Vol. 6, No. 10/October 1989/J. Opt. Soc. Am. A 1555
Analysis of coherent image of grating with rectangulartransmission profile
P. C. Uduh
Department of Physics, University of Jos, Jos, Nigeria
Received October 9, 1987; accepted April 4, 1989
The signal-to-noise ratio in the coherent image of a grating with a rectangular transmission profile is calculated anddisplayed as a function of the grating frequency relative to the cutoff frequency of the imaging system and as afunction of the grating frequency alone. It is shown that the signal-to-noise ratio in the image depends on the objectcontrast when the average size of the speckle in the image is small compared with the grating constant and that thesignal-to-noise ratio approaches asymptotically a constant value independent of the object contrast when theaverage size of the speckle is large compared with the grating constant. Furthermore, the speckle noise issuppressed when the size of the imaging aperture is increased, thereby enhancing the signal at a given frequency.
I. INTRODUCTION
The recognition of details in an optical image by a humanobserver and the extraction of information from an opticalpattern by making use of physically measured parametershave been the subjects of investigation for several decades. 1 -In physiological optics, simple test symbols embedded in auniform field are used almost exclusively. In this case, apartfrom physiological influences, factors such as the size andnature of the symbols, the light intensity, the adaptationstate of the observer, the presentation time, and the size andform of the field play important roles in detection. In addi-tion, the actual process of vision has not been explored andunderstood completely. All these factors tend to complicatethe subjective evaluation of an optical image or pattern.This difficulty can be circumvented by applying the formal-ism of optical transmission and information theory,5 6
whereby the nature of the light used and the properties ofthe imaging and detecting systems provide the parametersthat are most important for an objective evaluation. Byusing transmission and information theory, it is possible toanalyze optical images and hence to predict the performanceof, say, a tandem arrangement of imaging systems.
Optical imaging considered from the perspective of trans-mission theory is treated in Section 2. On the basis of thistheory incoherent and coherent imagery are discussed sub-sequently, and the relationship between them is establishedin Section 5. The effect of speckling that usually is associat-ed with coherent illumination is considered in Section 6.Since in the actual analysis diffuse illumination has beenassumed, this type of illumination and the approximationimposed on the diffuser are reviewed in Section 7. In Sec-tion 8 the principles and methods elucidated in Sections 2-7are applied specifically to the analysis of the coherent imageof a grating with a rectangular transmission profile.
2. OPTICAL IMAGING AND TRANSMISSIONTHEORY
An ideal optical imaging system would be one that images apoint source again as a point. Such a system does not exist
in practice. Owing to the effect of diffraction the image of apoint source is an extended light distribution (diffractionpattern). Every object can be regarded as comprising amultitude of points or diffraction gratings, and so light ema-nating from it can give rise to a diffraction pattern. Accord-ing to Abbe the fidelity of the image depends on the diffrac-tion spectrum that is formed in the focal plane of the objec-tive. This spectrum, which consists of diffraction maximaand is referred to as the primary image, is simply the diffrac-tion image of the object. The secondary image, i.e., the realimage of the object, is produced in the image plane by inter-ference of the diffraction maxima in that plane. The imagebecomes better as more diffraction maxima (orders) aremade to interfere with the zero-order maximum in the imageplane. The diffraction image is associated with the responseof the imaging system to the input light waves from theobject.
The response of most physical systems to a given signalcan be considered as a superposition of individual elemen-tary responses to the independent elementary componentsof the signal. Systems for which this condition holds aresaid to be linear; e.g., electrical networks composed of resis-tors, capacitors, and inductors are in a broad sense linear.On the other hand, the linearity of the wave equation thatadequately describes the propagation of light in many mediapermits the assumption that an optical imaging process is alinear operation by which the light distribution in the objectis mapped linearly into the light distribution in the image.The overall response function of a linear system to compli-cated input stimuli, be it voltage, currents, light amplitude,or light intensity, is therefore known when the elementaryresponse functions to the corresponding elementary stimuliare known. The system response is simply a linear combina-tion of the elementary response functions. A useful impor-tant mathematical technique for treating linear processes isthe Fourier analysis, a mathematical procedure popular incommunication engineering. Starting from the Fresnel-Kirchhoff equation, one can easily show that the process oflight diffraction under the Fraunhofer condition can be de-scribed by Fourier transformation.7 Fourier transforma-tion is used extensively in transmission and system theories.
1556 J. Opt. Soc. Am. A/Vol. 6, No. 10/October 1989
If the image is to be produced by spatially coherent light, it isappropriate to represent the light distribution in the objectand in the image by a spatial complex-valued amplitudedistribution. For incoherent imagery the light distributionsare portrayed as spatial real-valued intensity distributions.
3. INCOHERENT OPTICAL IMAGING
Suppose that i(x 0, ye) and i(xi, yi) are the light intensitydistributions in the object and the image, respectively.Then every image point P(xi, yi) is obtained by superpositionof the point images of all points i0(x0, yo). If, in addition, thecondition for linearity and isoplanism (the invariance condi-tion that the form of the point image be independent ofposition within the region being considered) are satisfied,the image intensity distribution is expressed as a convolu-tion integral. If we neglect a constant factor we obtain
ii(xi, yi) = JJ_ Ih(xi - x0 , yi - y.) 2i0 (x, y,)dx(dy 0 , (1)
where h(xi - xe, yi - ) is the complex-valued responsefunction (complex amplitude distribution) of the system.The square of the modulus Ih12 is the light intensity distribu-tion in the point image, commonly called the spread functionof the system. When images of the same motifs but differ-ent qualities are to be compared, it is more convenient andinformative to perform the analysis in the spatial-frequencydomain, as is the practice in transmission theory. The usualprocedure is to introduce the spectral functions of i0(x,, ye),ii(xi, yi), and h, which are normalized to unity at the zerofrequency. Hence we define
N(-fx, -fy) = H*(fx fy),
IY(fx, fy)l < O(, 0)1.
(4b)
(4c)
The spectral functions provide the possibility of describingthe structure of an optical image without explicit knowledgeof the individual details in the image. The functions aredefined for the positive and negative frequencies, and, byvirtue of their symmetry, the knowledge of one part impliesthe knowledge of the other. This can easily be ascertainedby a simple change of signs in Eqs. (2a) and (2c). The realproduct Ii(fx, fy)Ii*(fx, fy) = II(fx, fy)12 with the dimension of(energy)2 is the Wiener spectrum of the image. The Wienerspectrum of the object can be defined similarly. The con-servation of energy (Parseval's theorem) provides that
WO is the integrated Wiener spectrum. The relative differ-ence in WO in different spectral regions can serve as a mea-sure of the subjectively perceived structural difference intwo test images of similar motifs.8 9 Fourier transformsexist always for periodic processes. They do not, however,exist for random processes over an infinite interval or re-gion.10 For this reason the concepts of correlation functionsand energy density spectra usually are introduced in suchcases.11 12 By definition, the autocorrelation function(ACF) of the input signal io(xo, yo) is
'P(Q, 7) = (io(xo, y0)i(xo - t, Y0 - 7))
= JJ io(xo, yo)io(xo - s, yo - n)dxodyo, (6)
FTfi,(x 0 , y0)}
fE io(xo, y 0)dxodyo
FTii(xi, y)
f E ii(xi, yi)dxidyi
FTJlh(xi, y,)121
(2a) where (-) denotes an ensemble average.sion exists for the output signal i(xi, yi)-the following conditions:
(2b)
(2c)
00(O, ) = Ujox y))
1P(A, n) = CH0 (-, -7),
Po(c, °) > (i0o(x 7)1,
_0OsC) = (i"(X~, Y")) 2
A similar expres-The ACF satisfies
(7a)
(7b)
(7c)
(7d)
and FT {-1 denotes a Fourier transform. Applying the convo-lution theorem to Eqs. (2a)-(2b), we obtain the relationship
i (fx, fy) = (f, fy)'e0fx, fr) (3)
i(fx, fy), K(f., fy), and Ie(fx, fy) are complex-valued func-tions. If(h, fy) and Io(fx, fy) are the spectral functions of theimage and the object, respectively, whereas 11(fx, fy) is theoptical transfer function (OTF) of the imaging system andthe modulus IiI is the modulation transfer function. 11(fx,fy), which is associated with the system, is a complex-valuedweighting factor at the point (fx, 4y) relative to the weightingfactor at the zero frequency. Therefore the OTF satisfiesthe following conditions:
Y1(0, 0) = 1,
Here O(0, 0) gives the total energy in the signal. e(c, )can be regarded as the constant background, whereas thedifference iP(0, 0) - Vt(-, a) characterizes the variation.According to the Wiener-Khinchin theorem the energy den-sity spectrum (Wiener spectrum) is equal to the Fouriertransform of the autocorrelation function, i.e.,
Therefore the Wiener spectrum can be obtained either fromthe signal function io(xe, ye) or from the autocorrelationfunction q0(Q, n). If it is assumed that the imagery is inco-herent and that the corresponding Fourier transforms exist,then this process is summarized schematically in the objectspace as follows:
I0 (fx, fy) =
P. C. Uduh
1#X, fy =
Yi (fx, fy =
f f_ 1h(.xi, y,)12dXdy,
(4a)
Vol. 6, No. 10/October 1989/J. Opt. Soc. Am. A 1557
signal function(intensity distribution)
ie(Xo, Ye)
ACF
iI'(Q, A)
FT\
FT
spectral function
/ e f(fx, fy)
Wiener spectrumII(f fY)12
integrated Wiener spectrumWe
4. COHERENT OPTICAL IMAGING
A coherent optical imaging system is linear in the transmis-sion of the complex amplitude of the electric field of the lightwave. If a(xe, ye) and ai(xi, yi) are the complex amplitudefunctions for the object and the image, respectively, then, bya procedure similar to that described in Section 3, ai(xi, yi) isfound to equal a convolution integral,
A(fx f) and AL(fx, fy) are the spatial-frequency spectra ofthe system for the object and the image, respectively. H(f.,fy) is the coherent transfer function of the system. AlthoughHe and Ai are not observable (i.e., not measurable), thesquares of their moduli, the energy spectra IAeI2 and IAiI2, aremeasurable. The energy spectra are analogous to the Wie-ner spectra in coherent imaging and differ from them merelyin dimension. Whereas the dimension of the Wiener spec-trum is (energy density)2 , the energy spectrum has the di-mension of energy. The total energy in the signal is ob-tained by integrating the energy spectrum over the range offrequencies being analyzed. Corresponding to the defini-tion given in Section 3 for incoherent imaging, we have, forcoherent imagery,
oefxy 4) = FT1ae(xe, Ye)I2
= FTlae(xe, ye)ae*(X,, ye)}
= Jf__ A(fx, fy)Ae(Q - f- - fy)dtdn. (12)
The spectral function here manifests itself as the ACF func-tion of the spatial-frequency spectrum. The relationships
between the functions for coherent imaging can be exhibitedstepwise as follows:
signal amplitude function ae(xe, ye)
FTJspatial-frequency spectrum A(fx, fl)
energy spectrum A(fx, f,)2
I ftotal energy E
5. RELATIONSHIP BETWEEN OPTICALTRANSFER FUNCTION AND COHERENTTRANSFER FUNCTION
By definition, the OTF 5V(fx, 4y) is the Fourier transform ofthe square of the modulus of the impulse response functionIh(xi, yi)d2 normalized over the Fourier transform of thesquared modulus of the same function at the frequencypoint (0, 0), whereas the coherent transfer function H(fx, fy)is the Fourier transform of the impulse response function[Eqs. (2c) and (11)]. Hence
Using the autocorrelation theorem and a suitable coordinatetransformation, we can obtain a symmetric expression forthe OTF of the form
= Jat H( + &"7 + )Hf*( -& 7 d-J J IH(Q, ?)Pdtdn
(15)
from Eq. (14). Equation (15) represents a general and use-ful relationship between coherent and incoherent imagingsystems. For a diffraction-limited system with a circularimaging aperture of diameter D, the OTF is rotation sym-metrical 13 and can be expressed as a function of the radialfrequency u = (2 + F2)1/2:
2 {cosl
#(u) =
0
l2u\2ue) - U [ - (U )2]1/2}
for u < 2uo = DXdi
(16)
otherwise
u is the spatial frequency, and 2e is the incoherent cutofffrequency of the imaging system. X is the wavelength of thelight used, and di is the distance between the exit pupil of thesystem and the image plane.
P. C. Uduh
1558 J. Opt. Soc. Am. A/Vol. 6, No. 10/October 1989
6. EFFECT OF SPECKLING
Speckling always occurs whenever coherent illumination isused. The statistical properties of laser speckle were dis-cussed extensively and comprehensively by Goodman.'4
The influence of speckling on particular features of the ob-ject in the image of the object depends on the average size ofthe speckle relative to the particular object structures andtherefore depends on the spatial frequencies of the speckle.The knowledge of the Wiener spectrum of the light intensityin the image is necessary in order to be able to estimate theaverage diameter of the granulation. For a circular exitpupil of diameter D the ACF of the image intensity is [seeEqs. (6) and (7)]
f (r) = (I )2 + _ ) ] (17)
where, as is usually the case, because symmetry polar coordi-nates have been introduced into the pupil plane P(Q, -q) withr = (Q2 + 772)1/2. J, is the Bessel function of the first kind andfirst order. The first zero point of the second term in Eq.(17) defines the average diameter 0 of the speckle in theimage.
Accordingly,
Xd= 1.22 D (18)
An equivalent expression that takes into account the aper-ture number (f/D) of the system is'5
=1. 22 ( + m)X f (19)
m is the lateral magnification of the system, and f is the focallength of the objective. The spectral density function (Wie-ner spectrum) of the image obtained from Eq. (17) is
N= (Ii)2 6(U) + 2D.)_ {cos (? X )
- di u [I - ( ) 1})- (20)
Substituting D/Xdi = 2uo into Eq. (20) yields
N1(U) = (I,)2 (U) + 1 22 {cs1( )
- U [1 - (U )2]1/2})(21)
with u = (2 + f2)11
2. A comparison of Eq. (21) with Eq.
(16) reveals that the alternating component of the Wienerspectrum, which we denote by Ni., is given by
NiWju) =1 2 ' 2 cos- ( 2 [1-2Ue 2 r 2ueJ 2ue
U 2 1\2uo)
(22)
and differs from the incoherent transfer function (i.e., theOTF) by the factor (1/2)((Ii)/Ue) 2 . The integral of the alter-nating component is proportional to the variance.16 This isso also if the integration is performed over only a portion of
the frequency range being considered.17 It is obvious fromEq. (21) that the noise spectrum will remain the same, pro-vided that the system aperture D is kept constant.
7. DIFFUSE ILLUMINATION
An object that, because of the nature of its surface, scatterslight in several directions is a diffuser (light scatterer). Themost frequently used object of this nature, especially inholography, is ground glass. When a given object is illumi-nated by ground glass, we speak of diffuse illumination re-gardless of whether the light gets to the object by reflectionor transmission from the ground glass. Illumination with adiffuser has the advantage that the object is illuminateduniformly in the sense that every scattering center of thediffuser irradiates the entire object surface. In photograph-ic recording this advantage is particularly valuable, since therisk of exceeding the dynamic region of the film can beavoided by this means. In transmission geometry, as isassumed in the present analysis, if an object transparencythat is in direct contact with ground glass is to be imaged,specific postulations about the characteristics of the groundglass and the properties of the imaging system are necessaryin order to facilitate the mathematical analysis. In mostcases the imaging system is supposed to be linear invariant.For the distribution of the field amplitudes of the groundglass, stationary (i.e., the statistics are position indepen-dent) Gaussian statistics are assumed, with the phase fluctu-ation uniformly distributed in the interval [-7r, r]. Thebreadth of the ACF 'd(r), with r = (x, y), of the diffuseramplitude relative to the response function h(r) of the imag-ing system is important for the statistics of the amplitude aswell as for the intensity distribution in the image. If for thediffuser we suppose that the scattering centers are indepen-dent of one another, Ad(r) can be taken approximately as a 6function (white noise), and consequently the width of A1d(r)can be considered smaller than that of h(r) (Fig. 1).
All points within the impulse response function contrib-ute to the intensity at a point P in the image plane. Theamplitude correlation function of the image is then a convo-lution of the 6 function with the impulse response function ofthe imaging system. By Fourier transformation into thefrequency space, a spectrum that is proportional to the dif-fraction-limited transfer function of the imaging system isobtained, since the transformation of a delta function isconstant. The constant of proportion represents the totalenergy of the signal (i.e., of the diffuser). If the light iscoherent then the image will be a speckle pattern. On the
Diffuser \ 1 hCr) of the image(pround glass)od r) 6d(r) \ Smoi
lefunction
representationof 6 function
Fig. 1. Imaging of a diffuser. If the amplitude correlation functionAd(r) of the diffuser is narrow compared with the impulse responsefunction of the imaging system, the assumption of a delta functionfor the correlation function is valid.
P. C. Uduh
Vol. 6, No. 10/October 1989/J. Opt. Soc. Am. A 1559
other hand, if the diffuser is modulated by a transparency,which then serves as a signal, the proportionality constantgives the total energy of the signal.
8. COHERENT IMAGE OF A GRATING WITHA RECTANGULAR TRANSMISSION PROFILE
The grating transparency is illuminated coherently througha diffuser, which is in direct contact with it, and then isimaged with a diffraction-limited imaging system. The dif-fuser amplitude is thereby modulated by the transparency.Enloe considered the image of a uniform diffuse object.18
Lowenthal and Arsenault'9 extended and generalized En-loe's result by using a structured object diffusely illuminatedby a coherent source in their calculation. They assumed aGaussian distribution for the diffuser amplitude and a linearimaging system as described by Enloe, and they showed thatthe Wiener spectrum is the sum of two terms. The firstterm, which depends on the OTF of the imaging system,represents the Wiener spectrum of the signal and corre-sponds to the signal spectrum for an incoherent imagingsystem. The second term describes the speckle noise anddepends on the coherent transfer function of the imagingsystem. In addition, it was shown that the noise caused byspeckling in the image, with the exception of a multiplicativeconstant related to the total object energy, is independent ofthe object. From their calculation, the average Wienerspectrum of the image is given as
Ie(u) is the spectral function of the signal (object). V10(o),which is derived from the ACF of the intensity distributionin the object, is a constant equivalent to the total signalenergy.
The average total power of the signal can therefore beobtained by integrating the first term in Eq. (23), and theintegration of the second term gives the average power of thespeckle noise.
If a rectangular grating is used as a signal in a normalincoherent optical imaging system, the contrast of the im-age, which can be derived from the OTF of the system, is adeciding factor for the detection of the signal. For diffusecoherent imaging of the grating, the average image contrast(ensemble average) equals the image contrast obtained withan incoherent imaging system. It therefore follows that theimage contrast is adequate to a limited extent to describediffuse coherent imaging. More suitable for the character-ization of the information content of the image are eitherstatistical moments or Fourier coefficients.2 0,21 The com-monly used signal-to-noise ratio of the image can also bederived from these quantities.
The intensity profile of a rectangular grating can be syn-thesized from the profile of a cosine (or sine) grating bysuperposition. The intensity profile of a cosine grating isknown to have the form
I,,,(x) = I[1 + C cos(27rfgx)]. (24)
Here C is the contrast and fg is the spatial frequency of thegrating. Campbell and Robson2 2 showed that the contrastfor a rectangular grating differs by a factor of 4/7r from thatof a cosine (or sine) grating. Taking this into account in the
synthesis of the intensity function of a rectangular grating,we have
Irect(X) = IO 1 +-4\ o
{a( )nl cos[(2n - 1)2lrfgx]}l{ Z 2n -1 I(n=1
(25)
Thus, if the object is a rectangular grating, the intensitydistribution le in the spatial-frequency domain of a gratinghaving a fundamental frequency of fg is
e(fX) =IO (a(X)4 C (-'r h 2n-1
n=1
X ,1/2 [fx - (2n - l)fg] + 1/26[fx + (2n - )fg]i)
(u = fx fy = 0). (26)
The Wiener spectrum of the intensity distribution in theimage of the grating obtained with a diffraction-limited ob-jective with a circular aperture is, from Eq. (23),
Nig(fx) = Ife(fx)W(fx)2. (27)
The index g indicates that we are concerned with gratingimages. The average total powerPig of the signal is obtainedby integrating Eq. (27) over all possible grating frequenciesafter substituting Eq. (26), i.e.,
2uNPig = Nig(fx)dfx
= Ie2 {X2(0) + 8 1)2 2 [(2n - 1)fg]} (28)
where 2ue is the cutoff frequency of the imaging system and(2n - l)fg < 2ue.
To describe the total power of the noise, we must take thebandwidth of the detector (e.g., film, the eye) into account.However, if we assume that the detector has an infinitebandwidth, then the total noise power Png is obtained byintegrating the Wiener spectrum of the speckle, i.e., Eq. (22),over all frequencies u < 2 ue in the speckle pattern,
Png = J t Ni.(u)ududO,, ,U
(29)
where the arguments of the noise function have been trans-formed into polar coordinates. The signal-to-noise ratio S/N is then given as
Pig = °)2 6 + 2 E [(2n - )fg]SIN _=2o A (2n -1)
(2n - 1)fg < 2uo. (30)
The signal-to-noise ratio depends on the fundamental spa-tial frequency of the grating, the contrast of the grating, andthe cutoff frequency 2u of the imaging system. It is dis-played in Fig. 2(a) as a function of the relative frequency i= fg/2ue for five different grating contrasts between 0.2 and1. In Fig. 2(b) it is plotted against the imaging aperture for afixed contrast equal to unity. One can see that the signal-to-noise ratio decreases monotonically to a constant value atthe transmission limit, in agreement with the result obtained
P. C. Uduh
1560 J. Opt. Soc. Am. A/Vol. 6, No. 10/October 1989
S/N
0
'0
111
,P
S/No0
1.0
(a)
1.12
(b)
8 16 24 32 40 48 56
f-- '/ )
Fig. 2. Signal-to-noise ratio S/N in diffuse coherent images ofrectangular gratings as a function of frequency (a) for five objectcontrasts ranging from 0.2 to 1.0 and (b) for six apertures of diame-ters ranging from 0.6 to 3 mm, with an object contrast of unity.
by Enloe for a uniform object, despite the fact that hisapproximations were not used here.
9. DISCUSSION AND CONCLUSION
Qualitatively the relative frequency as defined here is theaverage size of the speckle in units of the grating constant[see Eq. (18)]. Figure 2(a) illustrates the dependence of thesignal-to-noise ratio on the average size of the speckle rela-tive to the grating constant, which is referred to as therelative size of the speckle kOrel. 'rel = 0/g, where p is the
average size of the speckle. We deduce from the same figurethat the signal-to-noise ratio in the image depends on therelative frequency for a fixed object contrast, provided that'rel is small. For large Morel the signal-to-noise ratio ap-proaches a minimum value asymptotically, independent of
the object contrast. Under this condition the validity of theassertion of Rafailov,2 3 that the contrast in a coherent imageis related uniquely to the spatial frequency of the specklestructure, appears to be limited.
The effect of the imaging aperture on the signal-to-noiseratio for a given grating frequency is depicted in Fig. 2(b).As the aperture size increases, a given signal frequency isenhanced compared with the speckle noise.
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