PCA-based Reconstruction of 3D Face shapes using Tikhonov Regularization

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1 Int. J. Advance. Sot Comput. Appl., Vol. 5, No. 2, July 213 ISSN ; Copyight SCRG Publication, 213 PCA-based Reconstuction o 3D Face shapes using Tikhonov Regulaization Asha Y. A. Maghai, Ibahim Venkat, Iman Yi Liao and Bahai Belaton School o Compute Sciences, Univesiti Sains Malaysia, Pinang, Malaysia myasha2@gmail.com School o Compute Sciences, Univesiti Sains Malaysia, Pinang, Malaysia ibahim@cs.usm.my School o Compute Science, Univesity o Nottingham Malaysia Campus Iman.Liao@nottingham.edu.my School o Compute Sciences, Univesiti Sains Malaysia, Pinang, Malaysia bahai@cs.usm.my Abstact Reconstucting a 3D ace shape om a limited numbe o 2D acial eatue points is consideed as an ill-posed poblem which can be solved using egulaization. Tikhonov egulaization is a popula method that incopoates pio inomation towads poviding the existence o closed-om solutions which we obtain as a esult o applying PCA, in ode to solve the ill-posed poblem. The common actos that geneally aect Tikhonov egulaization ae the egulaization matix, the numbe o eatue points, egulaization paametes and noise. In this study we epot ou indings on how vaious actos inluence the econstuction accuacy based on a case study peomed on the USF Human ID 3D database. Futhe, a statistical compaison between two Tikhonov egulaization matices viz., the identity matix and the diagonal matix compising o the eigenvalues (eigenvalue matix), has been peomed. We ound that, the econstuction eo can be educed signiicantly by using the late one. Finally ou study aids to detemine the most easible inteval in conjunction with optimal egulaization paametes which would lead towads achieving accuate and plausible solutions. Keywods: Tikhonov Regulaization, 3D ace econstuction, PCA, Tikhonov Regulaization.

2 Asha Y. A. Maghai et al. 2 1 Intoduction 3D shape povides popeties that ae invaiant to changes such as view point, illumination, backgound clutte and occlusion. These popeties can help in applications like ecognition, animation, medical application and allow pediction o appeaance unde new viewing conditions [1]. A majo poblem in compute vision is to extact such popeties given a single 2D image. Thee ae many appoaches o the econstuction o 3D aces om single images. One o such ealy techniques being utilized is Shape-om-Shading (SFS) [2-4], which capitalizes the idea that the depth inomation is elated to the intensity o a ace image acquied though a given/chosen electance model. SFS estimates the illumination diection in the 2D image to ine the 3D shape o the suace. It has been shown that SFS sues om poo global shape contol. Recently, Kemelmache-Shlizeman and Basi poposed an appoach that combines shading inomation with geneic shape inomation deived om a single eeence model by utilizing the global similaity o aces [1]. This method uses only a single eeence model o a dieent peson's ace to econstuct the 3D ace shape. It does not need a leaning stage to build a model o epesenting input aces. Consequently, not counting on a 3D eeence model which keeps shape similaities with the input image may esult in inaccuate 3D shape estimation. Thee ae also conventional leaning-based methods, such as neual netwoks [5, 6] and typical statistical leaning-based methods such as Hidden Makov Model (HMM) [7], Makov Random Field (MRF) [8] and analysis by synthesis using 3D Moph able Model (3DMM) [9]. Inteestingly computational intelligence based appoaches such as neual netwoks have also been applied in elated compute vision aeas such as medical image segmentation [1, 11]. In the last ew decades much inteest has been shown in the aea o extacting 3D suaces om obseved 2D images by using statistical models which allows moe ealistic ace econstuction than othe methods (e. g. [12] & [13]). These models can be used as pio inomation which can be incopoated with a itting algoithm to estimate the complete 3D ace shape om the given inomation such as a set o acial eatue points. One o the econstuction methods that uses pio knowledge to estimate the shape coeicients om a set o acial points is egulaization [14]. In [14], Jiang et al. use a egulaization equation that estimate the geomety coeicient in an iteative pocedue. Altenatively egulaization method has also been pesented in [15]. Reconstucting the 3D ace shape om a small set o 2D eatue coodinates is consideed as an ill-posed poblem which can be solved by using egulaization. One o the most appopiate methods that can be employed is the Tikhonov egulaization [16]. In the last ew decades, Tikhonov egulaization has been well addessed in the ield o invese poblems [17]. It is a popula and eective

3 3 PCA-based Reconstuction o 3D Face method that can be easily incopoated with pio inomation embedded in a closed-om solution which can be eadily obtained by applying PCA on the 3D ace shapes. Regulaization is necessay o obust, stable, and plausible econstuction. It inds a tadeo between itting the 3D shape to the available acial eatue points and poducing plausible esults in tems o pio inomation about the 3D ace object [18]. Impotantly econstuction by means o a Tikhonov egulaization method can be computed in one step (non-iteative way), theeby enabling aste 3D econstuctions. Hence, Tikhonov method is an eicient choice o seveal 3D oiented inteactive tools. The common actos that geneally aect Tikhonov egulaization ae the Tikhonov egulaization matix (stabilizing item), the numbe o eatue points, egulaization paametes and noise. The objective o this contibution is to study the eects o these actos on the econstuction accuacy and plausibility. Blanz and Vette [15] have studied the eect o the egulaization paametes and the noise on the accuacy with the aid o a Bayesian appoach based egulaization technique. The eect o eatue points on 3D ace shape econstuction has been studied by Maghai et al. [19] using the Tikhonov Method that uses the eigenvalues as the egulaization matix. In this study we aim to estimate the eect o the othe actos without vaying the numbe o eatue points. When the numbe o eatue points available ae lage with espect to the degees o eedom o the model, appoximate solution tends to the tue unction when the numbe o eatue points tends to ininity [2]. Howeve, in the eal application, in addition to the noise which depends on the acquisition system, the available eatue points ae limited. Hence, this study uses a small set o eatue points (25 2D points). By small numbe o eatue points, oveitting can easily occu. To avoid oveitting, egulaization that enoces the esults to be plausible accoding to the pio inomation is needed. The egulaization mechanism needs to ind a tadeo between itting 3D shape to the given 2D acial landmaks and poducing plausible solution in tems o pio knowledge [18]. Initially a PCA-based 3D ace shape model is incopoated to gain pio inomation about the 3D ace shape. Then the Tikhonov method is deployed to show how a complete ace shape can be estimated om a limited numbe o eatue points. The PCA-based 3D shape leaning model elies on examples o 3D scans om the USF Human ID 3D Face database [9]. In this pape we will study the eect o egulaization paametes and the noise on the Tikhonov egulaization with two dieent Tikhonov matices. We will ee to the Tikhonov method that uses the identity matix as STR (Standad Tikhonov Regulaization) and the one that uses the diagonal eigenvalue matix as ETR (Eigenvalue Tikhonov Regulaization). We will also ee to the matix that has the eigenvalues in its diagonal as eigenvalue matix. ETR has been compaed with STR in tems o accuacy, plausibility and obustness to noise. As pat o this

4 Asha Y. A. Maghai et al. 4 contibution, we demonstate how to detemine the best inteval o the egulaization paamete that poduces the best solution. This pape is oganized as ollows: Section 2 povides details on the PCA-based shape ace epesentation. Reconstuction appoaches based on egulaization algoithms ae bieed in Section 3. Section 4 descibes distance-based evaluation pocedues. In section 5, the egulaization paamete inteval is demonstated. Expeiments and statistical analysis ae epoted in section 6 and inally section 7 concludes the outcome o ou pape. 2 PCA-based 3D ace shape epesentation In ode to build a PCA based ace shape model, PCA decomposition has been applied to the 3D ace shapes oeed by the USF Human ID 3D database [1]. Each taining ace shape is epesented using the 3D coodinates o all its vetices in tems o a tiangulated mesh. Then these 3D shapes ae aligned with each othe in such a way that a 3D-3D coespondence o all the vetices ae obtained [9]. Each ace shape has p coesponding vetices which can be vecto zed as: s ) T i = ( xi 1, yi 1, zi 1,..., xip, yip, zip, (1) whee s i has the dimension n=3 p, n is the numbe o vetices and i = 1,, m, whee m epesent the numbe o taining shapes. Based on this vectoization, the mean ace shape among the taining ace shapes is computed and the deviation o each taining shape om the mean is calculated. Singula Value Decomposition (SVD) [21] is then used to peom PCA on the covaiance matix esulting in a mophable model. A new shape vecto s n ec R can be expessed as s ec = s + Eα = s + m i= 1 α e, (2) whee s is the mean 3D shape, E = [ e1, e2,..., em ] is a basis vecto matix and α i is the coeicient o the scaled basis vecto e i. Assuming x = s s Rn, Eq. (2) can be minimized to i i x ec = Eα. (3) Since E is an othogonal matix, the PCA-coeicients α o a vecto x can be deived om Eq. (3) as

5 5 PCA-based Reconstuction o 3D Face α = E T x. (4) A new 3D ace shape can be appoximately epesented by making use o Eq. (4) and Eq. (2). Taking into account the data uncetainties and the ew numbe o available eatue coodinates << p on the eal 2D ace images, Eq. (3) becomes x = Aα + ε. (5) A R l m is the coesponding subset o the matix o scaled basis vectos E R n m and ε Rl can be consideed as measuement eos with unknown popeties. x = s s, s R l ( l = 2 ) and s ae coesponding points on s (the aveage 3D ace shape). The pimay least squae (LS) method can be used to solve o α α = ( AT A) 1 AT x. (6) Howeve, in addition to the measuing eos, the measued eatue points captues only a small potion o the oiginal image x, which intoduces eos in the ecoveed model. This poblem is consideed as an ill-posed poblem. Theeoe, we see that, egulaization is necessay o a obust and plausible solution, which inds a tadeo between plausible esults (in tems o the pio inomation) and itting the 3D shape to the given acial points. x 3 Regulaization based Reconstuction The standad Tikhonov egulaized solution o Eq. (5) is to minimize the ollowing unction: { Aα 2 + λ α 2 } α eg = ag min x. (7) α Hee, α 2 is the stabilizing tem and λ > is called the egulaization paamete which is chosen to balance the data itting eo so as to get a plausible solution. A standad egulaized appoximation o α is given by whee I is the identity matix. α = ( AT A + λi ) 1 AT ( x ), (8) eg

6 Asha Y. A. Maghai et al. 6 Dieent egulaization matices have been employed in ode to impove the standad Tikhonov egulaization. Obviously, dieent econstuction accuacies will be obtained by using dieent egulaization matices. The objective unction in Eq. (7) can be genealized as: α { Aα 2 + λ α α 2 } = ag min x eg. (9) α Accoding to the Bayesian posteio pinciple, α is the expected value o α which can be computed as the aveage vecto compising o all the ace shape paamete vectos. Eq. (9) has an explicit optimal solution o α eg and is given by α eg = α + AT A + λlt L) 1 AT ( x A ). (1) ( α Owing to the Bayes theoem and unde the assumption that the data matix X = S S is a multivaiate nomal distibution and the eos in x ae independent with zeo mean, the Tikhonov-egulaized solution yields the most pobable solution [22]. Let C be the invese o the covaiance matix o α, then accoding to the maximum posteio pobability, the Tikhonov matix can be given as C = LT L. The covaiance matix is the diagonalised matix with eigenvalues. So, the stabilizing item is chosen to be the invese o the diagonal eigenvalue matix W. Assume that α =, then model paamete α can be estimated as ) = λ α ( AT A + W 1 1) A T x. (11) Then, a new ace shape s ec can be obtained by applying α ) to Eq. (2). In the next section we will discuss how to evaluate the Tikhonov method that uses the eigenvalue matix as a egulaization matix (ETR: Eq. (11)) compaed with the standad Tikhonov (STR: Eq. (8)). 4 Distance-based Evaluation The evaluation is based on computing the aveage distance between the points o econstucted ace shape vecto and the gound-tuth vecto 1 E = n n i= 1 si, oig. si, econst.. (12) To compae between the STR (Eq. (8)) and ETR (Eq. (11)), new test ace shapes ae econstucted om a set o eatue points using both the methods. Then, accoding to Eq. (12), o evey test ace shape, the minimum E is computed by

7 7 PCA-based Reconstuction o 3D Face the optimal egulaization paamete λ. The 3D aces in the database ae aleady aligned with each othe, which mean that the points can be easily selected by thei indices. A set o = 25 salient eatue points such as eye cones, eyebows, mouth cones, nose and ace contous have been selected om the ace model. These points can be diectly used to compute the 3D shape coeicient α using Eq. (8) o Eq. (11). Then, the coeicients ae used to econstuct the 3D ace shape using Eq. (2). The esulted ace shape is compaed with the oiginal shape by calculating E. ETR is evaluated quantitatively in ode to compae it to the STR. By quantitative evaluation, E o the econstucted 3D ace shape has been computed o all the test ace shapes. Then we have applied the t-test to compae the eect o the two matices on the econstuction eo. The t-test is a statistical test which allows the compaison o any two methods on objective tems with knowledge o isks associated with eaching the wong conclusion. On the othe hand, examples o econstucted 3D ace shapes have been visualized to clealy justiy the qualitative eatues o the econstucted ace shapes. This is to check how well the qualitative evaluation complies to the esults which have been obtained using quantitative techniques. Futhe, we compute the Euclidean distance between the econstucted ace shape and the mean ace to detemine how close is the solution to the mean ace using Eq Dm= n n i= 1 5 Regulaization Paamete Inteval s i, si, econst.. (13) In this section, we examine how to detemine the inteval o λ that educes the E between the econstucted ace shape and its gound tuth while poducing a plausible solution in tems o pio knowledge. Futhe, to avoid the excessive smoothness o the solution (the highly closeness to the mean ace), D m between the econstucted ace and the mean ace is computed. By this way, we can ind out the best inteval o λ that could poduce a plausible but not necessaily a vey smooth solution. The quality o econstucted aces can be contolled by λ as shown in Fig. 1. Low values o λ can lead to oveitting and high values leads to the mean ace. Fo example, by applying STR (Ps. see uppe ow o Fig. 2), when λ >.1, D m conveges to zeo and the esults becomes close to the mean ace; Howeve we see that E gows and oveitting occus when λ conveges to. Accoding to the esults shown in Fig. 1 (ight) whee the eatues ae noise ee, we can conclude that a good inteval o an optimal λ, in tems o the minimum aveage o E (o all 2 test aces) and plausible solution, is located in the inteval (.1,.1). Howeve we can see that, o noisy eatue points the

8 Asha Y. A. Maghai et al. 8 best inteval can be (.1,.1) accoding to the aveage econstuction eos E shown in Fig. 1 (let). Fig. 1. The eect o λ on the aveage econstuction eos E and the distance om the mean ace o 2 test aces, o a given set o 25 eatue points using the Standad Tikhonov Regulaization method (STR). Fig. 2. Fo a given set o 25 2D eatue coodinates o a pobe 3D ace (let), 3D ace shapes o the same pobe ace wee econstucted with dieent λ. The uppe ow shows econstucted shapes using STR while the lowe ow shows econstucted shapes using ETR. Fo ETR, the association to λ has the same manne as expeienced by the STR but with a dieent set o values. Fig. 2 (lowe ow) shows econstucted ace shapes o the same pobe ace shapes with dieent values o λ. Fo smalle values o λ, the ace shape becomes a om the oiginal shape, E gets lage and also the ace is distoted when λ conveges to. In contact, the shape becomes moe smooth when E deceases, and each its minimum at λ =1. Fo all 2 test aces, the values o λ in the inteval (1, 1) poduce the best aveage o E values with slight convegence to the aveage ace as λ inceases (see Fig. 3 (let)). I λ >

9 9 PCA-based Reconstuction o 3D Face 1, E inceases and D m emakably deceases until λ 1; uthe the econstucted ace becomes the mean ace o vey lage λ values wheeas D m conveges to. In tems o plausibility and closeness to the gound tuth, the best inteval has been suggested to be within the inteval (1, 1), wheeas the aveage E o all test aces has the minimum value. Fo noisy eatue points, the best inteval can be (1, 1) wheeas the aveage econstuction eos E has the minimum values and D m is lage enough o non successive smooth solution (see Fig. 3, ight). Fig. 3. Using ETR, the eect o λ on the aveage econstuction eos distance om the mean ace o 2 test aces, given 25 eatue points. E and the 6 Expeiments and Statistical Analysis Let us ecall that, this wok aims to exploe the inluence o the egulaization matices with espect to the accuacy o 3D ace shape econstuction by using Tikhonov egulaization method. We have evaluated the accuacy o econstuction o both o the standad identity matix and the Tikonov s eigenvalue matix and compaed them. The USF Human ID 3D Face database [9] which contains 1 3D aces has been used o ou expeimental validations. The PCA model has been tained with the 8 3D ace shapes. The emaining 2 3D ace shapes have been used o the pupose o testing. 6.1 Reconstuction om noise ee eatue points In this section, we analyze how the two egulaization matices peom on the test 3D ace shapes that have been chosen om outside the taining dataset. The xy coodinates o the selected eatue vetices have been used o econstuction. The esults in Fig. 4 shows that the econstuction eos yielded by econstuction o 2 test aces using the ETR is minimal when compaed to the STR. Moeove, the

10 Asha Y. A. Maghai et al. 1 aveage esults o the 2 test aces shown in Table 1 indicates that ETR outpeoms STR with a 95% conidence level, wheeas the P-value o the t-test coesponding to the two methods is less than.5 (level o signiicance). This indicates that thee is a statistically signiicant dieence between the econstuction eos o the two methods. This also justiies that using ETR yields bette accuacy than that o STR. Fig. 5 visualizes the eect o econstucted 3D ace shapes o the same oiginal ace (let) by using ETR and STR. Fig. 4. Compaison between the ETR and STR in tems o "econstuction om 25 noise ee eatue points". E is computed by the optimal egulaization paamete λ. Table 1. Compaison o esults between the econstuction eos yielded by ETR and STR. = 25 values 1 Noise ee eatue points Noisy eatue points ETR STR t-test against Mean Std. Mean Std. signiicant eigenvalues (P-value) E E-5

11 11 PCA-based Reconstuction o 3D Face Fig. 5. Reconstuction o 3D ace shapes om two dieent sets o 25 eatue points viz., noisy and noise ee eatue points using the dieent egulaization matices. The esults depend on the optimal λ which poduces the minimum E. 6.2 Reconstuction om Noisy eatue points In the pevious section, we have demonstated how the econstuction eo can be educed by using dieent egulaization matices. Howeve, i we add andom noise to all eatue points coodinates, α becomes moe diicult to estimate, and oveitting can easily occu. Fig. 6 shows the econstuction eos ound by econstuction o 2 test aces om 25 eatue points using both the matices. This uthe justiies that the eigenvalue Tikhonov (ETR) outpeoms the standad Tikhonov (STR) in tems o econstuction accuacy o all test aces. Moeove, the aveage eos o the 2 test aces shown in Table 1 indicates that the econstuction eos o ETR ae signiicantly lowe than those o STR at 95% conidence level. Fig. 5 visualizes the econstucted 3D ace shapes o the same oiginal ace (let) by optimal λ using STR and ETR. It can be seen that the quality o econstucted aces using ETR looks moe pecise and intuitive than those econstucted by the STR o noisy and noise ee eatue points. Fig. 6. Reconstuction eos geneated with the aid o choosing 25 noisy eatue points. E is computed by the optimal egulaization paamete λ.

12 Asha Y. A. Maghai et al Reconstuction o 3D Face Shapes om Images The CMU-PIE database [23] has been used o testing the visual eects o the poposed model. We intend to econstuct 3D models o the 2D images o CMU- PIE database. Fom small numbe o 2D acial landmaks, the poposed algoithm can ecove the 3D shape o the given 2D ace image. In this wok, the input 2D images ae in nea ontal pose with most o thei expession being neutal. The eatue points which have been manually selected, have been aligned with the eeence 3D model using Pocustes Analysis, which is the usual peliminay step beoe the econstuction stage. The aligned eatue points have been used to compute the model paametes α using ETR Eq. (11). Then, α has been used to econstuct the 3D shape using Eq.(2). Fig. 7 shows two typical 2D images (let) and thei coesponding econstucted 3D aces using ETR (ight). Fig. 7. Fo a given set o 25 2D eatue points o typical eal 2D images (let) thei coesponding 3D ace shapes have been econstucted with dieent distance measues o D m. Fo each o these 2D images, 6 econstucted aces with dieent distances D m have been shown. It can be seen that the quality o econstucted aces gets degaded i D m inceases. In othe wods, we can ine that the econstuction o aces gets gadually smoothe when D m deceases. Hence, we can employ the distance om the mean ace (D m ) to evaluate the 3D aces econstucted om eal 2D ace images. Accoding to the esults seen in Fig. 7, we can see that an optimal D m may be the value aound.1 (the aveage distances D m o all 3D aces in the taining data set). Assuming the eatue points ae available, ou MATLAB implementation o the ETR algoithm o econstucting the complete ace shape vecto takes about one second on a wokstation with Intel(R) Xeon(R) CPU 2.4GH which justiies that the ETR based 3D ace econstuction is quite ast and hence easible to be deployed o eal time applications.

13 13 PCA-based Reconstuction o 3D Face 7 Conclusion A majo poblem in compute vision is to extact 3D shape popeties om single 2D images. Example-based models allow moe ealistic ace econstuction than othe methods. In this pape, the standad PCA-model has been used as a holistic model to epesent the object class o 3D ace shapes, which aids in solving the ill-posed poblem o econstuction o complete 3D ace shapes om thei 2D images. This holistic model has been incopoated with the Tikhonov method to show how to estimate the 3D shapes o novel aces using a limited numbe o eatue points. The common actos that geneally aect Tikhonov egulaization ae the egulaization matix, the numbe o eatue points, value o egulaization paametes λ, and noise. This wok is basically an eective case study to show the eect o the egulaization matices on the quality o econstucted aces with espect to a small set o eatue points (25 2D eatue points). The eects o the egulaization paamete and noise have been also demonstated. Ou extensive expeimental esults justiies that by using ETR, the econstuction eos could be signiicantly educed o all o the test aces when compaed to the STR. Futhe we have shown that ETR outpeoms the STR with 95% conident level in tems o econstuction eos. In tems o plausibility and closeness to the gound tuth, ou esults demonstate that the quality o econstucted aces using ETR is bette than those econstucted using STR even in the pesence o noise. We have also shown a mechanism o detemining the best inteval o λ in ode to ind the optimal solution. Ou expeiments on eal 2D images demonstate that the ETR method was able to econstuct plausible 3D ace shapes om limited numbe o eatue points in an eective way. Impotantly as the ETR method incus less computation time, it can be eadily deployed in many eal wold inteactive applications. Fo ou utue wok, we intend to evaluate the moe challenging task o econstuction o images with vaying degees o poses and expessions. ACKNOWLEDGMENT The ist autho would like to thank the Ministy o Highe Education (MOHE), Malaysia o suppoting him with the MIS scholaship. This eseach is also patly suppoted by an RU Gant (11/PKCOMP/81755) oeed by Univesiti Sains Malaysia and an ERGS gant (23/PKCOMP/67375) oeed by MOHE.

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