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Degrees of Freedom of an Image inthe Presence of Noise and ‘a Priori’InformationS. Gupta a & S.P.S. Virdi aa Physics Department , Punjabi University , Patiala, 147002,IndiaPublished online: 01 Mar 2007.

To cite this article: S. Gupta & S.P.S. Virdi (1991) Degrees of Freedom of an Image in thePresence of Noise and ‘a Priori’ Information, Journal of Modern Optics, 38:10, 1927-1934, DOI:10.1080/09500349114552041

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JOURNAL OF MODERN OPTICS, 1991, VOL . 38, NO. 10, 1927-1934

Degrees of freedom of an image in the presenceof noise and `a priori' information

S. GUPTA and S. P. S . VIRDIPhysics Department, Punjabi University, Patiala-147002, India

(Received 15 June 1990 ; revision received 17 December 1990)

Abstract . The mathematical technique applied by Gori and Guttari forcalculation of degrees of freedom in images formed by point-like elemental pupilsis extended to continuous pupils . Eigenvalues of the imaging equation arecalculated for the sins as well as the sinc e kernel . The effect of noise and 'a priori'information are taken into account .

1 . IntroductionThe importance of calculation of the degrees of freedom in an optical system is

well realized [1-3] . We show in this paper that the mathematical technique employedby Gori and Guttari [4] for the calculation of the degree of freedom of images fromthe point-like elemental pupils can be applied to continuous pupils both for coherentas well as incoherent imagery . The degrees of freedom thus obtained may not all besignificant [7] . We determine the number of effective degrees of freedom when theautocorrelation function for the class of objects to be imaged is known . The methodthough approximate yields results to a reasonable accuracy . The technique should beuseful, in particular, for incoherent imagery where the eigenvalues of the sinc Z kernelhave till now only been estimated [5, 6] .

2 . TheoryWe consider a one dimensional object o(x) of spatial extent 2X centred at the

origin of the axis . The image i(y) is then related to o(x) through the lineartransformation

Xi(y)=

s(y-x)o(x)dx,

(1)-x

where s(y-x) is the impulse response of the linear shift invariant imaging system .The integral equation

Ix

xs(y- x)o1(x) dx= al4l(y)

(2)

defines an orthonormal set [4 1(x)] assumed to be complete in which the image and theobject distributions can be expanded with [a l] the set of associated eigenvalues of thefunction . If s(x) is assumed to be such that it vanished for large values of x, we mayapproximate it by an expansion in a Fourier series as

s(y-x)= Y Sk exp[27tiku0(y-x)],k=-ao

0950-0340/91 $3. 00 © 1991 Taylor & Francis Ltd .

(3)

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with 4X as the object extent. If s(y-x) is bandlimited, S k are zero beyond certainvalues of `k', say M. We may then write

Ms(y-x) = Y_ Sk exp [21tiku O(y-x)],

(6)k=-M

which is similar to the expression used in [4] for point-like elemental pupils having(2M+1) equidistant elements with transmittances given by Sk values . The inter-element distance in terms of the spatial frequency is u0 . Use of equation (6) in theintegral imaging expression then yields

1" Y_ Sk exp[27tikuo(y-x)]ol(x)dx=a lol ( y),X k= - M

(7)

Multiplying both sides of equation (7) by exp(-27timu oy) and integrating withrespect to y we have

M

X

XY_ Sk

exp [-21ti(m-k)u oy] dy ~- 4 l (x) exp (-21tikuox) dxk=-M

-x

X

Equation (9) states an eigenvalue problem whose eigenvalues, to a good approxim-ation, are the same as those for the original equation and the eigenvectors tk are thesampled Fourier transform values of eigenfunctions j l .

=a l f X0 l ( y) exp (-27tiku 0y) dy, (8)

or

MpI'mkOk

=!0401,

X

(9)

where

Nmk = f

k=-M

x

xexp [-27ti(m-k)u0y] dy

sin [21r(m-k)uoX]_ 'Sk (10)

7t(m-k)u0 '

and

(11)0k =x

0 1 (x) exp (- 27tikuox) dx.J x

1928

S. Gupta and S. P. S. Virdi

with

u0 =1 /4X, (4)

and

2XSk =1/4X f s(y-x) exp [-27tiku o(y-x)] d(y-x) . (5)

- 2x

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Degrees of freedom of an image

1929

3. Degrees of freedom in absence of noiseThis problem is solved for two cases of point spread functions

s l (y- x) =sinc [(y-x)/d],

(12)

and

s2(y-x)=sinc e [(y-x)/d],

(13)

where

sinc x = sin nx/7tx,

`d' is the Rayleigh resolution distance, assuming unit magnification . For an object ofextent 2X the system has a Shannon number

n,= 2X/d.

Figure 1 shows the eigenvalue behaviour against the order index for a sinc kernel fordifferent values of n. . Figure 2 (a) is the same plot for the sinc e p.s .f. Figure 2 (b) givesthe behaviour of first eigenvalue against 1 /n, for the same kernel . The resultsconform to those obtained by different methods [3-8] . As expected the eigenvaluesdrop to almost zero for l > n, for the coherent case and 1 > 2n, in the incoherent case .

1 . 0

015

IFigure 1 . Graph of eigenvalues against their order index for sinc p .s .f. for different Shannon

numbers ranging from 2(1)21 .

16

20

24 x

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1930

11

91

Y

1 .0

0.9

018

0 .7

0.6

015i I

0

01

0 .2

0 .3

0 .4

0 15

IA,

(b)

12

S. Gupta and S . P. S. Virdi

161

(a)

2 26 32 3

x

0

Figure 2 . (a) Graph of eigenvalues against their order index for sinc e p.s .f. for differentShannon numbers (n,) ranging from 2(2)40 . (b) Variation of first eigenvalue with inverseof Shannon number for a sinc e p .s .f.

The behaviour of the first eigenvalue in the incoherent case conforms to thatpredicted mathematically in [6] .

4 . Degrees of freedom in presence of noise and with `a priori' informationWe now calculate the effective number of degrees of freedom when noise is

present and when some `a priori' information in the form of object statistics isavailable. The use of explicit expressions for the eigenfunctions is not needed .

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Degrees of freedom of an image 1931

Following Saleh [7] the number of significant modes is given by the number ofsignificant D1, values where

fx x

Dim=

J

r(x-x')-O*~(x)¢m(x')dxdx',

(14)-x -x

and x(x-x') is the correlation function for the object irradiance . Again theassumption that x(x-x') is negligibly small for large x -x' enables us to expand itinto a Fourier series and write

f fx X

D,m =

Bk exp [2nikuo(x- x')] 0#(x)dm(x') dx dx',k

X 1 -x

where

fZx

B k =1/4X

x(x-x')exp[-2ttiku o(x-x')]d(x-x') .

(16)1 -2x

As earlier, using equation (11), we may reduce equation (15) to the following formM'

Dlm - L. Bkokok'

(17)k= -M'

where we have terminated the series for Iki > M assuming x(x-x') to be bandlimitedwithin (-M'uo , M'uo) . Normalization for Ok is affected using the relation

M'

Sk~k~k =a, .

(18)k=-M'

The Dim matrix thus obtained can further be diagonalized to obtain uncorrelatedmodes of Dim say G,, . We have calculated these G„ values assuming two forms of thecorrelation function x(x-x')

x l(x-x')=sinc [(x-x')/d,],

(19)

x2(x-x')=sinc e [(x-x')/d,],

(20)

that is, for objects with rectangular power spectrum and triangular power spectrumrespectively. The results depend on three lengths : X, the field size, d,, the objectdetail size and d, the system resolution length .

5. ResultsAssuming 2X= 2n, a constant value, we calculate the G„ values for different

values of

d,/d=n il/nc2 ,

( 21)

where n il coincides with the Shannon number of the system taken here as `20' andnc2 = 2X/d, .

We have chosen, for d~/d, the values 4, 2, 1, 0 . 5 and 0 . 25 and plotted the G„ valuesagainst l as shown in figures 3 (a) and (b) . To calculate the effective number of degreesof freedom in the presence of noise, following [7], we have drawn the noise factorN, =1 /(SNR x a,) (assuming noise to be band limited detection noise) for differentvalues of SNR, namely 5, 10, and 20 on the same graph . Figure 5 is the repetition ofthe above procedure for a sinc point spread function . From figures 3 and 5 we see thatgraphs for d~/d,1 are mainly dependent on the `a priori' information factor whereas

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S. Gupta and S. P. S. Virdi

Figure 3 . Variation of mode intensity factor G,, () and the noise factor N, (---) with themode index I for several values of ratio of object detail length to system resolution length(dd/d) and different values of SNR respectively, for a since p .s .f. ; (a) for objects withrectangular power spectrum ; (b) for objects with triangular power spectrum .

0N

(b)

Figure 4 . Variation of optimum number of d .o .f. (z op,) with d~/d for sinc e p .s .f. for differentvalues of SNR; (a) for the class of objects with rectangular power spectrum; (b) for aclass of objects with triangular power spectrum .

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Figure 5 . Variation of mode intensity factor G„ () and the noise factor N, (---) with themode index 1 for several values of ratio of object detail length to system resolution length(dd /d) and different values of SNR respectively, for a sinc p .s .f. ; (a) for objects withrectangular power spectrum ; (b) for objects with triangular power spectrum .

0 1I

(a)

z 3

Degrees of freedom of an image

X

y

20

o° 10N

00

1

2

3

4

d,/d

(b)

Figure 6 . Variation of optimum number of d .o .f . (zo,,,) with d~/d for sinc p .s .f . for differentvalues of SNR; (a) for the class objects with rectangular power spectrum ; (b) for a classof objects with triangular power spectrum .

(b)

1 93 3

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Degrees of freedom of an image

for d~/d < 1, on the system geometry factor, which is quite consistent . Since ford,/d,> I the object does not contain details more than the system is capable ofresolving and whatever details are present are passed through the system . On theother hand, when the object details become finer than the system resolution length,all the modes cannot be passed through the system and the system resolution sets anupper limit to the number of modes capable of being transferred to the output .

Figures 4 and 6 give us the variation of the number of degrees of freedom, z0 ,1 ,with d~/d for different SNR values for the sinc e p.s .f . and sinc p .s .f.'s respectively forthe two autocorrelation functions . The approximate triangular and rectangularbehaviour for triangular power spectral objects and rectangular power spectralobjects again endorses the views expressed by Saleh [7] . However, the limitingtendency of z0P, for low dd/d values is different from what he expects . Presumablybecause his predictions for d~/d < 1 values are based on certain assumptions made dueto the want of knowledge of exact eigenfunctions for 1 > 14 as required by hisalgorithm .

It should be noted that we have taken z0Pt =P+ 1 for the sinc e p.s .f. and 2(P+ 1)for the sinc p .s .f ., where P is that value of 1 where N, curve intersects G,, . That is thereason we are getting almost the same number of degrees of freedom for the twocases. The situation will not be the same for the two dimensional case, where weexpect almost two times the number of degrees of freedom in the incoherent casethan in the coherent case .

6. ConclusionsThe eigenvalue behaviour conforms to that expected both for sinc as well as sinc e

kernels. The values obtained by this method have been compared to those obtainedby Slepian et al . as in [8] . The agreement is better than 3% till the eigenvaluemagnitude drops to less than 1 % of the maximum . Thereafter the error is appreciablewhich is to be expected in view of the approximations of the procedure . For thecalculation of the degrees of freedom, higher order eigenvalues are not required andthe present technique is quite acceptable .

The main error in the method is that due to the numerical computation of theFourier transform . The minimum level beyond which the Fourier transform is to betaken to be zero determines the choice of the numbers M and M . Althoughcomputations for the sinc e kernel are not available similar levels of accuracy are to beexpected .

References[1] HELSTROM, C . W., 1967, J . opt. Soc. Am ., 57, 297 .[2] GoRI, F., and GUTTARI, G., 1973, Optics Commun ., 7,163 .[3] BENDINELLI, M ., CONSORTINI, A., RONCHI, L., and FRIEDEN, B . R ., 1974, J. opt . Soc . Am.,

64, 1498 .[4] GOBI, F ., and GUTTARI, G ., 1974, J . opt. Soc. Am ., 64, 453 .[5] GoRI, F ., 1974, J . opt . Soc. Am ., 64, 1237 .[6] GORI, F ., and PALMA, C ., 1975, J. Phys. A, 8, 1709 .[7] SALEH, B . E . A ., 1977, J. opt. Soc . Am., 67, 71 .[8] FRIEDEN, B . R ., 1971, Progress in Optics, Vol. IX, edited by E . Wolf (Amsterdam : North-

Holland), p . 311 .

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