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Sampling theorem and the number of degrees of freedom of an image
Volume
11, number
OPTICS COMMUNICATIONS
2
June 1974
SAMPLINGTHEOREMANDTHENUMBER OFDEGREESOFFREEDOMOFANIMAGE V. BLAiEK TESLA-I’opov Research Institute of Radiocommunications, 142 21 Praha 4 Bran/k. Novodvorskb 994, Czechoslovakia Received Revised manuscript
A sampling theorem in a form suitable for coherent aperture is presented. The theorem is used to calculate a disk or a ring aperture.
14 January received
Calculations of the number of degrees of freedom of an optical signal transmitted by an optical system are principally based on a sampling theorem [ 1, 21. As is well known, due to the analyticity properties of the Fourier transform of the finite object field, the image has an infinite number of degrees of freedom. It has been shown by Gori and Guattari [3] that the number of significant degrees of freedom is finite and that this statement is valid for an arbitrary two-dimensional geometry. The aperture of the optical system has very often a circular shape and it is therefore convenient to use the sampling theorem in polar coordinates. In sec. 2 we show that sampling circles instead of sampling points can be used in such cases. An approximation of the image function by a finite number of sampling circles is discussed in sec. 3. Applications of the sampling theorem in polar coordinates to calculations of the degrees of freedom of an image formed by an optical system with disk or ring aperture respectively, are presented in sec. 4. The results are compared with those of Toraldo di Francis [4] which are based on prolate spheroidal wave functions.
We consider a two-dimensional 144
function f(r, q)
1974
optical signals transmitted by an optical system with circular the degrees of freedom of an image if the optical system has
1. Introduction
2. The sampling theorem in polar coordinates
1974 27 March
whose spectrum of spatial frequencies is nonzero only in the closed interval (0, ,I@. This means that for its Fourier transform holds: Flf(r, cp)] = 0, for all p > pg. Polar coordinates in the object plane are denoted by r, cp,in the Fourier plane by p, 0, respectively. Following [5] we can write the function f(r, cp) in the form of the sampling series f(Y>cp) =
5 eimp ?g A, (“q)Grnn(r), m=--m
(1)
where
Amn denotes the nth root of the equation I,(r) = 0, where Z,(r) is the Bessel function of order m. The value Am(Xm,/po) represents a sampling circle, which is obtained by integrating the functionf‘(r, p) on the circle with radius r = Xm,/po. We see that in the sampling theorem in polar coordinates sampling circles instead of sampling points are used. The principal properties of the function G,,(r) are as follows:
Volume
11, number
OPTICS
2
Fig. 1. Range of influence of the function G,*(x) (for E = 0.5) which is given by the intervals J1 and J,; K = 2 [see eq. (7)]
I ifr=h,,/po, G,,(r)
=
whereK=1,2 Orthogonality
,..., n-l,n+l,.... condition:
s
0
(4)
Oifr = h,,lpu,
2 r dr = [po’;(x,,),2
Grrzn(r) G,,(r:l
%, I( .
June 1974
COMMUNICATIONS
(5)
The formula (1) is a special case of the more general so-called Lagrange interpolation series
Fig. 2. Range of influence of the function G13(x) (for E = 0.5) which is given by the interval J; K = 1 [see eq. (7)]
Eq. (6) expresses the influence of the “tails” of the function G,,( r ) m a neighbourhood of the points r=h mK/pO, where K = 1, 2, . . . . n. This fact must be kept in mind if an approximation of the function f(r, p) is made, i.e., if we take only a finite number of terms of the series (1) into account. It is easy to show that a similar expression can be derived for the spectrum of the spatial frequencies of a finite function.
3. Finite number of sampling circles where g(z) is an entire function with zeros at {zn}T. For the “canonical” functiong(z) holds: g’(z,) f 0. In our case it has the form 1, (x)/(x + X,,) where x = rp0. It is convenient for our purposes to consider the behavior of the function G,,(r) in a neighbourhood of the point r = hmn/pO. We find the intervals around the point r = hmn/pO such, that IG,,(r)l 2 E, E being a positive number << 1. This condition is fullfilled for all r in the interval J: J=
r: IZ,(r)l>
E
z*
(rZpi - A&,)
I}.
FL+,
~11= UCO, 0) =F(P,
(6)
interval
UJ,, k=l
whereK
These subintervals
are disjoint (see figs. 1, 2) if E > 0.
of the optical
1
0
.
F(p, 0) is a Fourier transform of the function f(r, cp). If the sampling theorem in polar coordinates for the function F(p, 0) is used, we get r
(7) and J3r
(8)
G@, 0
I for allp
o foral,p>p
K J=
0)
where for the optical transfer function system holds G@7e)=G(P)=
may be expressed as a union of a finite family of closed subintervals
This
Let us consider a band-limited optical system which is linear and space invariant. An input optical signal (object function) f(r, cp) is equal to zero for all r > ro. If these conditions are fulfilled, then the Fourier transform of an image function u(r, cp) becomes
E eime n2~,& m=--m
=&,,,/ru)
G,,(p), (9)
u(0)=
PQO
I 0,
;
P>POi 145
Volume
11, number
2
OPTIC’S COMMUNICATIONS
June 1974
where 2n B,, (P = &,/rO)
= s
RP
= hmn/ro, 0) cirn8
CM,
(12)
0 and this approximation calculations.
will be used in the following
4.1. Disk aperture The optical transfer function form
of the system has the
1, for all p
(13)
t 0, for p > pg.
This case was evaluated by Gabor [l] and Toraldo di Francis [4]. According to Gabor, the number of degrees of freedom of the coherent wave field forming an image is equal to the product of an object area and an accessible Fourier area. Both are formed by the regions where the functionsf(r, CJJ)and lJ(p, 0) are nonzero (the circles of radius r0 and p. 1211,respectively). Therefore the accessible Fourier area is identical with the domain of spatial frequencies transmitted by the optical system. The number of degrees of freedom, Ni, which has been obtained in this way is put equal to the number of elements of a certain index set
1= {(m,
n): h 111nGroPol
)
where
c
G((P,“)=
(111, n)t^i
eimsB,,,(p=X,,,./rg)
G,,,,(P)
(14)
approximates the function U(p, 0). These calculations (expressed in our notation) do not take into account an influence of the tails of the function G,,,,l(p) and therefore all sampling circles with a radius p > p. are excluded. We may therefore conclude that the number of degrees of freedom of the wave field at the output of the optical system is equal to the object area times an area of sampling. This area of sampling is a region in the Fourier plane where samples are taken for the construction of the function U(p, 0). Its dimensions are determined by the approximate formula (12) and it is equal to the circular plane of radius (p0/2n + l/xro). Equating the number of degrees of freedom to the number of elements of the index set I [denoted by Nj(l)] yields Nj(fj
146
=
(+ropo+ 1)“.
(15)
Volume
11, number
2
OPTICS
COMMUNICATIONS
For pOrO % z we obtain the same expression as has been derived by Gabor. This means that the area of sampling is equal to the accessible Fourier area. Because this assumption is in cases of practical interest fulfilled we need not take into account the influence of the tails of the functions G,,(p). Of course this statement holds only for the case of disk apertures and it is not valid for the case of a ring aperture. An approximate expression A$ = (~upu/rrr)~ was derived by Toraldo di Francis [4] on the basis of prolate spheroidal wave functions. It can be shown that this result can be considered as a certain minimum number of samples we need for a rough approximation of the function U(p, 0). 4.2. Thin ring aperture The optical transfer function l,forp=pu; G(P) =
0, for pfpn.
has the form
(16)
(17)
The area of sampling is in this extreme case formed by a ring in the Fourier plane, whose radius is equal to po/27r and its width is equal to 2/mo. It is evident that we must take samples of the function F(‘p, 0) in a relatively large neighbourhood of the ring p = p. to reconstruct a wave field at the output of the optical system. The corresponding slit aperture (with a length equal to 2np0) gives Ni = 2ropo + 1, if a conventional sampling theorem is used. Toraldo di Francis [4] has also solved this case and he obtained the value 2ropo/n. 4.3. Ring aperture The optical transfer function form
1974
l,forpEdoo~~-Ap,po+~Ap); G(P) =
0, for all other p.
Now applying the same procedure thin ring aperture we obtain NJZ) = 2rOp0 ($roAp
+ 1).
(18) as in the case of a
(19)
The area of sampling is formed again by a ring whose width is enlarged by Ap. It is worthwhile to point out here that the commonly used expression “bandwidth of spatial frequencies transmitted by an optical system” has no sense in the case of a ring aperture. Therefore the term “area of spatial frequencies transmitted by an optical system may be more useful. 5. Conclusion
It is obligatory for us in this case to take into consideration the influence of the tails of the functions G,,(p), because in the opposite case we are led to the wrong conclusion that the wave field at the output of the optical system has one degree of freedom. This means that we need only one sampling circle for its reconstruction but such a statement holds if and only if the object has circular symmetry. If the approximate formula (12) is used then the number of elements of the index set I and therefore the number of degrees of freedom is equal to Ni(fl = 2r()@).
June
may be written in the
The sampling theorem in polar coordinates is used to find the number of spatial degrees of freedom of optical wave fields at the outputs of optical systems with circular symmetric apertures of various shapes. The conclusions which have been obtained are also valid for a more general family of systems satisfying the conditions of linearity and space invariance. As a representative example may be considered a radiotelescope with circular symmetric transfer function. Thus, sampling can be artificially introduced to solve system analysis problems from the point of view of transmission of pictorial information. There is a significant difference between the sampling theorem which has been presented in this paper and Shannon’s one. The former one is based on sampling circles instead of sampling points. This implies that an integration of a function over a given circle is used instead of the value of this function at a given point. It is not excluded that such a system of sampling of image signals may have some practical applications. The sampling in this form is rather complicated but it may have an advantage of being less dependent on noise. Work is continuing in this direction. References [l] D. [2] [3] [4] [S]
Gabor, Progress in Optics, ed. E. Wolf (North-Holland, Amsterdam, 196 l), vol. 1. G. Toraldo di Francis, .I. Opt. Sot. Amer. 45 (1955) 497. F. Gori and G. Guattari, Opt. Commun. 7 (1973) 163. G. Toraldo di Francis, J. Opt. Sot. Amer. 59 (I 969) 799. L.M. Soroko, Osnovy golografii i kogerentnoj optiki, (in Russian), (Nauka, Moscow, 1971), 366.
147
11, number
OPTICS COMMUNICATIONS
2
June 1974
SAMPLINGTHEOREMANDTHENUMBER OFDEGREESOFFREEDOMOFANIMAGE V. BLAiEK TESLA-I’opov Research Institute of Radiocommunications, 142 21 Praha 4 Bran/k. Novodvorskb 994, Czechoslovakia Received Revised manuscript
A sampling theorem in a form suitable for coherent aperture is presented. The theorem is used to calculate a disk or a ring aperture.
14 January received
Calculations of the number of degrees of freedom of an optical signal transmitted by an optical system are principally based on a sampling theorem [ 1, 21. As is well known, due to the analyticity properties of the Fourier transform of the finite object field, the image has an infinite number of degrees of freedom. It has been shown by Gori and Guattari [3] that the number of significant degrees of freedom is finite and that this statement is valid for an arbitrary two-dimensional geometry. The aperture of the optical system has very often a circular shape and it is therefore convenient to use the sampling theorem in polar coordinates. In sec. 2 we show that sampling circles instead of sampling points can be used in such cases. An approximation of the image function by a finite number of sampling circles is discussed in sec. 3. Applications of the sampling theorem in polar coordinates to calculations of the degrees of freedom of an image formed by an optical system with disk or ring aperture respectively, are presented in sec. 4. The results are compared with those of Toraldo di Francis [4] which are based on prolate spheroidal wave functions.
We consider a two-dimensional 144
function f(r, q)
1974
optical signals transmitted by an optical system with circular the degrees of freedom of an image if the optical system has
1. Introduction
2. The sampling theorem in polar coordinates
1974 27 March
whose spectrum of spatial frequencies is nonzero only in the closed interval (0, ,I@. This means that for its Fourier transform holds: Flf(r, cp)] = 0, for all p > pg. Polar coordinates in the object plane are denoted by r, cp,in the Fourier plane by p, 0, respectively. Following [5] we can write the function f(r, cp) in the form of the sampling series f(Y>cp) =
5 eimp ?g A, (“q)Grnn(r), m=--m
(1)
where
Amn denotes the nth root of the equation I,(r) = 0, where Z,(r) is the Bessel function of order m. The value Am(Xm,/po) represents a sampling circle, which is obtained by integrating the functionf‘(r, p) on the circle with radius r = Xm,/po. We see that in the sampling theorem in polar coordinates sampling circles instead of sampling points are used. The principal properties of the function G,,(r) are as follows:
Volume
11, number
OPTICS
2
Fig. 1. Range of influence of the function G,*(x) (for E = 0.5) which is given by the intervals J1 and J,; K = 2 [see eq. (7)]
I ifr=h,,/po, G,,(r)
=
whereK=1,2 Orthogonality
,..., n-l,n+l,.... condition:
s
0
(4)
Oifr = h,,lpu,
2 r dr = [po’;(x,,),2
Grrzn(r) G,,(r:l
%, I( .
June 1974
COMMUNICATIONS
(5)
The formula (1) is a special case of the more general so-called Lagrange interpolation series
Fig. 2. Range of influence of the function G13(x) (for E = 0.5) which is given by the interval J; K = 1 [see eq. (7)]
Eq. (6) expresses the influence of the “tails” of the function G,,( r ) m a neighbourhood of the points r=h mK/pO, where K = 1, 2, . . . . n. This fact must be kept in mind if an approximation of the function f(r, p) is made, i.e., if we take only a finite number of terms of the series (1) into account. It is easy to show that a similar expression can be derived for the spectrum of the spatial frequencies of a finite function.
3. Finite number of sampling circles where g(z) is an entire function with zeros at {zn}T. For the “canonical” functiong(z) holds: g’(z,) f 0. In our case it has the form 1, (x)/(x + X,,) where x = rp0. It is convenient for our purposes to consider the behavior of the function G,,(r) in a neighbourhood of the point r = hmn/pO. We find the intervals around the point r = hmn/pO such, that IG,,(r)l 2 E, E being a positive number << 1. This condition is fullfilled for all r in the interval J: J=
r: IZ,(r)l>
E
z*
(rZpi - A&,)
I}.
FL+,
~11= UCO, 0) =F(P,
(6)
interval
UJ,, k=l
whereK
These subintervals
are disjoint (see figs. 1, 2) if E > 0.
of the optical
1
0
.
F(p, 0) is a Fourier transform of the function f(r, cp). If the sampling theorem in polar coordinates for the function F(p, 0) is used, we get r
(7) and J3r
(8)
G@, 0
I for allp
o foral,p>p
K J=
0)
where for the optical transfer function system holds G@7e)=G(P)=
may be expressed as a union of a finite family of closed subintervals
This
Let us consider a band-limited optical system which is linear and space invariant. An input optical signal (object function) f(r, cp) is equal to zero for all r > ro. If these conditions are fulfilled, then the Fourier transform of an image function u(r, cp) becomes
E eime n2~,& m=--m
=&,,,/ru)
G,,(p), (9)
u(0)=
PQO
I 0,
;
P>POi 145
Volume
11, number
2
OPTIC’S COMMUNICATIONS
June 1974
where 2n B,, (P = &,/rO)
= s
RP
= hmn/ro, 0) cirn8
CM,
(12)
0 and this approximation calculations.
will be used in the following
4.1. Disk aperture The optical transfer function form
of the system has the
1, for all p
(13)
t 0, for p > pg.
This case was evaluated by Gabor [l] and Toraldo di Francis [4]. According to Gabor, the number of degrees of freedom of the coherent wave field forming an image is equal to the product of an object area and an accessible Fourier area. Both are formed by the regions where the functionsf(r, CJJ)and lJ(p, 0) are nonzero (the circles of radius r0 and p. 1211,respectively). Therefore the accessible Fourier area is identical with the domain of spatial frequencies transmitted by the optical system. The number of degrees of freedom, Ni, which has been obtained in this way is put equal to the number of elements of a certain index set
1= {(m,
n): h 111nGroPol
)
where
c
G((P,“)=
(111, n)t^i
eimsB,,,(p=X,,,./rg)
G,,,,(P)
(14)
approximates the function U(p, 0). These calculations (expressed in our notation) do not take into account an influence of the tails of the function G,,,,l(p) and therefore all sampling circles with a radius p > p. are excluded. We may therefore conclude that the number of degrees of freedom of the wave field at the output of the optical system is equal to the object area times an area of sampling. This area of sampling is a region in the Fourier plane where samples are taken for the construction of the function U(p, 0). Its dimensions are determined by the approximate formula (12) and it is equal to the circular plane of radius (p0/2n + l/xro). Equating the number of degrees of freedom to the number of elements of the index set I [denoted by Nj(l)] yields Nj(fj
146
=
(+ropo+ 1)“.
(15)
Volume
11, number
2
OPTICS
COMMUNICATIONS
For pOrO % z we obtain the same expression as has been derived by Gabor. This means that the area of sampling is equal to the accessible Fourier area. Because this assumption is in cases of practical interest fulfilled we need not take into account the influence of the tails of the functions G,,(p). Of course this statement holds only for the case of disk apertures and it is not valid for the case of a ring aperture. An approximate expression A$ = (~upu/rrr)~ was derived by Toraldo di Francis [4] on the basis of prolate spheroidal wave functions. It can be shown that this result can be considered as a certain minimum number of samples we need for a rough approximation of the function U(p, 0). 4.2. Thin ring aperture The optical transfer function l,forp=pu; G(P) =
0, for pfpn.
has the form
(16)
(17)
The area of sampling is in this extreme case formed by a ring in the Fourier plane, whose radius is equal to po/27r and its width is equal to 2/mo. It is evident that we must take samples of the function F(‘p, 0) in a relatively large neighbourhood of the ring p = p. to reconstruct a wave field at the output of the optical system. The corresponding slit aperture (with a length equal to 2np0) gives Ni = 2ropo + 1, if a conventional sampling theorem is used. Toraldo di Francis [4] has also solved this case and he obtained the value 2ropo/n. 4.3. Ring aperture The optical transfer function form
1974
l,forpEdoo~~-Ap,po+~Ap); G(P) =
0, for all other p.
Now applying the same procedure thin ring aperture we obtain NJZ) = 2rOp0 ($roAp
+ 1).
(18) as in the case of a
(19)
The area of sampling is formed again by a ring whose width is enlarged by Ap. It is worthwhile to point out here that the commonly used expression “bandwidth of spatial frequencies transmitted by an optical system” has no sense in the case of a ring aperture. Therefore the term “area of spatial frequencies transmitted by an optical system may be more useful. 5. Conclusion
It is obligatory for us in this case to take into consideration the influence of the tails of the functions G,,(p), because in the opposite case we are led to the wrong conclusion that the wave field at the output of the optical system has one degree of freedom. This means that we need only one sampling circle for its reconstruction but such a statement holds if and only if the object has circular symmetry. If the approximate formula (12) is used then the number of elements of the index set I and therefore the number of degrees of freedom is equal to Ni(fl = 2r()@).
June
may be written in the
The sampling theorem in polar coordinates is used to find the number of spatial degrees of freedom of optical wave fields at the outputs of optical systems with circular symmetric apertures of various shapes. The conclusions which have been obtained are also valid for a more general family of systems satisfying the conditions of linearity and space invariance. As a representative example may be considered a radiotelescope with circular symmetric transfer function. Thus, sampling can be artificially introduced to solve system analysis problems from the point of view of transmission of pictorial information. There is a significant difference between the sampling theorem which has been presented in this paper and Shannon’s one. The former one is based on sampling circles instead of sampling points. This implies that an integration of a function over a given circle is used instead of the value of this function at a given point. It is not excluded that such a system of sampling of image signals may have some practical applications. The sampling in this form is rather complicated but it may have an advantage of being less dependent on noise. Work is continuing in this direction. References [l] D. [2] [3] [4] [S]
Gabor, Progress in Optics, ed. E. Wolf (North-Holland, Amsterdam, 196 l), vol. 1. G. Toraldo di Francis, .I. Opt. Sot. Amer. 45 (1955) 497. F. Gori and G. Guattari, Opt. Commun. 7 (1973) 163. G. Toraldo di Francis, J. Opt. Sot. Amer. 59 (I 969) 799. L.M. Soroko, Osnovy golografii i kogerentnoj optiki, (in Russian), (Nauka, Moscow, 1971), 366.
147