image correlation at low light levels: a computer simulation

Image correlation at low light levels: a computer simulation

G. Michael Morris

The effectiveness of photon-limited image correlation for recognition of realistic imagery is investigated.The correlation signal is obtained by cross correlating a low light-level input scene and a high light-level ref-erence image. The dependence of the probability density function of the correlation signal on the averagenumber of detected photoevents and on the number of gray levels in the images is illustrated. Monte Carlosimulations of image correlation using low light-level scenes are found to be in close agreement with the theo-retical predictions.

1. Introduction

The spatial coordinates of detected photoevents andthe number of detected photoevents in a given areaconvey information about the classical high light-levelirradiance of the scene. In this paper the 2-D spatialcoordinates of detected photoevents are used to calcu-late the cross-correlation function between a lowlight-level input scene and a high light-level referenceimage that is stored in computer memory. The fun-damental question that we address is: "How manydetected photoevents are needed in the input scene todistinguish a reference object from a set of background(or noise) images?" In the present treatment the ref-erence image is used as the system impulse response (seelater Fig. 1). Hence, the correlation signal representsthe output from a matched filter. Of course, in an ac-tual recognition system one can encounter changes inscale and orientation of the image as well. A possiblesolution to these additional complications is discussedin Sec. V.

Low light-level images arise in many applicationssuch as night vision, laser radar, radiological imaging,astronomy, and others. Rosel 2 has described experi-ments on the absolute sensitivity of the human visualsystem. A theoretical treatment of image correlationat low light levels has been reported by Morris.3 In Ref.3 expressions for the probability density function andthe characteristic function of the correlation signal forgeneral input scenes and references images were given.Barrett and Swindell4 have treated the problem of

The author is with University of Rochester, Institute of Optics,Rochester, New York 14627.

Received 16 April 1984.0003-6935/84/183152-08$02.00/0.© 1984 Optical Society of America.

photon noise in radiographic imaging systems. Burke 5

has applied classical estimation theory to low light-levelimage restoration. Goodman and Belsher6 have in-vestigated linear least-squares restoration of atmo-spherically degraded photon-limited images.Nowakowski and Elbaum7 have reported expressionsfor the minimum number of photons required to mea-sure the position of a light pattern on a noncoherentdetector array. The various photon-counting methodsused in stellar speckle interferometry have been sum-marized recently by Dainty.8

Advances in microchannel image intensifiers9 haveled to the development of several 2-D photoevent-counting detectors. The detectors differ primarily inthe anode assemblies that were used. These includeSIT TV cameras,1011 self-scanned detector arrays,1213

crossed-wire-grid anodes,14 multianode arrays,15 re-sistive anodes,16 -22 and gray-coded masks used with abank of photomultipliers.232 4 Spatial resolution ashigh as 500 X 500 pixels has been reported using a re-sistive anode.1 7

In this paper image correlation at low light levels isanalyzed. The emphasis is on recognition of realisticimagery. Theoretical expressions for the probabilitydensity functions of the correlation signal for generalreference and input images are summarized in Sec. II.High light-level scenes are input to the computer usinga video digitizer. The generation of the correspondinglow light-level (or photon-limited) images is describedin Sec. III. In Sec. IV the results of a Monte Carlosimulation, in which 3000 photon-limited images arecross correlated with a high light-level reference image,are compared with theoretical predictions.

II. Image Correlation Using Photon-Limited Scenes

A low light-level scene, (x',y'), can be representedas a collection of Dirac-delta functions,

N=E 6 - xi'y' - y),i=1

(1)

3152 APPLIED OPTICS / Vol. 23, No. 18 / 15 September 1984

POISSON V(x' IMPULSEV(x' y')N RANDOM NUMBER RESPONSE C(x y)

GENERATOR R(x'+x, y'+ y)

Fig. 1. System diagram for image correlation at low light levels.

where (xi,yi) denote the spatial coordinates of the ithdetected photoevent, and N is the number of photoe-vents in area A and time interval r. In Eq. (1) the po-sition coordinates (xiyj) and the number of photoeventsN are random variables.

From the theory of photodetection, the probabilityof emission of N photoelectrons in a time interval -r froma photocathode of area A is a conditional Poisson pro-cess: 2 5

PA (N) = (N)n exp( -N)/N!, (2)

in which

N = [/(hT)] fSA dx'dy'V(x',y'), (3)

where N is the average number of detected photoelec-trons, ij denotes the quantum efficiency of the photo-cathode, h is Planck's constant, v is the frequency of thequasi-monochromatic illumination, and V(x',y') rep-resents the classical high light-level irradiance of theinput scene. It is noted that Eq. (2) is the appropriatecounting distribution when illumination is provided bya well-stabilized single-mode laser. It is also the ap-propriate distribution for polarized quasi-monochro-matic thermal radiation when the integration time T ismuch larger than the coherence time.26

A linear system diagram for the correlation schemeis shown in Fig. 1. A low light-level image Q(x',y') canbe generated digitally using a Poisson random numbergenerator with the Poisson parameter N given in Eq.(3). V(x',y') is then cross correlated with a systemimpulse response to obtain the correlation signal C(xy).In the present treatment we will take R(x',y') to be ahigh light-level reference image.

Performing the cross correlation between R(x',y')and V(x',y') in Eq. (1) gives3

NC(x,y) = R(x + xi,y + yi),

i=l(4)

where (xi ,yi) denote the spatial coordinates of the ithdetected photoelectron, and (x,y) are the offset coor-dinates of the reference scene window within the inputscene. C(xy) is seen to be a 2-D spatial analog of a shotnoise process.

A convenient statistical quantity to calculate whenusing sums of independent random variables is thecharacteristic function of the process. The character-istic function of C(x,y) is found to be

flo) = (exp[iwC(xy)])

= exp([X/(hT)] ff dx'dy'V(x',y')

* jexp[iwR(x + x',y + y')] - 11), (5)

where ( .. ) indicates an ensemble average. Using thecharacteristic function, the mean value and variance of

the correlation signal are readily calculated. One findsthat

(C(x,y)) = N J3' dx'dy'p[x',y'I V(x',y')]R(x + x',y + y'), (6)

o2 = ([C(xy) - (C(xy))] 2 )

= N JT dx'dy'p[x',y'I V(x',y')R 2(x + x',y + y'), (7)

where

P[x',Y' V(x',y')] =V(x',y')

ff, dx'dy'V(x',y')A is the area of the reference scene, and N is given in Eq.(3).

For recognition of realistic scenes one would expectthat the average number of detected photoevents Nrequired to obtain reasonable error rates is relativelylarge. Since both (C(xy)) and o2 increase withN, oneshould consider the normalized process

@(x Y) = C(xy) - (C(x,y))

in the limit as N . As shown in Ref. 3 the charac-teristic function, and hence the probability densityfunction of @ (xy), tend to a Gaussian process as N getslarge, i.e.,

lim 4'e (co) = exp(- 2 /2),N-X

(8)

lim P[C(x,y)] = [1/(or-iJ2)] expj-[C(x,y) - (C(xy))] 2/(2 2 )j.

(9)

The detection problem can be conveniently formu-lated using the theory of hypothesis testing.27 With thistheory one calculates the probability density function(PDF) of C(x,y) when (1) the reference scene is theinput and (2) a noise scene is input; and then determinesthe area of overlap of the PDFs as a function of the av-erage number of detected photoevents. From the areaof overlap, the probability of false alarm and the prob-ability of detection can be calculated.

Ill. Generation of Low Light-Level Images

Low light-level images of a given high light-level scenecan be generated using a Poisson random number gen-erator. For a low light-level image the input Poissonparameter VLAMij is chosen as follows. Consider animage of area A, which has pixels of area AA, and sup-pose that the average number of detected photoeventsin the entire image is to be N. From photodetectiontheory, the Poisson parameter for the ijth picture ele-ment is

VLAMij = [/(hF)] SSAA i, dx'dy'V(x',y')

(10)

where V(xi,yj) is the high light-level irradiance of thepixel at spatial coordinates (xiyj). Combining Eqs. (3)and (10) gives

15 September 1984 / Vol. 23, No. 18 / APPLIED OPTICS 3153

-= [,qr1(hV)] V(xjyj)AA,

256 GRAY

LEVELS_

N =16K

N=4K

N =1K

N=-250*

Fig. 2. Computer-generated low light-level images: first column, Capitol Building; second column, Rush Rhees Library; third column, TheInstitute of Optics. The high light-level images contain 256 gray levels and have 256 X 256 elements. N is the average number of detected

photoevents over the entire image.


4 GRAY

LEVELS

N =16K

N=4K

N=1K

N =250

w1 w 1 BO Wi+ i7 t~ - 1, I

f - 1 la.*!st .J :. * i 1 4AMH

Fig. 3. Computer-generated low light-level images. Same as Fig. 2 except that the high light-level images have only four gray levels.


VLAMij =NV(xi,yj)AA

SSA dx'dy'V(x',y')

(11)

Hence, a low light-level image can be produced from acorresponding high light-level scene by calling a Poissonrandom generator with parameter VLAMij at each pixelin the associated high light-level image.

Images produced using this scheme are shown in Figs.2 and 3. High light-level, 256 X 256 images of theCapitol Building, the University of Rochester's RushRhees Library, and The Institute of Optics were pro-duced by digitizing photographs using a Dage MTImodel 68 TV camera and a Colorado Video model 274Dframe store that was interfaced to a PDP11/23-PLUSmicrocomputer. The photon-limited images were ob-tained using the IMSL28 Poisson random numbergenerator GGPON. With subroutine GGPON onespecifies the input Poisson parameter, a seed value, thenumber of random deviates to be generated, and theoutput vector which contains the random deviates.

In Fig. 2 the high light-level images were digitized to256 gray levels. In Fig. 3 the high light-level imageswere digitized to four gray levels. The average numberof photoevents N for the low light-level images in agiven row is indicated at the left.

It is interesting to note the effect of gray level on one'sability to recognize an object at low levels of illumina-tion. A human observer is particularly sensitive toedges in the image. In Fig. 3 the edges of zero-intensityregions tend to be relatively sharp; hence less informa-tion is required for recognition. In Fig. 2 the added graylevel tends to destroy edge sharpness.

IV. PDFs of the Correlation Signal

A. Theoretical Results

In this section probability density functions (PDFs)of the correlation signal in Eq. (4) are calculated usingthe high light-level images in the top row of Figs. 2 and

3. For realistic imagery one expects that for reliablerecognition the average number of detected photoeventsN will be relatively large; hence, the approximation ofa Gaussian PDF in Eq. (9) should be fairly accurate.

Since the reference and input images have the samearea A, we take the reference scene offset (xy) = (0,0).For this case the correlation signal will be denoted byC,

NC = C(O,O) = E R(xiyi).

i=1(12)

The expected value and variance of C are

(C) = J SSA dx'dy1V(x,Y')] fJfJ dx'dy'V(x',y')R(x',y)

(13)

C 1N1 JA dx dy V(x ,y)J fJA d'dy'V(x',y')R 2(X'y')y

(14)

respectively.The integrals in Eqs. (13) and (14) are calculated by

means of Simpson's rule. For both the 256-gray-leveland the four-gray-level cases, the Capitol Building ischosen as the reference scene R(x',y'). The values ofthe integrals in Eqs. (13) and (14) for the different inputimages (Capitol, Rush Rhees Library, and The Instituteof Optics) are given in Table I.

Using Eqs. (9), (13), (14), and Table I, theoreticalresults for the probability density functions of the cor-relation signal are calculated. These results are plottedin Figs. 4 and 5. Figure 4 shows the range of correlationvalues that one expects with the 256-gray-level imagesin which the average number of detected photoeventsfrom the entire image is (a) N = 250, (b) N = 500, and(c) N = 1000. Curve I is the PDF of the correlationsignal, C, in Eq. (12), when the input image is the Cap-itol Building. Curve II is the PDF when the inputimage is the Rush Rhees Library; and curve III is thePDF when The Institute of Optics is input.

Table I. Values for Integrals In Eqs. (13) and (14); Reference Image, R(x',y')-Capitol Building; Image Size (All): 256 X 256

Input Image Vx',y')

256-Gray-Level Images Four-Gray-Level Images

Capitol Rush Rhees The Institute Capitol Rush Rhees The InstituteBuilding Library of Optics Building Library of Optics

frdx'dy'V(x',y') 7.292E+06 4.859E+06 8.517E+06 3.909E+06 4.338E+06 3.398E+06A

rydx'dy'V(x',y')R(x',y') 1.258E+09 6.308E+08 8.674E+08 6.458E+08 3.500E+08 1.390E+08A

,Jldxdy'V(x',y')R2(x',y') 2.683E+11 1.169E+11 1.389E+11 1.142E+11 5.812E+10 2.288E+10

A


(a)7.0

'o

0.

3.75 5.5

C1x1041 (b)

3.5

5

,0

-0

(C)

m1 N1=UUV

I

'o. 1

2

n N=50

II

Df II '/ A . - I-

o 0.7 1.4C [x1041

.0 - nI N=100

II

.5 1

0__o 1,2 2A

C [x1041

1.5 III N=200

.75 - I

00 ~2.4 4..2

l8 12 16 20

c[xio 41

Fig,. 4. Probability density functions of the correlation signal whentle input image V(x',y') is (I) Capitol Building, (II) Rush Rhees Li-I)rary, and (III) The Institute of Optics. The average number of de-

lected photoevents is (a) N = 250, (b) N = 500, and (c) N = 1000.'['he reference scene R(x',y') for all cases is the Capitol Building. Thereference image and input images have 256 gray levels and contain

256 X 256 picture elements.

Corresponding results for the PDFs of the correlationsignal for images with four gray levels are given in Fig.5.

As one would expect, when the input scene matchesthe reference object (curve I), the correlation valuestend to be higher. Also, the Rush Rhees Library is moresimilar to the Capitol Building (the reference image)than The Institute of Optics; hence the correlationvalues obtained when the Rush Rhees Library is theinput image (curve II) tend to be higher than the cor-relation values obtained when The Institute of Opticsis the input image (curve III).

An important feature of the curves in Figs. 4 and 5 isthe area of overlap. For a given threshold value of thecorrelation signal CT, the overlap area of the probability

C J.1041

Fig. 5. Same as Fig. 4 except that the reference image and inputimages have only four gray levels, and the number of detected photons

is (a) N = 50, (b) N = 100, and (c) N = 200.

density functions determines the probability of makingan error. For example, when the average number ofphotons N is 1000 [fourth row of Fig. 2 and Fig. 4(c)] andCT is 1.5 X 105, the probability of detection with theCapitol Building as the input is 0.99990 and the prob-ability of false alarm when the Rush Rhees Library isthe input scene is 1.7 X 106. If the average of photonsN is 250 [last row of Figs. 2 and 4(a)] and CT = 3.75X 104, the probability of detection of the CapitolBuilding is 0.97, the probability of false alarm with theRush Rhees Library as the input is 0.02, and the prob-ability of false alarm with The Institute of Optics as theinput is 10-9.

B. Monte Carlo Simulation

To test the theoretical predictions, a Monte Carlosimulation of image correlation at low light levels wasperformed. For the simulation the Capitol Buildingwith 256 gray levels was chosen as the reference sceneand the average number of detected photoevents N wastaken to be 1000. For each building in Fig. 2, 1000 lowlight-level images with N = 1000 were generated using


(a)2.0

'01.0

0-2 02.0

(b)1.5

V'o 1.0

X- 0.5

(C)

1.0

so

-0 0.5

I

I

-

1.0-

I0.

0.5-

0

N =1000

i

1.2 1bC[x105

Fig. 6. Histogram of correlation values obtained from a Monte Carlosimulation of image correlation at low levels when the input imageV(x',y') is (I) Capitol Building, (II) Rush Rhees Library, and (111) TheInstitute of Optics. The average number of detected photoevents

N = 1000. The reference scene R(x',y') is the Capitol Building.

the method described in Sec. III, and the correlationsignal C, Eq. (12), was calculated for each low light-levelimage.

A histogram of the correlation values obtained in thesimulation is shown in Fig. 6. In Fig. 6 the solid curvesare the theoretical predictions for the PDFs given earlierin Fig. 4(c).

In Table II theoretical values for the expected value(C(x,y)) and standard deviation o- of the PDFs ob-tained from Eqs. (13) and (14) with N = 1000 and thecorresponding values that were calculated from theMonte Carlo simulation are summarized.

V. Discussion

The photon-correlation technique described hereinprovides a quantitative means for determining theminimum amount of information required for recog-nition of a given input image. From the results above,one notes that a sparse sampling of the input image isquite sufficient for accurate recognition.

This sparse sampling provides the potential for fastcomputation time. Commercially available 2-D pho-ton-counting detectors and position-computing elec-tronics, such as that manufactured by ITT Electro-Optical Products Division, Ft. Wayne, Ind., and SurfaceScience Laboratories, Inc., Mountain View, Calif., can

operate at rates of one photon every 10 ,usec with animage resolution of -400 X 400. The output of theposition-computing electronics is the digital address forthe x and y coordinates of the detected photoevent.These photon coordinates can then be sent to a digitalprocessor, which calculates C(x,y) in Eq. (4). Hence,the correlation signal can be built up photon by photon,and this simple summation can be performed at highrates. Using assembly language it should be possibleto process the photon coordinate information within 10pusec, i.e., before the arrival of the next photon. As anexample suppose that, on average, 1000 detected pho-toevents are required to give acceptable values for theprobability of false alarm and the probability of detec-tion. In this case it should be possible to detect, pro-cess, and make a recognition decision in -10 msec.From the point of view of processing time, in many in-stances it may actually be much faster for a computerto recognize an image using photon-counting techniquesthan to digitize and process the corresponding highlight-level scene directly.

As can be seen by comparing Figs. 4 and 5, it takesapproximately five times the number of photons todistinguish the images with 256 gray levels over thatrequired for the images with only four gray levels. Innatural scenes the gray levels are essentially continuous;hence, the digitized images with 256 gray levels give agood indication of the processing that would be requiredfor an actual scene.

Images with a reduced number of gray levels can beproduced by some sort of electronic preprocessing of thescene. From the discussion above, it follows that byreducing the number of gray levels it may be possibleto reduce significantly the amount of sampling requiredto make an accurate decision. However, further workon the effect of gray level is needed. The effect is def-initely scene-dependent. For example, if the gray-levelhistogram of the original image is spread over a widerange of values, the reduction of the number of graylevels seems to reduce the number of samples required.On the other hand, if the histogram of the original sceneis localized in a small range of gray levels, i.e., a lowcontrast image, the image may disappear completelywhen the number of gray levels is reduced.

Finally with regard to changes in image scale andorientation, one notes that the impulse responseR(x',y') in Fig. 1 can be any desired function. In thesimulation described above, R(x',y') was taken to be aclassical irradiance reference image, which corre-

Table II. Expected Values and Standard Deviations of the Correlation Signal

Input Image, I Theory Monte CarloV(x',.) <C> a <C> a

Capitol Building 1.726E+05 6.066E+03 1.728E+05 6.319E+03

Rush Rhees Library 1.298E+05 4.906E+03 1.306E+05 5.047E+03

The Institute of Optics 1.018E+05 4.038E+03 1.024E+05 3.975E+03


.1.

I3 .8

sponded to a matched filter case. However, if changesin image scale and orientation are present, one wouldprobably choose a different system impulse response.Several methods for scale- and/or rotation-invariantrecognition of classical images have been reported (see,for example, Refs. 29-34). Presently, wer are investi-gating the applicability of these techniques for imagerecognition at low light levels. The results will be re-ported in a subsequent publication.

VI. Summary

Image correlation at low light levels has been inves-tigated. In the method a low light-level (or photon-limited) image is cross correlated with a high light-levelscene stored in computer memory. The statistics of thecorrelation signal, Eq. (4), are given in Sec. II. Theexpected value and variance of the correlation are givenin Eqs. (6) and (7), respectively. The method used tocomputer generate low light-level images (see Figs. 2and 3) is described in Sec. III. Probability densityfunctions of the correlation signal using realistic imagesare given in Figs. 4 and 5. The results of a Monte Carlosimulation of image correlation at low light levels aregiven in Fig. 6 and Table II.

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This research was supported by the U.S. Army Re-search Office and the NYS Center for Advanced OpticalTechnology.


image correlation at low light levels: a computer simulation

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