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Necessary conditions for nonnegativity constrained spectral factorization
Volume 76, number 1
OPTICS COMMUNICATIONS
I ~Xpril1990
NECESSARY CONDITIONS FOR NONNEGATIVITY CONSTRAINED SPECTRAL F A C T O R I Z A T I O N ¢~
D.W. R I C H A R D S O N and N.K. BOSE Department q/'Electrical Engineering, The Spatial and Temporal Signal Processing ('enter, 7he I'ennsyl~'aniaState (?llver,s'ilt' University Park. PA 16802. US.4 Received 11 September 1989; revised manuscript received 5 December 1989
A set of necessary conditions involving the discrete autocorrelation samples is provided lbr checking into the existence of solution for the nonnegativity constrained spectral factorization problem.
1. Motivation and introduction
where
In the following development, we will assume that we are given a valid 2 N + 1 term real-valued autocorrelation sequence {r(l)}, / = - N . . . . . N, with r(-k)=r(k), for k = l , 2 ..... N. The power spectrum S ( z ) is defined as the z-transform of the autocorrelation sequence,
Y, r ( l ) z I.
S(z)= /
(1)
-N
After a change of index,
1 ) can be written as
2N
~ r(k-N)z
S(z)= k
-
-
k .4
(2)
0
Define the autocorrelation polynomial Q ( z ) as follows, 2N
Q(z)=zXS(z)=
Y, q ( k ) z k,
(3)
k--O
where q ( k ) = r ( k - N ) , k = 0 , I ..... 2N and r ( - i ) = r(i), i= 1, 2 ..... N i m p l i e s that q ( l ) = q ( 2 N - l ) , l=0, 1.....
N-1.
We wish to obtain a factorization of the form
Q(z)=F(z)
FT(z -' ) ,
This research was supported by the Office of Naval Research under the Fundamental Research Initiatives Program. /8
t..z(=
,)=zXF(_
i)
such that the coefficients of the Nth degree polynomial F ( z ) are real and nonnegative. This will be referred to as the nonnegativity constrained spectral factorization problem ( N C S F P ) , in contrast to the classical spectral factorization problem [1 ], where (4) holds with the nonnegativity constraint on the coefficients of F ( z ) replaced by the Schur property' on the polynomial F ( = ) . When F ( z ) has all its zeros inside the unit circle in the z-plane, it is defined to satisfy the Schur property, Unlike the classical spectral factorization problem lbr which various iteralive techniques exist for obtaining its unique solution [1, pp. 60-62, 2], the NCSFP has not been solved efficiently, in general. Since the autocorrelalion sequence does not contain phase information, a solution of NCSFP is equivalent to phase-retrieval, which is necessary in a variety of disciplines including radio astronomy, X-ray crystallography, electron microscopy and wave-front sensing. The n o n u n i q u c nature of the solutions to the NCSFP was first considered in ref. [3] and, more recently, in ref. [4]. In ref. [3], the discussion was restricted to the one-dimensional (1D) situation, while in ref. [4], 2D images were addressed. Previous attempts at solving the NCSFP are the enumerative procedure in rel: [5] and the recursive scheme in ref. [6]. The former is
0030-4018/90/$03.50 c(c(c@Elsevier Science Publishers B.V. (North-Holland)
Volume 76, number 1
OPTICS COMMUNICATIONS
a general procedure which is, however, computationally expensive, while the latter applies only to a special class of sub-problems. Since the problem of obtaining a set of necessary as well as sufficient conditions for the existence of a (not necessarily unique) solution to the NCSFP is known to be difficult, we provide here a simple set of necessary conditions which the autocorrelation samples must satisfy. To do this, we begin by comparing coefficients of like powers o f z in eqs. (3) and (4). We have the following system of nonlinear equations in the unknown coefficients of F(z),
qo =fofN,
q, =fofu_, +A fu-~+ ,... + f - , f u - ,
+f f u ,
qN=f2 +f2 + . . . + f 2 _ 1 + f 2 .
(5)
It is this system of equations which must be solved subject to the nonnegativity constraint on the coefficients of F(z). In the next subsection, we derive some necessary conditions for the existence of a solution to the above factorization problem.
Nodd Neven
1 April 1990
qi<~½qN, i = 0 , 1 , . . . , ( N - 1 ) / 2 , q~'-~qN, i = 0 , 1,..., ( N / 2 ) - - I ,
(iiib) IfN~<3,
q~<~½qN, i = 0 , 1 , . . . , N - - 1 . Proof (i) This can be verified immediately by direct computation and comparison with (5). (ii) The solution of system (5) may be interpreted geometrically as the intersection of the hypersurfaces defined by each equation of the system. We also have the constraint in condition (i), which can be viewed as a plane in ( N + 1 )-dimensional euclidean space formed by treatingfo, f~, ..., fu as variables. We can also notice that the last equation in (5) is the equation of a hypersphere of radius x/~N. In order for the system to have a real nonnegative solution, it is necessary that the plane intersect the hypersphere defined by the last equation in (5). This hypersphere is centered at the origin of the ( N + 1 )-dimensional euclidean space. We therefore compute the minimum distance from the origin to the plane by solving the following minimization problem,
2. Main results
min(f~ +f2 +...+f~-i +f2)
Given the system of equations (5), it will be shown that the following relationships are necessary conditions for the existence of a real nonnegative solution. It should be noted that condition (i) in the theorem below is a relationship between the unknown coefficients and the known autocorrelation sequence. Thus condition (i) may be used as a check once the coefficients have been determined.
subject to
Theorem 1. The following conditions are necessary for the system (5) to have a real nonnegative solution.
(i)
g~of =
qN +
i=o q i
.
N--I
(ii)
~ q,<~(N/2)qN. i=O
(iiia) If N > 3 the following cases hold:
)co+f~ +...+fN- l +fN=C, where N--I
c2=qJv + 2 ~ qi.
(6)
i=0
It is easy to show via use of the Lagrange multiplier that the minimum distance sought in this problem is c/v/N+l. In order for a proper intersection to occur, it is necessary that
cl
(7)
Substitution of c from (6) into (7) yields condition (ii). Condition (iii) may be established as follows. Assume N is odd and that i~< ( N - 1 )/2. The following minimization may be performed to find the distance from the origin to the hypersurface given below,
Volume 76, number 1
OPTICS COMMUNICATIONS
m i n t f ~ + f 2 , + . . - + f 7~,-, + f 2 )
q;, / = q2N~ - ~.2N2--1
f o r i = 0 , 1 ..... N,
subject to q, = f o f x
and
.j=0, 1..... N2.
It is desired to find a factorization of the form
, +... + f , f x •
by performing the constrained minimization, we can show that the m i n i m u m distance is x ~ , . Therefore, q,<~qN/2,
1 April 199(i)
i = 0 , 1 ..... ( N - - l ) ~ 2 ,
Q(z,,
22)F(27',
Z2)=I~'(ZI,
- ')z~' z~ '~
•2
(9)
= F ( z i , z 2 ) F T ( z l , Z2) ,
Nodd.
The result for the case when N is even is similarly obtained. It is noted that N~< 3 is a special case which can be determined directly.
such that the coefficients of the polynomial F(z~, z2 ) are nonnegative and F is given by N 1
N2
F(.~,,_~)= Z Z.,.,-,--. f -' -4
(10)
;=0 1=0
We now present a simple numerical example to verify the necessary conditions. E x a m p l e 1. Assume N = 4 and the following autocorrelation polynomial is given,
Again, by comparing coefficients of like powers of--~ and z2, we will obtain the following system of equations in the unknown coefficients of F(z~, -2 ),
k=0 /=0
Q( z ) = 12zS + 26zV + 84z6 + 93z 5 + 146z4+93z3+84zZ+26z+
for i=O, 1..... NI - l, j = O , 1..... 2N2
12.
It can be shown that the corresponding nonnegativity constrained spectral factor for this autocorrelation polynomial is
andi=Ni,j=0,
(11)
1..... N2 .
In the right-hand side of ( 11 ) it is understood that f , / = O outside the range for indices i, j covered in
F( z ) = 2 + 3z + 9z2 + 4z3 + 6z 4 .
(10). The 2D counterpart of theorem 1 is given below.
By performing the computations in (i) and (ii) and (iiia) of theorem 1 it can be shown that the coefficients of both the autocorrelation polynomial and the spectral factor satisfy the necessary conditions. This example also substantiates the fact that the condition in (iiib) does not hold, in general, when N > 3.
Theorem 2. The following conditions are necessary for the system ( 11 ) to have a real nonnegativc solution, Ni
(i)
N2
Z Z./;,; i=0 ]=0
NI---1 2~2
3. Two-dimensional nonnegativity constrained spectral factorization
=
qu,.u2 + 2
In this section, we briefly give the corresponding two-dimensional counterpart for necessary conditions (i), (ii) and (iii) in theorem 1. We shall proceed with the 2D case in a manner consistent with the 1D counterpart, namely, assume we are given a 2D autocorrelation polynomial 2NI 2)72
~
~ qiaz]zA,
i=0j=0
with 10
Z qi.:+2
i=0
:VI -- 1 2N2
Q(zl,z2)=
Z
(8)
(ii)
Z i=0
,,=0
,N2--1
Z
i 1'2
qN,.,
i=0
,%'2 - I
~ q,.;+ j=0
Z
q .....
1=0
~< ½(Nt N2 + N I
+N2)q~,iN2
(iii) ?V~, N2 even, q,.; <<,(qN,.x2)/2
f o r i = 0 , 1 ..... ( N I / 2 ) - I , / = 0 . 1..... ( N 2 / 2 ) - 1 .
.
Volume 76, number 1
OPTICS COMMUNICATIONS
propriate minimization problem will be - 2. If there exists mutual coupling between terms, then the value for the Lagrange multiplier is different from - 2, resulting in more complicated interactions between variables. By constructing an index map, it can be shown that coupling will not occur under the restrictions of condition (iii), and the result follows.
N1, N2 odd,
qi,j<~(qN~.N2)/2
for i = 1, 0 ..... ( N ~ - 1)/2, j = 0 , 1..... ( N 2 - 1 ) / 2 ,
NI even, N2 odd,
qi,j<~ (qNl,U2)/2
for i = 0 , 1, ..., ( N ~ / 2 ) - 1,
j=O, 1, ..., ( N 2 - 1 ) / 2 ,
Example 2. Assume we are given the following 2D autocorrelation array
NI odd, N2 even,
qi,j <~( qNI,N2) / 2,
for i = 0 , 1.... , ( N ~ - 1)/2,
i=O
j = 0 , 1,..., ( N z / 2 ) - I .
i=1
Proof (i) Condition evaluating
(i)
follows immediately after
Q(zl, z2) =F(zl, z2) FT(zl, Z2) at (z~, z z ) = ( 1 , 1). (ii) Condition (i) can be viewed as a plane in (N~ + 1 ) (N2+ 1 )-dimensional euclidean space. This plane must intersect the equation for a hypersphere obtained after setting i=N~ a n d j = N 2 in ( 11 ), in order for a solution to exist. This condition can be established by considering the following minimization problem, // NI
1 April 1990
i=2 Q=i=3 i=4 i=5 i=6
18 21 12 36 38 10 46 60 16 54 86 9 36 60 4 22 38 . 4 12 21 -
9
12 22 36 54 46 36 18
4-
4 9 16 10 12 9_
It can be shown that the corresponding object F will have coefficients as follows, i=1 F=i=2
4 4
i=3
3
"
In this case N1 = 3, N2 = 2 and it can be shown that the conditions ( i ) - ( i i i ) are indeed satisfied.
N2
mint,E=ojX=of ,J) 4. C o n c l u s i o n s
subject to
fi,j=C , where NI -- 1 2N2
c2=qN.,N2 +2 E i=O
N 2 -- 1
2 q,a + 2 2 j=0
i=O
qNl,i .
By performing the minimization via Lagrange multiplier, it can be shown that the distance sought is c/x/(nl + 1 ) (N2 + 1 ). In order for a solution to exist, it is, then, necessary that
c/x/(n , + l ) ( N 2 + l ) ~ < ~ .
(12)
Substituting c into (12) yields the result. (iii) Examination of ( 11 ) shows that if there is no mutual coupling between the terms on the right side, then the value of the Lagrange multiplier in the ap-
The principles used to derive the results in the 1D case are applicable not only to obtain the 2D counterpart, as done in this article, but also in the nD (n > 2) situation. The conditions which are simply to apply, will not only comprise a useful set of necessary conditions which must be satisfied for object recovery from autocorrelation data but might also provide a springboard for the complete satisfactory solution of the nonnegativity constrained spectral factorization problem [7, p. 30]. The complete solution has to come from the minimization problem, where all the constraints including nonnegativity have been simultaneously incorporated.
ll
Volume 76, number l
OPTICS COMMUNICATIONS
References [ 1 ] N.K. Bose, Digital filters (North-Holland Elsevier, New York, 1985). [2] G. Wilson, SIAM. J. Num. Anal. 6 (1969) 1. [ 3 ] Y.M. Bruck and L G . Sodin, Optics Comm. 30 ( 1979 ) 305.
12
1 April 1990
[4] J.R. Fienup, J. Opt. Soc. Am. Comm. 3 (1986) 284. [ 5] M. Nieto-Vesperinas and J.C. Dainty, Optics Commun. 58 (1986) 83. [6] Y.Q. Shi and N.K. Bose, Optics Comm. (1988) 251. [ 7 ] N.K. Bose, Multidimensional systems: progress, directions and open problems ( Reidel, Dordrecht, Holland, 1985 ).
OPTICS COMMUNICATIONS
I ~Xpril1990
NECESSARY CONDITIONS FOR NONNEGATIVITY CONSTRAINED SPECTRAL F A C T O R I Z A T I O N ¢~
D.W. R I C H A R D S O N and N.K. BOSE Department q/'Electrical Engineering, The Spatial and Temporal Signal Processing ('enter, 7he I'ennsyl~'aniaState (?llver,s'ilt' University Park. PA 16802. US.4 Received 11 September 1989; revised manuscript received 5 December 1989
A set of necessary conditions involving the discrete autocorrelation samples is provided lbr checking into the existence of solution for the nonnegativity constrained spectral factorization problem.
1. Motivation and introduction
where
In the following development, we will assume that we are given a valid 2 N + 1 term real-valued autocorrelation sequence {r(l)}, / = - N . . . . . N, with r(-k)=r(k), for k = l , 2 ..... N. The power spectrum S ( z ) is defined as the z-transform of the autocorrelation sequence,
Y, r ( l ) z I.
S(z)= /
(1)
-N
After a change of index,
1 ) can be written as
2N
~ r(k-N)z
S(z)= k
-
-
k .4
(2)
0
Define the autocorrelation polynomial Q ( z ) as follows, 2N
Q(z)=zXS(z)=
Y, q ( k ) z k,
(3)
k--O
where q ( k ) = r ( k - N ) , k = 0 , I ..... 2N and r ( - i ) = r(i), i= 1, 2 ..... N i m p l i e s that q ( l ) = q ( 2 N - l ) , l=0, 1.....
N-1.
We wish to obtain a factorization of the form
Q(z)=F(z)
FT(z -' ) ,
This research was supported by the Office of Naval Research under the Fundamental Research Initiatives Program. /8
t..z(=
,)=zXF(_
i)
such that the coefficients of the Nth degree polynomial F ( z ) are real and nonnegative. This will be referred to as the nonnegativity constrained spectral factorization problem ( N C S F P ) , in contrast to the classical spectral factorization problem [1 ], where (4) holds with the nonnegativity constraint on the coefficients of F ( z ) replaced by the Schur property' on the polynomial F ( = ) . When F ( z ) has all its zeros inside the unit circle in the z-plane, it is defined to satisfy the Schur property, Unlike the classical spectral factorization problem lbr which various iteralive techniques exist for obtaining its unique solution [1, pp. 60-62, 2], the NCSFP has not been solved efficiently, in general. Since the autocorrelalion sequence does not contain phase information, a solution of NCSFP is equivalent to phase-retrieval, which is necessary in a variety of disciplines including radio astronomy, X-ray crystallography, electron microscopy and wave-front sensing. The n o n u n i q u c nature of the solutions to the NCSFP was first considered in ref. [3] and, more recently, in ref. [4]. In ref. [3], the discussion was restricted to the one-dimensional (1D) situation, while in ref. [4], 2D images were addressed. Previous attempts at solving the NCSFP are the enumerative procedure in rel: [5] and the recursive scheme in ref. [6]. The former is
0030-4018/90/$03.50 c(c(c@Elsevier Science Publishers B.V. (North-Holland)
Volume 76, number 1
OPTICS COMMUNICATIONS
a general procedure which is, however, computationally expensive, while the latter applies only to a special class of sub-problems. Since the problem of obtaining a set of necessary as well as sufficient conditions for the existence of a (not necessarily unique) solution to the NCSFP is known to be difficult, we provide here a simple set of necessary conditions which the autocorrelation samples must satisfy. To do this, we begin by comparing coefficients of like powers o f z in eqs. (3) and (4). We have the following system of nonlinear equations in the unknown coefficients of F(z),
qo =fofN,
q, =fofu_, +A fu-~+ ,... + f - , f u - ,
+f f u ,
qN=f2 +f2 + . . . + f 2 _ 1 + f 2 .
(5)
It is this system of equations which must be solved subject to the nonnegativity constraint on the coefficients of F(z). In the next subsection, we derive some necessary conditions for the existence of a solution to the above factorization problem.
Nodd Neven
1 April 1990
qi<~½qN, i = 0 , 1 , . . . , ( N - 1 ) / 2 , q~'-~qN, i = 0 , 1,..., ( N / 2 ) - - I ,
(iiib) IfN~<3,
q~<~½qN, i = 0 , 1 , . . . , N - - 1 . Proof (i) This can be verified immediately by direct computation and comparison with (5). (ii) The solution of system (5) may be interpreted geometrically as the intersection of the hypersurfaces defined by each equation of the system. We also have the constraint in condition (i), which can be viewed as a plane in ( N + 1 )-dimensional euclidean space formed by treatingfo, f~, ..., fu as variables. We can also notice that the last equation in (5) is the equation of a hypersphere of radius x/~N. In order for the system to have a real nonnegative solution, it is necessary that the plane intersect the hypersphere defined by the last equation in (5). This hypersphere is centered at the origin of the ( N + 1 )-dimensional euclidean space. We therefore compute the minimum distance from the origin to the plane by solving the following minimization problem,
2. Main results
min(f~ +f2 +...+f~-i +f2)
Given the system of equations (5), it will be shown that the following relationships are necessary conditions for the existence of a real nonnegative solution. It should be noted that condition (i) in the theorem below is a relationship between the unknown coefficients and the known autocorrelation sequence. Thus condition (i) may be used as a check once the coefficients have been determined.
subject to
Theorem 1. The following conditions are necessary for the system (5) to have a real nonnegative solution.
(i)
g~of =
qN +
i=o q i
.
N--I
(ii)
~ q,<~(N/2)qN. i=O
(iiia) If N > 3 the following cases hold:
)co+f~ +...+fN- l +fN=C, where N--I
c2=qJv + 2 ~ qi.
(6)
i=0
It is easy to show via use of the Lagrange multiplier that the minimum distance sought in this problem is c/v/N+l. In order for a proper intersection to occur, it is necessary that
cl
(7)
Substitution of c from (6) into (7) yields condition (ii). Condition (iii) may be established as follows. Assume N is odd and that i~< ( N - 1 )/2. The following minimization may be performed to find the distance from the origin to the hypersurface given below,
Volume 76, number 1
OPTICS COMMUNICATIONS
m i n t f ~ + f 2 , + . . - + f 7~,-, + f 2 )
q;, / = q2N~ - ~.2N2--1
f o r i = 0 , 1 ..... N,
subject to q, = f o f x
and
.j=0, 1..... N2.
It is desired to find a factorization of the form
, +... + f , f x •
by performing the constrained minimization, we can show that the m i n i m u m distance is x ~ , . Therefore, q,<~qN/2,
1 April 199(i)
i = 0 , 1 ..... ( N - - l ) ~ 2 ,
Q(z,,
22)F(27',
Z2)=I~'(ZI,
- ')z~' z~ '~
•2
(9)
= F ( z i , z 2 ) F T ( z l , Z2) ,
Nodd.
The result for the case when N is even is similarly obtained. It is noted that N~< 3 is a special case which can be determined directly.
such that the coefficients of the polynomial F(z~, z2 ) are nonnegative and F is given by N 1
N2
F(.~,,_~)= Z Z.,.,-,--. f -' -4
(10)
;=0 1=0
We now present a simple numerical example to verify the necessary conditions. E x a m p l e 1. Assume N = 4 and the following autocorrelation polynomial is given,
Again, by comparing coefficients of like powers of--~ and z2, we will obtain the following system of equations in the unknown coefficients of F(z~, -2 ),
k=0 /=0
Q( z ) = 12zS + 26zV + 84z6 + 93z 5 + 146z4+93z3+84zZ+26z+
for i=O, 1..... NI - l, j = O , 1..... 2N2
12.
It can be shown that the corresponding nonnegativity constrained spectral factor for this autocorrelation polynomial is
andi=Ni,j=0,
(11)
1..... N2 .
In the right-hand side of ( 11 ) it is understood that f , / = O outside the range for indices i, j covered in
F( z ) = 2 + 3z + 9z2 + 4z3 + 6z 4 .
(10). The 2D counterpart of theorem 1 is given below.
By performing the computations in (i) and (ii) and (iiia) of theorem 1 it can be shown that the coefficients of both the autocorrelation polynomial and the spectral factor satisfy the necessary conditions. This example also substantiates the fact that the condition in (iiib) does not hold, in general, when N > 3.
Theorem 2. The following conditions are necessary for the system ( 11 ) to have a real nonnegativc solution, Ni
(i)
N2
Z Z./;,; i=0 ]=0
NI---1 2~2
3. Two-dimensional nonnegativity constrained spectral factorization
=
qu,.u2 + 2
In this section, we briefly give the corresponding two-dimensional counterpart for necessary conditions (i), (ii) and (iii) in theorem 1. We shall proceed with the 2D case in a manner consistent with the 1D counterpart, namely, assume we are given a 2D autocorrelation polynomial 2NI 2)72
~
~ qiaz]zA,
i=0j=0
with 10
Z qi.:+2
i=0
:VI -- 1 2N2
Q(zl,z2)=
Z
(8)
(ii)
Z i=0
,,=0
,N2--1
Z
i 1'2
qN,.,
i=0
,%'2 - I
~ q,.;+ j=0
Z
q .....
1=0
~< ½(Nt N2 + N I
+N2)q~,iN2
(iii) ?V~, N2 even, q,.; <<,(qN,.x2)/2
f o r i = 0 , 1 ..... ( N I / 2 ) - I , / = 0 . 1..... ( N 2 / 2 ) - 1 .
.
Volume 76, number 1
OPTICS COMMUNICATIONS
propriate minimization problem will be - 2. If there exists mutual coupling between terms, then the value for the Lagrange multiplier is different from - 2, resulting in more complicated interactions between variables. By constructing an index map, it can be shown that coupling will not occur under the restrictions of condition (iii), and the result follows.
N1, N2 odd,
qi,j<~(qN~.N2)/2
for i = 1, 0 ..... ( N ~ - 1)/2, j = 0 , 1..... ( N 2 - 1 ) / 2 ,
NI even, N2 odd,
qi,j<~ (qNl,U2)/2
for i = 0 , 1, ..., ( N ~ / 2 ) - 1,
j=O, 1, ..., ( N 2 - 1 ) / 2 ,
Example 2. Assume we are given the following 2D autocorrelation array
NI odd, N2 even,
qi,j <~( qNI,N2) / 2,
for i = 0 , 1.... , ( N ~ - 1)/2,
i=O
j = 0 , 1,..., ( N z / 2 ) - I .
i=1
Proof (i) Condition evaluating
(i)
follows immediately after
Q(zl, z2) =F(zl, z2) FT(zl, Z2) at (z~, z z ) = ( 1 , 1). (ii) Condition (i) can be viewed as a plane in (N~ + 1 ) (N2+ 1 )-dimensional euclidean space. This plane must intersect the equation for a hypersphere obtained after setting i=N~ a n d j = N 2 in ( 11 ), in order for a solution to exist. This condition can be established by considering the following minimization problem, // NI
1 April 1990
i=2 Q=i=3 i=4 i=5 i=6
18 21 12 36 38 10 46 60 16 54 86 9 36 60 4 22 38 . 4 12 21 -
9
12 22 36 54 46 36 18
4-
4 9 16 10 12 9_
It can be shown that the corresponding object F will have coefficients as follows, i=1 F=i=2
4 4
i=3
3
"
In this case N1 = 3, N2 = 2 and it can be shown that the conditions ( i ) - ( i i i ) are indeed satisfied.
N2
mint,E=ojX=of ,J) 4. C o n c l u s i o n s
subject to
fi,j=C , where NI -- 1 2N2
c2=qN.,N2 +2 E i=O
N 2 -- 1
2 q,a + 2 2 j=0
i=O
qNl,i .
By performing the minimization via Lagrange multiplier, it can be shown that the distance sought is c/x/(nl + 1 ) (N2 + 1 ). In order for a solution to exist, it is, then, necessary that
c/x/(n , + l ) ( N 2 + l ) ~ < ~ .
(12)
Substituting c into (12) yields the result. (iii) Examination of ( 11 ) shows that if there is no mutual coupling between the terms on the right side, then the value of the Lagrange multiplier in the ap-
The principles used to derive the results in the 1D case are applicable not only to obtain the 2D counterpart, as done in this article, but also in the nD (n > 2) situation. The conditions which are simply to apply, will not only comprise a useful set of necessary conditions which must be satisfied for object recovery from autocorrelation data but might also provide a springboard for the complete satisfactory solution of the nonnegativity constrained spectral factorization problem [7, p. 30]. The complete solution has to come from the minimization problem, where all the constraints including nonnegativity have been simultaneously incorporated.
ll
Volume 76, number l
OPTICS COMMUNICATIONS
References [ 1 ] N.K. Bose, Digital filters (North-Holland Elsevier, New York, 1985). [2] G. Wilson, SIAM. J. Num. Anal. 6 (1969) 1. [ 3 ] Y.M. Bruck and L G . Sodin, Optics Comm. 30 ( 1979 ) 305.
12
1 April 1990
[4] J.R. Fienup, J. Opt. Soc. Am. Comm. 3 (1986) 284. [ 5] M. Nieto-Vesperinas and J.C. Dainty, Optics Commun. 58 (1986) 83. [6] Y.Q. Shi and N.K. Bose, Optics Comm. (1988) 251. [ 7 ] N.K. Bose, Multidimensional systems: progress, directions and open problems ( Reidel, Dordrecht, Holland, 1985 ).