Eigendecomposition of Correlated Images Characterized by Three Parameters

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1 Eigedecompositio of Correlated Images Characterized by Three Parameters Kishor Saitwal ad Athoy A. Maciejewski Dept. of Electrical ad Computer Eg. Colorado State Uiversity Fort Collis, CO , USA {Kishor.Saitwal, Rodey G. Roberts Dept. of Electrical ad Computer Eg. Florida A & M - Florida State Uiversity Tallahassee, FL , USA rroberts@eg.fsu.edu Abstract Most eigedecompositio algorithms operate o correlated images that are characterized by oly oe parameter. Hece they lack the required specificatios of fully geeral 3D image data sets, i which the images eed to be characterized by three parameters. I this paper, a extesio of oe of the fastest kow eigedecompositio algorithms is successfully implemeted to improve the computatioal efficiecy of computig the eigedecompositio of such 3D image sets. This algorithm ca be used i patter recogitio applicatios such as fully geeral 3D pose estimatio of objects. Itroductio Eigedecompositio-based techiques play a importat role i umerous image processig ad computer visio applicatios. The advatage of these techiques is that they are purely appearace based ad require few olie computatios. Sigular value decompositio (SVD is oe such eigedecompositio techique that has bee used extesively i a variety of applicatios icludig patter recogitio [ 3]. All of these applicatios take advatage of the fact that a set of highly correlated images ca be approximately represeted by a small set of eigeimages. However, the offlie calculatio required to determie both the appropriate umber of eigeimages as well as the eigeim- This work was supported by the Natioal Imagery ad Mappig Agecy uder cotract o. NMA ad through collaborative participatio i the Robotics Cosortium sposored by the U. S. Army Research Laboratory uder the Collaborative Techology Alliace Program, Cooperative Agreemet DAAD The U. S. Govermet is authorized to reproduce ad distribute reprits for Govermet purposes otwithstadig ay copyright otatio thereo. ages themselves ca be prohibitively expesive. I particular, commo SVD algorithms to compute the complete SVD of a geeral m matrix require o the order of m flops. The computatioal complexity of the SVD has bee the subject of much research [4 0] with oe of the fastest kow algorithms described i [0]. Most previous eigeimage calculatios cosidered images that varied i oly oe dimesio, typically time. These image sets will be referred to as D image sets. However, certai patter recogitio applicatios require that differet views of a object take from differet 3D spatial camera locatios be cosidered i the traiig data set. Because they are characterized by three parameters, such image data sets will be referred to as 3D image sets. The goal of this paper is to exted Chag s algorithm [0] to effectively compute the SVD of 3D image sets. This paper is orgaized as follows. Sectio gives the fudametals of applyig the SVD to D image sets ad a brief overview of Chag s SVD algorithm. Sectio 3 explais the geeratio of fully geeral 3D image sets, while Sectio 4 explais how Chag s algorithm ca be exteded to compute the SVD of such image sets. Supportig results are show i Sectio 5, while Sectio 6 cocludes this paper. Eigedecompositio of D image sets Cosider a image set cotaiig correlated images (itesity ormalized betwee 0 ad with m pixels each (m. The images are row-scaed to obtai the correspodig image vectors x i ad the m image data matrix X =[x,x,,x ] is geerated. The SVD of X is give by X = UΣV T ( /06/$0.00/ 006 IEEE. 03

2 where U R m m ad V R are orthogoal, ad Σ=[Σ d 0] T R m where Σ d = diag(σ,,σ with σ σ σ 0 ad 0 is a by m zero matrix. The colums of U, deoted u i, i =,,m, are referred to as the eigeimages of X; the colums of V, deoted v i, i =,,, are referred to as the right sigular vectors of X, while the diagoal etries σ i of Σ give the correspodig sigular values of X. I practice, true eigeimages u i are ot computed exactly, ad istead estimates e,,e k that form a k- dimesioal basis are used. To quatify the accuracy of these estimates, the measure eergy recovery ratio, deoted ρ, is used here, which is give by k i= ρ(x,e,,e k = et i X X ( F where F deotes the Frobeious orm. Note that the true eigeimages give the optimal eergy recovery ratio, i.e., e i = u i with k = gives ρ =. I geeral, the itesity values of pixels i the correlated images i X vary slowly over time. Chag et al. [0] used this property to show that the right sigular vectors v i of X are spaed by a few low-frequecy siusoids ad the domiat frequecies i their power spectra icreased approximately liearly with their idex. Cosequetly, projectio of the row space of X (i.e., variatio i each pixel to a smaller subspace spaed by a few low-frequecy siusoids ca be used to improve the computatioal efficiecy of the SVD of X. I particular, cosider a sigle siusoid with a frequecy of α, deoted f α, elemets of which are give by f α (x = e jπαx/ (3 with 0 x. The the orthoormal basis that cosists of siusoids with icreasig frequecies ca be foud usig the basis of the D discrete Fourier trasform (DFT ad the correspodig real Fourier matrix H give by H = [ h 0 h h h 3 h 4 ] = [ ] f 0 Rf If Rf If c 0 s 0 c 0 s 0 c s c s = c s c 4 s c s c s (4 where c k =cos( πk ad s k =si( πk, while R ad I deote the real ad the imagiary part, respectively. Chag et al. showed that if p is such that the power spectra of the first k v i s of X are restricted to the bad [0, πp/], the the first k eigeimages e i of XH p are a good approximatio to those of X, where H p deotes the matrix cotaiig the first p colums of H. Now cosider the frequecy domai represetatio of a sigle row of X (deoted g, which ca be computed usig the D DFT as follows: G(α = = x=0 g(xe jπαx/ g(xf α (x. (5 x=0 The (3, (4, ad (5 clearly idicate that the multiplicatio XH ca be alterately performed usig fast Fourier trasform (FFT techiques for better computatioal efficiecy [0]. 3 Geeratig 3D Image Sets Fig. illustrates the setup for geeratig fully geeral 3D image sets, i which the correlated images are characterized by three parameters istead of oe. I this setup, camera locatios are defied i a spherical patch above the object with two cosecutive camera locatios separated by a equiagular distace i that patch. The rage of these camera locatios is characterized by two parameters, i.e., α l ad β m, while the third parameter γ characterizes image plae rotatio to capture differet views of the object i equal icremets. I practice, the required images ca be captured usig a video camera attached to a robot ed-effector. The robot movemet ca be cotrolled to positio the camera i oe of the specified locatios i the spherical patch ad the the robot ed-effector ca be rotated to rotate the image plae of the camera for capturig differet orietatios of the object from the same locatio. The image data matrix for the resultig 3D image sets is give by X 3 =[x,,x L,x, x L,,x LM, x,,x LM,,x ] (6 where a image vector x lm correspods to the rowscaed image of a object take from camera locatio (l, m at image plae rotatio, where l L, m M, ad N. 04

3 derig of x lm vectors i X 3 ito their respective colum vectors (deoted f αβγ so that the correspodig (L M N (L M N 3D Fourier matrix is give by γ αl βm F 3 =[f 000 f αβγ f (L (M (N ]. (9 Figure. This figure shows the simulatio setup for geeratig 3D image sets, i which the images are characterized by three parameters, i.e., α l, β m, ad γ. The crosses (x deote the simulated camera locatios that are placed i the spherical patch above the object. The rage of the parameters α l ad β m is from 45 o to 45 o with respect to adir, whereas γ rages from 0 to 360 degrees. 4 Extesio of Chag s SVD Algorithm to 3D Image Sets Cosider a three-dimesioal sigal g(x, y, z cotaiig L, M, ad N samples i x, y, ad z dimesios, respectively. The correspodig frequecy represetatio usig the 3D DFT ca be give by G(l, m, = L M N x=0 y=0 z=0 g(x, y, zωl lx ω my M ωz N (7 where ω L = e jπ/l, ω M = e jπ/m, ad ω N = e jπ/n. Thus, similar to D image sets, a orthoormal basis for the image data matrix X 3 ca be geerated usig the basis for the 3D DFT. I particular, the followig matrix represets oe 3D frequecy: F αβγ (x, y, z = ω αx L ω βy M ωγz N (8 where α, β, ad γ deote the desired frequecy compoets i three dimesios with 0 x L, 0 y M, ad 0 z N. All F αβγ matrices ca be lexicographically ordered (similar to the or- Note that the colums of F 3 give the 3D DFT basis for complex matrices. However, for real images i 3D image sets, the matrix X 3 i (6 will cotai all real values ad hece, similar to the basis give by colums of H i (4 for D image sets, X 3 will have a real basis. To fid this real basis, Euler s formula (e jx =cosx jsi x ca be used to rewrite (8 as follows: F αβγ (x, y, z = [(c αx c βy c γz c αx s βy s γz s αx c βy s γz s αx s βy c γz j(c αx c βy s γz c αx s βy c γz s αx c βy c γz + s αx s βy s γz ] (0 where c αx = cos( παx L, c βy = cos( πβy M, ad c γz =cos( πγz N, while s αx, s βy, ad s γz are the correspodig sie compoets. Let r deote the umber of o-zero α, β, ad γ frequecies. The there will be r differet sie-cosie combiatios for F αβγ. If these sie-cosie combiatios are lexicographically ordered ad are scaled by r to give orthoormal colums of H αβγ, the the real 3D Fourier matrix of size (L M N (L M N ca be give by H 3 = [ f 000 H αβγ ] ( where the first colum, f 000,ofH 3 refers to the DC compoet correspodig to r =0. Note that if ay of the three dimesios, for e.g., L, is eve, the oly the cosie (real compoet of the correspodig maximum real frequecy, i.e., L, is cosidered, otherwise both cosie (real ad sie (imagiary compoets of L are cosidered while geeratig the correspodig orthoormal colums i H 3. The resultig matrix H 3, which is geerated for a give X 3, ca be used to exted Chag s algorithm to compute the approximate SVD of X 3. I particular, the row space of X 3 ca be projected to the first few colums of H 3 ad the SVD of X 3 H 3(p ca be used to approximate the SVD of X, where H 3(p deotes the matrix cotaiig the first p colums of H 3. 05

4 ,,,M, Apprxomatio of left sigular vectors (first image set L,,,,N/ L,N,,M,N/ Eergy recovery ratio (ρ ρ(x,u,...,u p ρ(x,e,...,e k ρ(x T,h,...,h p L,,N/ L,M,N/ Dimesio of subspace Figure. This figure shows a few example sythetic images of a PUMA 560 Robot geerated usig ray-tracig. The first two letters i the headig idicate the simulated camera locatio coordiates, while the third letter idicates the image plae rotatio idex. Figure 3. This figure shows the typical relatioship betwee the true left sigular vectors, the computed estimates (as a fuctio of k, k p, for several fixed values of p, ad pure siusoids. The plots show here are for a image data matrix geerated for the robot i Fig. with L = M = N =0. 5 Experimetal results The proposed extesio of Chag s algorithm to efficietly compute the SVD of fully geeral 3D image sets was aalyzed usig a variety of objects. Two raytraced image sets of a PUMA 560 robot are used as illustrative examples here. The first image set had the same umber of images i all three dimesios with L = M = N =0, while the secod image set used L = M =0, N =36. Recall that the simulatio setup used i this study cosiders a rage of 90 degrees for both α l ad β m (refer to Fig., while the γ parameter is allowed to spa a full 360 degrees. Therefore, the agular separatio betwee cosecutive images i the first image set is differet, while it is kept the same i the secod image set. Fig. illustrates a few of the images (with = 4096 pixels each showig differet views of the robot take from simulated camera locatios that are at the four extreme corers of the spherical patch above the robot. I particular, images i the last two rows i Fig. are obtaied after rotatig the correspodig images i the first two rows through 80 degrees. Usig the two image sets, the correspodig image data matrices X 3 ad the real Fourier matrices H 3 were geerated. I particular, X 3 ad H 3 for the first image set were of size ad , respectively, while for the secod image set, they were of size ad , respectively. For both image sets, the colums of H 3 were placed i descedig order of their ability to recover eergy i X 3 usig ρ(x T,h,h, i (. It was observed that these ordered harmoics ca be effectively used to improve the computatioal efficiecy of the SVD of X 3. This is illustrated i Fig. 3 ad Fig. 4, where ρ is agai used to measure the quality of the estimates of the u i. The solid lie shows ρ(x,u,,u p as a fuctio of p, while the dotted lies show ρ(x,e,,e k for k =,,,p ad p =50, 00, 50, 00, 50. It is evidet that the e i give good estimates of the u i eve at small values of p idicatig a effective extesio of Chag s algorithm. Note that the differece betwee the results for both image sets is miimal cosiderig the differece betwee the umber of images used i the correspodig X 3 matrices. 06

5 Eergy recovery ratio (ρ Apprxomatio of left sigular vectors (secod image set ρ(x,u,...,u p ρ(x,e,...,e k ρ(x T,h,...,h p Chag s algorithm [0]. However, because the colums of H 3 are uordered ad because the correspodig ρ(x T 3,h,,h p gives a very loose lower boud for ρ(x 3,e,,e k, there is a eed to resolve the two aforemetioed issues. 7 Ackowledgmets The authors would like to ackowledge Bryce Eldridge for his help with the geeratio of ray-traced images for this paper Dimesio of subspace Figure 4. This figure shows the same plots as i Fig. 3 for a image data matrix geerated for the robot i Fig. with L = M =0ad N =36. 6 Coclusio ad Future Work This paper has illustrated a extesio of Chag s algorithm to compute a estimate of the SVD of 3D image sets, i which the images are characterized by three parameters istead of oe. The empirical evidece shows that this extesio ca be effectively used to improve the offlie computatioal efficiecy of calculatig the SVD of 3D image sets commoly used i 3D pose estimatio. The work preseted here lays a foudatio for a computatioally efficiet SVD algorithm for 3D image sets. However, there are still two issues that eed to be resolved. The first issue is related to the optimum orderig of the frequecies of 3D image sets based o their eergy recovery ability, while the secod issue ivolves the choice of the umber of frequecies, p that should be cosidered before computig the SVD of X 3 H 3(p. For D image sets, it was observed that the colums of H i (4 were already ordered i terms of their ability to recover eergy i X ad ρ(x T,h,,h p provided a very coservative lower boud for ρ(x,e,,e k ad hece the two issues were automatically resolved i Refereces [] S. K. Nayar, S. A. Nee, ad H. Murase. Subspace method for robot visio. IEEE Tras. Robot. Automat., (5: , Oct [] H. Murase ad S. K. Nayar. Detectio of 3D objects i cluttered scees usig hierarchical eigespace. Patter Recogit. Lett., 8(4: , April 997. [3] M. H. Yag, D. J. Kriegma, ad N. Ahuja. Detectig faces i images: A survey. IEEE Tras. PAMI, 4(:34 58, Ja. 00. [4] S. Shlie. A method for computig the partial sigular value decompositio. IEEE Tras. PAMI, 4(6:67 676, Nov. 98. [5] H. Murakami ad V. Kumar. Efficiet calculatio of primary images from a set of images. IEEE Tras. PAMI, 4(5:5 55, Sept. 98. [6] R. Haimi-Cohe ad A. Cohe. Gradiet-type algorithms for partial sigular value decompositio. IEEE Tras. PAMI, PAMI-9(:37 4, Ja [7] C. R. Vogel ad J. G. Wade. Iterative SVD-based methods for ill-posed problems. SIAM J. Sci. Comput., 5(3: , May 994. [8] H. Murase ad M. Lidebaum. Partial eigevalue decompositio of large images usig the spatial temporal adaptive method. IEEE Tras. Image Processig, 4(5:60 69, May 995. [9] S. Chadrasekara, B. Majuath, Y. Wag, J. Wikeler, ad H. Zhag. A eigespace update algorithm for image aalysis. CVGIP: Graphic Models ad Image Processig, 59(5:3 33, Sept [0] C. Y. Chag, A. A. Maciejewski, ad V. Balakrisha. Fast eigespace decompositio of correlated images. IEEE Tras. Image Processig, 9(: , Nov