- Email: [email protected]
Shannon number and degrees of freedom of an image
Volume
7, number
2
OPTICS
SHANNON
NUMBER
I:cbruary
COMMUNICATIONS
AND DEGREES
OF FREEDOM
1973
OF AN IMAGE
F. GORI and G. GUATTARI Istituto di Fisica-Facoltci di Ingegneria-University of Rome, Rome, Italy and Gruppo Nazionale Struttura della Materia de1 C.N.R., Italy Received Revised manuscript
9 November 1972 received 4 December
1972
The problem of the determination of the number of significant degrees of freedom equivalence with the Shannon number is shown to hold with a great generality.
1. Introduction In a series of fundamental papers [ 1, 21 Toraldo di Francis showed that the image of a finite object obtained through a finite pupil possesses only a finite number of degrees of freedom. Using the sampling theorem, Toraldo di Francis was able to show that such a finite number is given by the product of the object area times the accessible Fourier area. The resulting number is called the Shannon number. This result was also pointed out by Gabor [3]4. Subsequently, some authors questioned the exactness of the Toraldo’s result [6] . In fact, due to the analyticity properties of the Fourier transform of the finite object field, the image should have the same infinite number of degrees of freedom as the object. The problem is strictly analogous to the one-dimensional problem, arising in electrical communication theory, about the number of degrees of freedom of a signal of finite bandwidth observed for a finite time. Such a problem has been solved by Slepian, Pollak and Landau through the use of the prolate spheroidal wavefunctions [7V9]. These authors showed that the number of degrees of freedom of the signal can be idenf A mathematical
analysis of Gabor’s conclusions can be found in ref. [4]. The degrees of freedom of optical wave fields have also been treated by Winthrop [ 5 1 in connection with recording and propagation processes.
of an image is considered.
The
tified with the number of eigenvalues significantly different from zero. The thorough study of the pertinent eigenvalucs and eigenfunctions led to the conclusion that the eigenvalues, ordered in a decreasing manner, are nearly unity until a critical value of the index is reached, then nearly zero. Such a behaviour allows the assumption that the number of degrees of freedom equals the sum of the eigenvalues. It turns out that this sum is just equal to the Shannon number [9, p. 13 131 An extension of this result to the optical domain was made by Walther [IO]. Nevertheless, the generality of this extension could be questioned because the eigenfunctions and the corresponding eigenvalues arc not known for arbitrary two-dimensional geometry. Indeed, Toraldo di Francis has shown [ 1 11 that also in the case of circular geometry, whose eigenvalue problem has been solved [ 12, 131 , an estimate of the number of degrees of freedom based on the available numerical data about the eigenvalues is only in a rough agreement with the Shannon number. In this paper we show that the equivalence between the number of degrees of freedom and the Shannon number can be established with a great generality, by making use of some theorems proved by Landau in connection with the study of density conditions for sampling of entire functions [ 141
163
J. rua,!’ 1973
OPTICS COMMUNICATIONS
Volume 7, number 2
By introducing
2. Theory Let us briefly the object
eigenfunctions consider
recall how the relationship the image
and
of the imaging equation
an object
in terms of 1151. Let us
A imaged through
of finite extent
P. Under the usual assumptions
a pupil of finite extent
[ 161, the relation
between
be examined
cm
between
the object
eq. (4) into eq. (1) we obtain
and the image
As we can assign a degree of freedom to every @,2(x), comparison of eqs. (5) and (6) shows that all the object degrees
can be written
of freedom
are transferred
into the image.
However.
the effect of the imaging process is to multiby the correply the weight cil of each eigenfunction sponding eigcnvalues h,, It is a general feature of the eigenvalues
where X,J’ arc to be thought coordinates
for the one-dimensional
sional case respectively. ilnaget-y through the object
of as scalar or vectorial
Limiting
tegral equation
and two-dimenourselves
Lero with increasing
to coherent
clear pupils, I(x) and O(x) represent
and image field distributions
and S(x)
is
noise. very small eigenvalues carried
ofA
II. regardless
[ 171 For the unavoidable formation
of any in-
of the same kind as eq. (3) to tend to and P geometries
presence
of some kind of‘
imply
the loss of the in-
by the corresponding
eigenfunctions.
given by
This is why only a finite number
S(X) = lexp(
ale effective in the image formation process. In the oiie-diniensional case the object is limited
-2nivx)dv.
the interval
P
where
the frequency
v has the same dimensions
and _I’. Let us now consider
as x
the homogeneous
integral
equation (DJy).S(.x--y)dy
(3)
= A,, +,t(~).
of freedom
(~ X/2. X/2) and the pupil is a perfect
pass filter stopping as usual, (’ = nlVX,
to low
all the frequencies lvl> W. Putting. the pertinent eigenfunctions are
those corresponding
s
of degrees
to the kernel
S(x) = sin(2nWx)/7rx.
(7)
/I
It can be shown
that the kernel
is posi-
They are the well known
eq. (3) has a com-
tions +,2(~,, x) extensively
of this equation
[ 121 As a consequence.
tive definite
plete set of orthogonal
eigenfunctions
to a countably
set of real positive
infinite
unity eigenvalues. rem it follows formly S(x
c
development
A,, *,7(x>
and less than
franI the Mercer theo-
that the kernel S(x-~9)
convergent
y) =
Furtherlnore,
corresponding
admits
the uni-
[ 171
@,“;Cl’).
object
can be written
O(x) =
c=o
(‘,I
Q,)(X),
the complex conjugate. of the eigenfunctions.
studied
spheroidal by Slepian
wavefuncet al.
[7 -0, 181 *. These authors have shown that the corresponding eigenvalues X,, remain approximately unity until a critical
value II = jlcrit is reached.
For II > I[,,~~.
the X,, fall off to /era very rapidly. In other words. the h,, bebaviour is quite similar to that of a step function. This allows one to assume the eigenvalue SUIIIas a measure of the number of significant degrees of freedom. The eigenvalue
it=0
where the asterisk denotes Due to the completeness
prolate
sum turns out to be just equal to the Shannon
number I”. p. 131.31 Let us consider now the two-dimensional the
as
the simple
rectangular
geometry
the extension
case. For of the
preceding result is straightforward. Another geometr)’ of interest is that of circular object and pupil. In this
(5)
case the pertinent
eigenfunctions
91-c the circular
prolate
12
where (‘,, = s O(x) @;(x)dX.
164
’ Signlt’icanr applications of the prolate q~heroidal wave-func. tions to the optical domain have been made by Barnes 119) and by Kushfort and Harris [ 201. A complete review of the properties and the applications of the prolate spheroidal ~vavefunctions can bc found in ref. [ 211
Volume
7, number
Z!
OPTICS
COMMUNICATIONS
functions introduced by Slepian and Heurtley [ 12. 131 For these eigenfunctions the eigenvalue knowledge is not so detailed as for the prolate spheroidal wavefunctions [ 1% -2 I ). Toraldo di Francis has shown that the available data allow an estimate of the number of sig nificant degrees of freedom that is only in a rough agreement with the Shannon number. In fact, fcr a circular object of radius R and a circular pupil of radius W, an estimate of 4R2 W2 significant degrees of freedom can be made while the Shannon number is
n2R2W2 [II]. We want to observe now that if the step function behaviour of the eigenvalues were a feature shared by every geometry, the eigenvaiue sum could always be assumed as a measure of the number of significant degrees of freedom. This would eliminate the need for a detailed knowledge of the eigenvalues pertinent to every particular geometry. To show that this is the case, we have only to translate to the optical field some mathematical results established by Landau [ 141 in connection with the study of density conditions for sampling of entire functions. In ref. [ 141 it has been shown that, for any (sufficiently regular) geometry, eq. (3) admits a set of eigenvalues characterized by a step function behaviour. This allows one to assume the eigenvalue sum as a measure of the number of significant degrees of freedom. The evaluation of the eigenvalue sum requires only the application of well known mathematical properties [ 17, 141. It follows, in fact, from the Mercer theorem that for a positive kernel S(x,y) the eigenvalue sum is given by the kernel trace, i.e., E x,, = JS(x,x)dx, I?= 0 A
(8)
where the integral is extended to the object area A. As the kernel of eq. (3) depends on x and y only through their difference, from eq. (2) we have S(x,x) = S(0) = P,
(9)
where P is the pupil area. Then, eq. (8) becomes
5 x,, = JPdx n=O
= AP,
( 10)
A
that is. the eigenvalue sum turns out to be equal to the Shannon number for arbitrary geometry.
I:ebruary
1973
3. Conclusions We have shown that the equivalence between the Shannon number and the number of significant degrees of freedom of an image can be established in a general way. This is obtained by making use of some theorems proved by Landau, concerning the general behaviour of the imaging equation eigenvalues. This result does not require the solution of the particular integral imaging equation connected to the geometry under consideration.
Acknowledgement We thank the referee for useful bibliographic gestions.
sug
References [I] G. Toraldo di Francis, J. Opt. Sot. Am. 45 (1955) 497 [ 21 G. Toraldo di Francis, Trans. IRE AP-4 (1956) 473. [31 D. Gabor, in: Progress in optics, Vol. 1, ed. E. Wolf (North-Holland, Amsterdam, 196 I). J. Opt. Sec. Am. 50 (1960) 856. [41 K. Miyamoto, ISI J.T. Winthrop, IBM J. Res. Develop. 14 (1970) 501; J. Opt. Sot. Am. 61 (1971) 15. [61 H. Walter, in: Progress in optics, Vol. 1, ed. E. Wolf (North-Holland, Amsterdam, 196 1). [7l D. Slepian and H.O. Pollak, Bell System Tech. J. 40 (196 1) 43. [81 H.J. Landau and H.O. Pollak, Bell System Tech. J. 40 (1961) 65. 191 H.J. Landau and H.O. Pollak, Bell System Tech. J. 41 (1962) 1295. A. Walther, J. Opt. Sot. Am. 57 (1967) 639. [lOI G. Toraldo di Francis, J. Opt. Sot. Am. 59 (1969) 799. illI D. Slepian, Bell System Tech. J. 43 (1964) 3009. 1121 J.C. Heurtley. in: Proc. Symp. on Quasi-Optics, cd. J. 1:0x I131 (North-Holland, Amsterdam, 1970). H.J. Landau, Acta Mathematics 117 (1967) 37. [I41 J.W. Goodman, in: Progress in optics, Vol. 8, ed. E. Wolf [ISI (North-Holland, Amsterdam, 1970). J.W. Goodman, Introduction to the I:ouricr optic% [I61 (McGraw-liill, New York, 1968). 1’. Riesz and B.S7. Nagy, I;unctional analysis (Ungar, New [I71 York, 1955). D. Slepian and E. Sonnenblick, Bell System Tech. J. 44 [I81 (1965) 1745. C.W. Barnes, J. Opt. Sot. Am. 56 (1966) 575. 1191 1201 C.K. RusM‘ort and R.W. llarris, .I. Opt. Sot. Am. 58 (1968) 539. B.R. Flieden, in: Progress in optics, Vol. 9. ed. E. Wolf 1211 (North-Holland, Amsterdam, 1971).
165
7, number
2
OPTICS
SHANNON
NUMBER
I:cbruary
COMMUNICATIONS
AND DEGREES
OF FREEDOM
1973
OF AN IMAGE
F. GORI and G. GUATTARI Istituto di Fisica-Facoltci di Ingegneria-University of Rome, Rome, Italy and Gruppo Nazionale Struttura della Materia de1 C.N.R., Italy Received Revised manuscript
9 November 1972 received 4 December
1972
The problem of the determination of the number of significant degrees of freedom equivalence with the Shannon number is shown to hold with a great generality.
1. Introduction In a series of fundamental papers [ 1, 21 Toraldo di Francis showed that the image of a finite object obtained through a finite pupil possesses only a finite number of degrees of freedom. Using the sampling theorem, Toraldo di Francis was able to show that such a finite number is given by the product of the object area times the accessible Fourier area. The resulting number is called the Shannon number. This result was also pointed out by Gabor [3]4. Subsequently, some authors questioned the exactness of the Toraldo’s result [6] . In fact, due to the analyticity properties of the Fourier transform of the finite object field, the image should have the same infinite number of degrees of freedom as the object. The problem is strictly analogous to the one-dimensional problem, arising in electrical communication theory, about the number of degrees of freedom of a signal of finite bandwidth observed for a finite time. Such a problem has been solved by Slepian, Pollak and Landau through the use of the prolate spheroidal wavefunctions [7V9]. These authors showed that the number of degrees of freedom of the signal can be idenf A mathematical
analysis of Gabor’s conclusions can be found in ref. [4]. The degrees of freedom of optical wave fields have also been treated by Winthrop [ 5 1 in connection with recording and propagation processes.
of an image is considered.
The
tified with the number of eigenvalues significantly different from zero. The thorough study of the pertinent eigenvalucs and eigenfunctions led to the conclusion that the eigenvalues, ordered in a decreasing manner, are nearly unity until a critical value of the index is reached, then nearly zero. Such a behaviour allows the assumption that the number of degrees of freedom equals the sum of the eigenvalues. It turns out that this sum is just equal to the Shannon number [9, p. 13 131 An extension of this result to the optical domain was made by Walther [IO]. Nevertheless, the generality of this extension could be questioned because the eigenfunctions and the corresponding eigenvalues arc not known for arbitrary two-dimensional geometry. Indeed, Toraldo di Francis has shown [ 1 11 that also in the case of circular geometry, whose eigenvalue problem has been solved [ 12, 131 , an estimate of the number of degrees of freedom based on the available numerical data about the eigenvalues is only in a rough agreement with the Shannon number. In this paper we show that the equivalence between the number of degrees of freedom and the Shannon number can be established with a great generality, by making use of some theorems proved by Landau in connection with the study of density conditions for sampling of entire functions [ 141
163
J. rua,!’ 1973
OPTICS COMMUNICATIONS
Volume 7, number 2
By introducing
2. Theory Let us briefly the object
eigenfunctions consider
recall how the relationship the image
and
of the imaging equation
an object
in terms of 1151. Let us
A imaged through
of finite extent
P. Under the usual assumptions
a pupil of finite extent
[ 161, the relation
between
be examined
cm
between
the object
eq. (4) into eq. (1) we obtain
and the image
As we can assign a degree of freedom to every @,2(x), comparison of eqs. (5) and (6) shows that all the object degrees
can be written
of freedom
are transferred
into the image.
However.
the effect of the imaging process is to multiby the correply the weight cil of each eigenfunction sponding eigcnvalues h,, It is a general feature of the eigenvalues
where X,J’ arc to be thought coordinates
for the one-dimensional
sional case respectively. ilnaget-y through the object
of as scalar or vectorial
Limiting
tegral equation
and two-dimenourselves
Lero with increasing
to coherent
clear pupils, I(x) and O(x) represent
and image field distributions
and S(x)
is
noise. very small eigenvalues carried
ofA
II. regardless
[ 171 For the unavoidable formation
of any in-
of the same kind as eq. (3) to tend to and P geometries
presence
of some kind of‘
imply
the loss of the in-
by the corresponding
eigenfunctions.
given by
This is why only a finite number
S(X) = lexp(
ale effective in the image formation process. In the oiie-diniensional case the object is limited
-2nivx)dv.
the interval
P
where
the frequency
v has the same dimensions
and _I’. Let us now consider
as x
the homogeneous
integral
equation (DJy).S(.x--y)dy
(3)
= A,, +,t(~).
of freedom
(~ X/2. X/2) and the pupil is a perfect
pass filter stopping as usual, (’ = nlVX,
to low
all the frequencies lvl> W. Putting. the pertinent eigenfunctions are
those corresponding
s
of degrees
to the kernel
S(x) = sin(2nWx)/7rx.
(7)
/I
It can be shown
that the kernel
is posi-
They are the well known
eq. (3) has a com-
tions +,2(~,, x) extensively
of this equation
[ 121 As a consequence.
tive definite
plete set of orthogonal
eigenfunctions
to a countably
set of real positive
infinite
unity eigenvalues. rem it follows formly S(x
c
development
A,, *,7(x>
and less than
franI the Mercer theo-
that the kernel S(x-~9)
convergent
y) =
Furtherlnore,
corresponding
admits
the uni-
[ 171
@,“;Cl’).
object
can be written
O(x) =
c=o
(‘,I
Q,)(X),
the complex conjugate. of the eigenfunctions.
studied
spheroidal by Slepian
wavefuncet al.
[7 -0, 181 *. These authors have shown that the corresponding eigenvalues X,, remain approximately unity until a critical
value II = jlcrit is reached.
For II > I[,,~~.
the X,, fall off to /era very rapidly. In other words. the h,, bebaviour is quite similar to that of a step function. This allows one to assume the eigenvalue SUIIIas a measure of the number of significant degrees of freedom. The eigenvalue
it=0
where the asterisk denotes Due to the completeness
prolate
sum turns out to be just equal to the Shannon
number I”. p. 131.31 Let us consider now the two-dimensional the
as
the simple
rectangular
geometry
the extension
case. For of the
preceding result is straightforward. Another geometr)’ of interest is that of circular object and pupil. In this
(5)
case the pertinent
eigenfunctions
91-c the circular
prolate
12
where (‘,, = s O(x) @;(x)dX.
164
’ Signlt’icanr applications of the prolate q~heroidal wave-func. tions to the optical domain have been made by Barnes 119) and by Kushfort and Harris [ 201. A complete review of the properties and the applications of the prolate spheroidal ~vavefunctions can bc found in ref. [ 211
Volume
7, number
Z!
OPTICS
COMMUNICATIONS
functions introduced by Slepian and Heurtley [ 12. 131 For these eigenfunctions the eigenvalue knowledge is not so detailed as for the prolate spheroidal wavefunctions [ 1% -2 I ). Toraldo di Francis has shown that the available data allow an estimate of the number of sig nificant degrees of freedom that is only in a rough agreement with the Shannon number. In fact, fcr a circular object of radius R and a circular pupil of radius W, an estimate of 4R2 W2 significant degrees of freedom can be made while the Shannon number is
n2R2W2 [II]. We want to observe now that if the step function behaviour of the eigenvalues were a feature shared by every geometry, the eigenvaiue sum could always be assumed as a measure of the number of significant degrees of freedom. This would eliminate the need for a detailed knowledge of the eigenvalues pertinent to every particular geometry. To show that this is the case, we have only to translate to the optical field some mathematical results established by Landau [ 141 in connection with the study of density conditions for sampling of entire functions. In ref. [ 141 it has been shown that, for any (sufficiently regular) geometry, eq. (3) admits a set of eigenvalues characterized by a step function behaviour. This allows one to assume the eigenvalue sum as a measure of the number of significant degrees of freedom. The evaluation of the eigenvalue sum requires only the application of well known mathematical properties [ 17, 141. It follows, in fact, from the Mercer theorem that for a positive kernel S(x,y) the eigenvalue sum is given by the kernel trace, i.e., E x,, = JS(x,x)dx, I?= 0 A
(8)
where the integral is extended to the object area A. As the kernel of eq. (3) depends on x and y only through their difference, from eq. (2) we have S(x,x) = S(0) = P,
(9)
where P is the pupil area. Then, eq. (8) becomes
5 x,, = JPdx n=O
= AP,
( 10)
A
that is. the eigenvalue sum turns out to be equal to the Shannon number for arbitrary geometry.
I:ebruary
1973
3. Conclusions We have shown that the equivalence between the Shannon number and the number of significant degrees of freedom of an image can be established in a general way. This is obtained by making use of some theorems proved by Landau, concerning the general behaviour of the imaging equation eigenvalues. This result does not require the solution of the particular integral imaging equation connected to the geometry under consideration.
Acknowledgement We thank the referee for useful bibliographic gestions.
sug
References [I] G. Toraldo di Francis, J. Opt. Sot. Am. 45 (1955) 497 [ 21 G. Toraldo di Francis, Trans. IRE AP-4 (1956) 473. [31 D. Gabor, in: Progress in optics, Vol. 1, ed. E. Wolf (North-Holland, Amsterdam, 196 I). J. Opt. Sec. Am. 50 (1960) 856. [41 K. Miyamoto, ISI J.T. Winthrop, IBM J. Res. Develop. 14 (1970) 501; J. Opt. Sot. Am. 61 (1971) 15. [61 H. Walter, in: Progress in optics, Vol. 1, ed. E. Wolf (North-Holland, Amsterdam, 196 1). [7l D. Slepian and H.O. Pollak, Bell System Tech. J. 40 (196 1) 43. [81 H.J. Landau and H.O. Pollak, Bell System Tech. J. 40 (1961) 65. 191 H.J. Landau and H.O. Pollak, Bell System Tech. J. 41 (1962) 1295. A. Walther, J. Opt. Sot. Am. 57 (1967) 639. [lOI G. Toraldo di Francis, J. Opt. Sot. Am. 59 (1969) 799. illI D. Slepian, Bell System Tech. J. 43 (1964) 3009. 1121 J.C. Heurtley. in: Proc. Symp. on Quasi-Optics, cd. J. 1:0x I131 (North-Holland, Amsterdam, 1970). H.J. Landau, Acta Mathematics 117 (1967) 37. [I41 J.W. Goodman, in: Progress in optics, Vol. 8, ed. E. Wolf [ISI (North-Holland, Amsterdam, 1970). J.W. Goodman, Introduction to the I:ouricr optic% [I61 (McGraw-liill, New York, 1968). 1’. Riesz and B.S7. Nagy, I;unctional analysis (Ungar, New [I71 York, 1955). D. Slepian and E. Sonnenblick, Bell System Tech. J. 44 [I81 (1965) 1745. C.W. Barnes, J. Opt. Sot. Am. 56 (1966) 575. 1191 1201 C.K. RusM‘ort and R.W. llarris, .I. Opt. Sot. Am. 58 (1968) 539. B.R. Flieden, in: Progress in optics, Vol. 9. ed. E. Wolf 1211 (North-Holland, Amsterdam, 1971).
165