Transformation-invariant Collaborative Sub-representation

Transcription

1 Transforation-invariant Collaborative Sub-representation Yeqing Li, Chen Chen, Jungzhou Huang Departent of Coputer Science and Engineering University of Texas at Arlington, Texas 769, USA. Eail: Abstract In this paper, we present an efficient and robust iage representation ethod that can handle isalignent, occlusion and big noises with lower coputational cost. It is otivated by the sub-selection technique, which uses partial observations to efficiently approxiate the original high diensional probles. While it is very efficient, their ethod can not handle any real probles in practical applications, such as isalignent, occlusion and big noises. To this end, we propose a robust sub-representation ethod, which can effectively handle these probles with an efficient schee. While its perforance guarantee was theoretically proved, nuerous experients on practical applications have further deonstrated that the proposed ethod can lead to significant perforance iproveent in ters of speed and accuracy. I. INTRODUCTION Iage representation is an iportant proble in coputer vision and pattern recognition and it has gained a lot of attentions in past decade. While huge interest has been seen in iage representation, to date the ost popular approaches are sparse representation and collaborative representation. In Wright et al. s pioneer work SRC [2] on sparse representation, they odel the recognition proble as finding sparse representation of the test iage based on linear cobination of training iages. Furtherore, outlier pixels are also assued to be sparse. Favourable result has been achieved on face recognition application with occlusion and corruption. Zhang et al. proposed collaborative representation (CR) [3], [4] which relax the sparsity constraint on the representation coefficients. CR inspires any following works, such as [5], DLRD SR [6]. CR uses the l 2 -nor regularization instead of l -nor in SRC. Therefore, it is uch faster. It can achieve desired results for the clean data without corruption. However, if the testing iages are sparsely corrupted, l -nor regularization has to be used for constraining sparse occlusion, which will lead to ipressively degradation of speed. The high coputation coplexity is a big obstacle for the being used for high-diensional iages. Moreover, in practice, due to the registration error of the object detector or the otion of the target object, the test iage is usually not well-aligned with the training iages. To achieve effective representation, the test iage has to be aligned with training iages first by using iterative transforation estiation ethods [5], [7], [8]. This process has higher coputation costs because the transforation estiation step is usually solved in high diensional pixel-space and not able to utilize diension reduction techniques. Though existing ethods have achieved great success in various situations such as illuation change, occlusion and isalignent, their coputational inefficiency liits the being used in practical application involving high-diensional iages. To this end, we propose a robust sub-representation ethod, which can not only efficiently represent the iage but also effectively handle the probles of isalignent, occlusion and big noises. Its perforance guarantee can be theoretically proved. While cobining it with existing collaborate representation ethod, a Transforation-invariant Collaborative Sub-Representation (TCSR) algorith is proposed in this paper. Nuerous experients on practical applications have been conducted to further deonstrate its superior perforance in ters of both coputational coplexity and accuracy. The contributions of this paper are: ) To handle big noises, sub-selection ethod is generalized to robust subrepresentation. We have theoretically proved its benefit over sub-selection ethod; 2) Sub-representation is further extended to handle isalignent, occlusion and corrupted pixels, etc.. This extension is done by cobing it with soe existing techniques like transforation estiation algorith and collaborative representation. The resulting ethod can also be regarded as a case study of sub-representation, which shows its great potential in cooperating with other ethods. 3) Extensive experients are conducted to validate the efficiency and effectiveness of the proposed ethods, which deonstrates that the proposed ethod can significantly accelerate the collaborative representation with iperceptible loss of accuracy. II. RELATED WORK A. Sub-selection and Handling Incoplete Data Sub-selection [9] is an efficient way to reduce the diension of data. It projects the data onto lower diensional subspaces using a randoly chosen subset of the data features. An exaple of sub-selection on pixel features of an iage is shown in Fig.. In theoretical analysis, one situation that is siilar to sub-selection is handling incoplete data. Balzano et. al. have recently proved theoretical guarantees of subspace detection using incoplete data []. The theory is sketched as follows for convenience. Let v Ω be the vector of diension Ω coprised of the eleents v i, i Ω, ordered lexigraphically, where Ω denotes the cardinality of Ω. The energy of v in the subspace S is P S v 2 2, where P S denotes the projection operator onto S. Let U be an n r atrix whose coluns span the r-diensional subspace S. In this case, P S = U(U T U) U T. With this representation in d, let U Ω denote the Ω r atrix, whose

2 Iage Vectorize v Sub-selection Fig. : Sub-selection on pixel features of an iage. rows are the Ω rows of U indexed by the set Ω, arranged in lexigraphic order. Suppose we only observe v on the set Ω. One approach for estiating its energy in S is to assess how well v Ω can be represented in ters of the rows of U Ω. Define the projection operator P SΩ := U Ω (UΩ T U Ω) UΩ T, where denotes the pseudo-inverse. It follows iediately that if v S, then v P S v 2 2 = and v Ω P SΩ v Ω 2 2 =. Let the entries of v be sapled uniforly with replaceent. Again let Ω refer to the set of indices for observations of entries in v, and denote Ω = k. The following theore has been proved in []: Theore II.. (Theore in []): Let δ > and k 8 2r 3rµ(S)log( δ ). Then with probability at least 4δ, where α = k n ( α α ) v P S v 2 2 v Ω P SΩ v Ω 2 2 8rµ(S) 3k log( 2r v Ω ( + α) k n v P Sv 2 2 () 2µ(y) 2 k log( δ ), β = 2µ(y)log( δ ), γ = δ ), and α = rµ(s) k (+β) 2 γ. µ(s) is the coherence of a subspace S []: µ(s) := n r ax j P S e j 2 2, where e j represents a standard basis eleent. Note that µ(s) n r. We let µ(y) denote the coherence of the subspace spanned by y. By plugging in the definition, we have µ(y) = n y 2. Here, v = x + y, x S and y S. y 2 2 For general cases, Balzano et. al. have shown that if Ω is just slightly greater than rlog(r), then with high probability v Ω P SΩ v Ω 2 2 is very close to Ω n v P Sv 2 2. It eans that partial observations can efficiently approxiate the full observation with high diension. Balzano et. al. s result provides a useful starting point to analyse perforance of sub-selection representation. However, it only concerns clean data and can not directly extend to handle any real probles in practical applications, such as the earlier noted isalignent, occlusion and big noises. B. Transforation-invariant Collaborative Representation Yang et al [5] proposed a transforation-invariant collaborative representation which can handle the representation and the transforation estiation siultaneously. First, a sparse ter e is eployed to handle occlusion. The proble is solved by iizing the objective function: x,e F (x, e) = y Ax e λ x γ e, (2) where e represents the sparse big error, y R n is the query iage, A = [I, I 2,..., I p ] R n p is the dictionary, x R p denotes the representation coefficients. And then the transforation paraeter is introduced so that the objective function becoes: F 2(x, e, τ) = y τ Ax e λ x γ e, x,e,τ where τ is the transforation paraeters and y τ eans to apply the transforation on the query iage. The authors use a two-step strategy to accelerate the optiization process. Step one is to calculate SVD decoposition on the dictionary and use the singular vectors to solve x and τ: (3) F 3(β, e, τ) = y τ Uβ e γ e, (4) β,e,τ where A = USV T is the SVD decoposition of A and β = SV T x is teporal representation coefficient with big absolute values. Miniizing this F 3 (β, e, τ) will give an estiation of transforation paraeters τ, big sparse error e, and the representation coefficients. This approach is uch faster than the previous sparse representation approach RASR [8]. However, due to the l - nor optiization and transforation estiation, it is still slower for high-diensional iages in practical applications. III. METHODOLOGY In this section, we introduce sub-representation approach and provide the theoretical proof of its perforance guarantee for the case with sparse big noise. The resulting ethod is called as robust sub-representation. Finally, after cobining it with transforation estiation and collaborative representation, we propose the TCSR approach. In the following discussion, we shall interchangeably use τ to indicate the transforation paraeters and an operator to apply that transforation on a set of iages. Likewise, we ay also use Ω as a sub-selection atrix as well as a sub-selection operation using that atrix. So that y τ indicates applying the transforation on iage y, and A τ indicates applying the transforation on each iage (colun) in A. Siilarly, y Ω and A Ω represents applying sub-selection on y and A respectively. A. Sub-Representation Consider the proble of representing a query iage by linear cobination of a set of iages. Let y R n be the query iage, A = [I, I 2,..., I p ] R n p be the dictionary of p iages, x R p denotes the representation coefficients. This proble can be forulated as solving the equation Ax = y. The diension of this equation can be reduced using subselection that we discussed in Section II-A. Instead of solving Ax = y, we solve A Ωˆx = y Ω. That is ˆx = arg y Ω A Ω x 2 2, (5) x

3 where A Ω = A Ω denotes the dictionary under subselection, y Ω = y Ω denotes the query iage under subselection and ˆx is the representation coefficient vector. Fro Theore II., ˆx should be very close to original x when Ω satisfy certain conditions. Equation 5 is our basic idea of subrepresentation. The benefit of sub-representation is the solution of the original equation can be approxiated by the solution of the low diensional version of the equation. Hence, the the coputational cost is significantly reduced. Fro now on, we shall extend sub-representation to handle several challenging probles and finally reach a practical iage representation ethod. B. Robust Sub-Representation Occlusions and corrupted pixels are coon challenges in any practical scenarios. In previous literatures, they are odelled as sparse big noise. Instead of solving Ax = y, we need to solve Ax = y e, where e is the sparse error ter. Adding sparse regularization on e, the proble becoes iizing the following objective function: x,e F 4(x, e) = γ e + y Ax e 2 2 (6) where γ is the regularization paraeter. Solving this equation directly can be tie consug for even a ediu size iage, like , due to the l -nor iization. However, usually p n or the rank of atrix A is far less than p and n, a sub-selection operation (e.g. Ω = n) Ω can be applied on the representation. The new objective function is: G 4 (ˆx, e Ω ) = γ e Ω + y Ω A Ωˆx e Ω 2 2 (7) ˆx,e Ω where e Ω = e Ω is the error ter resulting fro acting sub-selection on e. Now we shall discuss the relationship between Eq. (6) and Eq. (7), which is issing in the previous literatures. We first prove the boundedness of l -nor ter under sub-selection. The follow theore is required in later discussion: Theore III.. (McDiarid s Inequality []): Let X,..., Xn be independent rando variables, and assue f is a function for which there exist t i, i =,..., n satisfying sup f(x,..., x n ) f(x,..., ˆx i,..., x n ) t i (8) x,...,x n,ˆx i where ˆx i indicates replacing the saple value x i with any other of its possible values. Call f(x,..., X n ) := Y. Then for any ɛ >, P [Y E[Y ] ɛ] exp P [Y E[Y ] + ɛ] exp ( ) 2ɛ 2 n i= t2 i ( ) 2ɛ 2 n i= t2 i (9) () With Theore III., the following lea can be proved: Lea III.. : Suppose δ >, y R n and Ω =, then ( α ) n y y Ω ( + α ) n y () with probability at least 2δ. Proof: We use McDiarid s inequality fro Theore III. for the function f(x,..., X ) = i= X i to prove this. Set X i = y Ω(i). Since y Ω(i) y for all i, we have i= X i i k X i ˆX Xk k = ˆX k 2 y. We first calculate E[ i= X i] as follows. Define {} to be the indicator function, and assue that the saples are taken uniforly with replaceent. E [ i= X i] = E [ i= y Ω(i) ] = [ [ n ]] i= E j= y j {Ω(i)=j} = n y. Invoking the Theore III., the left hand side is P [ i= X i E [ i= X i] ɛ] = P [ i= X i n y ɛ ]. We can let ɛ = α n y ( and then have ) that this probability is bounded by exp 2 ( n )2 y 2 2α 4 y Thus, the resulting probability bound is P [ ( ) y Ω ( α) n y ] 2 α exp 2 y 2 2n 2 y. 2 Substituting our definitions of µ (y) = n y y and α = 2µ (y) 2 log( δ ) shows that the lower bound holds with probability at least δ. The arguent for the upper bound can be proved siilarly. The Lea now follows by applying the union bound. Lea III.2. : Let δ >. Then with probability at least 6δ, F 4 (x, e) in Eq. (6) and G 4 (ˆx, e Ω ) in Eq. (7) satisfy: n ( α 4)F 4 (x, e) G 4 (ˆx, e Ω ) n ( + α 4)F 4 (x, e) (2) where α 4 is a sall positive constant for given proble. Lea III.2 can be proved using Theore II. and Lea III.. Due to liitation of pages, the detailed proof of this Lea and later Leas are put in the suppleentary aterial. With Lea III.2, it s easy to derive that the subrepresentation coefficient ˆx will be very close to the origin solution x. Lea III.2 provides the theoretical guarantee for quality of solution of Eq. (7). C. Sub-selection and Transforation Estiation Sub-selection can be used to accelerate transforation estiation. It can be forulated as: τ = arg y y 2 τ 2 2 (3) τ where y, y 2 are two iages without correct alignent, τ is the unknown paraeter of the transforation. For any kinds of transforation, like affine, siilarity, hoograph, translation, etc., the forula can be solved by an iterative approach [2]. To approxiate Eq. (3), a Taylor expansion is usually applied. The proble becoes iizing: τ F 5( τ) = y y 2 τ c J τ τ 2 2, (4) where τ c is the current estiation of the transforation, τ is the paraeter increent at each iteration and J τ is the Jacobian atrix. Eq. (4) is a least square equation and have close for solution. The diension of the transforation paraeters is usually very low copared to the iage diension. Hence, sub-selection is applicable here. Assue Ω is a sub-selection atrix(n choose ), y Ω, = y Ω, y Ω,2 = y 2 Ω and

4 J τ,ω = Jacobianˆτ (y Ω,2 ). The result objective function is as follow: ˆτ G 5( ˆτ) = y Ω, y Ω,2 ˆτ c Jˆτ,Ω ˆτ 2 2. (5) Then, we have: Lea III.3. : Let δ >. Then with probability at least 4δ, F 5 ( τ) in Eq. (4) and G 5 ( ˆτ) in Eq. (5) satisfy: n ( α 5)F 5 ( τ) G 5 ( ˆτ) n ( + α 5)F 5 ( τ) where α 5 is a sall positive constant for given proble. Lea III.3 can be proved using Theore II.. With this Lea, ˆτ should be very close to τ. This property allows us to address the unresolved proble of solving transforation estiation in the sae low diensional space of solving the representation coefficients. Lea III.3 provides theoretical guarantee for quality of solution of Eq. (5). D. Transforation Invariant Collaborative Sub- Representations (TCSR) Now, we coplete our discussion by cobining techniques we discuss above and reach our final proposed ethod. The basic idea is transfor Eq. (3) to low diensional space via sub-selection. There resulting forula is as follow: ˆx,e Ω,ˆτ G 2(ˆx, e Ω, ˆτ) = y Ω ˆτ A Ωˆx e Ω λ n ˆx γ e Ω, (6) where ˆx, ˆτ are the representation paraeters and transforation paraeters respectively. Then we have: Lea III.4. : Let δ > Then with probability at least 6δ, F 2 (x, e, τ) in Eq. (3) and G 2 (ˆx, e Ω, ˆτ) in Eq. (6) satisfy: n ( α 7)F 2 (x) G 2 (ˆx) n ( + α 7)F 2 (x) where α 7 is a sall positive constant for given proble. This Lea can be proved using Lea III.2, Lea III.3 and Theore II.. Lea III.4 provides theoretical guarantee for quality of solution of Eq. (6). Lea III.4 is useful to prove the bound of sub-selection version of Eq. (4): { ˆβ, } e Ω, ˆτ} = arg { y Ω ˆτ U Ω ˆβ eω γ e Ω ˆβ,e Ω,ˆτ (7) where U Ω is the singular vectors of sub-selection dictionary Ω A = A Ω = Ω USV T = U Ω SV T and other ters have the sae eaning as in the above discussion. The nuber of rows of Eq. (6) and Eq. (7) are uch saller than that of Eq. (3) and Eq. (4), which can effectively reduce the coputational coplexity. An object classification algorith based on TCSR is suarized in Algorith. Algorith Transfor-invariant Collaborative subrepresentation (TCSR) Input: Training data atrix A, query iage y, and initial transforation τ of y. : Generate l rando selection operator Ω,..., Ω l 2: for q =,..., l do 3: Copute A Ω = A Ω q 4: Copute y Ω = y Ω q 5: Set Û as the first η colun vectors of U Ω where A Ω = U Ω SV T 6: Solving Eq. (6) to get ˆx, e Ω, ˆτ q 7: Copute residue r q i = y Ω ˆτ q A Ωˆx i for each class i 8: end for 9: Copute r i = E[r i ] : Copute identity(y) = arg i E[r i ] : Copute transfor(y) = E[ˆτ q ] Output: identity(y) and transf or(y) IV. EXPERIMENTS In this section, we conducted extensive experients to validate the acceleration perforance of the proposed subrepresentation based ethods against transforation estiation, collaborative representation, and transfor-invariant collaborative representation, respectively. For fair coparisons, we download the code of these algoriths fro their websites and follow their default paraeter settings. Three databases are used for training or testing: Multi-PIE [3], Extended Yale B(EYB) [4] and AR [5]. All experients are conducted on a desktop coputer with Intel icore 7 3.4GHz CPU. Matlab version is 22a. A. Transforation Estiation First, we test our sub-selection for transforation estiation algorith (STE, i.e. Eq. (5)) using public database CMU Multi-PIE database [3]. The iages of subjects fro Session 2 are chosen and all iages are resized to The areas of huan faces are used as the region of interest (ROI) and an artificial transforation of x and y directions are introduced to the ROI. The reference face area is resized to 6 2. We copare STE with the hierarchical odelbased otion estiation (HMME, i.e. Eq. (4)) [2] algorith. Artificial translation in both x and y directions are added to the position of ROI. Suppose the groundtruth rectangle position is a 4-D vector R consisting of x, y coordinates, width and height of the rectangle, while R 2 is a siilar vector for the result rectangle. The accuracy of the estiation is calculated as: acc = R R 2 2 / R 2 The translation is set as, 2, 3, 4 pixels respectively. We set the saple rate as /5 (i.e. Ω n/5). The result is shown in Table I. In the table, the proposed algorith is 2 to 3 ties faster than the original HMME algorith while the loss of estiation accuracies is no ore than.%. These experients indicate that the sub-selection technique effectively reduces the coputational coplexity while preserving the accuracies. B. Face Recognition We use the proposed TCSR (i.e. Algorith ) for face recognition to validate its benefits. We copare the proposed

5 Translation HMME [2] proposed Acc. 99.6% 99.% 94.9% 74.3% Tie Acc. 99.5% 99.% 95.5% 75.3% Tie TABLE I: Result Coparison of Transforation Estiation. Acc. stands for accuracy and Tie stands for average execution tie for each iage x translation (a) y translation (b) TCSR with original collaborative representation classification(crc) [4] algorith on the Multi-PIE [3], Extended Yale B(EYB) [4] and AR databases [5]. For fair coparison, we follow the experiental setting in [4]. The initial position of all the testing and training iages are autoatically detected by Viola and Jone s face detector [6]. For the setting of Multi- PIE, the first subjects in Session with illuations {,, 7, 3, 4, 6, 8} are used as the training set while the first subjects in Session 3 with illuation {3, 6,, 9} are used for testing. The face areas in the training iages are all resized to 6 2. The saple rate is /5; For the setting of EYB, 2 subjects are selected. For each subject, 32 randoly selected frontal iages are used for training, with 29 of the reaining iages for testing. The face areas are resized to The saple rate is /; For the setting of AR, subjects and 7 iages per subject are selected as the training set, while subjects and 6 iages per subject for testing. There are 5 ale faces and 5 feale faces with various facial expressions and illuations. The face areas are resized to The saple rate is /8; CRC [4] proposed MPIE EYB AR 88.7% 99.4% 86.% Avg. Tie % 99.% 86.6% Avg. Tie TABLE II: Recognition rates and execution tie The experiental result is tabulated in Table II. The recognition rate of proposed TCSR algorith is alost the sae as the original CRC [4] algorith with difference less than.5% while the speed is 3 to ties faster than the origin version related to the saple rate. Also the variation of illuations of the training set and testing set affects the results. While the variation of the training set is sufficient to represent the query iages, the saple rate can be lower, otherwise ore data are needed. All these results have validated the proposed collaborative sub-representation has outperfored the collaborative representation in speed while achieving coparable accuracy. C. Transforation Invariant Face Recognition In this section, we conduct experients to validate the benefit of TCSR under isalignent. First, we copare the proposed algorith with state-of-the-art ethods, [5], RASR [8], TSR [7] on the Multi-PIE [3] database. The first subjects in Session with illuations {,, 7, 3, 4, 6, 8} are used as the training set while the first subjects in Session 2 with illuation {} are used for testing. Artificial transforation of 5 pixels translation in both x and y directions is added to the test iages. The saple rate is set to /5. The recognition rates of RASR, TSR, x translation (c) y translation Fig. 2: Recognition rates and Speed with Translation: Top row: (a) with only x direction translation; (b) with only y direction translation; Botto row: (c) Average Tie with only x direction translation; (d) Average Tie with only y direction translation; Angle of rotation (a) 5 5 Angle of rotation (c) (d) Scales (b) Scales Fig. 3: Recognition rates and Speed with Scale and Rotation: Top row: (a) with only in-place rotation; (b) with only scale variantion; Botto row: (c) Average Tie with only in-place rotation; (d) Average Tie with only scale variantion; and the proposed algorith are 9.8%, 89.% 95.9% 95.9% respectively, while the average execution ties are 95.9, 5.5, 4.9,.4 respectively. The RASR will try to fit all identities one by one, which akes it very slow in big training set. Although TSR is faster, it s less accurate. is significantly faster than RASR and TSR and also ore accurate. In the following (d)

6 experients we shall only copare our algorith with. Also note that in this experient, our proposed algorith is nearly 3 ties faster than while aintaining alost the sae accuracy. Next, we conduct experients on the Multi-PIE database with various kinds of transforations. The experiental setting is the sae as the previous experient. Artificial transforations include x direction translation, y direction translation, in-place rotation and scales. These experients copare the perforance of [5] and the proposed algorith. Here we use the saple rate /5. The experiental results are shown in Fig.2 and 3. The proposed algorith is 3 to 4 ties faster than the, while the difference of their accuracy is less than %. Due to the redundancy of data, the sub-selection ethod preserve the accuracy very well. All these experients validate that the proposed TCSR can handle various isalignents with uch less coputational cost than existing ethods. D. Recognition Despite Rando Block Occlusion Occlusion percentage (a) Occlusion percentage Fig. 4: and (a) The recognition rate(accuray) vs. occlusion percentage; (b) The execution tie vs. occlustion percentage In this section, we further validate the robustness of TCSR by coparing our ethod and [5] on iages with rando block occlusions. This kind of experient has been conducted in the and RASR [8], so we follow the setting of the experient and use the dataset as the sae as those used in the previous experient. The saple rate is /3. The training set and testing set are fro the Multi-PIE database. The first subjects in Session are used as the training set. And the testing set is subjects fro Session 2. Various levels of block occlusion are added to the testing iages. The testing results are shown in Fig.4. TCSR has alost the sae recognition rates as those of the under various levels of block occlusions. However, the TCSR is 2 to 3 ties faster than the algorith. These experients validate the benefit of proposed TCSR in handling occlusions. V. CONCLUSION This paper proposed a novel sub-representation ethod for iage representation. We have theoretically proved its benefit for handling sparse big noise over previous ethods. Cobining it with existing techniques, we proposed a transfor-invariant sub-representation, which can efficiently handle isalignent, occlusion and big noise probles in practical applications. The benefit of proposed ethods were (b) not only theoretically proved but also epirically validated by extensive experient results on practical applications. In the future, we would like to extend our approach in several directions. First, it is worth investigating beyond the siple sparse regularization to group sparsity [7]. Second, we would like to apply our approach to ore applications such as MR iaging reconstruction proble [8], [9]. REFERENCES [] L. Balzano, B. Recht, and R. 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