Rochester Institute of Technology RIT Scholar Works Theses 1990 Shift-invariant image reconstruction of speckle-degraded using bispectrum estimation Steven M. Wear Follow this and additional works at: https://scholarworks.rit.edu/theses Recommended Citation Wear, Steven M., "Shift-invariant image reconstruction of speckle-degraded using bispectrum estimation" (1990). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact ritscholarworks@rit.edu. ROCHESTER INSTITUTE OF TECHNOLOGYThis volume is the property of the Institute, but the literary rightsof the author must be respected. Passages must not be copied or closely paraphrasedwithout the previous written consent of the author. If the reader obtains anyassistance from this volume he must give proper credit in his own work.This Thesis has been used bv the following persons, whose signaturesattest to their acceptance of the above restrictions.Name Address DateCOPYRIGHT11/89THESIS RELEASE PERMISSION FORMRochester Institute of TechnologyCollege of Graphic Arts and TechnologyRochester, New YorkTitle of Thesis: SHIFT-INVARIANT IMAGE RECONSTRUCTION OFSPECKLE-DEGRADED IMAGES USING BISPECTRUM ESTIMATIONI. Steven M. Wear, hereby grant permission to theWallace Memorial Library of R.I.T. to reproduce mythesis in whole or in part. Any reproduction will not befor commercial use or profit.Signature5 3-UL- ^ODate-ill-ABSTRACTCoherent speckle noise is modeled as a multiplicative noise process that has anegative exponential probability density function. Using a homomorphic transformation, this speckle noise is converted to a signal-independent, additive process.The speckled images are randomly jittered from frame-to-frame against a uniformbackground to simulate image motion and/or platform jitter.Multiple images are logarithmically transformed and ensemble averaged in thebispectral domain. The bispectrum ignores this image motion so no blurring resultsfrom the ensemble averaging. Object Fourier magnitude and phase informationare also retained in the bispectrum so that the resultant image can be uniquelyreconstructed. This value is then exponentiated to complete the image reconstruction process.Since speckle masks the resolution of details in the noisy image and effectivelydestroys the object structure within the image, it is seen that image reconstructionusing bispectrum estimation results in images that regain their object structure.Both one-dimensional and two-dimensional images were tested using separatebispectral signal reconstruction algorithms for each.-IV-ACKNOWLEDGMENTSAs with any work that so completely consumes an individual, one must rely onmany people to fulfill that task that has been your life for so long. This is definitely the case for myself and the list of people who have helped me in some waywould be endless. Thus, I must confine my thanks to a few people who have beeninstrumental in seeing this task completed. My thesis committee members havebeen extremely helpful in this work.Professor Rao has given me excellent suggestions on improving the presentation ofthe material, expecially as regards the speckle modeling.I would like to thank Professors Dianat and Raghuveer for allowing me to use theirtwo-dimensional magnitude bispectrum reconstruction algorithm.Professor Raghuveer has been particularly helpful in that he continually helpedme search for ever-improving methods to overcome the problems I encounteredwhen implementing the bispectral routines.Similarly, early in this thesis work, Gopal Sundaramoorthy, a graduate student atR.I.T. working with the bispectrum, was very helpful in getting me started with thebispectrum and understanding its nuances.Major Paul Idell, Capt Dave Voelz, and Capt John Gonglewski of the Air ForceWeapons Laboratory were very helpful in helping me understand the parametersnecessary to properly model coherent speckle.Laura Abplanalp of Eastman Kodak played a pivotal role in helping me get thefinal product of this endeavor (the thesis itself) completed and published on a veryshort time schedule.The ultimate thanks goes to my family who showed the patience and understandingnecessary to allow me to complete thiswork.-v-DEDICATIONroJud\\ Marc, and Betham-vi-TABLE OF CONTENTSSection Title PageCERTIFICATE OF APPROVAL iiTHESIS RELEASE PERMISSION FORM iiiABSTRACT ivACKNOWLEDGMENTS vDEDICATION viTABLE OF CONTENTS viiLIST OF ILLUSTRATIONS viii1.0 INTRODUCTION 12.0 BACKGROUND 43.0 SPECKLE 83.1 Electromagnetic Field 83.2 Object Surface Statistics 103.3 Propagation And Imaging Of The Speckle Field 123.4 Nonstationary, Second-Order Image Statistics 143.5 Multiplicative Noise Model 213.6 Speckle Computer Models 223.7 Speckle Verification 364.0 BISPECTRUM ESTIMATION 604.1 Overview 604.2 Properties 624.3 Algorithms 654.3.1 One-Dimensional Algorithm 664.3.2 Two-Dimensional Algorithm 69-VII-TABLE OF CONTENTS (CONT'D)Section Title Page5.0 RESULTS 755.1 Non-Speckled 765.1.1 Phase Distortion Problem 775.1.2 Windowing the Data 915.2 Speckled 965.2.1 One-Dimensional Examples 1015.2.2 Two-Dimensional Examples 1125.3 Conclusions 1266.0 APPENDICES 129Appendix A Tabular Synopsis Of Speckle Models 129Appendix B Bispectrum Fourier Expression 130Appendix C Bispectrum Reconstruction Algorithm Flowchart 1337.0 REFERENCES 135LIST OF ILLUSTRATIONSFigure Title Page1 Typical Histogram of a Gaussian Random Number FunctionGenerated by Forsythe Routine 242 Flow Diagram for Direct Digital Model 253 Flow Diagram for Transfer Function Model 274 Flow Diagram for Random Phase Model 2S5 Flow Diagram for Correlated Window Model 306 Flow Diagram for Negative Exponential pdf Model 327a Flow Diagram of Additive Gaussian Model 347b Histogram of Zero Mean, Unitary Variance Gaussian Function . . 34V1I1TABLE OF CONTENTS (CONT'D)LIST OF ILLUSTRATIONS (CONT'D)Figure Title Page8 Plot of exp(-x-(exp(-x))) Function 379a Three-dimensional Plot of Sinc2(X/16,Y/16)Truth Object 399b Two-dimensional Contour Plot of Truth Object 39lOa-d Statistical Histograms and Phase for Direct Digital Model 40lOe Truth Object Speckled Using the DDM 4111a A Uniform"Wall"Speckled Via the CWM 42llb-e Statistical Histograms and Phase for the CWM 43llf Truth Object Speckled Via the CWM 4412a Truth Object Speckled Via the TFM. Coherent Transfer FunctionWas a 64 x 64 Rectangular Aperture 4612b Truth Object Speckled Via the TFM. Coherent Transfer FunctionWas a Cylindrical Aperture (Diameter = 64 pixels) 4712c Truth Object Speckled Via the TFM. Incoherent Transfer FunctionUsed. Cylindrical Aperture (Diameter = 64 pixels) 4813a Truth Object Speckled Via the RPM in Image Plane 4913b Flow Diagram for the Random Phase Model 5013c Truth Object Speckled Via the RPM in the Pupil Plane 5113d-g Statistical Histograms and Phase for the RPM 5214a Truth Object Speckled Via the NEM 5314b-e Statistical Histograms and Phase for the NEM 5415a Speckled"Wall"Using the AGM 5615b-e Statistical Histogram and Phase for Speckle"Wall"Using AGM . . 5715f Truth Object Speckled Via the AGM 5$-IX-TABLE OF CONTENTS (CONT'D)LIST OF ILLUSTRATIONS (CONT'D)Figure Title Page15g-h Truth Object Speckled Via the AGM 5916 Redundant Regions of the Bispectrum 6117a Three-Dimensional Plot of Bispectrally-ReconstructedSinc2(X/16,Y/16) 7817b Two-Dimensional Contour of (a) 7S17c Recentered Version of (b) 7818a One-Dimensional Truth Object 7918b Bispectrally Reconstructed Version of (a) 7919a One-Dimensional Truth Object 8019b Bispectrally Reconstructed Version of (a) 8020a"Battleship"Used for Magnitude/Phase Comparison 8420b"Lena"8421a Reconstructed Object Using"Battleship"Phase and"Lena"Magnitude 8421b Reconstructed Object Using"Lena"Phase and"Battleship"Magnitude 8422a Original Object 8622b Bispectrally Reconstructed Version of (a) 8623a Original Object 8723b Bispectrally Reconstructed Version of (a) 8724a Bispectrally Reconstructed Tri (X/7, Y/7) SS24b Centered Version of (a) 8$25a Original Objects 89-x-TABLE OF CONTENTS (CONT'D)LIST OF ILLUSTRATIONS (CONT'D)Figure Title Page25b Bispectrally Reconstructed Version of (a) 8926a Original Object 9026b Bispectrally Reconstructed Versions of (a) 9027a Original Object 9227b Representative Object Jittered Randomly In The Field Of ViewFor 10 Frames-No Noise 9227c Bispectrally Reconstructed Version of (a) 9228 Bispectrally Reconstructed Padded Tribar with Data Window .... 9429a Original Object 9529b Bispectral Reconstruction of (a) Using a Data Window 9530a Original Object 9530b Bispectral Reconstruction of (a) Using a Data Window 9531a Three-Dimensional Plot of Speckled Sinc2(X/16Y716) 9931b One-Dimensional Slice of (a) DDM Used 9932a Ensemble-Averaged Version of 31a After 150 Frames 10032b One-Dimensional Slice of (a) 10033a-d Example of Bispectral Averaging of Jittered, Speckled Objects . . 10233e-g Example of Bispectral Averaging of Jittered, SpeckledObjects (Cont'd) 10334a-c Result of Bispectral Averaging an Object in its LogarithmicDomain 10435a-d Comparison of the Fourier Representation of a Speckled Objectwith its Unspeckled Version 10635e-f Comparison of the Fourier Representation of a Speckled Objectwith its Unspeckled Version (Cont'd) 107-xi-TABLE OF CONTENTS (CONT'D)LIST OF ILLUSTRATIONS (CONT'D)Figure Title Page36a-d Example of Bispectral Averaging of Speckled Object 10836e-f Example of Bispectral Averaging of Speckled Object (Cont'd) ... 10937a-c Example of Bispectral Averaging of Jittered, Speckled Object ... 11037d-e Example of Bispectral Averaging of Jittered, Speckled Object(Cont'd) Ill38a-c Bispectral Reconstruction in Logarithmic Domain and FourierComparison of Speckled Object with Original Version 11338d-e Bispectral Reconstruction in Logarithmic Domain and FourierComparison of Speckled Object with Original Version (Cont'd) . . 11439a-b Bispectral Reconstruction of Speckled Object 11540a-b Bispectral Reconstruction of Speckled Object 1164la-c Bispectral Reconstruction of 2-D Object Using 1-D Algorithm -No Noise T T 11742a-c Bispectral Reconstruction of 2-D Object Using 1-D Algorithm -Additive Gaussian Noise (Variance = 0.001) 11843a-b Bispectral Reconstruction of 2-D Object Using 1-D Algorithm -Additive Gaussian Noise (Variance = 0.01) 12044a-d Bispectral Reconstruction Usine 2-D Algorithm ObjectSpeckled Via CWM ". 12145a-c Bispectral Reconstruction Using 2-D Algorithm DDMUsed to Speckle Object 12246a-d Bispectral Reconstruction Using 2-D Algorithm AGMUsed to Speckle Object ". 12347 Autocorrelation-Averaged Reconstruction 12548 Bispectral Reconstruction of Speckled Object withoutLogarithmic Transformation 125C-l 2-D Imaee Reconstruction Overview 134-xii-1.0 INTRODUCTIONWhen an object is illuminated by a coherent source of electromagnetic radiation,and this object has a surface structure that is rough on the order of a wavelength ofthe incident radiation, a speckle pattern results. A fully developed speckle patternwill appear chaotic and disorganized. Even when the speckled object is imaged, thepresence of speckle noise is seen as a collection of spots superimposed on theactual image. Thus, speckle is considered as noise that degrades an image wherebyinformation present in the image is masked. This noise can be present in either thepupil plane (farfield) or the image plane.Studies related to the occurrence of speckle phenomenon are not new. Dainty(1984) relates a brief history of observations of speckle patterns, beginning withNewton s attempt to explain star twinkling. Exner, in the late 19th century.sketched a speckle pattern created by a candle seen through fogged glass. Dehaas.in the early 20th century, actually photographed such a pattern. In the early 1960"s.with the advent of the laser, the investigation of this phenomenon accelerated.Coherent speckle phenomenon can occur is such diverse imagery as synthetic aperture radar (SARj. acoustic imagery, ultrasound, x-ray scattering, electron scattering, microwaves, and obviously laser illumination.What really is important is how speckle degrades animage. Speckle increases thesize of the minimum resolution patch obtained with a given aperture as comparedto the same size aperture using incoherent illumination (Kozma and Christensen.1976). The resolution of the image is reduced making certain feature identificationdifficult. The Maximum Likelihood Estimate (MLE) of a speckle image is obtainedfrom ensemble averaging. This technique is a highly effective way of removingspeckle noise (Sadjadi, 1990; Jain and Christensen, 1980; Lim and Nawab, 1981).Multiple images are necessary to perform this procedure. Perfect registrationofeach image with the other is required so that the resultant, averaged image is notblurred. This concern often precludes using ensemble averaging due to thedifficulty in registering images exactly. This is especially true when the image is changing its position from frame-to-frame.A signal processing technique called bispectrum estimation can perform ensembleaveraging on multiple, shifted, speckled images. The bispectrum is defined in thispaper to be the Fourier transform of the triple correlation of an image. Imagereconstruction with the bispectrum has the following beneficial properties:1. The bispectrum is insensitive to linear phase shifts;2. The bispectrum retains both Fourier magnitude and phase information:3. The bispectrum is insensitive to certain additive noise processes.This technique is a spectral domain technique as opposed to a spatial domaintechnique, where much of the current digital speckle reduction techniques operate.This thesis will review the statistics of speckle, speckle as noise, and how this noiseaffects the image. A thorough examination of speckle computer modeling will beconducted and several methods of generating speckled objects digitally will be reviewed in detail with examples of speckle imagery given. I will also review thebispectrum and how its use with homomorphic processing will aid in image reeon-struction of speckle-degraded images. Two separate bispectral reconstruction algorithms are used for one-dimensional and two-dimensional images.2.0 BACKGROUNDThe technique I will use to reconstruct the image of a speckled object will be awholly digital technique. By this I mean that each speckled image will be in adigitized format and will be operated on totally using a computer on apixel-by-pixel basis. I state this at the beginning so as to differentiate this technique fromphysical methods of reducing speckle. For example, McKechnie (1984) lists several methods one can do to reduce speckle before it is imaged, such as:1. Illuminating the object with a temporally partially coherent source;2. Illuminating the object with a spatially partially coherent source;3. Observing the speckled object through a moving aperture that performs time-averaging as it moves across the pattern:4. Observing the pattern through a finite aperture that acts as a low-pass filter.Guenther. et a!. (1978) ha? one of the earliest papers listing several different digital filters that could be applied to reduce speckle. They describe the basic ensemble averaeinc technique (where several images are averaged together to reconstructthe image) as well as the spatial processing technique using an averaging window(which moves across the image spatially). Two simple, digital, non-linear filter-(square-root and squaring) are also described. Jain and Christensen (1980; reviewsimilar techniques as well as a homomorphic Wiener filter for digital speckle removal. Lim and Nawab (1981) compare other digital techniques for speckle reduction such as low-pas- filtering in the frequency (Fourier) and density (logarithmic)domains as well as a method they call the "short space spectral subtraction imagerestorationtechnique."They discuss briefly the homomorphic approach to imagerestoration using a filter built specifically for this method but do not describe thisfilter in any detail, though they do discuss the benefits of using the homomorphicapproach. Sadjadi (1990) reviewed the before-mentioned averaging techniques, amedian filter, local statistical filters, an adaptive filter, and a sigma filter andcompared these with the homomorphic approach to speckle reduction. Thehomomorphic approach allows additive-noise reduction techniques to be applied tomultiplicative noise conditions. A better description of a local statistics filter usedin homomorphic processing is given by Arsenault and Levesque (1984). A generaldescription of signal-dependent noise is presented in this article, also.A comparison of phase-retrieval algorithms was performed by Fienup (1982) andsome aspects of these algorithms are used to restore speckled images (Idell, et al.1987). Cederquist, et al. (198S) used a phase retrieval algorithm for farfield. computer-generated speckle. These algorithms are useful because phase information islost when spectral processing a recorded image using the power spectral density(PSD). Recovering the phase helps one reconstruct the image uniquely. Bispectralprocessing retains the phase of an object and it is this idea that makes it a possibletechnique to perform image reconstruction.Kuan, et al. (198~). derived an adaptive restoration filter for speckled images. Thisarticle also brings up some importantpoints. One key point is the difference between a noisv object and a noisy imaging system. A noisy object occurs when anincident coherent wavefront is scattered by the object such that random interfer-ences occur among the dephased, but still coherent, reflected waves. This is thespeckle case I am investigating. The noisy imaging system results from a randomlyvariable transmitting medium, such as the atmosphere, and this is the case forstellar speckle. Instead of being object-related, it is medium-related, i.e., a point isindividually speckled throughout the image due to system effects. Crimmins (1985)used a geometric filter on speckled images and then compared his results with a3X3 median filter used on the same images. Safa and Flouzat (1988) used morphological techniques on remotely-sensed speckled SAR images.All of these previous noise-reduction methods operate on the speckled image inthe spatial domain (except for the homomorphic Wiener filter). Recently.Marathay, et al. (1989) used computer-simulated speckle patterns and appliedthird and fourth order intensity correlations to restore the image. The speckle patterns generated were not imaged but rather found in the pupil plane. Newman andVan Vracken (1989) combine the bispectrum with photon-bias correction techniques (see Northcott. et al, 1988) to recover multiple, shifted objects in a uniformbackground of photon noise (e.g., Poisson). The signal-to-noise ratio (SNR) ofthese objects and background noise was less than one! Freeman, et al. (1985;. usedthe bispectrum on one-dimensional (1-D). infrared, astronomical speckle data torecover the phase of a stellar object. Avers, et al. (1988) compare the triple correlation with the Knox-Thompson phase retrieval technique for astronomical speckleand their results indicate that the triple correlation is more robust for recoveringphase than the Knox-Thompson. Again, as a reminder, astronomical speckle isdifferent than the speckle considered in this thesis. Astronomical speckle can also-6-be termed "incoherentspeckle"while the speckle modeled in this thesis is coherent. Mavroidis, et al, (1990) found that the bispectrum could not be used for recovering coherently imaged objects through atmospheric turburlence. This was basedon the theory of phase closure and its relation to the bispectrum. What they did notdo was perform the homomorphic transformation before they attempted imagerecovery. As will be shown later, this is a key parameter to successfully reconstructan image.The goal of this thesis is to use recently developed algorithms for bispectrum estimation to reconstruct multiple, shifted images degraded by speckle noise. This is aconceptual study to determine if the bispectrum is robust enough to handle theimage degradation resulting from speckle. The aim is to implement solutions froman engineer's viewpoint so that practical solutions can be obtained. Therefore, insome cases, a median filter may be applied, after bispectral averaging and theimage reconstruction has been implemented, to further smooth the image.-/-3.0 SPECKLETo properly model speckled images, I must attempt to review the underlying physics and mathematics behind speckle theory. Since speckle is random, it lends itselfquite well to statistical analysis and the literature is very complete regarding thisaspect of speckle. By using key concepts from this area, it will be shown thatspeckle obeys certain constraints that are conducive to computer modeling.3.1 Electromagnetic FieldThe general method I will use to describe the electric field vector of the propagating source is defined by several authors (Gaskill. Haus) and is reviewed here.Obviously any representation of the electric field vector must also obey Maxwell'sequations, which in turn leads to the wave equation:V-E - pe = 0 (1)5 t"where:E is the electric field intensity vector E(x.y.z;t)u. is the permeabilitye is the dielectric constanta: a2 a:v-is the Laplacian operator: + + .o x:3 y~3 z2One solution to the wave equation is found by writing the vector E as (withr = (x.y.z) being a \ector component):E [r; t] = A(r ) exp[j2i7Vt] (2)where:A(r) is the amplitude functionv is the mean frequency.In imaging I am concerned primarily with spatial quantities and not temporal ones.Thus, I can suppress the time dependence in the previous equation. Furthermore.since the electric field is, in general, a complex quantity, I can express it as:A(r) =Re[A(r)l +jIm[A(r)] =|a(i)| exp[jd>(r)] (3)where:Re denotes the real part of the fieldIm denotes the imaginary part of the field|A(r) | denotes the magnitude of the field, where < > denotes the ensembleaverage. Notice that the value within the <> is simply the individual phasor intensities, therefore 1 can rewrite Equation 7 in the more familiar form:pll) =exp-I(S)Another parameter is called the contrast of the speckle pattern and is defined as aninverse SXR. For fully-developed speckle, the contrast should equal one (Goodman, 19S5: Cederquist. 1988). The contrast is given by:C(9)All of these first-order statistics will be important when I create computer-generated speckle because they will be used to verify that the speckle patterns areactually"speckled."3.3 Propagation And Imaging Of The Speckle FieldSince I am dealing with images, their representation must of necessity be real andnon-negative because image intensity (or its field modulus) can never be less thanzero. Second-order statistics will help in describing the image formation process in-12-this case because these statistics define the spatial structure of the field (Dainty.1976). I have reviewed several cases and will relate key results here. Goldfischer(1965) derived the autocorrelation (ATC) and the PSD for a uniform diffuse objectviewed in the farfield at the pupil plane, thus not imaged. The farfield is thatttL2region a distance z from the object that satifisfies the constraints of zA.(Gaskill), where L is the size of the aperture and X is the wavelength of the incident radiation.The farfield, or Fraunhrfer. diffraction pattern intensity is proportional to the complex ATC of the object and is thus a measure of the spatial frequency spectrum ofthe power. In other words, the ATC and the PSD are Fourier transform pairs(Yariv). Goldfiseher"s model used a uniform, and hence a statistically stationaryobject. Since images are. in general, non-stationary, it will be shown that his results are a special case of the more usual case of non-uniformity. Enloe (196~)later extended these results to an imaged, uniform object.There exists a Fourier transform relationship between the speckle field and itsfarfield propagation, given by (Enloe,:AjU.ti ) = //a0(x.v) expz\(X t + XT) ) dx dv (10)where:An(x.y) is the object speckle fieldA,( oil ) is the farfield distribution.13-At the farfield an aperture is located at the pupil plane. This aperture could be apupil stop some distance from a lens or the lens itself. The lens then focuses thiscoherent radiation onto a detector located at the image plane. The resultant field atthe image plane is given by:A,(v.w) = //H(:,Ti)A1(t,Ti)exp ^j^-(vt + a)Ti)JdS dr, (11)where:HfC.T)) is the System Transfer Function (Fourier transform of pupil function)f is the focal length of the lensi'.c merely denote a different coordinate system from that of the object.The lens acts as a Fourier transformer (Goodman. 1968 1. Thus the field at theimage plane is the Fourier transform of the field in front of the lens, mod, lied bythe aperture. The intensity a: the image plane is found from Equation 11 times itscomplex conjugate (denoted by ") by:I( v. w ) = A, (v, co )(A, (x\ u ))'(12)Notice that the i:v.er.s.~y at the image plane is nothing more than the autocorrelation of the obiect convolved with the imaging system.3.4 Nonstationary. Second-Order Image StatisticsThe previous development was for stationary objects. Lowenthal and Arsenault(19~0) studied the case of an imaged, nonstationary object being speckled. Theirdevelopment is important because several important relationships regarding the-14-object structure and the second order statistics of the speckle field arise. To beginwith, the object f (r) is composed of two components: t(r) is the actual object function that acts to modulate the rough surface, which is denoted by d(r). This nonuniform object is represented by (neglecting the vector representation for r fornotational simplicity):fu") = t(r)d(r) (13)The image plane field, g(r). is related to the object field via the convolution operator (") with the system impulse response, h(r):g(r) = f(rrh(r) (14)Relating dir) to the statistics discussed previously. d(r) is stationary and a circular,complex Gaussian random variable with independent real and imaginary parts having zero means and equal variances. The function can also be thought of aswideband, random noise (Lim and Nawab. 1981). An important point here is that.even though the object is non-stationary, its noise function is stationary and thu=for all cases of tin. we can find the parameters of gir). Referring to Equationl4. Ican express the mean of the image as (hfr) is the same function throughout theensemble):< g(r) >= < f \') > hd') (15)Since < 6>r) > = 0. then < br) > = < on > < din >= 0. and therefore < gfn > = 0The variance of the amplitudefor field) is given for real, non-negative images as(from the standard relationshipo"= (Friedan):(o.ir))"=<|gir)|">- (16)-15-Since the image mean is zero and the magnitude squared of the amplitude equalsthe intensity, I can rewrite the variance of the image amplitude as:(og(r))'= (17)This interesting result says that the variance of the image amplitude is equal to themean of the image intensity. For the variance of the intensity I can write:(^(r))2= _ 2(18)The definition of a statistical ATC function is given by Goodman(19S5) for theintensity (since intensity is real) as:Rn(rr r: > = (19;This function allows us to compare the intensities at i'j and n over the ensemble.By letting ij = r^ = r. can rewrite Equation IS as:(o(r)):= R ( r.r ) - :(20)It can be shown (Goodman. 1985) that another way to write Equation 19. withrl=r2=r. is given by:Rn(r.r )=< ](r) >< I(r) > + |Rc.(r,r)|: ^21)where R0,, (r.r) is the ATC of the image amplitude and. from the relationship between Equation 12 and Equation 19. is equal to < g(r) (g(r))*> (since amplitudecan be complex, in general). For t(r) and h(r). though, this will always be a real-16-function, so I can write Rce (r,r) =< | g (r) p>= . Equation 21 can bewritten as:Rn ( r,r ) = < I(r)>:+ < I(r)>2=2< I(r)>2(22)Using this result in Equation 20 gives ( ou(r))2=2, which, upon takingthe square root of both sides and dividing by the ensemble averaged intensity,gives us the same result as Equation 9 for the contrast. So the rms of theintensity-equals the mean intensity even for non-stationary objects. The difference here isthat, in general, the variance is not a constant but is a result of the ensembleaverage and the image coordinates. Consider the mean intensity of the image givenas. by definition. < I(r) > = < g(r) g'r)*>. However, as shown above, this j?nothing more than the ATC of g(r). given as < I(r) > = R2: (r.r)Another important property is the ATC of the image. Returning to my originalnotation, where d(n is assumed stationary, its ATC can be written asR, , (r., x -. ) = R ir. -r-, ) = b (r.-r^) since I assume the ATC peak to bevery-narrow tor the wideband noise (b represents the Dirac delta function). The ATC ofthe speckled object f(r) then becomes:Rn C'T r: ) = < T ,ri ) d Cri )x*(r: ) d*lr : ) >R;^ri-r: ) =Rdd(ri. r; ) (2?)R.f(ri. r. i = tdo ) t fr;)'8 (r2 -r: )-1"-Note that Rff (rr r2) is still a function of r1 and r2 and not just r1-r2 . eventhough d(r) is stationary, since I have retained the nonstationarity of t(r). So theimage amplitude ATC can be expressed in integral form as:Rcc(rl'r2 ) = /kro)|2h(ri -r0)h(r2 - r0 )*dr0 (24)n0where the integration is over the object plane n0 and a linear filter operation isassumed with the impulse response h(r). Again letting r1=r?= r. and using therelation from a previous paragraph of = R cc (r,r) in the expressionabove (Equation 24) I can write(*denotes convolution): =|t(r)|:*|h(r)|:(25)From Goodman (1965) this equation is of the form of the transfer function for asystem illuminated by incoherent light (where|t(r)|:is the source intensity andhir) is the system impulse response).Thus, the transfer function that determines the mean intensity in the image is theincoherent transfer function of the system, even for an optical system illuminatedby a coherent diffuse object field! One can see from Equation 25 that the PSD of aspeckle pattern has the shape of the ATC of the squared modulus of the pupilfunction (Goodman. 1985).Labeyrie and Dainty (1973) use the principle of Equation 25 to analyze atmospheric speckle. I mention this because it relates a practical application of this expression. By measuring the image intensities and summing their individual Fourier-18-transforms from each realization, and then dividing this result by the time-averaged modulus of each transfer function, the Fourier magnitude of the object'sintensity can be reconstructed.Miller, et al. (1975) use a more rigorous approach to come to the same conclusionsas given above. They find that the intensity is, in general, a nonstationary randomprocess so the ATC must be a function of two variables. They make the point thatthese second-order results are good for both Fresnel (so-called near field) andFraunhofer diffraction but that the usual ATC/PSD Fourier relationship does notnecessarily hold. For nonstationary images, the PSD of the intensity is given as:SjU) = <| J I (r) exp [-j27rtr] dr|2> (26)where Sj ( C ) the PSD. is the average value of the modulus squared Fourier-transformed image intensity and C is a spatial frequency vector.This relationship was extended by April and Lowenthal (1984). They showed thatEquation 26 is really only one of two parts of the PSD of the image intensity Thecomplete PSDis-given as Sj(c)=Sj(c)i +Sj(t ): (eq. 26 is Sj( t )1 ) . This isderived simply by taking the PSD of Equation 21. The first term is the object andthe second term is the speckle noise. Sj ( l ): is written as:oosi( ): =11 l< (ri )S *('':) >T exP [-J277t( ri ~r2 ' ] dri dr2 (2")-19-The key point in all of this is that the noise spatial frequencies are across the samefrequency region as the incoherent transfer function relationship given by Equation26. This was proven by April and Lowenthal experimentally. Using the speckledimage of a Ronchi ruling, they showed that it is the incoherent cutoff frequency(which is twice the coherent cutoff frequency (Goodman (1968) and Gaskill)) thatdetermines the distribution of the spatial frequencies present in the diffuse coherent light. Note that all indication of the object's presence is still lost in a speckledimage if the incoherent cutoff frequency is less than the fundamental frequency ofthe Ronchi ruling. Kozma and Christensen (1976) also experimentally verified thatthe aperture of a coherently illuminated system must be twice as large as anequivalent incoherently illuminated system to achieve the same resolution for each.They compared bar targets of varying spatial frequencies illuminated by both incoherent and coherent sources.To summarize all of these points, it was shown that, even using nonstationarystatistics, the rms intensity still equals the mean intensity and so. for fully developed speckle, the contrast is one. Also, nonstationary statistics do not change thefact that the rough surface is a complex Gaussian random variable leading to anegative exponential distribution in intensity. Finally, the signal was shown to bethe object intensity attenuated by the incoherent transfer function of the system.Therefore, information transmission with coherent diffuse illumination is dependent on the incoherent bandwidth of the optical system.3.5 Multiplicative Noise ModelThese results are the basis for the noise model used by most researchers (Kuan, etal, 1987; April and Arsenault, 1976, 1984, 1986; Guenther, et al, 1978; Jain andChristensen, 1980). They describe speckle noise as multiplicative in the sense thatthe speckled object's intensity is a product of the incoherent signal and a noisefunction.Therefore, the amount of speckle noise is proportional to the image. The signal-dependent nature of speckle is due to the fact that the rms amplitude of the specklenoise is proportional to the image. This model is described by (Tur. et al, 1982) as:I5(r) = aoFigure 9a. Three-dimensional Plot of Sinc2(X/'16,Y/16)Truth ObjectFieure 9b. Two-dimensional Contour Plot of Truth Object39-HISTOGRAM OF SPECKLE INTENSITY-I ; 1r-o 5iot52casaoxDIBCE'OIcrTM. MODELao m *cHISTOGRAM OF SPECKLE FIELD MODULUS-1 1 n 11(a) (b)HISTOGRik' OF LOG (SPECKLE INTENSITY, 1-D SLICE OF S=ECKLE PHiEEK K 6: ti(c) (d)Figure lOa-d. Statistical Histograms and Phase for Direct Digital Model-40-DIRECT DIGITAL SPECKLE METHODFieure lOe. Truth Object Speckled Using the DDM-41-CORRELATED WINDOW MODELo.c 10.0 20.0 30.0 40.0C-l. 027807850.0 60.0 70.0Ficure 11a. A Uniform"Wall"Speckled Via the CWM-4:HISTOGRAM OF SPECKLE IKTENSITY-1 1 rHISTOGRAM OF SPECKLE FIELD KODULUS(b) (c)HISTOGRAM OF LOG (S"=EC<_E ISTE'.= :1-D SLICE OF SPEC(m,n))ia|m(m,n) = \/l(m,n) sin(4>(m,n))ia(m,n) = aRe(m,n) + j a,m(m,n)iA(i,j) = 3{a(m,n)}ig(m,n) = | A(m,n)I2Ficure 13b. Flow Diaeram for the Random Phase Model-50-RANDOM PHASE METHOD10.0 2D.0 30.0C-l.001926so.o 60. 0 70.0Figure 15c. Truth Object Speckled Via the RPM in the Pupil Plane-51-HISTOGRAM OF SPECKLE IK-p.sntI 'O 5 LClKflS3CK.C439G93fCHUGO PHAK ^-.C3HTS-OSRAM OF SPECKLE FIE-j MODULUS!r~~r~-ii0 L-10 13KlC9E0<5XSSCiBIUM3 KUC MOOE.(d) (e)HIS-0G=4 of L0C. ;s=e:kle ISTE'.'S 1-0 S^ICE 0r SPECKLE PHA5E. i!I ...C 3 ID Z; 2f 3C *S*' * KRWO* PHASE SQ2"1-e.s --.e-HUMi! inil"' i-J ; l_X30*C:!CS6Cte(f ) (g)Ficure 13d-e. Statistical Histograms and Phase for the RPMNEG EXP PDF MULT METHODFigure 14a. Truth Object Speckled Via the NEM-53-HISTOGRAM OF SPECKLE INTENSITY[T ' ' l I r"l--ri1-HISTOGRAH OF SPECKLE FIELD MDDULUS0 5 10 IS 20 ES 3C S- .0 *5 5C 55(b) (c)40 -HIS-OGBiV Cc LOG (SICKLE INTENSITY)/ i0 5 10 it EC SC 5b 6CKEGa'I'.E Ei^CKr'.llA^ POr }.:...Li1-0 SLICE OF SPECKLE PHASE-3 S L_ -I !: ! I1 u5 10 IS M E 3; R *; SC Bs 60W6ATIVE EiPONENTI*i. PtV OD0_.(d) (e)Figure 14b-e. Statistical Histograms and Phase for the NEM-54-Figures 15a through 15h demonstrate the AGM. Figure 15a is the AGM using a"wall"for an object (C=l.214454) and Figures 15b-e exhibit its statistics. Noticethat the logarithm of the speckle intensity has a Gaussian histogram(skew-ness=7.562X10"4as it should given that the object had a completely singular intensity (ln(l)=0). Figure 15f shows the speckled sine2 using the AGM (C=l. 102836).Figure 15g show its logarithm of the speckle intensity histogram and Figure 15hshows the same statistics but for the logarithm of the object intensity subtractedfirst (skewness=6.56X10~4). This simply verifies that my model was implementedcorrectly (see Figure 7a).From these results, it is clear that a coherently speckled object can be modeled ineither the pupil or image plane. For my bispectral analysis, I will use the DDM.CWM. and the AGM for the speckle modeling. As will be shown in the 2-D reconstruction, the bispectrum works particularly well for the AGM case. Appendix Ahas a tabular synopsis of all of these speckling techniques.ADDITIVE GAUSSIAN MODELo81S1oS1oo o - ** "i; \-> v* ^* fcr-, fs O'Z !*?** .< ^r <.- U-T"I i '0.0 10. 0 20.0 30.0 40.0 SD.O 60.0 70.0Figure 15a. Speckled"Wall"Using the AGM-56-HISTOGHAH OF SPECKLE ISTEKSITY-I THISTOGRAM OF SPECKLE FIELD MOOl^'JSlr-* wADDITIVE ftAUCSXAM MODEL(b) (c)HISTOGRAM OF LOG (S^CCk^ IMEnSTT;-] 1 1 1 1 1 1 11-0 SLICE OF SPECKLE PHASE(d) (e)Fieure 15b-e. Statistical Histogram and Phase for Speckle"Wall"Using AGMGAUSSIAN IN DENSITY DOMAIN METHODFieure 15f. Truth Object Speckled Via the AGM-58-HISTOGRAM OF LOG (SPECKLE INTENSITY)GJ ISO-Xa.fc 300-1 1 r -i 1 1 1 1 r_i i I St l \ l I0 5 10 15KBMJSM45S0S5UUSSUH IN DBUITV OOUIN ICTHOO(15g)HISTOGRAM OF LOG (SPEK OBJ) -LOG (OBJ)1C IE 2C 85 SO 35> = 5 MADOITIVE SAUSSIAN IN THE DENSITYKMilN(15h)Fieure 15e-h. Truth ObjectSpeckled Via the AGM-59-4.0 BISPECTRUM ESTIMATION4.1 OverviewThe definition of the bispectrum, as stated previously, is given as the Fourier transform of the triple correlation. A more rigorous definition than this is given byNikias and Raghuveer (1987), where the bispectrum is the Fourier transform of thethird order cumulant sequence of a random process. This definition satisfies abroader range of statistical criteria, using cumulants, than the correlation does. Butfor purposes of this thesis, the correlation definition will suffice. Having stated thisin words I now relate its mathematical description, using the notation of Lohmannand Wirnitzer (19S4j. The triple correlation is given as:oo$3>(xvx2) =Ji(x)i(x + x1)i(x + x2)dx (41)COwhere i(.$ is a 1-D object (or a two dimensional (2-D) object with x being a 2-Dvector).The bispectrum is then formally given as:CO ooI^tfj.f.)=J Ji(xrx2) exp [-2i7j(f1x1 + f2x2) ] dxj dx2 (42)OO OOThis can be written as (see Appendix B):I'-(fl.f2 ) =l(f1)I(f2)U(fi+ f2)(4?'The bispectrum. then, becomes the triple productof the object Fourier transformover spatial frequencies fi and f2- From Equation 43 it can be seen that the-60-bispectrum is generally complex and has redundant regions providing hexagonalsupport (Nikias and Raghuveer, 1987). This is shown in Figure 16 (Bartlelt, et al,1984). Note that, from Equation 43, a 1-D Fourier object results in a 2-Dbispectrum and thus, by inference, a 2-D Fourier object will give a four dimensional (4-D) bispectrum. This can be a concern for large dimensions due to thememory space required in a computer. Therefore, any algorithm which applies thebispectrum must attempt to resolve this by sampling only portions of thebispectrum, since it has redundant information built into it.* p(i) I(3)(p,q) =I(3)(q,p)(ii) I(3)(p,q) =I(3)(p, -p-q)(iii) I(3j(p,q) =|l(3)(-p. ~q) Tif i (x) is realFicure 16. Redundant Regions of the Bispectrum-61-4.2 PropertiesSignal processing with the bispectrum has several important advantages. Unlikethe power spectrum density (PSD), Fourier phase information is retained in thebispectrum so that unique image reconstruction can occur. Fourier phase is lost inthe PSD due to the complete spatial symmetry of the autocorrelation while thebispectrum's support retains this object phase. This is quite easy to see from thedefinition of the phase of the bispectrum (see Appendix B). This property is alsouseful in identifying whether a system is maximum, minimum, or mixed phase(Nikias and Raghuveer, 1987).Another advantage of the bispectrum is that it is shift-invariant, i.e., it is insensitive to linear, translational motion. This property allows one to use ensemble averaging to reduce noise present in the image without concern for registering images.When averaging M images together, the noise decreases as the yM (Jain andChristensen, 1980). If these images are not perfectly aligned, the resultant noise-reduced image will be blurred (Gonzalez and Wintz). Since the bispectrum is insensitive to this misregistration, ensemble averaging can be performed without thisimage blur if the averaging is performed in the bispectral domain. In this thesis.this property will be tested by randomly shifting the image from frame-to-framewhile bispectral averaging is being performed.-62-Another property of the bispectrum has to do with its third order correlation. If thenoise present in the image has a symmetric pdf, then its third order moment iszero in the average. This is a property of all odd-order correlations. If the noise,n(x), is signal-independent, additive and stationary, then the noisy image, j(x), canbe written as(Lohmann and Wirnitzer, 1984):j(x) = i(x) + n(x) (44)Taking the ensemble average of the triple correlation of Equation 44 gives (Loh-mann and Wirnitzer, 1984; Bartelt, et al, 1984; Freeman, et al. 1988):=i(3)(Xl,x2) ++[i(2)(Xl)+ i(2J(x2)+ i(2)(x2-Xl)] (45)+ T[++nr2)(x2-x1)]where:< > denotes the ensemble averagen (x) is the autocorrelation (ATC) of the noisen (xj, x2) is the triple correlation of the noisei is the image mean is the noise meani <-xl> x2) is the triple correlation of the image.Three undesired terms exist that need to be dissolved. If the noise is assumed to bezero mean, all values multiplied by that term go to zero. Thus the ATC of theimage is of no concern. If the noise has a symmetric pdf, its third order momentgoes to zero and thus the n (x1( x2) term vanishes. Finally, the i term can bezero if the image mean is subtracted from the image before processing, or if the-63-image mean is zero initially. The terms multiplied by that value are ignored if thisis the case. All that is left is: = i(3)(Xl,x2) (46)From this relationship the object Fourier magnitude and phase can be recovered(via the bispectrum) and then the original object can be reconstructed. Sun-daramoorthy, et al, (1990), proved this same result in the Fourier domain. Theyfound that as long as the bispectral axes were not used in the reconstruction,unbiased estimates would be obtained for the original object (if it was in an additive noise process that had a zero bispectrum).Two key points should be considered here before I continue. One is that the noiseabove is additive and that the noise reduction is performed in the ensemble. Asstated previously, speckle noise is multiplicative. The task is to convert this multiplicative process into an additive one. To do this a homomorphic transformationcan be taken on the speckled image. The transformation is called homomorphicbecause a homomorphism from the set of multiplicative numbers to an additivegroup of numbers has occurred. A homomorphic transformation is one in whichthe natural logarithm of the image intensity is taken before any further processingis performed. The multiplicative noise process then becomes additive due to thislogarithmic operation. Several authors review this technique in general (Oppen-heim and Schafer, 1975; Lim, 1987; Arsenault and Levesque, 1984; and Gonzalezand Wintz, 1987) while others have reviewed this transformation for speckle inparticular (Jain and Christensen, 1980; Lim and Nawab, 1981; and Sadjadi, 1990).Arsenault and April (1976) have shown that an object in speckle noise becomes-64-additive and signal-independent with an approximately Gaussian pdf whenundergoing the transformation of:D =-ln(I) (47)where D is the density (a term left over from the film grain studies) and I is thespeckle intensity. Therefore, theoretically at least, performing a homomorphictransformation on Equation 34 will give me the result of Equation 44. After thehomomorphic transformation is performed for each speckled image, these imagesshould be ensemble averaged. Guenther, et al, (1978) shows that this should reduce the noise level by the \ N, where N is the number of images processed. Howthis will be done using the bispectrum is discussed next.4.3 AlgorithmsIn this section I will review the algorithms used to reconstruct multiple, speckle-degraded, jittered images. To begin I refer to the bispectrum definition of Equation43, written here as:I(3,(f1,f2) = |l(3J(fi,f2)| exp[j*(flff2)] (4S)This function has both magnitude and phase, broken into their respective components here as:|l(3)(ii,f2)| = |l(fi)l U(h)\ |l*(fi + f2)l (49)\|/(f!,f2;= 4>(fi) + (fi + h ) (50)-65-For a noisy sequence j(k)=i(k)+n(k), from 0 < k < N-l, its Fourier transform isgiven as:J(M= E j(n)exp[-j27rnx](51)n = - ooFor M images, the ensemble-averaged bispectrum of Equation 51 becomes:j(3)Ui>M =17 S JI(X1)JI(X2)JI*(X1 + X2) (52)i =iAwhere j(3)( X^, \2) is the bispectrum estimate of the noisy image. The ultimateAgoal is to obtain a bispectrum estimateJ(3)that is as close as possible to theimage bispectrum I^3) (\lt \2) , the Fourier transform of Equation 46. Once thisis attained image recovery can be performed by the following algorithms.4.3.1 One-Dimensional AlgorithmFor 1-D images I use a different algorithm each for the magnitude and the phasereconstruction. The magnitude algorithm is from Sundaramoorthy, et al, (1990)while the phase algorithm is from Bartelt, et al, (1984). Matsuoka and Ulrych(1984) discuss several other phase algorithms that could be used for 1-D images(seismic wavelets) but the recursive algorithm of Bartelt appears to be sufficient. Iwill first develop the 1-D algorithm for a 1-D sequence from 0 < m.n < N-l,letting B(m,n) denoteI(3)( \1 , X 2 ) and H denote I. With these substitutions I get:B(m,n) = H(m)H(n)H*(m+n) (53)-66-where B(m,n) is the bispectrum of H ( the Fourier transform of the 1-D sequence).Letting m=l, Equation 53 becomes:B(l,n) =H(l)H(n)H*(l+n) (54)Since the object is assumed to be real its Fourier transform is Hermitian, i.e.,f(x) = f *(-x) (Gaskill). Hermitian means that the complex function has a realcomponent which is even (e(x) = e(-x)) and an imaginary component which is odd(o(x)=-o(-x)). This implies I only have to sample the bispectrum from Equation 54for 1 < n < 1 and then I can later extend the reconstructed object from these~2Jsamples for ^ to N-l. The value of H(0) is the image mean (Gonzalez andWintz). If this value is zero, or the mean is subtracted from the image beforeFourier transformation, this term could cause magnitude reconstruction problems,since many magnitude algorithms have this term as a divisor (see Bartelt, 19S4).Once the bispectrum values of Equation 54 have been calculated and summed forM images, I can reconstruct the Fourier magnitude and phase of the original object. For the magnitude, I simply write Equation 54 in terms of its magnitude,letting n=j-l, as (for real objects):|H(j)|-,I^-"I(55,|H(1)| I H(j-l) |where the value of j eoes from 2 to For values of j > , |H( j) = H(N-j) .2 2Notice that I have the H( 1)| term as a constant to determine before I can recur--67-sively find the other values of |H(n)|. This value is given from Sundaramoorthy, etal, as:6/H(l)| =B(l,l)|3|B(1,3)|B(1,2)| |B(2,2)|(56)Note that an additional bispectrum term has arisen in this formulation. Thereforethe constant B(2,2) must be calculated and summed during the calculation of theother bispecturm terms. From Equation 53 this becomes:B(2,2 ) =H(2)2H*(4) (57)From this I can find its magnitude and use its value in the recursive routine (B(2,2)is not needed in the phase recursion routine). Since this technique requires only-^- samples of the bispectrum along the m=l line and the B(2,2) sample (for animage of size N), considerable savings can be had both in memory allocation andcomputation time.The phase portion of the bispectrum reconstruction is also recursive. The calculated values of the bispectrum phase are an extension of Equation 50 used inEquation 54, written as:v>(l,n)= (n)-(l+n) (5S)where, as before, t|f is the bispectrum phase and $ is the object phase. Lettingi = n-l, these values are calculated only from 1 as:-68-(i)=-(l,i-l) +<()(1) + 4>(i-l) (59)where: 4>(1) = -^ $ (l, n) (60a)4>(i)=-4>(N-i) for i >-E+1 (60b)Values for both the magnitude and the phase for n > +1 to N-l can be found bytaking advantage of the Hermitian property of the Fourier object. The value of4>(^-) is set initially at zero since it does not affect the reconstruction but merelythe placement of the reconstructed object (Matsuoka and Ulrych, 1984; Bartelt,1984). The final values of |H(n)| and 4>(n) can then be combined using Euler'srelationship and the inverse Fourier transform taken to obtain the reconstructedobject.4.3.2 Two-Dimensional AlgorithmThe 1-D algorithm developed previously worked extremely well on all types ofnoise, as well as speckle (see Results section) but, being a 1-D algorithm, it didnot extend well to 2-D images. By extension I mean continuing to use this 1-Dapproach on 2-D images by first "raster-scanning", or peeling, the image beforeobtaining its bispectrum. If the 2-Dimage was noise-free, this technique workedperfectly for jittered, imagereconstruction. All 2-D objects of any extent, constraint, or phase type could be reconstructed with this method. Once a noise wasadded, thoush. even in small amounts, reconstruction would not occur. Since noise-69-essentially creates phase errors in the object phase (Marron, et al, 1990), thesephase errors are propagated and therefore increased when scanning an image recursively over its NXN extent. This is because you have given the phase errors agreater amount of data to accumulate in as you scan the array as opposed to a1XN image.The 2-D magnitude reconstruction algorithm I used is from Dianat and Raghuveer(1990). Their approach is developed as follows. Write Equation 53 for 2-D imagesas:B(i,j;m,n) = H(i,j)H(m,n)H*(i+m,j+n) (61)where the B(i,j:m,n) is the 4-D bispectrum of the 2-D Fourier object H for pixelsm,n,i,j. To simplify to a 2-D problem, let i = m, j = n, to get, only for 2-D sub-plane:B(m,n) H*(2m,2n) (62)Taking the natural logarithm of the magnitude of both sides of Equation 61 gives:In [|B(m,n)|] = 2 In [|H(m,n)|] + In [ | H (2m, 2n)|] (63)By describing the above equation in terms of its discrete Fourier transform (DFT,see Equation 51) and equating like coefficients on both sides, the following relationships can be seen as (where the ~ term denotes the DFT of the natural logarithm of the magnitude):-70-1. For m,n odd:H(m,n) B(m,n) (64)2. For 0r ^ odd:2 2(65)3. Repeat Step 2 recursively for cases when -21 or is odd for L = 2,3 .2L 2Llog2(N)-l.4. Finally, compute:H(0,0)=^[5(0,0)-h(0,-|] -H^,0]-h(^,^]] (66)Once I have calculated all values of H(m,n) recursively from the bispectrum usinsthe above relationships, I exponentiate this value to obtain the Fourier magnitude|H(m n) |. I have found this algorithm to work practically flawlessly for magnitudereconstruction. As will be discussed in detail later, it is the 2-D phase that willdetermine the true shape of the reconstructed object. The above routine also canbe extended to the phase but it is not as robust in phase recovery as the magnitudealgorithm is. Thus, for phase reconstruction, I used a 2-D recursive routine similarto the 1-D in development but with some added calculations.-71-The two-dimensional, recursive, phase reconstruction algorithm is a 2-D extensionof the Lii-Rosenblatt technique described by Matsuoka and Ulrych (1984). Thisalgorithm requires the H(0,1) and H(1,0) subplanes as a minimum because thesesubplanes represent the lowest spatial frequencies in the 2-D object. This being thecase, I let i=0, j=l in Equation 61 to get:B(m,n) = H(0,l)H(m,n)H*(m,n+l) (67)Similarly, for i=l, j=0 I get:B(m,n) = H(l,0)H(m,n)H*(m+l,n) (68)Finally, I need to calculate a single bispectrum constant to complete the calculatedvalues necessary to reconstruct the image. This value represents the limit of thesubplanes given above that I will need to sample:B (f ,0; Q,f)- H (f ,0) H (0,f )H*(f,f) (69)To calculate the Fourier phase I use the following relationships, once I have ob-tained the bispectrum for 2 < m,n < - 1:1. Usine tl/ calculated from Equation 67 for m=0, 2 < n <- 1," Y24>(0,n) = cb(O.l) + 4)(0,n-l)-\|/(0,n-l) (70)N2 2--^iwhere: ^,(1.0) = -^ " ^ (0,n)Nm=l-72-2. Using \|/ calculated from Equation 68 for n=0, 2 < m < -1,24>(m,0) = cb(l,0) + 4>(m-l,0) - \|/(m-l,0) (71)where: 4>(L 0) = -2- i|r(m,0)Nm=l3. Using Equation 70 for 1 < m,n < -1,2(m+N/2) =-(m,n)=(0,l) + <|)(m,n-l)-\J/(m,n) (74a)(0,0) = 0 cd t.c u.o ls.c s.a17b. Two-Dimensional Contourof (a)17c. Recentered Version of (b)-78-THUTH OBJECT: 1-0 SLICE OF S/C (GHY)1 I5o s leuaoasaoMJOAMBtouFigure 18a. One-Dimensional Truth ObjectaECONSTR'JCTEO 1-D SLICE OF S/C (GHY)J 1 1 1 Lo 9 >o a M as jo x *\i sc 55 eo t;Figure 18b. Bispectrally Reconstructed Version of (a)-79-ORIGINAL OBJECT (TRISAfl. H-32!'''' 11 L_J 1 I I I 1_I < It It W II II tsFigure 19a. One-dimensional Truth ObjectBISPECTRUM RECONSTRUCTED TRIBAR (NO PADO|H)I i i r0 > ' > M II U tC tl t< M U 1C UFigure 19b. Bispectrally Reconstructed Version of (a)-80-As we have seen, each Fourier quantity has both magnitude and phase since theFourier expression is, in general, complex. Thus, I can write Equation 76 as:G(M = |F(\)||H(X)|exp[-j{a>F(X) + H(M"] (78)If ^h(^-) = 2ifcv\, then the output function, g(x) would merely be a shifted version of the input, since, by definition, h(x) = 8(x - a) in Equation 75 (where 8 isthe Dirac delta function (Papoulis)):g(x) = f(x)* 8(x - a) = f(x-a) (79)There is no phase distortion, then, if the phase transfer function is a linear function of \. Also, if $H(V) is a constant no phase distortion will occur (Gaskill).For example, if (A.) =-5-, the real-valued input becomes an imaginary-valuedH 2output. Similarly, for (X) = t\2tt (n is an integer), g(x)= f(x) and forH (X,) =(2n +1)tt. gfx) = -f(x). The 1-D tribar target of Figure 18a did not showHwa phase shift upon reconstruction probably because of the spatial frequency of thetribar was such that O (X.) = n2it.HThe importance of phase information cannot be overstated when reconstructing anobject. In general, the magnitude gives you information regarding the strengths ofthe signal but the phase gives you information about the structure of the signal.Reconstructing an object merely from itsFourier magnitude is similar to taking theautocorrelation of the object (Goodman, 1985). This is the result of the square-law-81-detection aspect of image recording. As shown above, phase records the locationof the image whereas the magnitude does not. A simple example of this is seen bythe following three objects where (Dainty and Fienup):f(x,y) is the original objectf(x - xo, y-yo) is the object shifted by xrj, xo-f(x-xo, y-yo is the object shifted and reflected about origin.All three objects have the same Fourier modulus but all 3 are not located at thesame position or orientation. The phase problem becomes, then, as posed by Batesand McDonnell, does there exist more than one object belonging to the set of allpossibilities for the phase? If the phase can be retained exactly, there is no ambiguity in the object reconstruction (Oppenheim, et al, 1982).An illustration that compares the relative importance of phase versus magnitudefor image reconstruction is given by Oppenheim and Lim (1981). They provide theexample of the reconstruction of the atomic structure of a L-tyrosine HCL crystalimage from its exact phase data. They compare three succeedingly worse cases forthe magnitude: 1) magnitude of one; 2) magnitude equal to an average of severaldifferent L-tyrosine HCL atoms; and 3) magnitude from a totally different atom!In all cases the original L-tyrosine HCL atomic structure was reproduced from thecorrect phase but erroneous magnitude information.-82-For further proof of the importance of phase in reconstructing an object, I performed my own experiment using the truth objects of Figure 20a ("battleship")and Figure 20b ("Lena"). I combined the battleship's phase with Lena's magnitudeand obtained Figure 21a. Similarly, I combined Lena's phase with the battleship'smagnitude and created Figure 21b. Clearly, the phase is key when reconstructingan image.Returning to the phase filter, if the recovered phase is distorted, the reconstructedobject will be distorted. Phase distortion occurs when the"phase-filter"aspects ofthe operation cause the spectral components of the original signal to be misplacedfrom their original spacing (i.e., the amount of shift of the spectral components isnot linear with increasing frequency (Gaskill)). When phase distortion occurs, youdo not have the simple, shifted, delta function convolution described in Equation79for integer values of ex. This relationship is merely a special case of the moregeneral result:g(x)= f(x)*sinc(x-o;) (80)where sinc(x)-sin^7TX/(see Gaskill) and a delta function results at the limit (seePapoulis, p. 280-281). If | a | S-- (Dianat and Raghuveer, 1990) then nonin-teger values of the phase filter exist and thus nonlinear increases in the phase shiftoccur. For values of a > y upto multiple integer values, the shift is merely a it or2tt phase shift and further distortion does not occur. The reconstructed object appears to be blurred by a convolution with a sine function. Examples of this areseen in Figures 22 through 26.-83-Figure 20a."Battleship"Used for Magnitude/PhaseComparisonFigure 20b."Lena"OflPFigure 21a. Reconstructed Object Using"Battleship"Phase and"Lena"MagnitudeFigure 21b. Reconstructed Object Using"Lena"Phase and "Battleship"Maginitude-84-Figure 22a is the same tribar as in Figure 19a except that now it is zero-padded.Its reconstructed image is in Figure 22b. Notice that this image exhibits the phaseshifting and the sine blurring (which causes the negative-valued artifacts seen, forexample, in Figure 17). For the 1-D case I removed the sine blurring with datawindows, which I will discuss shortly.For 2-D images, the sine blurring was much worse. I tried several different phasereconstruction algorithms (Dianat and Raghuveer, 1990; Swami and Giannakis,1988) before settling on the one described previously and it still shows sine blurring for certain constrained objects. For example, Figure 23a is a tri(-r-,-^-) trutho oobject reconstructed in Figure 23b. Notice the severe sine blurring of the object. Inow change the constraint of the object to be tri(y,-r) and the reconstructedobject of Figure 24a is exact, save for the introduced phase shift. My shift-imageprogram then recenters the object for viewing (Figure 24b).Not only is object constraint a problem but also the object shape can cause sineblurring in the reconstructed image. Figure 25a is what I call a"binary"spacecraftbecause its gray levels change value abruptly. Figure 25b is the reconstructed object. Notice that the sine blurring has actually distorted the object and makes itappear noisy. Even for smoothly varying objects, such as my"gray"spacecraft ofFigure 26a (so-called because there are several gray levels in the image thatvary-smoothly), you can see a ripple of sine blurring in the reconstructed object(Figure 26b).-85-I .5TRIBAR IN ZERO-PADDED ARRAY 1N-64)nn T -| 1 1 1 1 i10 15 80 85 30 X 40 10 U C ISFigure 22a. Original ObjectflECONSTRUCTED TRIBAR SHOWING SINC BLUR~~' l T ' 1 T 1 1 11-!_j i i ' i i i i i i i: 10 15 80 85 30 5 40 45 SC B5 SO t5Fisure 22b. Bispectrally Reconstructed Version of (a)-86-TRUTH OBJECT. TtOff/i.X/S)Figure 23a. Original ObjectTRIff/e.VS) *TTH 3INC BUTR AOT PHASE SHOTFigure 23b. Bispectrally Reconstructed Version of (a)-87-HaereTRUCTED TXI&7.V) with phase shutFigure 24a. Bispectrally Reconstructed Tri {XII, Y/7)CESTERD RECOWniCUWliO TRICXAX'T)Figure 24b. Centered Version of (a)-88-\--\Figure 25a. Original Objects/r*n.s^.Figure 25b. Bispectrally Reconstructed Version of (a)-89-V>AFigure 26a. Original Object,,\\\*>Figure 26b. Bispectrally Reconstructed Versions of (a)-90-One way to avoid the sine blurring in the 2-D case was to use the 1-D algorithmand raster-scan the image before calculating its bispectrum. Figures 27a shows thebinary spacecraft, its jittered representation is in Figure 27b from and the reconstructed object is in Figure 27c. Notice that there is no sine blurring of the reconstructed image (I have recentered it after image reconstruction). No noise waspresent so there was no reconstruction problems when applying the 1-Draster-scan method to the 2-D image.5.1.2 Windowing the DataIn the 1-D case certain bispectrum reconstructions resulted not only in sine blurring but also in incomplete reconstructions, i.e., it would lead to an indeterminatecondition (divide by zero). For the 1-D case, I believed these occurrences werehappening because of errors in sampling the spectrum. This was because I noticedthat whenever I zero-padded an array, the reconstructed object often had sineblurring or it would not reconstruct at all. To overcome this, I decided to use datawindows and they worked as advertised.To review, data windows are used to relieve leakage or"ringing"artifacts whenspectrum data is aperiodically sampled. Data windows help by forcing the sampledspectra to have a period of N extent (where N is the length of the data record) eventhough the object itself may have an aperiodic spectra. The formulas used for thedata windows were taken from Stearns and David. In particular, I used the Bartlett(triangular) and the tapered rectangular data windows.-91-0.0 10.0 X.O 10.0 40. 0 B.O K.O 7GJ3Figure 27a. Original ObjectK.O 70. 0Figure 27b. Representative ObjectJittered Randomly In The Field OfView For 10 Frames-No Noiseo.o m.cFigure 27c. Bispectrally Reconstructed Version of (a)-92-Figure 28 is an example of using a tapered rectangular window on the object inFigure 22a before calculating its bispectrum. Compare this result to that of Figure22b. This implies to me that the sine blurring in the 1-D case is actually ringingdue to aperiodic sampling and not the same phenomenon seen in the 2-D reconstruction.Figure 29a is an example of an object that would not reconstruct with the 1-Dalgorithm unless a data window was used. By using a tapered rectangular window,though, I obtained the reconstructed object of Figure 29b. I noticed that if a certain(N7X*NT \V'^T'V' etc-)'then it would not reconstruct at all, either. Figure 30a is an example of this objectand Figure 30b is its reconstructed shape after windowing by a Bartlett window. Inall cases where an indeterminate reconstruction arose, data windowing allowed forits reconstruction (though the window function is visible in the reconstruction).Finally, in the 1-D case, objects placed in very large data records (1X256, 512.1024, etc.) would not reconstruct but give an indeterminate condition unless datawindows were used.I attempted to use data windowing in the 2-D reconstruction but was never successful in removing the sine blur. Thus, I conclude that the cause of the sine blur ismore fundamental in the 2-D case then mere aperiodic sampling errors. It appearsthat, since my 2-D phasereconstruction algorithm only samples the two lowestspatial frequency bispectral subplanes [(1,0), (0,1)], I have bandlimited the resultto these lower spatial frequencies. Higher spatial frequencies present in therecon--93-structed image would give me sharper edges and not the sine-blurred edges I amgetting now.cozLUA.J.|1.0 - 1.9 h> 1 1 1 1 1 1 1 1 1 1II ni-.Bt- ..7-!i6 j-u5,3.4-l\ P.3-M.2-.1- II II hn 1 | ! 1 1 1 1 1 1 ! 1 15 10 15 20 25 30 35 40 45 50 55 60 65Figure 28. Bispectrally Reconstructed Padded Tribar with Data Window-94-1.41 T 1 r 1 i 1 1 1 1 1.t-.1-~.4-.1-ff *- , , ,l_ iii,,0 uFigure 29a. Original ObjectI iri 1 1 1 |Ti-i -i i iIns u>Figure 29b. Bispectral Reconstruction of (a) Using a Data Window-l'""T '! ) f n "i' 't t* is m~l ' 1 1 ! 1 1 rJ I J Lf ix *i m *c m u m ic m%urrLm uicickwFigure 30a. Original Object Figure 30b. Bispectral Reconstruction of (a) Using a Data Window-95-5.2 SpeckledTo begin the speckled section I should review at the outset the flow of the processing that I performed to reconstruct a speckled object. Before I do that I will brieflydiscuss it. The overall methodology will be, in general, the same whether usingthel-D or 2-D bispectrum algorithm.The truth object is created and stored in an array that does not change. For aselected number of images, M, each one is speckled independently using one of thespeckle algorithms. The speckled object varies from frame-to-frame by using adifferent random number seed to create the speckle for each frame. I then randomly jitter the speckled object in x and y for each frame. This models imagecapture. I then perform a homomorphic transformation before I Fourier transformthe image. I then calculate the bispectrum of this function and sum it with previously calculated values. Once all images have been summed, I normalize the resultant function and calculate the Fourier magnitude and phase from their respective1-D or 2-D algorithms. Combining these quantities and inverse transforming, thenexponentiating that result, gives the final answer. A median filter is included, ifdesired. See Appendix C for the flow diagram of this entire procedure.As noted previously several times, the MLE of an undegraded version of the speckled image is the ensemble average of the multiple speckled images. To find theMLE of the undegraded image using multiple observations of the multiplicativenoise model, I can write:g,(m,n)= I(m,n)uk(m,n) (81)-96-where the values take the same meaning as Equation 34 except that it is thekthestimate for N observations (k=l,2,3 ,N). Performing the homomorphic transformation, I get:ln(gk(m,n))= ln(I(m,n)) + ln(uk(m,n)) (82)From Raghuveer, let these values equal, respectively,:Zk = 0 + nk (83)To find the MLE, I want to maximize the probability that the N observations produce the original object. For N independent observations I can write:P(z|e) =P(zJe )p(z2|e )p(z3|e ) p(zN|e) (84)where z is a vector. Looking only at one observation, I can express it in terms ofn1=z1- 9 as:p(zje) = p(n2) =p(Zl-6) (85)From the pdf for the logarithm of the intensity, I can be write:p(zi-Q)= exp[-(zi-e)-exp[-(zi-0)]] (86)therefore, for all N observations, taking the logarithm of Equation 86 gives:N r ,ln(p(z,-6)) = E(-(z -6)-exp[-(zk-6)J ) (87)k=l-97-ln(p(z-6))= Sz + N0 - Eexp [-(z.-8)] (88)k=l*k=lKdifferentiating this with respect to 0 and setting it equal to zero gives:N = exp [0] S exp [-zJ (89)k=lkreturning to the original definitions, I can substitute in for 0 and zk and divide by Nto cet:I(m, n ) =-1 g, (m, n) (90)N k=iKThus, averaging N observations of the noisy image will provide the MLE of thedegraded image. This result is similar to the result given by Lim and Nawab (1981)except they kept the noise expression in the non-logarithmic domain during theproof.To see what this implies for image reconstruction, notice the speckled version of asine2seen in Figures 31a and b, both in 2-D and 1-D, respectively. Figures 32aand b are the resultant images after 150, independently-speckled images wereensemble averaged. Notice that you do not reconstruct the image to obtain Figure 9but, hopefully, to see what the object was in the first place! This is a result of thesevere image degradation caused by the fully-developed speckle.-98-Figure 31a. Three-Dimensional Plot of Speckled Sinc2(X/16,Y/16)^y\VA'n / lAM-V 5 "Ato.c u sFieure 31b. One-Dimensional Slice of (a) DDM Used-99-Figure 32a. Ensemble-Averaged Version of Figure 31a After 150 FramesFigure 32b. One-Dimensional Slice of (a)-100-5.2.1 One-Dimensional ExamplesThe first example used showing image reconstruction is found in Figures 33a-g.Figure 33a is the truth object and Figure 33b is a typical speckled object before thebispectrum routine begins. The DDM was used to speckle the object. Figure 33c isthe ensemble averaged reconstruction of the jittered images and Figure 33d is thereconstructed object after 200 images were bispectrum averaged. Figure 33e is theobject after a 1X3 median filter has passed over the reconstructed image. Figure33f is the reconstructed object if no homomorphic transformation is taken. As onecan see, no object reconstruction will occur for speckled objects unless the naturallogarithm is taken first. Figure 33g is the reconstructed object after only 30 imageswere bispectrum averaged, showing how rapidly bispectral averaging can convergeto a solution.One reason the bispectrum rapidly converges to a solution is shown in Figures34a-c. Figure 34a is the natural logarithm of a sinc2(x/8) function. Wherever thereal function goes to zero I set its density domain representation to -150. Figure34b is the log of the speckle intensity shown in Figure 33b. Notice how the noisenow appears to"ride"on top of the function, essentially improving the SNR of theresultant image. Figure 34c is the bispectrally averaged estimate of the speckledimage prior to exponentiation (which then gives Figure 33d). Notice how the noisehas smoothed considerably.-101-TRUTH OBJECT: 6INC*2 (X/B)SPECKLED SINC**2 (X/B) IN WIDE FIELD1 "A ' 1 1 1 "I 1 1 r-1^ 1 1 1 i i0 B UlfltOaBIO3E4C40BOeBtOIH DUE FIELD FTJB JTTTER MVDEKT-i 1 1 1 1r~_i i S^f\0 5i0 15K253036 40 45CI5&C65(a) (b)-r--r 1RECONSTRUCTED SlNC-2 [A/B) . NC WINDOW\ \ A Kl'IU \A: \\J \ft i*cIKI- so9 10 19 20 St 30 35 40 45 SO 55 M 6WW ITERATIONS. LUR-lMG-AYE'D20 -iI J/ \ J/ \1\-l10 IB SO M 3C M -C 4S bO i 6J KNO MEOIAN FILTER. JITTESEI RAXD3_I(C) (d)Figure 33a-d. Example of Bispectral Averaging of Jittered, Speckled Objects-102-C20KM2 icS- 120J1005--i 1 1r-0 15 2C 23 901X3 METJIAN FILTER USED50 96 60 65 10 IS tO 25 30 35 40 45 5C 55HO HOMOMORPHICTHAIS'-OBMiT10* US=;(e) (f)I i 1 1 1 1 1 1r-2j:220IK _ ^WO h i120 - i-1C0 "Mi _SCi isci/ \i402Cui.\. . . .Ai i15 SO 25 SC 35 40 45KO KEDIiK 30 IHABES RiKDOHLY JIT^EBEO5C 55 60 65(g)Fieure 33e-g. Example of Bispectral Averaging of Jittered, Speckled Objects(Cont'd)-103--lao j-ILOG (SINC2 (X/8)r 1.J l 1 l_I l I. l J l_10 IE K EC 0 40 4S SC 90 MPUtt KONOMOOPHIC TUNSFOfWATlOM IKfTLOG (SPECKLED SINC2)-! 1r--11i r-4(-II-w h! iI T0 S 10 15 K SO 35 JO 45 50 55 60(a) (b)soLOG (RECONSTRUCTED SINC**2)1Vrc~~-4C200i_-20--40--601 1 1 1 __l L iiiip, 15 80 25 3C K 4C 5 50 556C BEJUST ffllOR TO EXPONENTIATION [200 IWA6EE)(c)Figure 34a-c. Result of Bispectral Averaging an Object in its LogarithmicDomain-104-Another reason the bispectrum can handle speckled images is seen in Figures35a-f. I use the same sine2 again but with no zero padding so as to view the resultsbetter. I show the logarithmic representation but I also include the Fourier transform of the log of the speckle intensity (Figure 35f). Compare this to Figure 35e.Thus, the bispectrum, which uses the function in Figure 35f to obtain its values,basically sees a very close approximation of the original object when processing inthis domain (the Fourier of the log domain).The next three sets of Figures use a tri(x) object to describe three key points.Figures 36a-f show a non-jittered object with its bispectrally-averaged rendition.Notice that, even using the ensemble average (Figure 36c) I only get a form of thetri back. Figure 36d is what a 1X3 median filter would do if used directly onFigure 36b. Figures 36e-f are the bispectrally reconstructed image without andwith the median filter, respectively.The next set of Figures (37a-e) are of a tri that is jittered randomly in a wider field200 times. Since averaging is done in the bispectral domain, the shifting is ignoredby the bispectrum and reconstruction occurs (Figures 37c-d). Figure 37e shows theresult if no homomorphic transformation occurs.-105-TRUTH OBJECT SINC*S (X/8)~l I 1r-o a 4 e to 12 14 ie u ao 22 24 a so(a)bHtUKLfcU blNU#ei lA/tSJ , NU KJ.INUUKi 1 1 1 1 I i 1 1 1 1 1 1 1r-\J IJ d L.0 2 4 e e IC 12 14 16 IB 2C 22 2* 2 28 30 32(b)NATURAL LOG (SINC2 (X/8) )_1 I 1 L. _1 I I 1_LOG (SPECKLED SINCHK2 (X/B) )0 f 4 e | 10 IE 1. IC it ec 22 tw KB n 10 K 0 t 4 t t 10 It U 16 It 0 22 34 C t V J.(c) (d)Figure 35a-d. Comparison of the Fourier Representation of a Speckled Objectwith its Unspeckled Version-106-FT (LOG (SINCtwE (X/8) ) )"> t i i i i i i i i i i i i r-i 1 i i i ' '10 12 14 16 16 M EC 24patxal rmauemyM JO M(e)FT (LOG (SPECKLE SINOttZ (X/8) ) 110 12 14 16 11 eo 22 24SPATIAL FREQUENCYft n ao 32(0Figure 35e-f. Comparison of the Fourier Representation of a Speckled Objectwith its Unspeckled Version (Cont'd)-107-TrTJTH OBJECT: THI (X/U)1| 1 1 1 1 r 1 r__nIr'Ti'ri'I-/;r /0 I 1 1 1 L_\o a 4 io n i4 is ib to as 24 a m to seSPECKLED TRI (X/14)i 1 1 1r--i i 1 1r-r A1 M-J 1 1 I ' ' j L_0 t 4 t I 10 li U |l || K ti 34 21 K(a) (b)EKSEMBl.E AVERAGED TBI (X/14) MEDIAN FILTEHEO SPECKLED TRI(X/14itec p_1 r , j j {*40?5 iL.TT"\J.j i i i i i_\10 12 14 II IB 2C E |4 M 2B 3C1X3 MEDIAN FILTER(c) (d)Figure 36a-d. Example of Bispectral Averaging of Speckled Object-108-RECONSTRUCTED TRI {X/U) . NO WINDOWa 10 12 14 16 16 20 22 24NO MEDIAN FILTEH NO JTTTERtt 10 St(e)RECONSTRUCTED TRI (X/H) . NO WINDOW0 2 4 6 | 10 12 14 IE 16 20 2224 261X5 MEDIAN flLTtP. USED: NO JITTEP(f)Figure 36e-f. Example of Bispectral Averaging of Speckled Object (Cont'd)-109-THKX/14] IN WIDE FIELD-i 1 nr 1 1 1 1 1 1_L 1 l_SPECKLED TRI IX/14) IN HIDE FIELD-1 1 1 1 1 1 r40 4B Be0 5 10 1SIOH3eX4046S0(a) (b)RECONSTRUCTED TRI (X/14) : 200 IMAGES_l I L10 IS 20 22 30 35 40 4 50 55NO MEDIAN FILTER OH KlkDOM USEC(c)Figure 37a-c. Example of Bispectral Averaging of Jittered, Speckled Object-110-RECONSTRUCTED TRI (X/14) IN JITTER~i 1 1 1 1 1 1 1 1 1 1 rO 1 1 I L-J I 1 I I I 11 I I L_0 S 10 1B20283036404SBOB601XB MEDIAN FILTER USED; NO KINDOU(d)RECONSTRUCTED TRI (X/14): 200 IMAGES-i 1 1 i 1 rJ 120I I I I i-"| |iou(ca30aS404JBOIBKKMS NOOOKC TIUNSnmATION USED(e)Figure 37d-e. Example of Bispectral Averaging of Jittered, Speckled Object(Cont'd)-111-The third set of Figures (38a-e) show why bispectral averaging works for the tri.The log of the speckled tri (Figure 38b) has such an improved SNR (visually) thanthe speckled version. After averaging (Figure 38c), the noise is considerablysmoothed. Figure 38d is the Fourier transform of the log of the tri. Compare thisto Figure 38e: there is no difference! For the scale factors used, the noise is practically nonexistent in this domain. This is what the bispectrum sees when it is beingcalculated.Finally, Figures 39 and 40 are speckled and reconstructed versions of Figures 18aand 22a, respectively. Notice how well we can discriminate the form of these objects which was lost in the speckle (see Korwar and Pierce, (1981) for a discussionof this loss of discrimination) after bispectral averaging.5.2.2 Two-Dimensional ExamplesI was so pleased with the results of the 1-D algorithm that I decided to extend it to2-D images. To do this, I would raster-scan the 2-D image, after jittering andspeckling, to reconstruct the object. Before attempting this, though, I tested the1-D algorithm on Figure 41a, which is the object"E"jittered in a'"smearing"action (Figure 41b). Figure 41c is the result: perfect reconstruction!I then tried it on a very small amount of additive Gaussian noise. It is so small(variance + 0.003 about the image) that you can barely see it on a 3-D plot (Figure42a). After 32 iterations, an"E"results that is slightly distorted (Figure 42b). By150 iterations, the noise is gone and the original image results (though it is phase-shifted) (Figure 42c).-112--i 1 r 1 1 r! '-IX -0 3 10 13 to 28 30 36_l I I60 K K(a)LOG (S?EC-~L if^ ri' '.i.JL60 B K 66UMXM.Y JITTKCD(b)Figure 40a-b. Bispectral Reconstruction of Speckled Object-116-0RK2NJU. OR/ECT (H-32) 32 ttanUosUL 4iraot-ta4f wrg& If*-l ,0C, MN-O.OO-0 - ci(a) (b)32 itttnUeu, bupcctrum T+aorwniI! If.I .e .c(c)Figure 41a-c. Bispectral Reconstruction of 2-D Object Using 1-DAlgorithm - No Noise-117-(a)32 lUntioni. \>tip#drum-rcovrXj ISO iUrUonj, tmp^ctnun ncovtrei(b) (c)Figure 42a-c. Bispectral Reconstruction of 2-D Object Using 1-DAlgorithm - Additive Gaussian Noise (Variance = 0.001)-118-I next increased the variance of the zero mean, additive Gaussian noise to 10X theprevious value to see the results. Figure 43b is the result: unrecognizable reconstruction! When noise was present, a 2-D extension of the 1-D algorithm was notpossible. This is due to the phase errors propagating throughout the reconstructionalgorithm from the recursion formula used. Larger noise means larger phaseerrors.Obviously, speckle was an even worse problem than this so I had to find a 2-Dalgorithm to work on 2-D images. That 2-D algorithm, developed previously inthis thesis, was used on these 2-D images.Figure 44a is the speckled version of Figure 27a using the CWM with a contrast of1.053862. Figure 44b shows 30 speckled images ensemble-averaged together. Figure 44c is the reconstructed object after 30 images were bispectrally averaged, andFigure 44d is the same image but with a median filter passed over the image.-119-(b)Figure 43a-b. Bispectral Reconstruction of 2-D Object Using 1-DAlgorithm - Additive Gaussian Noise (Variance = 0.01)-120-MftCCXD "YXiXrVtCtOJTT s/cJ t*J(a) (b)RKOOTTBUCTn> EDvAKT S/Z (CTlO CEttERiD. RXCOKSTHDCTD BCvUTT 5 ^13 M " " L"" "''30 1WB3, K IOI* riLTWUSE(c) (d)Figure 44a-d. BispectralReconstruction Using 2-D Algorithm. ObjectSpeckled Via CWM-121-IRCXLID ^rt (BQ TU CDU BCSUBIX-AVBLLCZX). iTTTTRXD. BDCUTf S^C(a) (b)ncCbSSTBUCTn) S/C (KM); SO TyAfT!UJ S-* X-I .C KJ .w wife* w ro:* tilto use;(c)Figure 45a-c. Bispectral Reconstruction Using 2-D Algorithm. DDM Used toSpeckle Object-122-SWaOJLO KKAJTMPtOOXTtpi^gl/W?irrwn ""***rT ^r(OOITtVC BrlOSlW fCTKE USE*(a)ta-t j(CDI7IVC OUS31W UDO. UK=BtJ -e(b)EBSKSTKUCTB) BOOST VC (30 DUOZS)WLUk3iKU.Hl; BDUiT %K (30 DIAIZ9)KstTIvc mjssir* pcthx usesHOIIIrt OtSSIW IfXZL use(c) (d)Figure 46a-d. Bispectral Reconstruction Using 2-DAlgorithm. AGM Used toSpeckle Object-123-Figure 47 is the reconstructed spacecraft from 30 speckled images using a correlation technique I devised to keep track of the shift of the object fromframe-to-frame. Basically, I use the first speckled image as a reference with which I correlate all of the following images. Since speckle is wideband, random noise, theircorrelations have a distinct peak when they overlap during the correlation procedure. I keep track of this peak's location and align the images with the referenceimage so that ensemble averaging can be performed. The 2-D contour plot showsa gradual sloping toward the center of each object structure and the edges are a bitblurred. No homomorphic transformation is taken with this method.A very important point has to be made here. When using the bispectrum to estimate a speckled object, the logarithm must be taken prior to the estimation. Thiswas seen in the 1-D case and Figure 48 is the proof in the 2-D case for a speckledbinary spacecraft.In using the homomorphic transformation, I also discovered that care must betaken prior to exponentiation to obtain optimum results. In some cases, the reconstructed Fourier object had values that would cause an arithmetic overflow if notnormalized by some constant. The choice of this constant also was an integral partof the quality of the reconstruction. Different objects required different normalizing factors and the choice was purely subjective based on what the reconstructedobject needed to enhance its appearance, i.e, edges. Since the goal of speckle-image reconstruction is the return of the object form destroyed by the speckle, this isthe only necessary metric by which to base the decision on the choice of the normalizing factor.-124-RECONSTRUCTED SPECKLED BINARY S/tb -oJ*^>:_-kia0Lti*LO U.O X.C XJ 40.0 V.ORIC-HYERflSE rCTHOD USEDFigure 47. Autocorrelation-Averaged ReconstructionRECONSTRUCTED"BINARY"S/C (NO LOGARITHM)7C-Figure 48. Bispectral Reconstruction of Speckled Object without LogarithmicTransformation-125-5.3 ConclusionsThis thesis has shown that, conceptually, bispectrum estimation is robust enough torecover speckle-degraded images when a homomorphic transformation is takenand the objects are averaged in the bispectrum. This is especially true in the 1-Dcase, where any object could be reconstructed from any form of the speckle model.The necessary object conditions were that: 1) it had to stay consistently within thefield of view of the array when jittering and, 2) even though it could change position within the field of view, it could not change orientation.The phase recovery problems with the 2-D case were not totally resolved. Phaseunwrapping was thought to be a solution but trying to unwrap a 2-D object's phaseto obtain a unique solution in 2-D is difficult, especially if the phase changes bymodulo 2tt every pixel! A 2-D object has phase terms going in the x and y directions; phase unwrapping in one direction may not be what is needed in the otherdirection at each m,n pixel location. This is because phase unwrapping attempts toimpose continuity on the phase function to remove any ambiguities present in thephase (Fiddy and Stark). One ambiguity that arises is from the introduced phaseshift of the reconstructed bispectrum. If I could write the reconstructed estimate asan integer phase shift then I would get (after Kaveh and Soumekh):F(\) = F(U) exp[j2Trn]= |f ( \ ) |exp [j ( ( X ) +2tt n)] (91)AThe argument of F ( k. t ) for the phase, then, is given by:r^ i flm(F(\, )MARG{F ( X, ) } = 27m + arctan [^p^^ J (92)-126-The ambiguity for the value of the integer, n, is no problem since^(^'O and F(A., ) + 2,m give the same function for the given modulus,| F(^ t ) I The term 2irn carries no information thus no physical means exist todetermine it anyway (Takajo and Takahashi, 1988). But if I take the logarithm ofthe function, F (*.,{;), we need to know n. This can be seen from the logarithm ofthe complex function, F (K), which is written as:In (F ( \, ) ) = In (j F( \, ) | ] + j (2 + arctan^^^* }) ) (93)Note that the log of the reconstructed object has the same real part as the log ofthe real object but it has an imaginary part that differs by integrals of 2tt. Thelogarithm of the spectrum is no longer unique, as in the real object's case, due tothis phase ambiguity. Phase unwrapping becomes a process whereby we attempt toretrieve this term.Finally, phase can be described physically as the wavefront surface of a propagating complex wave field. If the amplitude of the propagating wave is zero at anypoint, its phase is indeterminate. Moving about that point causes a phase change of2tt (1 wavelength) implying wavefront dislocations (Fiddy and Stark). Sincespeckle is caused by modulo 2tr phase excursions, there are as many wavefrontdislocations as there are speckles. Phase unwrapping in this case would be indeterminate since the chance of obtaining an unambiguous phase unwrap would beslim.-127-In summation, this thesis was an attempt to apply spectral techniques to removemultiplicative noise from an object jittered against a uniform background in bothone and two dimensions. It was also an exercise in determining the utility of thebispectrum as a signal processing tool. The homomorphic transformation creates asituation where you have a signal plus noise and this noise is small compared tothe signal in the bispectral domain. Averaging in this domain allows us to obtainan estimate of the original object. Finally, this thesis attempts to review speckletheory and then model it in different fashions to fully understand the mechanismsbehind coherent speckle. Further work could attempt to use actual speckled imagesand also increase the robustness of the 2-D phase reconstruction routine. Onecould also look at using additional enhancement routines (aside from the medianfilter used in this thesis) as post-processing techniques to improve image reconstruction.-128-6.0 APPENDICESAppendix ATabular Synopsis Of Speckle ModelsModel Description Assumptions Transfer Function?DDM Circular, complexGaussian r.v. with zeromean, i.i.d. real andimaginary componentscreated as field.Magnitude squared forspeckle intensity.TFM Speckle field as in DDM,then Fourier transformedfor propagation. Coherenttransfer function multipliedby this and then inversetransformed beforemagnitude squaring.CWM Speckle field as in DDM.Complex spatial windowof equal strengthscorrelates complex field.Magnitude squaringresult gives intensity.RPM Modulo 2tt random phasecreated from uniformrandom number generator.This used with imagefield via Euler's notationto create speckle field,then magnitude squaredto give speckle intensity.Can be imaged orpupil-plane speckle.NEM Noise function createdthat has a negativeexponential pdf. This isthen multiplied by theincoherent image ofthe truth object.AGM Zero mean, unity variance,Gaussian noise added tothe logarithm of theobject intensity.Large number ofsurface scatterers;no multiple scattering; no depolarization uponreflection.Coherent opticalsystem can resolveall details in object.High quality opticswould correlatethe speckle locally.Object surface isrough on the orderof the wavelengthof the incidentcoherent source,up to multiplesof2tt until coherencelength of source isreached.Speckle noise is asignal-dependentand multiplicativenoise process.Log of multiplespeckle intensityis approximatelyGaussian.NO-assumed embeddedin incoherent image.YES-user defined. Akey player in ultimateobject appearance andstatistics.YES-in the form of a3X3, 5X5, etc. spatialwindow. Window actsas averaging filter.YES-is optional but wouldbe implemented similar toTFM once speckle field hasbeen created.NO-field statistics are neverderived.NO-field statistics are neverderived.-129-Appendix BBispectrum Fourier ExpressionThis appendix is proof of Equation 43 in the text. Rewriting Equation 42, thebispectrum is given as the Fourier transform of the triple correlation:l(3)(fi- f2) = J /i(3)(xrx2)exp[-277j(f1x1 + f2x2)] dxt dx2 (Bl)oo oooowhere:i(3)(xr x2) =Ji (x) i( x + Xj)i(x + x2 ) dx (B2)oois the triple correlation of the spatial image i(x) and x is a 2-D vector. SubstitutingB2 into Bl gives the bispectrum definition in terms of the real object, i(x) and itsshifted versions, i(x + x1 ) and i(x + x2) as:oo oo ooI(3)(flt f2) =/ / Ji(x)i(x +xx )i(x +x2)(B3)exp [-27TJ(f1x1 + f2X2)j dx dx1 dx2let (3 = (x + x1), oi = (x + x2), therefore d(3 = dxi and da= dx2 since x isconstant and (x1+ x2)are the shifted, changing values. Substitute these into B3above to get:oo oo ool(3)(fi. h) = J J /i(x)i(3)i(a)(BA)exp [-27tj(f1(p-x) + f2 (a -x ))]dx d(Bda-130-Collecting terms, noting that J f(x) exp [- j2ttx] =F() is the continuousoodefinition of the Fourier transform of f(x) (Gaskill), I get:p oo ooI(3)(f1,f2)= Ji(x)dx/i(3) exp[-j27rf1f3]d(3 Ji(a)OO_oo ooexp[-j 2tt f2 a] da exp [j 2.7^ x] exp [j27rf2xj(B5)ooI(3)(frf2) =/i(x)dx exp[j2Trx(f1 +f2)]l(f1)I(f2) (B6)_ooExpanding the exponential term in Equation B6 into its Euler's notation, exp[-j9] =cos(6)- jsin(6), I can write the Fourier transform definition in terms of its real andimaginary components, respectively, as F(t)= FRe(b)-JFjm(t). Thus, I canwrite Equation B6 as:I(3)(flf f2) = I(fi )I(f2)[lRe(fl + h) + J Ilm(fl+ f2)] (B7)ooNoting that F*() = Jf(x) exp [j277x] Equation B7 then becomes:I(3;(fi, f2) = I(f! ) I(f2)I (fl+f2)(BS)-131-From this expression I can simply write the bispectrum magnitude and phase relationships as functions of the object's Fourier representation as ( I I I is the objectmagnitude and 4> is the object phase):I(3)(fi,f2) = |l(3)(fi, f2)| exp[ jiKfi,f2)] (B9)where:|l(3)(fi , f2) | = | I(fi) | | I (f2) | |l(-fi -f 2) | (B10a)4> (fi,f2) = 4>(fi) + 4>(f2) -4>(f1 + f2) (B1ob)To compare this result with the power spectrum density (PSD), which is theFourier transform of the autocorrelation, it is defined as:IC2)(fi) = |l(2)(fi)| exp [jtKfi)] (BH)where:|l(2\fx ) | = | l(f1 ) | ^(-f^ | = | I(fj_ )|2(B12a)^(fl) = (fi) -4>(fi) = 0 (B12b)The PSD gives us magnitude information regarding the object but phase is lost.-132-Appendix CBispectrum Reconstruction Algorithm FlowchartThis top level block diagram gives the logic flow of how thealgorithm was implemented to perform shift-invariant, speckle-degraded imagereconstruction. 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