American Journal of Computational Mathematics, 2012, 2, 287294 http://dx.doi.org/10.4236/ajcm.2012.24039 Published Online December 2012 (http://www.SciRP.org/journal/ajcm) Face Recognition from Incomplete Measurements via ℓ1Optimization Miguel Argaez1*, Reinaldo Sanchez2, Carlos Ramirez2 1Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, USA 2Program in Computational Science, The University of Texas at El Paso, El Paso, USA Email: *margaez@utep.edu, rsanchezarias@miners.utep.edu, caramirezvillamarin@miners.utep.edu Received May 15, 2012; revised August 5, 2012; accepted September 12, 2012 ABSTRACT In this work, we consider a homotopic principle for solving largescale and dense underdetermined problems and its applications in image processing and classification. We solve the face recognition problem where the input image contains corrupted and/or lost pixels. The approach involves two steps: first, the incomplete or corrupted image is sub ject to an inpainting process, and secondly, the restored image is used to carry out the classification or recognition task. Addressing these two steps involves solving large scale minimization problems. To that end, we propose to solve a sequence of linear equality constrained multiquadric problems that depends on a regularization parameter that con verges to zero. The procedure generates a central path that converges to a point on the solution set of the underde termined problem. In order to solve each subproblem, a conjugate gradient algorithm is formulated. When noise is pre sent in the model, inexact directions are taken so that an approximate solution is computed faster. This prevents the ill conditioning produced when the conjugate gradient is required to iterate until a zero residual is attained. 1 1 1 Keywords: Sparse Representation; Minimization; Face Recognition; Sparse Recovery; Interior Point Methods; Sparse Regularization 1 1. Introduction Over the last years, new developments in the area of computational harmonic analysis have shown that a wide class of signals can be well represented by linear combi nations of only few elements of an appropriate basis. The benefits and applications of this new advance abound and are of extensive research in the present. The new sampling theory of compressed sensing has unified several insights about wavelets and sparse repre sentation, benefiting several disciplines in sciences in cluding image processing. Practical compressed sensing problems involve solving an optimization problem of the form 0 min subject to , x Ax b (1) for decoding a sparse signal that has been sig nificantly subsampled by a sampling matrix n x mn with Here .mn0 counts the number of nonzero entries of the vector . Solving (1) is equivalent to finding the sparsest vector x such that . xb Nevertheless, finding such a vec tor x is by nature a combinatorial and generally NPhard problem [1]. Efficient numerical algorithms to recover signals under this framework have been developed [24], and extensions of this theory have been explored for solving general problems in different areas including statistics, signal processing, geophysics, and others. Sig nificant progress toward understanding this problem has been made in recent years [57], and its study has be come stateoftheart interdisciplinary research. One of the most important characteristics of problem (1) is that under some mild conditions, the input vector can be recovered by solving an norm underde termined problem 1 1 min subject to . x Ax b (2) This decoding model in compressed sensing is known as the basispursuit problem, first investigated by Chen, Donoho and Saunders [8] and theoretically studied by Donoho and Huo [9]. Candès and Tao [5] proved for mally this equivalence provided that is sufficiently sparse, and that A possesses certain properties. The compressed sensing theory is extended to solve a more general problem that considers noise on the meas urements, and almost sparsity for the input vector to be recovered. That is, *Corresponding author. C opyright © 2012 SciRes. AJCM
M. ARGAEZ ET AL. 288 1 min subject to , xxAxb (3) where the energy of the noise vector is upper bounded by . Problem (3) can also be reformulated as an uncon strained regularized linear leastsquares problem. One strategy consists of replacing the Tikhonov regularization with one that uses the 1norm [2,4]. Another equiva lent formulation is aimed at minimizing the 1 norm function regularized by a linear leastsquares problem [3], known as unconstrained basis pursuit denoising. All these approaches have proved to be successful for solv ing compressed sensing problems. We reformulate problem (3) by 1 min subject to , x Ax b (4) where is bounded by m . In this work we propose a new strategy to obtain an optimal solution to problem (4), and present an applica tion in robust face recognition to demonstrate the effect tiveness of our algorithm. The idea consists of relaxing the nondifferentiable objective function by a sequence of multiquadric, continuously differentiable, strictly convex functions that depend on a positive regularization pa rameter . More precisely, we solve 2 1 min subject to . n xi Ax b (5) The main accomplishment of this idea is that it leads to the generation of a path that converges to an optimal solution of problem (4). This leads us to a pathfollowing algorithm similar to the ones used in primaldual inte riorpoint methods. The pathfollowing strategy that we are proposing uses inexact fixedpoint directions to ob tain approximate solutions to problems of the form (5). Such inexact directions are computed via a conjugate gradient algorithm. In order to prevent the procedure from becoming costly, a proximity measure to the central path is introduced for each regularization parameter. The regularization parameter is defined in a dynamic manner that converges to zero as in interiorpoint methods. 2. Problem Formulation We study the underdetermined problem (4), where 1 m , n x ,b1 1, n i i x and is a fullrank matrix with . The Lagrangian function associated with problem (4) is mn T 1 ,,lxyxAxb y where is the Lagrange multiplier for the equal ity constraint. The optimality conditions for (4) are given by m y T T * T 1 if 0, 1 if 0, ,: and 1,1 if 0, i i i nm i i i Ay x Ay x Xxy Ax b Ay x . Notice that the main role in the characterization of the optimal conditions for problem (4) is not played by the Lagrange multiplier , but by . T y Using this fact, the complementarity conditions associated with the pri mal variables i are determined by T T 0 with 1 if 0 0 with 1 i . f 0. ii ii i ii ii i xzzA yx xzzA yx Therefore a necessary condition for a feasible point ,nm xy to be an optimal solution of (4) is T 1 0,0and 0. n ii i xzxxzA y 3. A Regularization Path The nondifferentiability of (4) is overcome by regulariz ing the 1norm with a sum of continuously differenti able functions in the following way: for 0 suffi ciently small, 1 1 n i i gx where is the scalar function defined by 2,. ii i gx xx 3.1. Optimality Conditions We propose to obtain an optimal solution to problem (4) by solving a sequence of subproblems of the form (5) as 0. Since each subproblem (5) is strictly convex, then the optimal solution is obtained by solving the associated KKT conditions. The Lagrangian function associated with (5) is T 2 1 ,, n i i lxyxAxb y (6) where m y is the Lagrange multiplier for the equal ity constraint. Therefore, the KKT conditions are given by the following square system of nonlinear equations: 12 T0 ,, 0 Dx xAy Fxy Ax b (7) where 2 diag for 1,2,, and 0. i Dx xin 3.2. A Fixed Point Problem for the KKT Conditions We propose to solve (7) using a fixedpoint method. To Copyright © 2012 SciRes. AJCM
M. ARGAEZ ET AL. 289 that effect, we rewrite these nonlinear equations as the augmented system 12 T0, 0 x Dx A yb A (8) where , n x, m y 2 diag , i Dx x 1, 2,,,in and 0. The matrix associated with (8) is nonsingular since has full rank and 12 Dx is positive definite. In this manner, the nonlinear Equation (7) is posed as a fixed point problem. In order to solve (8) for a fixed 0, we proceed by taking an initial point 0 and iteratively compute , kkk y for until two con secutive iterations are less than some stopping criteria. 0, 1,k 3.3. Inexact Directions for the Augmented System For a current point , the following system 12 T0 _, 0 x Dx A yb A (9) is reduced to a weighted normal equation. The first block of equations gives 12 T 0,xDx Ay and since , xb we obtain the weighted normal equation 12 T .ADxA yb (10) With this reduction, we move from an indefinite sys tem of order to a positive definite system of or der Moreover, the conjugate gradient algorithm ap plied to (10) converges in at most iterations in exact arithmetic. The solution nm . m m of (9) is computed directly by 12 T _ Dx Ay once is obtained. y Taking into account that the values of T y charac terize the optimality set *, we formulate a conjugate gradient algorithm that finds an approximation of T y rather than y, see Figure 1. At each iteration, the CG algorithm satisfies the first block of equations 12 T 0,DxxAy therefore controlling the stopping criteria for solving the aug mented system (9) is equivalent to controlling the stop ping criteria for solving the linear system 0. Ax b we define the vector 0, nm r Based on this, as the or for the augmented system, where .rAxb residual vect Note that 0r implies .Ax b Now, since is bounded by , then the conjugate hmgradient algorit stops when .Ax b This implies the stopping criterio zero, overcoming the illconditioning of the Figure 1. Conjugate gradient algorithm. 4. Path l path associated with problem (4) Following Method 4.1. Central Path We define the centra by 12T 0DxxAy C ,: . 0 nm x yAx b (11) The set consists of all the points that are sol of the sublem (5) for C bpro utions 0. ) as t This set defines a smooth cur called the central path that converges to an optimal solution of problem (4he regularization pa rameter ve tends to zero [10]. 4.2. Promity Measures xi ollows in the direction sequencef regularization Our pathfollowing method f generated by a decreasing C o parameters Since moving on C obtain an opti mal solution for (4) could be computationally expensive, we restrict thiterates to some neigorhoods of the cen tral path given by to e hb 1 :. 1 k jj nm jk j (12) 4.3. Updating the Perturbation Parameter Since T0xz is a necessary condition for obtain optimal solution of problem (4), then following th ing an e same primalidea ofdual methods we define the regularization parameter by n does not need to be weighted normal equation close to the solution. T , z (13) n where T ,, xz Ay 1 is the centering pa Copyright © 2012 SciRes. AJCM
M. ARGAEZ ET AL. 290 0,1 , rameter in andis the able n size of the primal vari . Tguarantee the decreo ase of the parameter , we update it by rev where mi n , 0, 1 and rev des the pvious value for the regularize tion parameter. Now, we present a globalorithm for so the convex and nondifferentiable problem (4). The methodology consists in following the central path C to note re ization alglving obtain an optimal solution. To prevent the algorithm from becoming computationally expensive a series of neighborhoods, , k around the central path are de fined to be used as proximity measures of the path for a decreasing regularization parameters . To obtain an approximate solution on the path for a given 0, an inexact fixedpoint procedure is applied to (8) until a point ,.xy If an optimal solution to (4) is not found, we decrease , specify a new neighborhood and repeat the fixedpoint procedure. An optimal solutiofound as , n for (4) is approaches to zero. For the primal variables we define their corresponding complementarity variables z such that 0 ii xz and 0 i z for 1, ,in at the optimal solution of prob lem (4). This allows us to define the regularization pa rameter in the same manner as in interiorpoint methods. 4.4. Path Following Algorithm In this sec ry” ( tion we PFSR prese t our “Path ) algo ithm. The pseu 2. n r Following Recove docod Sign e for al m of the algorithm is presented in Figure The PFSR algorithm generates two sequences of iter ates. The first sequence (inner loop) generates a series of iterates for obtaining an approximate solution of sub problem (5) for a fixed regularization parameter 0. The second sequence (outer loop) generates a series of approximate solutions for the subproblem (5) that con verges to an optimal solution of problem (4) for a se quence of decreasing regularization parameters 0. 5. Sparse Signal Recovery In this section, we present a set of experimentltal res AB im us plethat illustrate the performance of th m. e MATL mentation for the proposed algorith In the implementation of Algorithm 1 the initial points and the parameters are chosen as follows. In Step 1a, the initial points for T Ay and are the ndimensional zero vector. In Step 2, we fix the maximum number of CG iterations by cg_maxiter = 10. In the implementation of Algorithm 2 the parameters are chosen as follows. The initial regularization parame ter is given by T 1, n ii 1i Ab Figure 2. PFSR algorithm. where is the minimum energy reconstruction. Our numperimentation suggests erical ex0.008 0 as the (5) to m as a In Step 3, we axi umber o solve. good choice. num set maxiter = 1 f subproblems of the formm The new regularization parameter is updated in Step 10 by min ,gap with 0.9. 5.1. Sparse Signal Recovery Example This experimentation has the objeive of investigating the capg sparse ct ability of recoverin signals by our PFSR algorithm. The goal in this test is to reconstruct a re m < n. lengthn sparse signal from m observations, whe We start with a classical example also considered in [2,4,11]. The problem consists of recovering a signal 4096 x with 160 spikes with amplitude ±1, from m = 1024 noisy measurements. We use a partial DCT matrix A whose m rows are chosen randomly from the nn discrete cosine transform, without having access to it in explicit form, but using A as a linear operator on . n The same for the matrix T. Partial DCT matrices are fast transforms for which matrixvector multiplications cost just logOn n flops, and storage is not required. This case is common for compressed sensing [3,4]. this problem, we have , m Ax where the noise vector In b is set according to a Gaussian distribu tion with mean 0 an standard deviation 0.01. The origin nal and reconstructed signal are shown in Figure 3. Moreover, the algorithm was successfully run one hun dred times wit an average CPU time of 0.2895 seconds, and 0.0491 average relative 2norm error defined by d h 2 2 2norm error , xx x being the true solution and x the solution reached by m. the PFSR algorith Copyright © 2012 SciRes. AJCM
M. ARGAEZ ET AL. 291 Original signal x 0 1 0.5 0 –0.5 –1 1 0.5 0 –0.5 –1 0 1000 2000 3000 4000 PFSR Solution 0 1000 2000 3000 4000 Figure 3. Signal reconstruction. 5.2. The Effect of Noise For the same test problem described above, we run the algorithm considering the noiseless case, and zero mean Gaussian noarying from 0.01 to 0.04. The stopping criterion for the conjugate ed by the noise level as ex ro blems for instance, an input sample can usually be ex at is, be expressed as a linear ise with standard deviations v gradient residual is determin plained in Section 3.3. In all cases we successfully re cover the original signal after a process of thresholding. Table 1 reports the results for this experimentation showing that the algorithm is also effective for solving noiseless sparse recovery problems. Moreover, success ful recovery was obtained with noise level up to 0.04. 6. Sparse Representation in Classification and Image Processing There exist a number of areas where the sparse represent tation model (4) emerges naturally. In classification p plained from few other previously trained samples. Th an incoming sample b can combination of only few columns ,1,,, i in where the matrix , mn A ,mn is a matrix whose columns are previously trained samples. On the other hand, natural images can be efficiently encoded in an appropriate basis that exploits the hight correlation present between pixels. Among many emerg ing class of transformations, the Di crete Cosine Trans form (DCT) and Discrete Wavelet Transform (DWT) are a s 0.01 0.02 0.03 0.04 standard basis where natural images can be sparsely represented. This property has been extensively studied in recent years, leading to a construction of new models for images where the sparsity is incorporated in the model as a regularizer. 6.1. Classification In pattern recognition and machine learning, a classifica Table 1. Performance considering different setups for noise. Noise 0.00 2norm error0.016 0.063 0.121 0.208 0.294 Recovery %100 %100 %100 %100 %100 Time [s] 0.40560.3432 0.3744 0.40560.4836 tion m consists ofng rior  innpa tofal rie ma andaf labels/classes proble findian algothm fassign g a given iut dato one severcategos. For 1 w , n w, ,W lly, given input taset, a set o , , n t 1,tT trg and a ainindataset ,: 1,ti n ii i of the sample , i u a classifier is a mapping f from W to T, assigning the correct label tT tout w, that is, Du such that is the label/class a given inp t ,. Dw t Consider a tra inin ,: 1,,, ii ut in g data set ,1,2,, d ii ut N N , with n being the number of samples and the number of classes. The vector , d i u represents the ith sample, and i t is the corre sponding label. The sparse representation p ple , d b find the sparsest vector T 12, ,, n roblem is formulated as follows: For a testing sam xx x such that 112 2. nn bxu xuxu (14) sampl erefore inducing arse representation. rrange the given samples from the same ith class as the colums of a submatrix We show that indeed a valid test sample can be repre sented using only the traininges from its same class, th natural spa Let us reatraining i n n ,1 ,2, ,,, . i i dn iii in uu u In other words, we group all of those samples with the same label into a matrix . i A Any test sample b from the same class will be rep resented as a linear combination of the training samples associated with class i: ,1 ,1,2,2iii i bxuxu,, , inin xu ii (15) for some values of ,,1,,. ij i jn Now, making use of the whole training dataset, we define a dn matrix A by concatenating all of the n training samples of the different N classes, that is 12 ,,, . N AA entation of the test sample A b t Then, the linear repres : hat belongs to class i is written by ,bAx () where T ,1, 2 0, ,0,,, ,,0 ii xxx Thus, the test sample b is expressed by a sparse linear combi nation of the training 16 samples, more specifically, as a linear combination of only th longing to the same class. This motivates us to formulate the following problem: ,, ,0. i n in x ose training samples be Copyright © 2012 SciRes. AJCM
M. ARGAEZ ET AL. 292 1 minsubject to , Ax b (17) follows the same structure as (2). . 6.1.1. Discrimi n a nt Functions and Classifier s computed, we which One of the advantages of this formulation is that the lack of robustness with respect to outliers can be over come. Furthermore, we do not need to care for model selection as in support vector machine approaches for classification problems [13] Once the sparse representation vector x i identify the class to which the testing sample b belongs. The approach consists in associating the nonzero entries of x with the columns of A corresponding to those train ing samples having the same class of the testing sample b. The solution vector x is decomposed as the sum of ddimensional vectors ˆ, k where ˆk is obtained by h class k keeping only those entries in x associated wit and assigning zeros to all the other entries. Then, we de fine the N discriminant functions 2 ˆ,1,,. kk bbAxkN (18) Thus, k represents the approximation error when b is assigned to category k. Finally, we assign b to the class with the smallest approxition error. That is, ma ˆarg min,1,,. k tgbkN (19) In this manner, we identify the class of the test sample b based on how effectively the coefficients as with the training samples of each class recreate b. red by its error rate on the entire population. Cross Valida statistical method for evaluating performance i th eing validated. sociated 6.1.2. Cross Valid ation A classifier performance is commonly measu tion is a n which e data is divided in two sets: one used for the training stage, and the second one used for testing (validation). Both training and testing sets should crossover in con secutive rounds in such a way that each sample in the data set has a chance of b In the case of Kfold cross validation, a Kfold parti tion of the dataset is created by splitting the data into K equally (nearly equal) sized subsets (folds), and then for each of the K experiments, K − 1 folds are used for training and the remaining one for testing. A common choice for KFold cross validation is K = 10. The work in [12], compares several approaches for estimating accu racy, and recommends stratified 10fold crossvalidation as the best model selection method because it provides less biased estimation of the actual accuracy. In our nu merical experimentation we follow this validation ap proach to test the performance of the classification algo rithm. 6.2. Image Processing In many practical image processing applications, we are interested in recover a target image , n u that has been subject to a degradation processes modeled as ,bHu (20) where m b is the degraded image, is additive noise of certain distribution, and H is a linear operator that acts on the target image u. For instance, H can be a odels atmospheric turbulence, n process due to movement. whic the image. Consequently, addition in orde convolution operator that m or defects on the acquisitio From a point of view of parameter estimation, problem (20) corresponds to an inverse problemh is very difficult to solve due to the illconditioning nature of H and the large number of degrees of freedom present in al information is re quired r to obtain a meaningful solution to (20). This is accomplished with the 1 norm regularization, and our problem becomes 1 minsubject to , x Hx b (21) where is an sparsifying matrix for u, and x is a sparse vector of coefficients. That is, .ux When H models a process of missing information, it receives the name of mask, and is constructed by remov ing from the nn identity matrix, the n − m rows asso ciated with the missing data. In this setting, we consider that n − m pixels has been lost, leading to an incomplete image . m b 7. Robust Face Recognition rrupd. man faces from the ambridge University In this section, we demonstrate the effectiveness of the proposedlgorithm by showing a rea al application in ro bust face recognition. The challenge consists in auto matically identify an input human face image within an existing database of several individuals [14]. Further more, we assume that the input image has been subject to a data loss process, or that several of its pixels has been severely cote We consider the database of hu AT&T laboratories in Cambridge (C Computer Laboratory)1 which consists of 400 images of 40 individuals each of which with ten different images. For some individuals, the images were taken at different times, lighting and facial expression. The size of each image is 112 92 pixels with 256 gray levels in a pgm format. The proposed approach involves two steps. First, the test image containing several corrupted or lost pixels is reconstructed via inpainting. To that end, we solve prob lem (21) where b is the corrupted test image, is the wavelet Daubechies level 7 matrix, and H is the matrix 1http://www.cl.cam.ac.uk/research/dtg/attarchive/facedatabase.html Copyright © 2012 SciRes. AJCM
M. ARGAEZ ET AL. Copyright © 2012 SciRes. AJCM 293 f the tra e. Both processes are carried out by solving an ro ach of which so 2)s and 3) Corruptive vertical lin ) re sp e tation that shows that our efficiently when recov ering large signals with noisy data. The sparse represent tation approach is applied to a face recognition problem where the data is incomplete and/or corrupted. An in painting procedure is carried out for reconstructing the input image, and then a classification process is per formed in order to identify the correct individual. Both processes are accomplished with the proposed algorithm for 1 minimization problems, achieving a high recog nition rate. (mask) associated with the missing pixels. Secondly, we exhibit the reconstructed test image to a training dataset in order to find the sparsest linear com bination oining samples that better represents the test imag 1 minimization problem. Figure 4 depicts the recognition process where the test image is corrupted by two different kind of masks. In our numerical experiments, we perform a css vali dation scheme in order to assess a recognition rate for a total of 10fold cross validation runs, e lves 40 test problems. None of these experiments in cluded the test image in the training dataset. We consider three different types of corruptive masks: 1) Random missing pixels uniformly distributed over the test image, Corruptive horizontal line 9. Acknowledgements The authors want to thank the financial support provided by the US Army Research Laboratory, through the Incomplete test image Inpainted test image PFSR recognition es. Despite Figure 4 shows two corrupt test images with a total of 34.12% (top) and 15.18% (bottom ectively, in our numerical experiments we consider a percentage of lost data ranging from 5 to 40%. The av erage recognition rate obtained after the cross validation experiments was 97.25%, and Table 2 reports each of these rates per fold. In Figure 5 one can notice that the solution vector x for problem (17) is sparse, and its nonzero components mark those training samples i u in the dictionary A that are from the same class of the test sample b, that is, those other pictures of the individual b in the training dataset. 8. Concluding Remarks In this work we present a novel methodology for solving largescale and dense underdetermined problems. Fo Figure 4. Classification scheme. 1 r solving the largescale and dense linear systems as sociated with the problem, a conjugate gradient algo rithm is formulated. The regularization parameter is im plemented in the same fashion as in interiorpoint meth ods by characterizing the complementarity variables as sociated with the primal variables of the problem. W Table 2. Recognition rate per fold. Fold 1Fold 2 Fold 3 Fold 4Fold 5 Recognition rate100%97.5% 95% 97.5%95% Fold 6Fold 7 Fold 8 Fold 9Fold 10 Recognition rate present a numerical experimen algorithm is capable to perform %95 %100 %95 %100 %97.5 x b 0.4 0.3 0.2 0.1 0 –0.1 A 0 50 100 150 200 250 300 350 400 Figure 5. Sparse linear combination.
M. ARGAEZ ET AL. 294 Army High Performance Computing Research Center, Cooperative Agreement W911NF0720027. The au thors also thank the Mathematical Science Department and Computational Science Department at UTEP. REFERENCES [1] B. K. Natarajan, “Sparse Approximate Solutions to Linear Systems,” SIAM Journal on Computing, Vol. 24, No. 2, 1995, pp. 227234. doi:10.1137/S0097539792240406 on Information Theory, Vol. 52, No. 4, 2006, pp. 1289 1306. doi:10.1109/TIT.2006.871582 [8] S. Chen, D. Donoho and M. Saunders, “Atomic Decom position by Basis Pursuit,” SIAM Review, Vol. 43, No. 1, 2001, pp. 129159. doi:10.1137/S003614450037906X [9] D. Donoho and X. Huo, “Uncertainty Principles and Ideal Atomic Decomposition,” IEEE Transactions on Informa tion Theory, Vol. 47, No. 7, 2001, pp. 28452862. doi:10.1109/18.959265 [10] M. Argáez, C. Ramirez and R. Sanchez, “An Algo rithm for Underdetermined Systems and Applications,” IEEE Conference Proceedings on North American Fuzzy Information Processing Society, El Paso, 1820 March 2011, pp. 16. doi:10.1109/NAFIPS.2011.5752016 [2] M. Figueiredo, R. Nowak and S. Wright, “Gradient Pro jection for Sparse Reconstruction: Application to Com pressed Sensing and Other Inverse Problems,” IEEE Journal of Selected Topics in Signal Processing, Vol. 1, No. 4, 2007, pp. 586597. ] E. Hale, W. Yin and Y. Zhang, “A Fixed [4] S. Kim, K. Koh, M. Lustig, S. Boyd and D. Gorinvesky “A Method forrized Least Squares Problems with rocessing and Sta [3 Point Continua tion Method for 1 Regularized Minimization with Ap plications to Compresses Sensing,” Technical Report TR0707, Department of Computational and Applied Mathematics, Rice University, Houston, 2007. , LargeScale 1 Regula Applications in Signal P tistics,” IEEE Journal of Selected Topics in Signal Proc essing, 2007. www.stanford.edu/˜boyd/l1_ls.html [5] E. Candès, J. Romberg and T. Tao, “Robust Uncertainty Principles: Exact Signal Reconstruction from Highly In complete Frequency Information,” IEEE Transactions on Information Theory, Vol. 52, No. 2, 2006, pp. 489509. doi:10.1109/TIT.2005.862083 [6] S. Chen, D. Donoho and M. Saunders, “Atomic Decom position by Basis Pursuit,” SIAM Review, Vol. 43, No. 1, 2001, pp. 129159. doi:10.1137/S003614450037906X [7] D. Donoho, “Compressed Sensing,” IEEE Transactions 1 . Nowak and M. Figueiredo, “Sparse Recon Separable Approximation,” IEEE Interna [11] S. Wright, R struction by tional Conference on Acoustics, Speech and Signal Proc essing, ICASSP 2008, Las Vagas, 31 March4 April 2008. doi:10.1109/ICASSP.2008.4518374 [12] R. Kohavi, “A Study of CrossValidation and Bootstrap for Accuracy Estimation and Model Selection,” Proceed ings of International Joint Conference on AI, Quebec, 2025 August 1995, pp. 11371145. [13] R. Sanchez, M. Argáez and P. Guillen. “Sparse Repre sentation via 1 Minimization for Underdetermined Sys tems in Classification of Tumors with Gene Expression Data. IEEE 33rd Annual International Conference Pro ceedings of the Engineering in Medicine and Biology So ciety, Boston, 30 August3 September 2011, pp. 3362 3366. [14] J. Wright, Y. Yang, A. Ganesh, S. Shankar and Y. Ma, “Robust Face Recognition via Sparse Representation,” IEEE Transactions on Pattern Analysis and Machine In telligence, Vol. 31, No. 2, 2009, pp. 210227. doi:10.1109/TPAMI.2008.79 Copyright © 2012 SciRes. AJCM
