# New Robust Regression Method for Outliers and Heavy Sparse Noise Detection via Affine Transformation for Head Pose Estimation and Image Reconstruction in Highly Complex and Correlated Data: Applications in Signal Processing

Authors: Peidong Liang; Habte Tadesse Likassa; Chentao Zhang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this work, we propose a novel method for head pose estimation and face recovery, particularly to solve the potential impacts of noises in signal processing to get an efficient and effective model that is more resilient with annoying effects through adding affine transformation with the low-rank robust subspace regression. Consequently, the corrupted images can be correctly recovered by affine transformations to render more best regression outcomes. Thereby, we need to search so as to get optimal parameters which can be regarded as convex constrained optimization techniques. Afterward, the alternating direction method for multipliers (ADMM) approach is considered and a new set of updated equations is well established so as to update the optimization parameters and affine transformations iteratively in a round-robin manner. Additionally, the convergence of these new updating equations is well scrutinized as well. Thus, the experimental simulations reveal that the proposed method outperforms the state-of-the-art works for head pose estimation and face recovery on some public databases.


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## Body

## 1. Introduction

Images, processing particularly for the head pose estimation and image recovery, have been important research potential topics and can have applications in a variety of areas such as surveillance systems [1, 2], signal processing [3, 4], image denoising [5–9] and recovery [10, 11], communications [12], computational imaging [13, 14], and computer vision [15–19]. However, analyzing visual data is a difficult task due to miscellaneous adverse effects such as illuminations, outliers, and sparse noises. It is thus of importance to develop robust face recovery and head pose estimation algorithms, which are resilient to various annoying effects.After the inception of the pioneering baselines of robust principal component analysis (RPCA) by Candes et al. [20], a myriad of methods has been considered for robust sparse-low-rank image recovery, e.g., [21, 22]. However, these methods do not work well when the outliers and heavy sparse noises are not normally distributed.To tackle this drawback, Oh et al. [23] proposed a new partial singular value thresholding (PSVT) method, which replaced the nuclear norm in RPCA [8, 20] with the partial sum of singular values to improve the recovery of the low-rank part. Lu et al. [24] proposed a tensor robust principal component (T-RPCA) approach to find the clean tuber low-rank component. However, T-RPCA is not scalable and robust when the number of tensors becomes large. Currently, several algorithms, which combined regression with RPCA [25], were proposed to further enhance the performance. For instance, Ji et al. [26] addressed a regularized sparse regression via combining RPCA [22, 27] with lasso regression [28] to mitigate the influence of outliers in the head pose estimation. But, the performance of this relaxation method degrades when the fitting error is grossly increasing. Wang et al. [29] developed a robust regression method via self-scaled regularization to boost the performance in the presence of gross outliers. Huang et al. [30] proposed a low-rank robust regression (LR-RR) algorithm to clean the outliers and sparse errors from highly contaminated data. Although LR-RR can mitigate the impact of sparse errors inside and outside subspaces, its sensitivity to sparse errors and outliers lying in the disjoint subspaces jeopardizes its performance in some severe scenarios. Yin et al. [31] considered a robust multinomial logistic and binary regression to remove the sparse noise and outliers from the contaminated data. Yang et al. [32] proposed a matrix regression scheme for face image representation based on the nuclear norm. Zhang et al. [33, 34] proposed a low-rank-sparse subspace representation for robust regression (LRS-RR) method to find the clean low-rank part by low-rank subspace recovery along with regression to deal with errors or outliers lying in the corrupted disjoint subspaces. To resolve this, Zeng et al. [35] addressed labeled-robust regression, but its performance is not yet promising to denoise the high dimensional images, particularly in signal processing; to tackle this, Wu et al. [36, 37] addressed the sparse prior information.This work proposes a new robust method for head pose estimation and image recovery to denoise the potential impacts of outliers and heavy sparse noises in signal processing. To develop a method that is working well with various annoying effects, the new approach incorporates affine transformations taken from [38–40] into the robust regression methods [30, 33] with the robust regression for more faithful low-rank-sparse image representation. Consequently, the corrupted high dimensional images can be recovered correctly by affine transformations to achieve more promising regression outcomes in statistical signal processing. The newly developed algorithm is first cast as a convex optimization programming, in which the affine transformations, low-rank subspace recovery, and regression are carried out simultaneously. Afterward, the alternating direction method for multiplier (ADMM) method is applied and a new set of equations is established to update the optimization variables and affine transformations iteratively in a round-robin manner. Moreover, the convergence of the entire newly developed equations is scrutinized as well. Conducted simulation results reveal that the proposed method excels the state-of-the-art works for head pose estimation and face recovery on some public datasets. The main contributions of this paper include the following:(1)
The affine transformations are incorporated into the low-rank-sparse decomposition to correct the illuminated and highly distorted or misaligned images to attain more precise low-rank image decomposition(2)
The ADMM method is proposed to solve the convex constrained optimization problems and a new set of updating equations is developed to iteratively update the optimization parameters and affine transformations(3)
The convergence of the derived iterative equations that considers more updating parameters is investigated(4)
The proposed method outperforms to the baselinesThis work is organized as follows. Section2 gives an overview of the related works. Section 3 addresses the formulation of the new problem. Section 4 depicts the new set of updating equations to solve the formulated convex optimization problem and Section 5 analyzes its convergence characteristics. Experimental simulation results are provided in Section 6 to verify the proposed method. Section 7 draws some concluding remarks to summarize the paper.


## 2. Related Works

A number of robust methods have been reported for image recovery [41–45]. For instance, Wei et al. [14, 43, 45] addressed the least trimmed squares to alleviate the gross errors in the regression to explicitly search a data subset that reduces the square of the errors. The discriminatory least square regression [46] and the worst-case linear discriminant analysis [47] were proposed to solve the least square loss function that influences the correlation between the explanatory and response variables. These methods, however, are not that robust for big data. Bunea et al. [48] scrutinized a rank selection criterion to select the best rank estimator of the coefficient regression matrix in the multivariate regression approach. To deal with linearly structured matrices, Zachariah et al. [49, 50] addressed an iterative algorithm via least square estimation for low-rank matrix reconstruction, but it required the prior knowledge of the matrix structure. Chen et al. [51] proposed an iterative reweighted least squared method for sparsity recovery, which incorporated the structure of sparsity along with an orthogonal basis and the total variation. However, it requires conducting matrix inversion at each iteration, leading to high computational complexity. Instead of using the handcrafted least square regressions as in [13, 52–55], some more recent methods [56–61] have improved the visual quality in face reconstruction via the low-rank approximation.Head pose estimation has also received a considerable amount of research attention. For instance, linear regression methods were considered in [62, 63] for head pose estimation. These methods, however, are sensitive to occlusions and uncontrolled illuminations. Lathuiliere et al. [64] proposed a deep mixture regression approach to replace the supervised manifold learning in [65, 66] to perform head pose estimation. Recently, Sun et al. [61, 67, 68] proposed a probabilistic method for head pose estimation by directly mapping the feature vectors onto the yaw angles. Diaz-Chito et al. [69] addressed an algorithm to narrow down the gap between the head yaw angles and the regression by combining manifold embedding methods with linear regression. Meyer et al. [70] considered a three-dimensional head pose estimation method to handle large pose angles and partial occlusions. However, it cannot prune out outliers from disjoint subspaces in head pose estimation.


## 3. Problem Formulation

Givenn images, Ii0∈ℜw×h,i=1,…,n, where w and h denote respectively the weight and height of the highly illuminated and corrupted images, all of which consists the same objects. In many dilemmas, these images are highly correlated and corrupted by occlusions and outliers. We can stack these images into a matrix: M=vecI10|vecI20|…|vecIn0∈ℜdm×n, where vec⋅ denotes the vector stacking operator. In light of the fact that the subspaces may not be independent from each other or the data are illuminated and contaminated by large illuminations and noises and outliers, we decompose the original data matrix into a low-rank component and sparse errors, i.e., M=AC+E [71–73], where A∈ℜdm×n is the low-rank component, C∈ℜn×n is a reconstruction coefficient matrix used to represent M, and E∈ℜdm×n denotes a sparse error matrix incurred by some adverse effects.In reality,Ii0 are generally not well corrected and aligned, which makes the issue of the low-rank and sparse separation to be imprecise. To mitigate the issue of the misalignment, inspired by [74, 75], we apply affine transformations τi to Ii0 to get the transformed images Ii=Ii0oτi, where the operator o indicates the transformation applied to the potentially misaligned input images. After taking the affine transformation on M, we can obtain Moτ=vecI1|vecI2|…|vecIn∈ℜdm×n, where Ii=Ii0oτi is an aligned version of the ith image after the transformation. The aligned images can be treated as samples taken from a union of low-dimensional subspaces, which exhibits a low-rank subpace structure as the rank of the transformed images is as small as possible, up to some outliers and heavy sparse errors. To improve the issue of the nonlinearity in Moτ, we can further represent that the change produced by these affine transformations τ is small and an initial affine transformation of τ is known, then we can further linearize it by using the first-order Taylor approximation as Moτ+Δτ≈Moτ+∑i=1nJiΔτviviT, where Moτ∈ℜdm×n is the transformed image, Δτ∈ℜp×n, where p indicates the number of parameters, Ji=∂vecIioτi/∂τi∈ℜdm×p denotes the Jacobian of the ith image with respect to τi, and vi denotes the standard basis for ℜn.The main goal is to learn a regression model, denoted as the regression matrixF∈ℜdm×dm+1, which maps Moτ to the regression output Y∈ℜdy×n by minimizing the fitting error WY−FβF2 [30, 33], where W∈ℜdy×dy is the diagonal regression matrix that adjusts the regression output dimension and β=AC;1T∈ℜdm+1×n denotes the augmented noise-free data matrix with the extra dimension accounting for the regression bias and dm denoting the dimensional samples. The main objective of this work is to reduce the reconstruction error through extracting the low-rank component from complex highly correlated data in statistical signal processing. The overall problem can thus be posted as the following indicating the constrained convex optimization problem:(1)minA,F,β,E,C,Q,Δτγ2UF2+A∗+C∗+λ1Q1+λ2E1s.tMoτ+∑i=1nJiΔτviviT=AC+E,β=AC;1T,C=Q,Q≽0,where U=WY−Fβ, 1n∈ℜn indicates a vector of dimension n with all one entries, A∗=∑i=1mindm,nσiA is the nuclear norm of A, in which σiA indicates the singular values of A, UF2=TraceUTU, λ1, λ2, and γ are the regularization parameters, X,Y=TraceXTY, and Q1=max1≤j≤n∑i=1nQij.The second termA∗ in (1) represents the low-rank component of A. The second and the third terms, C∗ and Q1, are to constrain the low-rank and sparse representation, respectively. The last term E1 constrains and regularizes the outliers and heavy sparse noises modelled by E to be sparse. In the constraints of (1), the affine transformations are used to alleviate the impact of outliers and heavy sparse noises, and Q is constrained to be positive semidefinite to bound the regression errors.


## 4. Proposed Method

To get the optimal solution of the convex constrained optimization problem in (1), take into consideration the augmented Lagrangian function given by(2)ℒF,β,A,C,E,Q,Δτ=γ2UF2+A∗+C∗+λ1Q1+λ2E1+Z1,Moτ+∑i=1nJiΔτviviT−AC−E+μ12Moτ+∑i=1nJiΔτviviT−AC−EF2+Z2,β−AC;1T+μ22β−AC;1TF2+Z3,C−Q+μ32C−QF2,where Z1∈ℜdm×n, Z2∈ℜdm+1×n, and Z3∈ℜn×n are the Lagrangian multipliers, and μ1, μ2, and μ3 are the penalty parameters. Considering the linearized alternating direction method with an adaptive penalty (LADMAP) [33, 76], (2) can be rewritten as(3)ℒF,β,A,C,E,Q,Δτ=γ2UF2+A∗+C∗+λ2E1+λ1Q1+μ12Moτ+∑i=1nJiΔτviviT−AC−E+Z1μ1F2+μ22β−AC;1T+Z2μ2F2+μ32C−Q+Z3μ3F2.Solving (3) directly is its computational load and complexity is very expensive, thereby we consider to iteratively updating the optimization parameters and affine transformations via ADMM [77], which decomposes the minimization into several subproblems. To make the mathematical formulation easy, we additionally assume W=I in the following derivations.Firstly, we need to get the optimal updates ofF, we fix A, E, Δτ, β, C, and Q as constants in (3), and Fk+1 can be determined by(4)Fk+1=argminFℒF,βk,Ak,Ek,Ck,Qk,Δτk,where k is the iteration index and ignoring all the irrelevant terms of F in (4), equation (4) can be reexpressed as(5)Fk+1=argminFγ2WY−FβkF2,this is exactly the same as a standard least square regression. Consequently, we can obtain(6)Fk+1=βkβkT+γIdm+1−1YβkT,where Idm+1 is a dm+1×dm+1 identity matrix.Secondly, to find the update ofβ, we fix A, E, F, C, Q, and Δτ and decide βk+1 by(7)βk+1=argminβℒFk+1,β,Ak,Ek,Ck,Qk,Δτk.Ignoring all of the irrelevant terms ofβ in (7), we can get(8)βk+1=argminβγ2UkF2+μ2k2β−AkCk;1T+Z2kμ2kF2,where Uk=WY−Fkβk. Thereby, βk+1 can be determined by(9)βk+1=γFkTWTWFk+μ2kIdm+1−1γFkTWTWY−Z2k+μ2kAkCk;1T.Similarly, to updateA, we fix E, F, β, Q, C, and Δτ and decide Ak+1 by(10)Ak+1=argminAℒFk+1,βk+1,A,Ek,Ck,Qk,Δτk.By ignoring all of the irrelevant terms ofA, (10) can be simplified as(11)Ak+1=argminAA∗+μ1k2Moτ+∑i=1nJiΔτkviviT−ACk−Ek+Z1kμ1kF2+μ22βk+1−ACk;1T+Z2kμ2kF2Considering the linear augmented direction method and the singular value threshold operator [72, 78–80], we can obtain an update of Ak+1 given by(12)Ak+1=Ω1/κAAk−PAkκA,where Ω is the singular value thresholding operator [79], PAk=μ1k+μ2kAkTCk−μ1kMoτ+∑i=1nJiΔτkviviT−Ek−Z1k−μ2kβk+Z2kCkT, κA=μ1k+μ2kτA/2, and τA>σCTC is the proximal parameter, in which σCTC denotes the spectral radius of CTC.To obtain an update ofE, by keeping A, F, β, Q, C, and Δτ as constants, then Ek+1 is can be determined by(13)Ek+1=argminEℒFk+1,βk+1,Ak+1,E,Ck,Qk,Δτk.Again, by ignoring all of the irrelevant terms ofE, (13) can be simplified as(14)Ek+1=argminEλ2E1+μ1k2Moτ+∑i=1nJiΔτkviviT−Ak+1Ck−E+Z1kμ1kF2.By employing the linearized alternating direction method,Ek+1 is updated by(15)Ek+1=Γλ1/μ1Moτ+∑i=1nJiΔτkviviT−Ak+1Ck+Z1kμ1k,where Γλ1/μ1Θ=sgnΘmaxΘ−λ1/μ1,0 is the soft shrinkage thresholding operator [78, 79], in which sgnΘ denotes the sign function.Next, to get an update ofC, we again keep A, E, F, β, Q, and Δτ as constants and Ck+1 can then be determined by(16)Ck+1=argminCℒFk+1,βk+1,Ak+1,Ek+1,C,Qk,Δτk.By ignoring all of the irrelevant terms ofC, (16) can be simplified as(17)Ck+1=argminCC∗+μ1k2Moτ+∑i=1nJiΔτkviviT−Ak+1C−Ek+1+Z1kμ1kF2+μ2k2βk+1−Ak+1C;1T+Z2kμ2kF2+μ3k2C−Qk+Z3kμ3kF2.To solve this subproblem, we can apply a linearized augmented Lagrangian multiplier with the singular value threshold operator and updateCk+1 by(18)Ck+1=Ω2/κcCk−PCkκC,where κC=μ1k+μ2k+μ3kτC/2 with τC>σAAT and PCk=Akμ1k+μ2kAk−μ1kMoτ+∑i=1nJiΔτkviviT−Ek−Z1k−μ2kβk+Z2k+μ3kCk−Qk+Z3k.To get an update ofQ, we keep A, E, F, β, C, and Δτ as constants and determine Qk+1 by(19)Qk+1=argminQℒFk+1,βk+1,Ak+1,Ek+1,Ck+1,Q,Δτk.By removing all of the irrelevant terms ofQ, (19) is reduced to(20)Qk+1=argminQλ1Q1+μ3k2Ck−Q+Z3kμ3kF2.Similarly, employing the soft threshold operator and the augmented Lagrangian multiplier, we can updateQk+1 by(21)Qk+1=Γλ2/μ3Ck+1+Z3kμ3k.To update the affine transformations, by keeping all other variables as constants, we can get(22)Δτk+1=argminΔτμ1k2Moτ+∑i=1nJiΔτviviT−Ak+1Ck+1−Ek+1+Z1kμ1kF2,Solving (22) with the threshold operators [74], we can get an update of Δτk+1 as(23)Δτk+1=∑i=1nJi+Ak+1Ck+1+Ek+1−Moτ−Z1kμ1kviviT,where Ji+ denotes the Moore–Penrose pseudoinverse of Ji [81], in which Ji denotes the Jacobian of the ith image with respect to τi as defined in Section 3.Finally, following the same steps as above, the Lagrangian multipliersZ1,Z2, and Z3 are updated by(24)Z1k+1=Z1k+μ1kMoτ+∑i=1nJiΔτk+1viviT−Ak+1Ck+1−Ek+1.(25)Z2k+1=Z2k+μ2kβk+1−Ak+1Ck+1;1T.(26)Z3k+1=Z3k+μ3kCk+1−Qk+1.Likewise, the regularization parametersμ1, μ2, and μ3 are updated respectively by(27)μ1k+1=minμmax,ρμ1k.(28)μ2k+1=minμmax,ρμ2k.(29)μ3k+1=minμmax,ρμ3k,where ρ and μmax are appropriately chosen parameters adjusting the convergence speed of the new approach.The overall updating equations of our proposed approach can be summarized as follows. First, the regression matrixF and the regression coefficients β are updated by (6) and (9), respectively. Next, A, E, C, Q, and the affine transformation, Δτ, are updated by (12), (15), (18), (21), and (23), respectively. Finally, the Lagrangian multipliers Z1, Z2, and Z3 and the regularization parameters μ1, μ2, and μ3 are updated by (24) and (29). The above updating equations proceed in a round-robin manner until convergence. For an easy understanding of the manuscript, the summarized algorithm of this work is given in Algorithm 1.Algorithm 1: ADMM for the proposed algorithm.
Input
Data matrixM∈ℜm×n, A0∈ℜm×n, E0∈ℜm×n, Δτ0∈ℜp×n, C0∈ℜn×n, Q0∈ℜn×n, λ1, λ2, ρWhile not converged do(1)
Update:Ak+1 by (12)(2)
Update:Ek+1 by (15)(3)
Update:Ck+1 by (18)(4)
Update:Qk+1 by (21)(5)
Update:Δτk+1 by (23)(6)
Update:Z1k+1 by (24)(7)
Update:Z2k+1 by (25)(8)
Update:Z3k+1 by (26)(9)
Update:μ1k+1 by (27)(10)
Update:μ2k+1 by (28)(11)
Update:μ3k+1 by (29)End whileOutputs
:A,E, C, Q, Δτ


## 5. Convergence Analysis

In this section, we consider the convergence behavior of the updating equations, by addressing two theorems related to the convergence of ADMM. We first consider the following two propositions:Proposition 1.
Ifμk is nondecreasing and upper bounded by τA>σCCT and τC>σAAT‖, the subgradients are then defined as(a)
−μkτAAk+1−Ak−α∗Z1k+1∈∂fAk+1(b)
−μkηEEk+1−Ek−Ψ∗Z2k+1∈∂gEk+1(c)
−μkτCCk+1−Ck−ω∗Z3k+1∈∂hCk+1(d)
−μkηQQk+1−Qk−φ∗Z3k+1∈∂pQk+1(e)
−μkηΔτΔτk+1−Δτk−δ∗Z3k+1∈∂qΔτk+1,
Here,∂f, ∂g, ∂h, ∂p, and ∂q are the subgradients of A, E, C, Q, and Δτ respectively, and α∗, Ψ∗, ω∗, φ∗, and δ∗ are the adjoints of the linear identity mapping operators corresponding to A, E, C, Q, and Δτ, respectively.
The proof is similar to [76].Proposition 2.
Ifμk is nondecreasing and upper bounded by τA>σCCT and τC>σAAT, then A∗,E∗,Z1∗,Z2∗ and C∗,Q∗,Δτ∗,Z3∗ are any Karush–Kuhn–Tucker (KKT) point of problem (1). Also,(a)
τAAk−A∗2−αAk−A∗2+ηEEk−E∗2−ΨEk−E∗2+μk−2Z1k−Z1∗2+μk−2Z2k−Z2∗2 is nonincreasing.(b)
Ak+1−Ak⟶0, Ek+1−Ek⟶0, Z1k+1−Z1k⟶0, Z2k+1−Z2k⟶0.(c)
τCCk−C∗2−ωCk−C∗2+ηQQk−Q∗2−φQk−Q∗2+ηΔτΔτk−Δτ∗2−δΔτk−Δτ∗2+μk−2Z3k−Z3∗2 is nonincreasing.(d)
Ck+1−Ck⟶0, Qk+1−Qk⟶0, Δτk+1−Δτk⟶0, Z3k+1−Z3k⟶0.
The proof is similar to [76].Theorem 1.
Ifμ1 and μ2 are nondecreasing and upper bounded by τA>σCCT, then the sequence Ak, Ek, Z1k, and Z2k generated by ADMM converging to a KKT point of (11) and (14).Proof.
See Appendix.
Theorem1 implies that the optimization variables Ak+1 and Ek+1 are guaranteed to converge to the global optimum with an appropriate choice of Z1k and Z2k and sufficiently large penalty parameters μ1 and μ2.Theorem 2.
Ifμ1, μ2, and μ3 are nondecreasing and upper bounded by τC>σAAT, then the sequence Ck,Qk, Δτk, and Z3k generated by ADMM converges to a KKT point of (17), (20), and (22).Proof is similar to that of Theorem1.Theorem2 shows that the updating variables Ck+1, Qk+1, and Δτk+1 are guaranteed to converge to the global optimum with an appropriate choice of Z3k and sufficiently large penalty parameters μ1, μ2, and μ3.


## 6. Experimental Results and Discussion

In this section, we first verify the correct recovery guarantee and evaluate the effectiveness of the new method image recovery and head pose estimation based either on synthetic data or on some public databases. Four baseline methods, including T-RPCA [24] +LSR, PSVT [23] +LSR, LR-RR [30], and LRS-RR [33] and the proposed one are conducted for comparison, where T-RPCA + LSR and PSVT + LSR first perform T-RPCA and PSVT on the illuminated and corrupted input data, respectively, and then conduct regression on the error free data using the standard least square regression. Then, first we try to evaluate the effectiveness of the proposed method based on the synthetic datasets. Following this, several public datasets are taken into account to verify the effectiveness of the proposed method.


### 6.1. Synthetic Data Recovery

First, we further assess the proposed algorithm on the generated synthetic data for prediction.As [30, 33], we generate 400 three-dimensional samples, in which the first 2 parts of the samples are obtained from a uniform distribution in −6,6, while the 3rd one is from two different joint subspaces given by z=u−v and z=u+v. In addition, we add zero-mean white Gaussian noise with unit variance into the second dimension, which simulates the in-subspace noise. Similarly, we add zero-mean white Gaussian sparse noise with unit variance in the 3rd subspace to simulate the noise outside subspaces. 200 data samples are randomly chosen for the training and the other 200 samples for the testing. As a fair comparison, we use the relative absolute error RAE between the true regression matrix F and the one learned, F^, i.e., RAEF=F^−FF/FF, and the RAE between the true regression output Y and the predicted one Y^, i.e., RAEY=Y^−YF/YF, as the performance measure of accuracy for regression.The comparison ofRAEF and RAEY using the aforementioned methods based on the generated synthetic data is shown in Table 1, from which we can see that PSVT + LSR yields better performance than T-RPCA + LSR, as it employs the truncated nuclear norm instead of using more tensors to deal with the outliers and heavy sparse noises.Table 1
Comparison of relative absolute error and standard deviation for the recovery of synthetic data.
MethodsRAEFRAEYT-RPCA [24] + LSR0.0740±0.10400.0530±0.1650PSVT [23] + LSR0.0697±0.08890.0150±0.0060LR-RR [30]0.0350±0.01500.0150±0.0060LRS-RR [33]0.0050±0.00050.0100±0.0030Ours0.0047±0.00010.0042±0.000038LR-RR is superior to PSVT + LSR and T-RPCA + LSR, as it cleans noises and outliers in and outside subspaces in a supervised manner to yield more precise prediction. LRS-RR provides the second best performance, as it can cope with the outliers coming from inside and outside subspaces, and the disjoint subspaces. We can also see that our method outperforms all of the baselines in both ofRAEF and RAEY. This is because it incorporates the affine transformations with the robust regression for low-rank subspace recovery, so it can handle the aggregated outliers from various subspaces and heavy sparse noises to produce more precise results.


### 6.2. Face Image Recovery

In this subsection, we assess the performance of the proposed method for face image reconstruction in terms ofRAEF and RAEY based on the images from the YaleB [82] database.First, we conduct simulations on the YaleB database [82] for face image recovery. The YaleB database consists of over 2300 frontal face images from 38 subjects with varying illuminations. First, the cropped face images with 196×128 pixels are taken from the first fifteen subjects. Next, we compute the twenty-dimensional eigenfaces based on the training images which are taken as M.In each test face image, ten blocks of30×30 of the corrupted pixels are generated randomly and added as a synthetic corrupted data. To assess the accuracy, we decide the true regression model based on the unblocked tested face images via the eigenfaces, then we learn the regression matrix by using various algorithms with eigenimages as the input M and the blocked face images as Y.The comparison ofRAEF and RAEY based on the proposed method as well as the aforementioned baselines are given in Table 2, from which we can see that LR-RR outperforms PSVT + LSR and T-RPCA + LSR as it cleans the intrasample in a supervised manner to reduce the overfitting error. LRS-RR is superior to LR-RR, as it considers the outliers and heavy sparse noises coming from disjoint subspaces into consideration, thereby reducing the fitting errors and the model errors. We can also notice that the proposed method provides the best performance. This is because the incorporation of affine transformations with robust regression enables our approach to tackle the impact of outliers and heavy sparse errors better compared with the other methods.Table 2
Comparison of relative absolute error and standard deviation for face recovery on the YaleB database.
MethodsRAEFRAEYT-RPCA [24] + LSR1.3000±0.01770.2274±0.0080PSVT [23] + LSR1.2783±0.01970.2278±0.0066LR-RR [30]1.0715±0.04300.1854±0.0056LRS-RR [33]1.0457±0.04960.1659±0.0061Ours0.2265±0.00100.1482±0.0055As an illustration, we provide some corrupted images recovered by the aforementioned methods, as depicted in Figure1, from which we can observe that the proposed algorithm recovers the corrupted data better compared with the other four baselines. As shown in Figure 1(g), the recovered image provides a clearer visual quality by removing the corruption. This is in agreement with the results in Table 2 and further justifies that the new approach is more resilient to outliers and heavy sparse noises.Figure 1
Face image recovery: (a) original; (b) corrupted; (c) T-RPCA + LSR; (d) PSVT + LSR; (e) LR-RR; (f) LRS-RR; (g) ours.


### 6.3. Head Pose Estimation

In this section, we conduct simulations for head pose estimation based on the CMU multi-PIE database [83], which consists of more than 5000 faces taken from 53 different subjects. In this experiment, nine images are chosen with yaw pose angles that vary from θ=−90o,90o with an increment of 22.5°. The face images are cropped around the faces and resized to 48×48. We also readjust the face images after applying linearization to M and the yaw angles are considered as the output, denoted as Y=cosθ;sinθ [30, 33].The comparison of the averaged yaw angle errors by the proposed method and the four baselines is given in Table3, from which we can see that PSVT + LSR yields smaller yaw angle errors than T-RPCA + LSR. This is because the performance of T-RPCA is influenced by the number of tensors, so it cannot work well when there are lots of outliers and heavy sparse noise.Table 3
Comparison of the average of the yaw angle errors and its standard deviation on the CMU database.
MethodsYaw angle errorT-RPCA [24] + LSR24.1636o±20.2201oPSVT [23] + LSR23.2429o±19.0697oLR-RR [30]1.97o±5.77oLRS-RR [33]1.03o±5.65oOurs0.95o±5.26oWe can also notice that LR-RR outperforms the above two methods, as it is based on a supervised learning to better tackle the impact of outliers and heavy sparse noises. LRS-RR produces even smaller yaw angle errors, as it also takes into account the outliers and sparse errors lying in disjoint subspaces. Our approach is superior to all of the baselines because it combines the affine transformations with more robust low-rank-sparse representation, entailing better resilience to outliers and heavy sparse noises.To further verify the performance of the proposed method as an illustration, some projections of the head pose images onto the output spaceY are also furnished in Figure 2, from which we can find that the head poses predicted by the new novel method are close to the true ones compared with the other baselines, as shown in Figure 2(e). The superiority of the new approach is to combine the affine transformations with more robust low-rank-sparse representation, so it is more robust against the cumbersome noises, outliers, and heavy sparse noises. This is again in agreement with the results in Table 3.Figure 2
Projection of the head pose images, where (“.”) denotes the output space and the red (“+”) is the ground true location of pose angles. (a) T-RPCA + LSR; (b) PSVT + LSR; (c) LR-RR; (d) LRS-RR; (e) ours.
(a)(b)(c)(d)(e)


## 6.1. Synthetic Data Recovery

First, we further assess the proposed algorithm on the generated synthetic data for prediction.As [30, 33], we generate 400 three-dimensional samples, in which the first 2 parts of the samples are obtained from a uniform distribution in −6,6, while the 3rd one is from two different joint subspaces given by z=u−v and z=u+v. In addition, we add zero-mean white Gaussian noise with unit variance into the second dimension, which simulates the in-subspace noise. Similarly, we add zero-mean white Gaussian sparse noise with unit variance in the 3rd subspace to simulate the noise outside subspaces. 200 data samples are randomly chosen for the training and the other 200 samples for the testing. As a fair comparison, we use the relative absolute error RAE between the true regression matrix F and the one learned, F^, i.e., RAEF=F^−FF/FF, and the RAE between the true regression output Y and the predicted one Y^, i.e., RAEY=Y^−YF/YF, as the performance measure of accuracy for regression.The comparison ofRAEF and RAEY using the aforementioned methods based on the generated synthetic data is shown in Table 1, from which we can see that PSVT + LSR yields better performance than T-RPCA + LSR, as it employs the truncated nuclear norm instead of using more tensors to deal with the outliers and heavy sparse noises.Table 1
Comparison of relative absolute error and standard deviation for the recovery of synthetic data.
MethodsRAEFRAEYT-RPCA [24] + LSR0.0740±0.10400.0530±0.1650PSVT [23] + LSR0.0697±0.08890.0150±0.0060LR-RR [30]0.0350±0.01500.0150±0.0060LRS-RR [33]0.0050±0.00050.0100±0.0030Ours0.0047±0.00010.0042±0.000038LR-RR is superior to PSVT + LSR and T-RPCA + LSR, as it cleans noises and outliers in and outside subspaces in a supervised manner to yield more precise prediction. LRS-RR provides the second best performance, as it can cope with the outliers coming from inside and outside subspaces, and the disjoint subspaces. We can also see that our method outperforms all of the baselines in both ofRAEF and RAEY. This is because it incorporates the affine transformations with the robust regression for low-rank subspace recovery, so it can handle the aggregated outliers from various subspaces and heavy sparse noises to produce more precise results.


## 6.2. Face Image Recovery

In this subsection, we assess the performance of the proposed method for face image reconstruction in terms ofRAEF and RAEY based on the images from the YaleB [82] database.First, we conduct simulations on the YaleB database [82] for face image recovery. The YaleB database consists of over 2300 frontal face images from 38 subjects with varying illuminations. First, the cropped face images with 196×128 pixels are taken from the first fifteen subjects. Next, we compute the twenty-dimensional eigenfaces based on the training images which are taken as M.In each test face image, ten blocks of30×30 of the corrupted pixels are generated randomly and added as a synthetic corrupted data. To assess the accuracy, we decide the true regression model based on the unblocked tested face images via the eigenfaces, then we learn the regression matrix by using various algorithms with eigenimages as the input M and the blocked face images as Y.The comparison ofRAEF and RAEY based on the proposed method as well as the aforementioned baselines are given in Table 2, from which we can see that LR-RR outperforms PSVT + LSR and T-RPCA + LSR as it cleans the intrasample in a supervised manner to reduce the overfitting error. LRS-RR is superior to LR-RR, as it considers the outliers and heavy sparse noises coming from disjoint subspaces into consideration, thereby reducing the fitting errors and the model errors. We can also notice that the proposed method provides the best performance. This is because the incorporation of affine transformations with robust regression enables our approach to tackle the impact of outliers and heavy sparse errors better compared with the other methods.Table 2
Comparison of relative absolute error and standard deviation for face recovery on the YaleB database.
MethodsRAEFRAEYT-RPCA [24] + LSR1.3000±0.01770.2274±0.0080PSVT [23] + LSR1.2783±0.01970.2278±0.0066LR-RR [30]1.0715±0.04300.1854±0.0056LRS-RR [33]1.0457±0.04960.1659±0.0061Ours0.2265±0.00100.1482±0.0055As an illustration, we provide some corrupted images recovered by the aforementioned methods, as depicted in Figure1, from which we can observe that the proposed algorithm recovers the corrupted data better compared with the other four baselines. As shown in Figure 1(g), the recovered image provides a clearer visual quality by removing the corruption. This is in agreement with the results in Table 2 and further justifies that the new approach is more resilient to outliers and heavy sparse noises.Figure 1
Face image recovery: (a) original; (b) corrupted; (c) T-RPCA + LSR; (d) PSVT + LSR; (e) LR-RR; (f) LRS-RR; (g) ours.


## 6.3. Head Pose Estimation

In this section, we conduct simulations for head pose estimation based on the CMU multi-PIE database [83], which consists of more than 5000 faces taken from 53 different subjects. In this experiment, nine images are chosen with yaw pose angles that vary from θ=−90o,90o with an increment of 22.5°. The face images are cropped around the faces and resized to 48×48. We also readjust the face images after applying linearization to M and the yaw angles are considered as the output, denoted as Y=cosθ;sinθ [30, 33].The comparison of the averaged yaw angle errors by the proposed method and the four baselines is given in Table3, from which we can see that PSVT + LSR yields smaller yaw angle errors than T-RPCA + LSR. This is because the performance of T-RPCA is influenced by the number of tensors, so it cannot work well when there are lots of outliers and heavy sparse noise.Table 3
Comparison of the average of the yaw angle errors and its standard deviation on the CMU database.
MethodsYaw angle errorT-RPCA [24] + LSR24.1636o±20.2201oPSVT [23] + LSR23.2429o±19.0697oLR-RR [30]1.97o±5.77oLRS-RR [33]1.03o±5.65oOurs0.95o±5.26oWe can also notice that LR-RR outperforms the above two methods, as it is based on a supervised learning to better tackle the impact of outliers and heavy sparse noises. LRS-RR produces even smaller yaw angle errors, as it also takes into account the outliers and sparse errors lying in disjoint subspaces. Our approach is superior to all of the baselines because it combines the affine transformations with more robust low-rank-sparse representation, entailing better resilience to outliers and heavy sparse noises.To further verify the performance of the proposed method as an illustration, some projections of the head pose images onto the output spaceY are also furnished in Figure 2, from which we can find that the head poses predicted by the new novel method are close to the true ones compared with the other baselines, as shown in Figure 2(e). The superiority of the new approach is to combine the affine transformations with more robust low-rank-sparse representation, so it is more robust against the cumbersome noises, outliers, and heavy sparse noises. This is again in agreement with the results in Table 3.Figure 2
Projection of the head pose images, where (“.”) denotes the output space and the red (“+”) is the ground true location of pose angles. (a) T-RPCA + LSR; (b) PSVT + LSR; (c) LR-RR; (d) LRS-RR; (e) ours.
(a)(b)(c)(d)(e)


## 7. Conclusions

In this work, we considered affine transformation for image recovery and head pose estimation to remove the potential impacts of annoying effects in statistical signal processing. This approach is very useful to correct the distorted or misaligned images. The determination of the affine transformations as well as the optimization parameters are formulated as a convex optimization problem. Thereafter, the ADMM approach is considered and a new set of parameters and equations is derived to update the parameters and affine transformations iteratively in a round-robin manner. Additionally, the convergence of the developed updating equations is addressed as well. The experimental conducted simulations show that the new approach outperforms the state-of-the-art method for head pose estimation and face recovery on some common databases.



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*Source: 2054546-2022-02-18.xml*