Imaging system design for improved information capacity

Imaging system design for improved information capacity

Carl L. Fales, Friedrich 0. Huck, and Richard W. Samms

Shannon's theory of information for communication channels is used to assess the performance of line-scanand sensor-array imaging systems and to optimize the design trade-offs involving sensitivity, spatial re-sponse, and sampling intervals. Formulations and computational evaluations account for spatial responsestypical of line-scan and sensor-array mechanisms, lens diffraction and transmittance shading, defocus blur,and square and hexagonal sampling lattices.

1. IntroductionFellgett and Linfoot1 2 first applied Shannon's3

theory of information for communication channels tothe assessment of image quality. In their formulationthey accounted for degradations caused by blurring andrandom noise in photographic images. Huck and Park4

extended these formulations to include degradationscaused by aliasing and quantization in images that arereconstructed from digital data.

Information theory treats the reconstructed imageas a received message giving information about the in-cident radiance field and accounts for degradations asloss of information. Although the information contentof the image cannot be directly measured for an objec-tive experimental evaluation of image quality, the re-sponses of imaging and reconstruction systems areroutinely characterized by Fourier and noise analyses.Information theory can, therefore, be regarded as alogical extension of the analyses currently used in sys-tem design and evaluation.

Formulations based on information theory are inev-itably constrained by the assumptions that the systemis linear and isoplanatic, which is inherent in Fourieranalyses, and that the signal and noise amplitudes areGaussian, ergodic, additive, and statistically indepen-dent. However, these assumptions, which are requiredfor a rigorous derivation, lead to a robust expression thatyields generally valid results for a wide range of appli-cations.

Clearly, image quality requirements are so varied thata single design criterion cannot reasonably be expectedto be valid over the whole range of practical applica-tions. For example, as discussed by Fellgett and Lin-foot,1 if the aim of the design is to produce an imagewhich is directly similar to the object without extensiveimage processing and interpretation, then it may bemore appropriate to use the image fidelity criterionwhich assesses the mean-square difference betweenobject and image. However, if the aim of the design isto produce an image containing the highest realizable

R. W. Samms is with Information & Control Systems, Inc.,Hampton, Virginia 23666; the other authors are with NASA LangleyResearch Center, Hampton, Virginia 23665.

Received 27 September 1983.

amount of information without regard to the complexityof any subsequent processing (e.g., edge enhancement)and interpretation, then it would seem appropriate touse the information content criterion which dependsnot only on the OTF and sampling passband charac-teristics of the imaging system, as does the image fidelitycriterion, but also on the statistical properties of thepresumed object set and noise sources. Thus, the se-lection of an image quality criterion for the design of animaging system must be consistent with the intendedapplication.

Results of initial applications5 6 of information con-tent as figure of merit to the assessment of line-scanmechanisms were found to be consistent with the resultsof widely recognized analytical and experimental studiesby Schade7 8 for the design of television cameras. Thisagreement seems appropriate since Schade was con-cerned mostly with resolving small spatial detail nearthe resolution limit of the imaging system.

In this paper we continue the earlier- 6 mathematicaldevelopments of information content as figure of meritand use it to assess the performance of line-scan andsensor-array imaging systems and to optimize the designtrade-offs involving sensitivity, spatial response, andsampling intervals. The earlier studies were limited tothe assessment and optimization of spatial responsestypical of line-scan mechanisms and did not account forcomplete systems with optics. Here we add a detailedtheoretical treatment of the relationship between thesensitivity and normalized spatial response (i.e., PSFand OTF) of imaging systems and evaluate the effectsof lens diffraction and transmittance shading, defocusblur, square and hexagonal sampling lattices, andsampled noise. We address ourselves, in particular, tothe inevitable trade-off between aliasing and blurringand show how the optimization of this trade-off dependson the sensitivity of the imaging system.

11. Model of Imaging ProcessFigure 1 illustrates a linear imaging process which

converts a continuous spectral radiance field inputL(x,y;X) into a discrete signal output s(x,y). We as-sume that the spatial and spectral properties of the ra-diance field L(x,y;A) and the imaging system r(x,y;X)are separable so that L(x,y;X) = L(xy)L(I\) and

872 APPLIED OPTICS / Vol. 23, No. 6 / 15 March 1984

NoiseRadiance lgI

field Signal

L e n s h o o s e n r S a m p li n g

Fig. 1. Model of imaging process.

, X

x.S

Spatial domain

,T

X,

L. I VXs

Bs

X

tx?

2/

;1Xr4--2-

,3Xr

Frequency domain

(a) Square. (b) Regular hexagon.

Fig. 2. Sampling lattices and passbands. Equal sampling densityoccurs (i.e., s = B) when the sampling intervals X of the squarelattice and the sampling intervals X, of the regular hexagonal lattice

are such that X = V/3/2 Xr = 0.93X,.

T(x,y;X) = r(x,y)T(X). The conversion of the contin-uous radiance field L (xy) into the discrete signal s (xy)is defined by the expression

s(x,y) = KL(x,y) * T(X,y) + N(x,y) 111(x,y), (la)

where K is the steady-state gain of the conversion pro-cess, T(x,y) is the spatial response or point-spreadfunction (PSF) of the imaging system, N(x,y) is noise,* denotes spatial convolution, and II (x,y) denotessampling. Salient properties of the process defined byEq. (la) are often more convenient to evaluate in thefrequency domain than in the spatial domain. TheFourier transform of Eq. (la) is

§(v,) = KL(v,w)T(v,w) + N(v,w)) * 111(vw), (lb)

where L (vw) and I(v,w) are the spectral radiance andnoise spectra, respectively, and f(v,w) is the spatialfrequency response or optical transfer function (OTF)of the imaging system. The OTF N(v,w) is the productof the OTFs of the objective lens and photosensor ap-erture.

The steady-state gain K of the conversion process canbe accounted for by the expression

K = kA 1Q J L(X),(X)dX,

where k is the effective transmittance of the objectivelens, Al is the area of the lens aperture, Qp is the solidangle of the instantaneous field of view (IFOV) formedby the photosensor aperture, L(X) and (X) are thespectral components of the radiance field and systemresponse, respectively, and X is wavelength.

Appendix A demonstrates the validity of the nor-malization of the PSF and OTF in Eqs. (1) and theirrelationship to the gain constant K given by Eq. (2).The effective lens transmittance k is given by Eq. (18),and the solid angle Qp of the IFOV by Eq. (23).

Whatever noise is contained in the radiance field (e.g.,photon noise) or generated by insufficient sampling (i.e.,aliasing) can be treated as an undesired component ofthe radiance field. Appendix B uses the 2-D functionN(x ,y) in Eq. (la) as an artifact for structuring a theoryto account for electronic and quantization noise whenthe continuous signal L(x,y) * r(x,y) is sampled withdelta functions.

Figure 2 illustrates the two sampling lattices of in-terest: the conventional square lattice given by thefunction

(3a)

J1. 8(v,a,) = E E 6 - Co-- (3b)m=-- =- X, X,

and the regular hexagonal (or 120° rhombic) latticegiven by the function

JI,(Xy)=X E E2

X {X YiX,(m +n), y -!X,(m -n)} (4a)

mlI(-, = nE=E V _ (m + n) X, rJ (m- 01mn=- n=-@ Xr f

(4b)

The sampling passband P, for the square lattice is theset of all spatial frequencies (vw) with v < 1/2X,, I w I< 1/2X,; and, similarly, the sampling passband Br forthe regular hexagonal lattice is the set of all (v,$) withlvI < 1/'i3Xr and Ivi/2 + -V31 1/2 < l/V\Xr. Thecorresponding sampling passband areas are

I& = 1/X2,

lfBr = 2X r2

(5a)

(5b)

The areas Is I and I-er , and hence the sampling den-sities of the square and regular hexagonal lattice, areequal to each other when the parameters X,,Xr thatcharacterize the sampling density are such that X, =,/3/2 Xr = 0.93Xr.

Conventional comparisons of the square and hexag-onal lattice have been based on the premise that thesignal spectrum is circularly symmetric and bandlim-ited.8-10 Given this premise, it can be shown that theregular hexagonal lattice optimizes the density withwhich the circularly symmetric sidebands can be ar-ranged in the spatial frequency (i.e., Fourier) domainwithout overlap, permitting exact reconstruction of the

15 March 1984 / Vol. 23, No. 6 / APPLIED OPTICS 873

(2)

y

I

4

I

2L..'(X'Y)=X' i �, 5(x-Xmy-Xn),m=-- n=--

11"Y

signal with a minimum of samples. In particular, Pe-terson and Middleton9 have shown that the contiguoussampling sidebands cover 90.8% of the spatial frequencydomain for the regular hexagonal lattice compared with78.5% for the square lattice, and Legault8 and Merser-eau'0 have shown that, as a consequence, regular hex-agonal sensor arrays can have 13% fewer detectors (andcorrespondingly larger detector areas) than squaresensor arrays for the exact reconstruction of circularlysymmetric, sufficiently sampled signals.

However, because the spatial response of opticalapertures can never be negative, OTFs inevitably de-crease gradually with frequency, generally without afinite cutoff (except for lens diffraction). Sufficientsampling by line-scan and sensor-array mechanisms istherefore usually impractical,5-8 so that it is moreappropriate here to compare the performance of the twosampling lattices on the premise that their samplingdensity and photosensor aperture size are both equal.

We assume that the image reconstruction filterfl (v,w) is matched to the sampling passband so that thereconstructed image r(x,y) is the inverse Fouriertransform of

P(V,co) = s(uW) fl( ,),

where

'V' <-II fl (Vc) = ( 2X, 2X, (6a)

0, elsewhere,

[, jvj l J vj I3wl 1II,(v,co) = 2 J,+ 2 V X' (6b)

[, elsewhere.

This reconstruction would be optimum in the sense oftransferring information from the output of the imagingsystem to the displayed image without loss of infor-mation. The actual shape of the reconstruction filteris not important as far as information transfer is con-cerned, providing that the filter extends out to thesampling passband. For example, the reconstructionfilter may be shaped to provide edge enhancementwithout loss of information. However, practical imagereconstructions usually cause some loss of informa-tion.12

4DL(VW)jr(Vw)l2 * .1 (vc; m,n 0,0)

= E E ADL (v __ n )v m n2m=- n-- X5 X5 X., X.5(mn) ;e (0,0)

for the square sampling lattice, and by4.L(VW)T(VW)12 * 111r(V,W; m,n # 0,o)

E E AL [V_(m + n), (m -n)lm=- n=-- V Xr Xr J(mn) i f (0,0) X [ (m + n) ___-____

Ir [ Xr X j 2

(8a)

(8b)

for the regular hexagonal sampling lattice.The term DN(V,W) * 111(vw), which accounts for

electronic noise, is given by4

'N(V,W) * (V,w) = X Z AN ( -)lf ( ) 2

for line-scan mechanisms, where is the spatial fre-quency component in the line-scan direction, lf (w) isthe frequency response of the electronic filter, X is theinterval between line scans, and Y is the interval be-tween electronic samples. If the effects on electronicnoise by filtering (which reduces it) and undersampling(which increases it) are neglected, the degradationcaused by electronic noise in line-scan as well as sen-sor-array mechanisms is accounted for by the expres-sion

d'N(V,W) * mN(V,@) = jBI a, (9)

where iN is the variance of the electronic noise, and [ 9 is the area of the sampling passband given by Eqs. (6)for the square and hexagonal sampling lattices. (Thistreatment of sampled noise corrects that in Refs. 4 and5; see appendix B for details.)

The mean-square difference between the incidentradiance field L(xy) and the reconstruction r(xy) fromthe sampled signal s(xy), using the ideal reconstructionfilter fl (v,), is

0x2= r + 4L(vw)dvdw + + N, (10)

where A$L (W) is the Wiener spectrum of the radiancefield. The first term, Fi2, accounts for the image deg-radation caused by blurring within the sampling pass-band and is given by

lo

Ill. Aliasing, Blurring, and Information DensityTreating the radiance field L(x,y) and electronic

noise N(x,y) as independent Gaussian random (sto-chastic) processes, it can be shown that the Wienerspectrum of the sampled signal s(xy) given by Eqs. (1)is 4

'5

4(V,w) = lL(V,Co)Ir(V,w)12 + N(V,0 * t(V,@)

= 'L(V,W) 1T(V,W)12 + 4L (V,w)jT(V, )j12

* 11(vw; m,n F- 0,o) + N(Uw) * ii(u,), (7)

where L(v,w) and 4?N(v,w) are the Wiener spectrumof the radiance field and electronic noise, respectively.The term L(vw.) (V,w)i2 * 1 1 (v,(; m,n --z 0,0), whichaccounts for the sampling sidebands, is given by

b1z

<ano

ce'

100

10-

10-2

.. , ' lo' lo-' l o, .

0-4 i0-3 t10 101 100° 0p2 =2 + C 2

Fig. 3. Wiener spectrum of radiance field.

3, \

\\.

. K'


.i n -3

b = fJ 4'L(VW)I - (V,CO)I

2dudw. (1)

The second integral term accounts for the degradationcaused by the loss of spatial frequency componentswhich fall outside the sampling passband B. The thirdterm, aa, accounts for the degradation caused by theintrusion of displaced spatial frequency componentsfrom the sampling sidebands into the sampling pass-band P and is given by

a= jff &L'L(V,w)Ir(V,w)j2 * J11(v,cv; m,n #7 0,O)dvdo. (12)

The information gained about a random radiancefield can be regarded as a reduction in the uncertaintyof the probable state of that field. In this sense, it canbe shown that the average amount of information perunit area or information density hi of the sampled signal

(x,y) is4' 5

the (expected) mean separation or spatial detail , andwhose magnitude obeys the Gaussian probability den-sity function with the (expected) mean AL and variance

13

V. System Response

A. Objective LensThe normalized OTF fi (vw) of a diffraction-limited

lens with a circular aperture is given by14-' 6

SS< Pj(v + T/2,w)PI ( - T/2,w) exp(iupv)dvdw

If P1 (v w)PJ (v,w)dvdw

(16)

hi=1 109 1 L (V,W) IT(V,@o) 2 1 dd

2 Jff I L L(VW 1.(V') 1)2 * 111 (v,wo; m,n 5# 0,O) + 4'N(VW) * 1 (V dud

2 1 19 [ 4tL(Vw)Ir(VuC)12 1 dudw,

,J2 Of 2 *i 111(v,@; m~ # 0,0) + K-L) -2-I l4(v~w)I~v~w)1 * I(v,c;mr,n Fl 0,0) + f-J B'

where 4'L(v,w) = aL2L (hvc) is the normalized Wienerspectrum of the radiance field, and oa2 is the variancegiven by

UL = ,ff L(v,w)dvdw. (13c)

The sampling sideband components (L (v,W) (v,w) 12* I(vw; m,n id 0,0) are given by Eqs. (8).

The corresponding average information per sampleor the efficiency with which information is gained is

hi JI -1 = X2 (14a)

hi IB l - = d X 2hi/2 (14b)

for the square and regular hexagonal lattice, respec-tively.

IV. Radiance Field

We assume that the radiance field L(x,y) is bothhomogeneous and isotropic so that the variance ofL(x,y) is independent of (xy). Furthermore, we as-sume that the autocorrelation of L (x ,y) is

4'L(X,Y) = a' exp(-rLr), (15a)

where r2 = x2 + y2. The associated Wiener spectrum,which is the circularly symmetric Fourier transform(i.e., the Hankel transform) of 4 L(X,y) is

4PL (V,WL) = 7aa 1b

[1 + (27rirp)2

]3/2 (15b)

where p2 = v2 + w2. Figure 3 illustrates the normalizedWiener spectrum 4'L(v,w) = +L4 +L(v,W).

Equations (15) can be derived by assuming thatL(x,y) is a random set of 2-D pulses whose separationr obeys the Poisson probability density function with

(13a)

(13b)

where Pi (v,w) is the pupil function given by P (v,w) -T(v,w) for the geometrical coordinates (Dv/2, Dw/2) ofthe lens within the field stop and by P (v,w) = 0 outsidethe field stop, and T(v,w) is transmittance. The nor-malized spatial-frequency variable p- is related to thespatial frequency p = (V2 + W2)1/2 by the approximateexpression - 2XFp, where F = fID is the lens f/No.,and f and D are the lens focal length and diameter, re-spectively. The defocus parameter u is approximatelygiven by

7r D\2 7rAu -AlI-

2X l 2XF 2

where Al = I li - lp l, and i and lp are the image andphotosensor plane distances from the lens, respectively.The normalized coherent cutoff frequency is fic = 1, andthe coherent cutoff frequency for a lens with f/No. F isPc - ;c/2XF = 1/2XF.

The OTF iz(v,w) of a defocused diffraction-limitedlens has been formulated by Hopkins14"15 for a clear lens[i.e., for T(v,w) = 1] and by Mino and Okano16 for twocircularly symmetric lens transmittance shadings. Forone of these shadings, the lens transmittance increasesfrom the center of the pupil toward the edge, as givenby T(v,w) = ( 2 + w2), resulting in an OTF shape sim-ilar to the familiar OTF shape of optical systems witha central obscuration, namely, a response that is de-pressed at mid-spatial frequencies and raised at high-spatial frequencies. For the other shading, the lenstransmittance decreases from the center to the edge asgiven by T(v,w) = 1 - (V2 + w2), resulting in an OTFwith an increased response at low spatial frequencies,a depressed response at high spatial frequencies, and,as was desired, an improved tolerance to defocus or in-creased depth of field.


T (V,(S) =

.8

3 .8

3 .4

< W .2.0~~~~~~~~~~~) .

-.2 clear aperturl..2 .4 .e .8 .0 1;2 1;4 1.6 iL.A 2.0

(a) Clar apertur.

I I I I I I I I.2, . .8 .8 1. 1.2 1.4 1.8 1.8 .0_aO .2 .4 .6 .B 1.0 1.2 1.4 1.6 1.8 2.0

(b) Shaded aperture, a = 2.

'.2 __~~~~~~~~~~J..

.20 .2 .4 .6 .8 1.0 .2 1.4 1.6 1.8 2.0

(c) Shaded perture, = .

Fig. 4. Optical transfer function of diffraction-limited lens with clear and shaded apertures for a coherent cutoff frequency 1/2XF = 1 andseveral values of defocus u. The effective lens transmittance is k = 0.33 for a = 2, and k = 0.17 for a = 1.

(a) Square apertures and sampling -(b) Regular hexagonal apertureslattice, and sampling lattice.

Fig. 5. Sensor-array patterns. Sensor apertures are of equal areawhen ^ys = \fM2Yr = 0.93 ^Yr.

To investigate OTFs for other lens transmittanceshadings, we performed numerical integrations of Eq.

(16) for

T(v,w) = 1 - (v2

+ W2)a/

2(17)

and a = 1 and 2. The results are plotted in Fig. 4 to-gether with the OTF of a clear aperture obtained withthe Hopkins equation. The results for a = 2 are inagreement with OTF curves obtained with equationsby Mino and Okano. Aperture transmittance shading,of course, reduces the amount of light transmitted to thephotosensor. The ratio k of light transmitted throughthe shaded aperture to a clear aperture, or briefly theeffective transmittance used in Eq. (2), is given by

k = 2 XfzIT(z)I2dz,

where z2= V2 + W2.

(18)

B. Photosensor

The conversion of radiance into a discrete signal in-volves some sort of photon detection and samplingmechanism. The most common mechanisms are sen-sor-array and line-scan devices. Sensor arrays usuallyhave square apertures and sampling lattices as illus-trated in Fig. 5(a). Sensor arrays with regular hexag-onal apertures and sampling lattices, as illustrated inFig. 5(b), are also evaluated for comparison. Optical-mechanical scanners usually have either a circular orsquare photosensor aperture, and TV camera tubesusually have an approximately Gaussian spot intensity

profile. Figure 6 characterizes the four spatial re-sponses given by the following equations:(a) Square aperture:

Ts(XY) = Y | 2 2

0, elsewhere,

T8(U,w) = sincysL sinc),w.

(b) Regular hexagonal aperture:

Tr(X,y) = |V3Y 1 2 2 2 2

0, elsewhere,

(19a)

(19b)

(20a)

^ ( ) = 1 sin (y) [Cos 7 Yr ( s si 1 i Yr + )

+ Cos 2Yr (I + sine- rI-a))]2 bv/ 2 / b IJ

+ Cos (7rYrV) sine- Yr (V--) sin 1 Y (~ + o) .

(20b)(c) Circular aperture:

2 c'y,Tc (Xy) = |N3y 2

O. elsewhere,

= 2J,(7rcyp)

rc'yrp

(d) Gaussian spot intensity:

1Tg(X,Y) = -~ exp(-7rr 2/'y2),

'YS

(21a)

(21b)

(22a)

~g (U,(0) = exp(- rSySp 2 ), (22b)

where r2 = X2 + y 2 , p2 = V2 + 2 , and c = 2-f/711.05. The spatial frequency responses of the hexagonaland circular aperture shown in Figs. 6(b) and (c), re-spectively, are very similar to each other; thus, the cir-cular aperture can be used as an approximation for thehexagonal aperture.

The three apertures have the same area AP= =(-x/4)(c/yr) 2 , and the Gaussian spot intensity has thesame equivalent area, when y = /_/2-yr = 0.93 Yr.The solid angle Qp of the IFOV in Eq. (2) is thereforegiven by (see Appendix A)


aS3 :s- -- -- ----

i

---------

I I

QA = LeS)2 = r (qy, 2(3_p = ,(23)

It is convenient for characterizing spatial responsesand comparing their performance to let y, = 0.93 yr =1 be dimensionless, thereby reducing the number ofunits to be accounted for by one. Thus, if I1- and [-]represent magnitude and unit, respectively, -y = I [m]

= 1 (unit) or [m] = yI -1 (dimensionless units). Forexample, the dimensionless forms of X and v are ob-tained as follows:

X= IX -171

V = Il [min] = III1VI.

All quantities calculated using dimensionless variables

1.0

y

4 1.0

.80.5

0.53:X4 .4-

0

-2.0 -1.5 -1.0 -.5 0 .5 1.0x.y

1.5 2.0

(a) Square aperture.

y

0.537

/I \0.620

y

0.564

_.58x i

3.

10 _

-.2

-. 0 - 1. 10 -5 0 .

1.0

.8

.8

3 .4

<. .2

0

-.2

-1.0 -.5 0 .Xy

7 - \

--__ __ ___- _ ____ _ _

1 I I .3 .6 .9 1.2 1.5 1.8 2.1 2.4 2.7 3.0

V., )

1.2 1.5 1.8 2.1 2.4 2.7 3.0V, Co

1.0

.8

.6

3 .4

.+ .2

-.2

I I - I I I 1 -.4.5 1.0 1.5 2.0 I

E -

I I I I I I I I I I.3 .6 .9 1.2 1.5 1.8 2.1 2.4 2.7 3.0

V, 0

(d) Gaussian spot intensity.

.Fig. 6. Spatial response of photosensor apertures and spot intensity profile.


(b) Regular hexagonal aperture.

1.0

.8

.8

.4-

.2

-.2, -2.0 -1.5 -1.0 -.5 0 .5 1.0 1.5 2.1

X.y

(c) Circular aperture.

T(X.y)

1.0

.8

0.5

.6

y i .4

I- .'2

X.

-2.0 -1.5 -1.0 -.5 0x. y

_,;.

_,z

.

I., ,0-2. 0 -1.5 1.U I.D

.0

-------------------

.

I ---------I I -I I I I I I

_

t I I I I I

0 1/9A A 1/3

03- b 09 a Ig S S

6 A A A A A

10 I 4D 9 9 9 9W 32 84 128 256 512

(a) Square array.

I I I I I I

h Q 99 Q9 9id 32 64 128 256 512

Kaja,(b) Regular hexagonal array.

0 square

0 hexagon

16 32 64 128 256 512

Kagoal .(c) Square vs. hexagonal arrays, AL, = 1.

Fig. 7. Information density hi vs SNR KaL/aN for sensor-array imaging systems. Results are given for an infinite lens cutoff frequency(i.e., 1/2XF = -), contiguous photosensor apertures (i.e., Ys = 0.93 Yr,

are, of course, also dimensionless. To restore theproperly scaled units, one multiplies or divides bypowers of unity. For example, the PSF T(xy) properlyhas units [- 2 ] and the dimensional form is obtainedby dividing the dimensionless form by y12[iM2] = 1.

Previous analyses,5 '6 which did not include the effectsof lens diffraction (i.e., spatial filtering by the objectivelens), showed that the Gaussian spot intensity profileof TV cameras provides better performance (i.e., lessaliasing and higher information capacity) than the cir-cular and square photosensor apertures commonly usedin optical-mechanical scanners, and that the perfor-mance of optical-mechanical scanners can be improvedin this respect to essentially equal that of TV camerasby properly shaping the photosensor aperture normalto the line-scan direction and the electronic filter alongthe line-scan direction. Thus, by selecting the circularaperture and Gaussian profile, we encompass the rangeof spatial responses typically encountered with line-scanmechanisms.

VI. Performance and Design

The basic performance and design trade-offs forline-scan and sensor-array imaging systems involvesensitivity, spatial response, and sampling intervals.First, we present a general assessment of the perfor-mance of imaging systems in terms of aliasing, blurring,and information density as a function of sensitivity, lenscoherent cutoff frequency, defocus, sampling intervals,and the statistical properties of the radiance field.Subsequently, we present a series of performance anddesign trade-off curves for optimizing the informationcapacity and efficiency of line-scan and sensor-arrayimaging systems.

A. Performance Characteristics

1. Computational Results

Figure 7 illustrates information density vs signal-to-noise ratio (SNR) for the square and hexagonalsensor arrays. The results show that informationdensity, which might have been expected to increasewith increasing SNR, soon levels off to a nearly constantvalue. The reason for this is that the spatial responseof the contiguous sensor-array apertures insufficientlybandwidth limits the radiance field prior to sampling,permitting aliasing noise to exceed electronic noise ex-

= 1), and several radiance fields with different mean spatial detail Pr.

cept at very low sensitivities. Thus, increases in sen-sitivity provide little or no increases in information orrecoverable spatial detail. Furthermore, the value ofthe SNR at which information density reaches a plateaudepends little on the statistical properties of the radi-ance field or the geometry of the sampling lattice; in-stead, it depends mostly on the relationship betweenspatial response and sampling intervals.

Since the sampling intervals of sensor-array devicesare fixed (with the apertures, at best, contiguous), it isnecessary to spatially filter the radiance field prior tosampling by either limiting the lens cutoff frequency,shading its transmittance, defocusing, or a combinationthereof. Of course, the selection of lens cutoff fre-quency and transmittance shading also affects the SNR.In fact, the two objectives-to reduce the lens cutofffrequency and increase the SNR-are inherently con-tradictory. This contradiction arises because a re-duction (or increase) in the lens aperture diameter re-,duces (or increases) both the lens cutoff frequency andthe SNR. The solution would be to increase the lensdiameter (with or without transmittance shading) asneeded to attain the desired SNR and then to defocusthe optical system to suppress spatial frequency com-ponents outside the sampling passband.

Figures 8 and 9 illustrate the variation of aliasing,blurring, and information density with lens cutoff fre-quency and defocus, respectively. The basic trade-offis between aliasing and blurring: aliasing decreases andblurring increases as the lens cutoff frequency is reducedor defocus is increased. As before, the optimum com-promise between aliasing and blurring depends little onthe statistical properties of the radiance field or thegeometry of the sampling lattice; instead, it dependsmostly on sensitivity or SNR.

Since the sampling intervals of line-scan devices arevariable and independent of photosensor aperture sizeand shape, it is possible to divorce the selection ofsampling intervals from the design trade-offs betweensensitivity and spatial response (or resolution). Figure10 illustrates the variation of information density withsampling intervals for line-scan imaging devices. Asmight be expected, while the effect of changes in sam-pling intervals on blurring is small, it is large on aliasing;for example, a decrease in sampling intervals by a factorof 2, from X = 1 to X = 0.5, decreases aliasing noiseby a factor of 5 for the circular aperture and 40 for


I

-~~i--

K 8l ^ l l l

-0P

- 0 9

I I I I

o 0

~~ [1 s qua re -- g O~~~ hexagonal

= I I I I I0~~~~0 0

I I I I I I

I II

.2 .3 .4 .5 .6 .7

I /2XF(a) Square array.

.2 .3 .4 .5 .f .7

1/2XF(b) Regular hexagonal array.

- a

I I I I I

@ a0000

I I 1. - L__I I I _.2 .3 .4 .5 .6 .7

1/2XF(c) Square vs. hexagonal arrays. /, = 1.

Fig. 8. Aliasing a,,, blurring ofb, and information density hi vs lens coherent cutoff frequency 1/2XF for sensor-array imaging systems. Resultsare given for a clear lens, contiguous photosensor apertures (i.e:, 'Ys = 0.93 Yr = 1), a SNR KUL/GN = 128, and several radiance fields with

different mean spatial details ,.

b' 10--

10--

I I I I I I -

0

0 0~~

$ I I_

I I I I I I

- o n~~~~q

- 0 0 A A

__ . _i 8 I I I

lal-- I_ I '--T----T --

:S ' a a 0 R 1A A A

10° _ __ _ |0 2 4 6 8 10

(a) Square array.

F - IT _ - - T--17__ '--T---

At L 0 1

I I L__I__0 2 4 6 8 10

(b) Regular hexagonal array.

0 2 4 6 8 10u

(c) Square vs. hexagonal arrays, , = 1.

Fig. 9. Aliasing as blurring Ub, and information density hi vs defocus u for sensor-array imaging systems. Results are given for a clear lenswith coherent cutoff frequency 1/2XF = 1, contiguous photosensor apertures (i.e., Ys = 0.93 yr = 1), a SNR KL/laN = 128, and several radiance

fields with different mean spatial details pr.


I I

- O 1/9- A 1/3

_ 0 10 3 A

- nA 9 z

I 4)

10-'

bs 10-'

10'

10°

b

10-'

I II I

0

c0 8 1 I 1 I

Co.

I I I I

PI ~I, 0 1/9

1/3 1 - 0 1 _ 0 3 0

, 9I

0 * Q

0

10°

10-'

_ 6 I I I I0

_ _

0 0 square __0 hexagonal

I I I I I I

- _ _ - F I I I I -

00

00

t _L I I I j I

- I I I I I

0 0 00 9 @1

I I I I I I

1 1

7

9 n� n

z

I I I I

. A i \

l ' W W t l lIX-

I

FQH

10'

10''

10'

10-

10* I l l I

t- 10-' I I I I

101 I l l.9 £V. .1 .0

X.(a) Circular.

I I I I I I

I l I

.5 .8 .7 .8 .9 1.0X.

(b) Gaussian.

o circularGaussian 0 0

0 no 0 0 LIl

I I I I I I

I II

~I I I I I

I I I I

.5 .6 .7 .8 .9X.

(c) Circular vs. Gaussian, A, = 1.

1.0

Fig. 10. Aliasing a, blurring Orb, and information density hi vs sampling intervals X, = Y for line-scan imaging systems. Results are givenfor an infinite lens cutoff frequency (i.e., 1/2XF = ), a SNR KUL/UN = 128, and several radiance fields with different mean spatial details

hr -

the Gaussian spot intensity, and increases the infor-mation density by a factor of -5 for both PSFs if theSNR is very high (i.e., K c-L/N 256).

2. Information Density s Spatial DetailSignal information density varies comparatively little

with large variations in the mean spatial detail of therandom radiance field. For example, variations of themean spatial detail by nearly 2 orders of magnitude(from 1/9 to 9 times the photosensor aperture dimen-sion) yield information density variations by less thana factor of 3.

Information density tends to be maximum when themean spatial detail is approximately equal to the pho-tosensor aperture dimension. This result is consistentwith the conditions that lead to the theoretical upperlimit of information density. These conditions requirethat the OTF and sampling passband be matched to theWiener spectrum of the radiance field. 5

3. Aliasing vs Blurring

Whereas blurring is a source of degradation thatpermits the retrieval of information about all spatialfrequency components for which the SNR is sufficientlyhigh, aliasing is a source of noise that causes irretrie-vable loss of information. Thus, if it is desired toreconstruct the smallest possible detail within thesampling passband, aliasing should be minimized byalmost completely suppressing the OTF outside the

sampling passband, and the SNR should be increasedto large values so that the desired detail can be recon-structed from the extensively blurred signal within thesampling passband by image enhancement usinghigh-pass filtering.

4. Hexagonal vs Square Sampling LatticeThe hexagonal sampling lattice provides 5% higher

information density than the square lattice. The im-provement ranges from 3% at very low SNRs to 6% athigh SNRs and is relatively independent of lens cutofffrequency, defocus blur, and statistical properties of theradiance field. This improvement appears modestcompared to the previously discussed assessments basedon bandlimited and noise-free signal spectrums.

B. Design Optimization

1. Computational ResultsFigures 11-14 present various performance and de-

sign trade-off curves. The radiance field statistics arelimited to a mean spatial detail Ar = 1; as can be seenfrom the previous results, design optimization is ratherindependent of these statistics. The lens aperture iseither clear or has a transmittance shading character-ized by the shape parameter a = 2 (with an effectivetransmittance = 0.33). Shadings for which a > 2would lead to a higher effective lens transmittance butless high-frequency suppression, while shadings forwhich a < 2 would lead to OTFs with slightly more


b

A

b

j

high-frequency suppression but lower effective lenstransmittance. It is not intended here to optimize lenstransmittance shading but to indicate potential ad-vantages and disadvantages.

2. OTF vs Sampling PassbandThe relationship between OTF and sampling pass-

band which maximizes information density and effi-ciency is a function of sensitivity.

As sensitivity is increased in sensor-array imagingsystems, the trade-off between aliasing and blurring

should change in favor of reducing aliasing at the costof increased blurring. This can be accomplished by theproper selection of defocus as well as lens cutoff fre-quency and transmittance shading. As demonstratedin Figs. 12 and 13, information density can be increasedmost effectively by increasing both lens F/No. and de-focus blur together.

As sensitivity is increased in line-scan imaging sys-tems, the sampling passband should also be increased.However,- as demonstrated in Fig. 14, while informationdensity increases monotonically with increasing pass-band, information efficiency begins to decrease as the

.3 .4 .5 .6 .7 .8 .9 1.0 1.1

1/2>.F.3 .4 .5 .6 .7

1/2..8 .9 1.0 1 1 . .3 .4 .5 .8 .7

1 /2XF

1 /2XF0.53

-0.1----- 0.41___-- 0.36

___- 0.35

I I I I I I.9 1.0 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

V. C

(b) Clear lens, square vs. hexagonal.

. _,- - -- - -- -

1/2).F

0.36-0._---- 0.45

I I I I I I I I I I I0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

V, C

(c) Square, clear vs. shaded lens.

Fig. 11. Information density hi vs lens coherent cutoff frequency 1/2XF for sensor-array imaging systems, and relationship between OTFand sampling passband which maximizes information density.

1/2IF I - 0.5- ----- 1.0- -- 2.0

I I I I4 a 12 l2 20

(a) ala = 128.

_ 1/2 KajarI I I I- 0.5 32- 1.0 128 _

- 2.0 512 /

I I V I I I0 4 8 12 i8 20

u(b) Kajar, adjusted for lens F-number

.1

8 _ I I ------ 8

-.2

0 > .1 .2 .3 .4 .5 .

(c) Optical transfer functionsand sampling passband.

Fig. 12. Information density hi vs defocus u for sensor-array imaging systems with a clear lens and square array, and relationship betweenOTF and sampling passband which maximizes information density.

1/2).F Ka,10

0.5 111.0 422.0 169 _

0 4 8 12 18 20

(b) Kajar adjusted for lens F-number

and shading.

1.0

.8

p.a~

u

0

__---- 10--- 18

-CCXeI

I I I I I I I I I I._ .. .2 .3 .4 .5 .8 .7 .8

V. Ca)

(c) Optical transfer functions

and sampling passband.

Fig. 13. Same as Fig. 12 except for a shaded instead of a clear lens.


C.0

.6

10 I I I KoJaa -

32_~~~~~~~ 64 -

- ------- 128 - X 5

I I I I I I I I.2

I I I I I I I I -

- ~ square--------- hexagonal

I I I I I I I I

- I I I I I I I I -

- clear--------- shaded

I I I I I I I I.8 .9 1.0 1.1

I I I I.1 .2 .3 .4 .5 .6 .7 .8

V.1a

(a) Clear lens, square aperture.

10'

5

0'

D:5.6

I 1/2),F I I I I- - 0.5

-- 1.0--- _ 2.0

I I I I I0 4 8 12 18

(a) Kaja, = 128.

20 .9 1.0

-

I

a

I-

.7 .8 .9 1.0

sampling passband approaches the first zero crossing It is interesting to compare these results to recom-of the OTF. Thus, as a compromise between infor- mendations made by Schade,7,8 based on extensive andmation capacity and efficiency, most design and per- widely recognized studies of TV systems. He definesformance trade-offs should lead to a (normalized) line-scan intervals as a function of the OTF of thesampling interval between 0.5 and 0.7: a sampling in- imaging system and recommends that the lower andterval near 0.5 if the sensitivity is very high and near 0.7 upper limits on the line-scan intervals, expressed in ourif the sensitivity is low. notation, be

._ - , - --

A ---------- 64

z ow - 1-'- 2564

.2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1X.

I_ F I I I I I I I .

I I I I

I I I I I i I I I I

circular------- Gaussian

I I . I I I I II . - . . .8 .7' .8 .9 1.0 1.1

X.

(i) r = . = (, V < , 1 < X,{

I I I I I I I I 1

I I I I I I I I I

I fIJ Iaj

- I I I I I I I

- 4

clear--------- shaded

I I I I I I .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1

L~~~~~~X

-_FFIFl-TFFF I I rT1

.- I I I I I - I

:: 10 ,

E

X. .. .3 .4 .5 .6 .7 .8 .9 1.0 1.1

X.

0, = > 1.0.V < 1.07 < < 1O. v > 1.07, co > 1.07

I I I I I I I I L W J I.2 .3 .4 .5 .6 .7 .8 .9 1.0 I.I

X,

O.7D

1. _ -------- T---- -1

.8_ ,= ,0.7 & 0.5.8 _ I

1. -~~~~~~~~~~~~~~~~~~~~~~~~~~

s.4 1 1,

.2-

-.21 I 1 1 1 1 1 1 1 1 I0 .2 .4 .8 .8 1.0 1.2 1.4 1.8 1.8 2.0

V, 1

_ - - - - -- - - -_I _ _ _

X\\\ ,~~~~~~~~~~~~~~~~~~~~~~~~~~I

_ "'S~~~~~~~~~~~~~~~

O .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.1

F _____ I___________ I

I , I

----- -

0 0 I I .8 I I I I . IV .10 V .10

(a) Clear lens, circular aperture.

(iii) Optical transfer functions and sampling passbands.

(b) Clear lens, circular vs Gaussian. (c) Circular, clear vs shaded lens.

Fig. 14. Information density hi and efficiency hi B |- 1 vs sampling interval Xs for line-scan imaging systems, and relationship between OTFand range of sampling passbands which provide a favorable compromise between high information density and efficiency. The lens coherentcutoff frequency is 1/2XF = 1, and the SNR is KL/aN = 128. The reconstruction passband Ps is equal to the sampling passband A,: (i) forall sampling intervals, and (ii) for all sampling intervals larger than 0.47. The sampling passband exceeds the OTF zero crossing when X,

< 0.47.


10 I I I I I I r I I

_10. I - I I I I

- I I I I I I I I I I

_~~~~~~~~~~~~~

_~~~~~~~~~~~ZZ -

I I I I I I I

I I I I I I I I I I

1 1

2v o=0.05 2vT=0.4

where vT = 0.05 and vT = 0.4 are the frequencies at whichthe OTF normal to the line-scan direction is 0.05 and0.4, respectively. That is, he recommends that thevalue of the OTF at the limit of the sampling passbandshould range from 0.05 to 0.4, depending on imagequality requirements and communication bandwidthconstraints.

Our results lead to design recommendations similarto those made by Schade but more explicitly as a func-tion of sensitivity. For line-scan imaging systems thevalue of the OTF at the sampling passband limit shouldrange from 0.03 to 0.2: the low value of 0.03 should beapproached if the SNR is high and the major designobjective is high information capacity; the high valueof 0.2 should be approached if the SNR is low or themajor objective is high information efficiency. Forsensor-array imaging systems the value of the OTF atthe sampling passband limit should range from 0.1 to0.2: the low value of 0.1 should be approached for highSNRs; the high value of 0.2 should be approached forlow SNRs. In addition to SNR, the shape of the OTFshould also be considered, as can be observed by com-paring the results in Figs. 12 and 13.

High SNRs are desirable in applications in which itis important to extract fine spatial detail with imageenhancement techniques. Image enhancement thatcan be attained by high-pass digital filtering is com-monly limited by photosensor and aliasing noise.Furthermore, as the results in Figs. 11(a) and 14(b)demonstrate, performance and design trade-offs are lesscritical if sensitivities are low.

3. Lens Aperture ShadingShading of the lens aperture transmittance would

usually not be advantageous for line-scan imaging sys-tems. It is usually difficult enough to obtain both highangular resolutions and high SNRs even without theabsorption caused by the transmittance shading. Thereare two other more effective approaches to suppressingaliasing: decreasing sampling intervals and shaping thesensor aperture.

However, neither of these two alternate approachesis applicable to sensor-array imaging systems. Fur-thermore, it is easier to obtain high sensitivity becauseof the relatively long dwell times permitted by thesensor-array mechanisms. Thus, aperture shadingcould be advantageous. The optimization of thisshading, involving a trade-off between sensitivity vssuppression of aliasing and increase in depth of field,would differ with application.

4. Sensor Aperture ShapingThe Gaussian spot intensity of TV cameras can, be-

cause of its shape, attain -24% higher informationdensity than the circular photosensor aperture com-monly used in optical-mechanical scanners if the lensdiffraction (i.e., spatial filtering) is negligible and thesensitivity is high (i.e., Kc-L/oN 128). If these con-ditions exist, it is advantageous to shape the photo-sensor aperture and electronic filter5'6 of optical-me-

chanical scanners to obtain an effective spatial responsesimilar to the Gaussian spot intensity.

However, if lens diffraction provides effective spatialfiltering and the sampling intervals are within the rec-ommended range, there is no advantage to be gained byphotosensor aperture shaping; in fact, the performancewith the circular photosensor aperture can then exceedthat with the Gaussian spot intensity.

VIl. Concluding RemarksBy tying together the statistical properties of the

radiance field and the sensitivity, spatial response, andsampling interval of imaging systems into a single figureof merit, information theory provides an obviously at-tractive approach for optimizing performance and de-sign trade-offs. A particular advantage of this approachis the ease with which the effects of changes in designparameters can be assessed. Such assessments wouldbe much more difficult and expensive to obtain fromexperimental evaluations.

In our study we represented the radiance field by aset of Wiener spectrums that range, within the OTF ofthe imaging system, from a nearly constant magnitudeto a magnitude which decreases at the rate p-3 (wherep is spatial frequency). This spectrum set accounts formost natural scenes. However, a significant departurefrom this set could result from periodic components inthe radiance field. The Wiener spectrum of a radiancefield with a periodic as well as random component is thesum of two parts-the continuous curve for the randomcomponent (accounted for in our study) and a series ofimpulses for the periodic component' 7 (not accountedfor in our study). The effect of periodic components,usually referred to as moire patterns in optical systemanalyses, has been illustrated in numerous publications(e.g., Schade7 8).

Our analysis is primarily concerned with the inevi-table trade-off between aliasing and blurring, and hencethe relationship between OTF and sampling passband,in those applications which aim to maximize informa-tion content. Our design recommendations are con-sistent with those made by Schade7,8 who was concernedprimarily with the detection of spatial detail near theresolution limit of television cameras. However, ourrecommendations are more explicit in terms of type ofimaging system (i.e., line-scan and sensor-array), sig-nal-to-noise-ratio, shape of OTF (e.g., lens apertureshading and sensor aperture shaping), and geometry ofsampling lattice (i.e., square and hexagonal). They alsodifferentiate between emphasis on information ca-pacity for applications in which data transmission is nota major constraint and information efficiency in whichdata transmission is a major constraint.

To relate the results obtained by information theory,or any other criterion, to a specific application, it wouldbe usually desirable and often necessary to augmentanalytical assessments with experimental evaluations.It is then, of course, important that these evaluationsare consistent with the intended application, the pre-sumed object set, and the subsequent data processingand interpretation.


Appendix A: Sensitivity and Spatial ResponseThis appendix formulates the relationship between

sensitivity and normalized point-spread function (PSF)and optical transfer function (OTF) as used in Eqs. (1)to model the line-scan and sensor-array imaging pro-cesses. The formulations include the various transferfunction norms and their relationship to the gain con-stant K, and the explicit presentation of the coherentOTF which is needed to determine defocus.

It can be shown that for incoherent and quasi-monochromatic radiation the van Cittert-Zernike the-orem for the mutual coherence function' 5 and Fourieroptics' 8 lead to an expression for the radiant power perunit area at the photosensor plane given by

P(x,y) = 2 Jf L(x',y')JhI(x_-x',y _y')J2dx'dy', (Al)

where L(x,y) is the geometric image of the radiancefield, and h (x,y) is the coherent PSF of the lens givenby

hi(x,y) = SI Pl(Xliv,Xli) exp[iWp 2 + i2r(vx + wy)]dudw.

(A2)

To allow for the possibility of absorptive shading ofthe lens characterized by a real amplitude transmittanceT(x,y), we let the pupil function of the lens PI(x,y) =T(x,y) for the geometrical coordinates (x,y) of the lenswithin the field stop, and P1(x,y) = 0 outside the fieldstop. When the normalized coordinates (v,w) = (V/Pc,w/Pc) are used, it is understood that the functionPj(v,w) in Eq. (16) is related to PI(x,y) discussed hereby the relation

P1(v,w) Pi(Xlipv ,Xlipw) P1P , )

The parameter W - 7rXAl is a measure of defocus whereAl = I li - p I, and li and lp are the image and photo-sensor plane distances from the lens, respectively.

The coherent transfer function hI (v,w) is the Fouriertransform of h(x,y) or

hl(v,c) = Pl(A\liuxiw) exp(iWp2 ). (A3a)

For clear optics with circular symmetry, Eq. (A3a) re-duces to

hl(vco) = hi(T) = exp(+iu 2/2), < 1, (A3b)

where the normalized frequency ; is defined by p- P/PcThe defocus u then becomes

2 7r (D12 rAlu 2Wp-Al I- c c (A42X Mli 2XF 2

where p, is the coherent cutoff frequency given by

P, = D/2Xli 1/2XF. (A5)

Equation (Al) can be rewritten as

P(x,y) = X2 L(xy) * hl(xy)12.

The normalized PSF for the objective lens is then de-fined by

Tr(X,y) =Ihi(x,y)12

(A6)

SfS Ihi(x,y)I2dxdy

Since

J}' Ihi(xy)I2dxdy = fJ' Ihi(v,W)I2dvw,

the corresponding normalized OTF becomes

h1(vw) * h(-V,-c)

jiJ hI(V,0)J2dvdw

(A7)

The OTF Tl (v,w) is normalized in the sense that T (0,0)= 1.

Furthermore,

ff - Jh,(,,,@)J2dvd = f IpI(XliuXliC0)J2dvdco = kA1

where Al is the area of the lens aperture, and k is theeffective power transmittance of the objective lens givenby

k =-,ff P(x,y)I2dxdy.

AlHne

Hence,

P(xy) = kA- L(x,y) *rj(xy).

Generalizing this result yields

P(x,y;X) = kA L(xy;X) * T(xy;X),JL

(A8a)

(A8b)

where L (x,y;X) is the spectral radiance and P(x,y;X) isthe incident radiant power per unit area per unitwavelength.

Formulations of the imaging process given by Eqs. (1)are based on the assumptions that L(x,y) = f ' L(x,y;X)dX, and that each spatial location in the scene has thesame spectral content, so that

L(x,y;X) = L(x,y)L(X), (A9)

where Sfo L(X)dX = 1.For uniform spatial response, the quantum efficiency

n\(X) is defined as the number of electronic carriersavailable for conduction per incident photon of wave-length X. The rate at which photons are incident perunit area of the photosensors which reside in thewavelength interval AX is

P(x,y;X)hc/X

If e is the electronic charge, the electronic current perunit area due to photons in the wavelength interval AXis

AJ(xY;X) e ) 7(X)P(xy;X)AX,

and hence the total current per unit area is

J(xy) = f e ( (X)P(xy;X)dX. (A10)

Let Pp (x ,y) be the photosensor pupil function which isunity in amplitude for (xy) inside the active photo-


TM v,Wt) =

sensor area and vanishes outside. Then the total sensorcurrent becomes

s = J -j P,(x,y)J(x,y)dxdy. (All)

In a similar manner as before, we define a normalizedphotosensor PSF as

Tp(X,Y) =

SSL Pp(x,y)dxdy

P,(X,Y) Pp(xy)

where AP is the area of the photosensor aperture.Fourier transform defines the OTF:

Tp(U'c) =55 -¾,(x,y) exp[-i2r(vx + y)]dxdy,

(A12)

The

Tp(OO) = JJ p(x,y)dxdy = 1. (A13)

An effective PSF for the objective lens is introducedby

° e (A) (X)L(X),,(x,y;X)dX

e -) (X)L(X)dX

Therefore,

s = K J'-f rp(x,y)[L(x,y) * r(x,y)]dxdy,

where

K - AAp L(X)T(X)dX,

and the spectral property of the sensor is

(hc)

Defining Qp - AP/l as the solid angle of the instanta-neous field of view formed by the photosensor aperture,Eq. (A15a) becomes

K = kAQp | f L(x),(X)dX.JO0

(Al5b)

From Appendix B, the signal value that is associatedwith the scene coordinate (x,y) is given by

s(x,y) = KL(x,y) * 1(X,y) * Tp(X,Y)

= KL(x,y) * -(X,y).

(Al6a)

(Al6b)

1. Sensor-Array Mechanisms

For a stationary scene and signal-independent pho-tosensor noise, the sensor signal at location (mX,nY)is given by (K = 1):

s(mX,nY;t) = ffS [L(x',y') * TI(X',Y')ITp,n(X,y')dX'dy'

+ Ne(mX,nY;t) * rf(t), (Bi)

where TI(x,y) and Tp,mn(x,y), respectively, are thepoint-spread functions (PSFs) of the objective lens andphotosensor aperture located at position (mX,nY), andrf(t) is the impulse response of an electronic filter fornoise reduction. Consequently, assuming inversionsymmetry Tp,mn(X,y) = rp(x - mX, y - nY) = rp(mX

- x, nY - y), and Eq. (Bi) becomes

s(mX,nY;t) = L(mX,nY) * i-(mX,nY) + Ne(mX,nY;t) * Tf(t),

(B2)

where r(x,y) r l(x,y) * rp(x,y). The signals(mX,nY;t) is sampled at time tmn in such a mannerthat the set of all tmn) falls within the same frame pe-riod. Suppressing the time of sampling, Eq. (B2) canbe written as

s(mX,nY) = L(mX,nY) * r(mX,nY) + Ne(mX,nY) (B3a)

(A14) or, equivalently, as

s(m,n) = Ls(m,n) + Ne(m,n). (B3b)

The electronic noise is assumed to be independent ofthe signal. In addition, it is also reasonable to assumethat the electronic noise originating from distinctphotosensors in an array is stationary and uncorrelated.(This latter assumption cannot be exactly true since anysensor array is finite. For a discussion see section online-scan mechanisms.) Hence the autocorrelationfunction for a sufficiently large array is

RNe(m,n) = ElNe(m' + m, n' + n)N(m',n')}

= 5(m,n)e,

where

5(m,n) = 1, m,n = O

= O, m,n 7 O,

and the variance is

¢e -- EJN'(-,n)J.

(B4)

Again, the lens and photosensor aperture OTFs f1 (v,co)and fp (v,co), respectively, are normalized so that (0,)= -?T (0,O) = p (O,O) = 1.

Appendix B: Sampled Noise

This appendix examines the connection betweenanalog and discrete formulations pertaining to the in-formation content of images and formulates the effectof electronic and quantization noise in sensor-array andline-scan imaging systems. For ease and clarity ofnotation, we use a rectangular sampling lattice for theanalysis. Results may then be stated simply in termsof the sampling lattices used in the main part of thispaper.

(B5)

The Fourier transform of a 2-D discrete process isdefined here as

X(v,w) = _ X(m,n) exp[-i27r(mvX + nwY)], (B6a)m,n

where X (v,w) is a periodic function in terms of the twospatial-frequency variables (v,w) with periods 1/X and1/Y, respectively. The inverse transform is then givenby

X(m,n) = I PI -1 12X 51/2Y X(vv)f-1/2X f-1/2 Y

X exp[i27r(vmX + conY)]dvdw, (B6b)


-

where A is the area of the sampling passband and regionof integration; B = (XY)-' for the rectangular samplinglattice.

Let the discrete function X(m,n) be a (wide-sense)stationary random process with autocorrelation R(m,n).The corresponding Fourier transform pair is

J(v,co) = E R(m,n) exp[-i2r(vmX + wnY)J,m,n

R(m,n) = XY -r J(v)

J-_/ 2 X J-1/2Y

(B7a)

X exp[i2ir(vmX + wnY)]dudw. (B7b)

For m,n = 0, the total power is

11/2Xr 1/2YEJX

2(Mn)J = (0,0) =c [ XYe(v, )] dvd, (B8)

S-1/2X4 -1/2Y

and the power spectral density or Wiener spectrum is

4(uw) XY(v,). (B9)

The power R (0,0) resides entirely inside the frequencyplane region (Ivl S 1/2X, IwI s 1/2Y) of the samplingpassband. This result is a natural consequence of thediscrete process. By analogy, the power spectral den-sity for the electronic noise is

'Ne(CU@) =c |fl| le = XY,4Je. (B10)

Let sa (x,y) be a stationary analog random processsuch that s(m,n) = sa(mX,nY). The signalLs(x,y) =L(mX,nY) * (mX,nY) is an immediate example,whereas electronic and quantization noise have no suchphysical analog. Furthermore, let R(m,n) be the au-tocorrelation of s (m,n) and a(V,W) the power spectraldensity of the corresponding analog process. It can beshown that

R(m,n) = R0(mX,nY), (Blla)

40'co) = V X y)W (Bllb)

where

4)" (v) =54' R.(xy) exp[-i27r(vx + wy)]dxdy.

Applying these relationships to the signal LS (x ,y), wehave for the power spectral density of the discrete pro-cess Ls (m,n):

4~(v~w)=z~L(~-M n) (M n) 12

(V^@) = E (V'W X 2 1 ( xl (V1

_DL (V,W) | (Vw)| 2* i (uw) (B12)

At this juncture, it is appropriate to briefly discusssome aspects of information theory associated with thediscrete process X(m,n). Let X(m,n) represent anycombination of the Ne(m,n), L(m,n), and (later)N (m,n) which are discrete variables in their spatialcoordinates but with continuous amplitudes. X(m,n)is finite for

IN,,( - 1 ll Ny - 1

and vanishes otherwise. This is sufficient to allow thediscrete Fourier transform pair representation ofX(m,n) given by

1 (N. -1)/2 (N,- 1)/2X(m,n) = E E X(vk,wl)

N,,Ny k=-(N,,-1)12 =-(Ny-1)12

X exp(i27rUkmX + i27culnY),

(N.-1)12 (Ny- 0/29(vk,W,) = E E_ X(m,n)

m=-(Nx-1)/2 n=-(Ny-1)/2

X exp(-i27rkmX - i27rw1 nY),

(B13a)

(Bl3b)

where

Vk = k/NIX -- k/lx, col = 1/Ny Y -- 1/1,,

X(v/,,cl) = for IkI > 2 -l ,l > Y2 2

Note that X(vkw 1) is the value of the Fourier transformat (k,col). Therefore, the N.Ny values of X(m,n) arecompletely specified by the N.Ny values of X(Mk,c01).Since the X(m,n) are real, the truly independent setcould be chosen for

- I~~~kl < 2 ,1'0,lk• 2

which shows up as a factor of 1/2.The relation [Eq. (B13)] is identical in form to that

used by Fellgett and Linfoot [Ref. 1, Eq. (3.31)]. Theyargued that for a bandlimited image (I vI s 1/2X, I I< 1/2 Y in our notation), the image is completely spec-ified by a discrete set of its values at the sampling points(mX,nY). Furthermore, the image is characterized toa good approximation by its values at the finite set ofN.Ny sampling points inside the region of area A = lX Iywith all others set to zero. Reversing the above argu-ments, the Fourier transform of the image is essentiallyspatially bandlimited (Ix I ' I/2, I < y/2) and it (theFourier transform) is well characterized by a discreteset of values at NxNy frequency sampling points (m/lx,n/lY). Even though Fellgett and Linfoot are dealingwith a truly continuous system, it is reduced to a dis-crete system before their analysis of information ismade. They did not, however, consider the effects ofundersampling or discrete noise sources.

Equation (B13) represents a linear transformationbetween the NNy variables X(m,n) and X(Vk,W1)For the change of variable

Y(VkMl) = X (kc1),

the maximized entropy in terms of the 9(vk,,c) repre-sentation is, up to a constant, given by! 1~~~ (N,-1)/2 (Ny )/2

- Y ZY 1092(Vk,W1),2 k=-(N.-1)/2 1=-(NY-1)/2

where the maximization is under the constraint 4 (Vk ,w1)-El.(vk, 1)l

2 } and 4 (kWl) can be shown to be thepower spectral density of the X(m,n) process. FromVk = k/l, c1 = I/ly, the density of the states is 1xly = A;and for a sufficiently smooth function, the entropy isfound to be


A f 1/2X 1/2Y 109 2 (vco)dvdco.

2 _J-1/2X J-1/2Y

When the aliasing and electronic noises are identified,the same assumptions used by Fellgett and Linfoot re-sult in Eq. (B13). It should be noted that Eq. (13) is interms of the power spectral densities of the discreteprocess. When multiplied by A, this is the amount ofinformation of equivalently the log2 of the number ofdistinguishable discrete images.

Equation (B13) is identical to an analog systemsampled with delta functions as long as the electronicnoise term is properly defined, and a reconstructionfilter which vanishes at no more than a negligiblenumber points and vanishes outside the Nyquist planeis used. As noted in the main text, the information isnot affected by the detailed shape of the filter. Equa-tion (1), as written, implies that N(xy) is to be treatedlike the radiance field. Indeed, noise in the objectshould be treated in the same way. On the other hand,the electronic noise has no such physical analog. Weconstruct the artificial analog process N(xy) for theelectronic noise by the reconstructed discrete processby

N(x,y) = Y Ne(m,n) sinc ( m) sinc( - n)

'N(VW) = | 2X 2Y

O. otherwise.

(B14a)

raster dimensions, and M. = /X and My = y/Y be thenumber of samples per raster length. Furthermore,let

q = n + mMy,

n =0,:1,..., M 1) m =O, . (X )

where MxMy are odd for symmetrical limits. Thisnotation places the 1-D data stream into a 2-D array.Consequently, s(qY) = s[(n + mMy)] s(mX,nY).

Separating the convolution integral given by Eq.(B16) into vt' segments for which x = mX is a constantand introducing the transformation vt = y + mly, weget

(M -1)/2 /2s,(mX,nY) = F ~ (mX,)

m'=-(Mx-1)/2 -Iy/2

+ Ne(y' + m'ly)]Tf[nY - y' + (m - m')ly]dy'.(B17)

If the characteristic distance of the filter is of the orderof the sampling interval Y, except for some error withinseveral Ys around line switching points at the edges ofthe image frame, the integral inside the sum is negligibleexcept when m = m'. Thus Eq. (B17) can be simplifiedto

s(mX,nY) = L,(mX,nY) + N(m,n),

(B14b)

The analog correlation function is given by

RN (XY) = 55 N(v,w) exp[i2r(vx + y)]dvdw

= aNe sinc () sinc ( Y . (B14c)

2. Line-Scan Mechanisms

When the image of a scene undergoes coordinatetranslations as a result of the line-scan imaging process,the signal from a single photosensor is given by

s[x(t),y(t);t] = L[x(t),y(t)] * r'[x(t),y(t)] + Ne(t), (B15)

where '(x,y) = (x,y) * Tp(X,y). After passingthrough an electronic filter, the signal becomes

(B18)

where

L (x,y) L (x,y) * T(Xy),

7(x,y) r'(xy) * rf(y) = rj(x,y) * Tp(X,y) * f(y),

N(m,n) [Ne(mXxy) * Tf(Y)Jy=nY,

Ne(mXy) N,(y + mly).

The last definition is not intended to suggest that anoise function Ne (xy) exists in the way that the signalfunction L, (xy) does; it is simply intended as a lineindex.

Let us now further investigate the discrete noise termN(m,n). Clearly, the process N(m,n) is not truly sta-tionary. We are thus motivated to define an averagecorrelation function R(m,n) by

1R(m,n) =- EN(m' + m, n' + n)N(m',n')}

MMy m',n'(B19a)

(Mx - ml) (My-, - Il) r I/ 2Iy/2

R(m,n) = MXMy Jlx/2 -I 2 RNe

0, I > MxJInJ>My.

s(t) -- s(vt) =f J 1L,[x(t'),y(t')] + Ne(vt')l

X -f(vt - vt')d(vt'), (B16)

where L'(x,y) = L(x,y) * r'(xy), and v is the line-scanvelocity. The signal s (t) is sampled at each interval T= Y/v along the line-scan direction, so that vt = qY,where q is an integer. Let ,ly be the rectangular image

- y + nY+ mMYY)Tf(y)f(')dYdy', Iml < M, Inl < Ny (B19b)

It may be assumed that RNe (z) 0 O for z not satisfyingthe condition I z I << My Y or, equivalently, that the onlynon-negligible terms of R (m,n) are those for which Iy'- y + (n + mMy)YI << MyY. Furthermore, since thecharacteristic distance of Tf (y) is of the order of Y, it isonly for small y,y' (S several Y) that Tf(y) contributesto the integral, so that I n + mM I << Mys. The onlypossibilities are R(0,n), R(-1,InI), R(1,-Inl). Ofthese terms, R(O,n) contributes only for In I << My, and


.

R(-l,InI) and R(l,-InI) contribute only for My - InI<< M,. The integrand encompasses all three terms, butthe multiplier (My - In )/My selects R (0,n) as the onlynon-negligible term.

In reality, a more stringent condition, I z L< Y isfrequently valid so that as M,,,My -o only R(O,n) isfinite. Intuitively, one expects that the discrete noiseprocess becomes wide-sense stationary as the raster sizebecomes sufficiently large, in which case the true andaverage correlation functions are the same; for theseconditions

R(mnn) =c 3(m) f- X RNY -y + nY)rf(y)Tf(y')dydy'.

(B20)Since

RNe(Z) c (4Ne(W) exp(-i27rcz)dco,

the 2-D power spectral density is given bycc,(~w XY Fj R(m,n) exp[-i27r(mvX + nwY) (20a)

m,n

=x ( I( n'1 ( n,~12 (B20b)

The corresponding analog power spectral density orWiener spectrum assumes a similar form as that givenby Eq. (B13) for sensor-array mechanisms, namely,

( l)N(Vf) = | n Nf ( Y Yf ) 2

0, otherwise.

1 1

2X 2Y

If the electronic filter response Tf(w) does not suffi-ciently bandlimit the noise which is generated along theline-scan direction, it is possible for noise to be aliased.Such an effect cannot occur in the v direction. If

)N (W) Iff(CO)12 is of constant magnitude (i.e., white)and limited to the sampling passband 1wI 1/2Y,then

2 = 1/2Y0Ne =c X N ()d =-

1 -/2Y Y

fXYCN2,V 1<1 IW 4)N(V, ) |U| - 2X ' s(B22)

O. otherwise.

This result for the sufficiently passband limited elec-tronic noise power density spectrum N(VW) corre-sponds to that given for sensor-array mechanisms byEq. (B13).

It should be emphasized for clarity that the analognoise term is correctly expressed by Ne (x ,y) and not (aswas previously done in Refs. 4 and 5) by Ne (y). Al-though it cannot be explicitly written down, the analogcorrelation function corresponding to Eq. (B22) is theinverse Fourier transform which is identical to thatgiven by Eq. (B14) for sensor arrays.

3. Quantization NoiseQuantization of the signal s (m,n) introduces another

source of noise referred to as quantization noise. Thequantized signal s (m,n) is expressed as

s,(m,n) = s(m,n) + N,(mn), (B23)

where s(m,n) is the true signal value and Nq(m,n) is arandom error variable. Following, for example, Op-penheim and Schafer' 9 with the usual assumptions ofsignal independence, stationarity, and whiteness, theautocorrelation function RN, (m,n) of the quantizationnoise is

RNq(m,n) = EIN,(m' + m, n' + n)N,(m',n')l = (mn) 2

where the variance is defined by o-N = EINM(m,n)).Consequently, by Eq. (B10) the discrete power spectraldensity becomes

UNq(VW) =c Al 1aNq= XYN (B24)

and the corresponding analog power spectral densityis

IXYr2 | v < 1Iq(V),() c otherwi 2X

j0, otherwise.

l1@ S 2Y '2Y' (B25)

Since the quantized signal Sq (m,n) consists of a finitediscrete set of amplitudes, the question naturally ariseswhether a discrete rather than the continuous approachthat led to Eqs. (13) should be taken to formulate theinformation density of quantized signals. However,since it is certain that the sum of the continuous (in thedistribution of their amplitudes) variables LS, Ne, andNq is discrete, the discreteness that results fromquantization has no further effect on information den-sity than the degradation accounted for by the (inde-pendent) quantization noise term.References

1. P. B. Fellgett and E. H. Linfoot, Philos. Trans. R. Soc. London247, 269 (1955).

2. E. H. Linfoot, J. Opt. Soc. Am. 45, 808 (1955).3. C. Shannon, Bell Syst. Tech. J. 27,379 (1978); or C. Shannon and

W. Weaver, The Mathematical Theory of Communication (U.Illinois Press, Urbana, 1964).

4. F. 0. Huck and S. K. Park, Appl. Opt. 14, 2508 (1975).5. F. 0. Huck, N. Halyo, and S. K. Park, Appl. Opt. 20, 1990

(1981).6. F. 0. Huck et al., Opt. Laser Technol. 15, 21 (1982).7. 0. H. Schade, Sr., J. Soc. Motion Pict. Telev. Eng. 56,131 (1951);

58, 181 (1952); 61, 97 (1953); 64, 593 (1955); 73, 81 (1964).8. L. M. Biberman, Ed., Perception of Displayed Information

(Plenum, New York, 1973).9. D. P. Peterson and D. Middleton, Inf. Control 5, 279 (1962).

10. R. M. Mersereau, Proc. IEEE 67, 930 (1979).11. F. 0. Huck, N. Halyo, and S. K. Park, Appl. Opt. 19, 2174

(1980).12. S. K. Park and R. A. Schowengerdt, Appl. Opt. 21, 3142 (1982).13. Y. Itakura, et al., Infrared Phys. 14, 17 (1974).14. H. H. Hopkins, Proc. R. Soc. London 231, 91 (1955).15. M. Born and E. Wolf, Principles of Optics (Pergamon, New York,

1965).16. M. Mino and Y. Okano, Appl. Opt. 10, 2219 (1971). Expressions

for P2 and q2 in Eq. (9) contain a typographical error.17. Y. L. Lee, Statistical Theory of Communications (John Wiley

and Sons, New York, 1964).18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,

New York, 1968).19. A. V. Oppenheim and R. W. Schafer, Digital Signal Processing

(Prentice-Hall, Englewood Cliffs, N.J., 1975).


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