM AND RECTIFICATION
/
159
\o
Fig.8. The determinant of the transformation is: m2O 2 [: - cos and the inequality (eq.4) gives:
m20'-' -/.~
(1 + cos :~v~)
~
(m2V2
1+ \
a9
cos
as the required condition. This analysis is given as a simple example of the use of the inequality, and it is not suggested that the problem could not be, nor has not been, solved in a more direct way.
Note 6 Very briefly, KLEIN (1939) argues as follows. In the present context an affine transformation can be considered as transforming a circle with centre at the origin into an ellipse also with its centre at the origin. This ellipse will either intersect the circle in four (real) points or will lie wholly inside or outside the circle (apart, that is to say, from borderline cases). In the first case we can take the line joining a pair of opposite points of intersection as the axis of perspective but in the last two cases we cannot do this. Klein's criterion is then simply whether the transformed circle intersects the original circle in four real points or not. Klein's argument seems to fail in the case of a similarity transformation, for the circle now projects into a circle of a different size. We take our transformation to be:
Where: A ~r
all
a12]
La x Photogrammetria, 20 (1965) 143-16l
160
E . H . THOMPSON
Let the circle be:
(XY) E X - q =a2 Then the ellipse is: (XY) A T - 1 A - I [ - y X
3
=a 2
That is: (a212 + a222)X2- 2(alia21 +
a~2a22)XY + (aaa 2 + a122)y2 = a 2 1 A 12
To obtain the points of intersection of the circle and ellipse we substitute for Y from the equation of the circle, putting: y2 = a 2 _ X 2 and: y
= -+- (a 2_X2)%
in the equation of the ellipse. The resulting equation, after the surd has been removed, is: (p2 + 4Q2)X4 +
2a2(pR_ 202)X 2 + a4R2 = 0
Where: P = (a212 + a222 - a a l 2 -a122) Q = (alia21 + a12a22) R = (a112 + a122 - I A
12)
If we write the quartic:
pX 4 + qX 2 + r = 0 we have:
X = ± ~ q +- (q2-4pr)~/~-~ and the conditions that all four values of X shall be real are:
q2 ~ 4pr q ~ r
0
(i.e., q is zero or negative)
~> 0
(i.e., r is zero or positive)
when we have taken p positive. In the present case the first condition gives:
(PR 202) 2 ~> R2(P 2 + 4Q 2) -
That is:
Q2 ~ R(R + P) If this is satisfied then:
Q2 ~ RP
(a fortiori)
and: 202 ~
RP
so that the second condition is also satisfied. Moreover since r = (a2R) 2 the third condition
Photogrammetria, 20 (1965) 143-161
A PERSPECTIVE THEOREM AND RECTIFICATION
161
is always satisfied. When expressed in terms of the elements of the transformation matrix it is easily verified that:
Q2 ~ R(R ÷ P) is our inequality (4), viz: all 2 -4- a122 d,-a212 Jr- a222 ~
1 ~+- I A 12
Photogrammetria, 20 (1965) 143-161