Journal of Computational Information Systems 11: 19 (2015) 7103–7112 Available at http://www.Jofcis.com
Depth Estimation using Hybrid Optimization Algorithm Punnam Chandar. KOTHAPELLI 1,∗, 1 University
Satya Savithri. TIRUMALA 2
College of Engineering, Kakatiya University, Warangal 506002, India
2 Jawaharlal
Nehru Technological University, Hyderabad 999008, India
Abstract In this paper, depth estimation of important features from one or more multi-view 2D face images i.e., 2D to 3D problem is considered as a global continuous optimization problem and solved by means of novel hybrid optimization algorithm comprising of Differential Evolution (DE) and Improved Iterative Soft Thresholding algorithm (IISTA). Our proposed algorithm consists of two stages. In the first stage head pose is estimated using continuous Differential Evolution Optimization. In the second stage, depth of the important features are estimated using our proposed IISTA. Our approach is validated by computing the Pearson Linear Correlation Coefficient of the estimated depths and true depths on 3D Bosphorus Database. Extensive experimental results on this database show that our proposed algorithm can estimate the sparse 3D face satisfactorily in comparison to the state-of-the-art. All the simulations were carried out in Matlab. Keywords: Candide Model; Differential Evolution Optimization; Iterative Soft Thresholding Algorithm; 3D Reconstruction; Structure from Motion
1
Introduction
Reconstruction of 3D model from 2D images is a classic problem in computer vision. The reconstructed shape can be expressed in several ways: depth Z(x, y), surface normal (nx , ny , nz ), surface gradient (p, q), and surface slant, φ, and tilt, θ. The depth can be considered as the relative distance from camera to surface points, or the relative surface height above the x − y plane. The surface normal is the orientation of a vector perpendicular to the tangent plane on ∂z ∂z the subject surface. The surface gradient, (p, q) = ( ∂x , ∂y ), is the rate of change of depth in the x and y directions. The surface slant, φ, and tilt θ, are related to the surface normal as (nx , ny , nz ) = (l sin φ cos θ, l sin φ sin θ, l cos φ), where l is the magnitude of the surface normal [1]. The reconstructed 3D models, expressed in depth Z(x, y) are useful in face image processing, like face recognition, face tracking, face animation, etc. [2, 3] as they are invariant to viewpoint, pose and illumination [4]. Presently there are two main stream approaches to create 3D face models. The first one is to use specialized 3D cameras (Konica Minolta, Hammamatsu) capable of capturing depth Z(x, y) in ∗
Corresponding author. Email address: k
[email protected] (Punnam Chandar. KOTHAPELLI).
1553–9105 / Copyright © 2015 Binary Information Press DOI: 10.12733/jcis15526 October 1, 2015
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addition to texture. The second one is to use 3D reconstruction algorithms capable of retrieving the depth information which is mathematically erased during the projection process from one or more multi-view 2D images. Recovering the depth information based on the second way is an important tool that can be used in surveillance applications to alleviate the problems of face recognition in low quality multi view 2D images [5] and video sequences [6]. In recent years 2D to 3D reconstruction research has received much attention and many algorithms for reconstructing the 3D model from 2D images have been proposed, such as Shape-from shading [7-9], Structured Light [28, 29], 3D Morphable Model [10-12], structure from motion [5, 13]. Structure-from-motion (SFM) is a popular approach to recover the 3D shape of an object when multiple frames of an image sequence are available. Given a set of observations of 2D feature points, SFM can estimate the 3D structure of the feature points. After proving the rank-3 theorem, i.e., the rank of the observation matrix is 3 under an orthographic projection, a robust factorization algorithm was proposed in [14] to factor the observation matrix into a shape matrix and a motion matrix using the singular value decomposition (SVD) technique. By considering the observations as mixing signals, a novel algorithm for maximizing the posterior shape was developed in [15] to estimate the shape from a perspective of blind source separation (BSS). In [13], a 3D object is assumed to be non-rigid, and the observed shapes are represented as a linear combination of a few basis shapes. The 3D face shape is predicted from a single image in [16] by using PLS to explore the relationship between the intensity images and 3-D shape. In [17], a Gaussian prior is assumed for the shape coefficients, and the optimization is solved using the expectation-maximization (EM) algorithm. In [5], the 3D to 2D projection process is assumed to be orthographic and the parameters of projection process are recovered using Genetic Algorithm (GA). In recent years Differential Evolution (DE) algorithm emerged as a very competitive form of optimization algorithm. Compared to the GA the control parameters are very few and the optimization is very robust to noise which makes DE as a choice of many researchers to solve real world engineering problems. DE has been widely used in various fields e.g., signal processing, artificial neural networks, Bio informatics, Pattern Recognition and Image Processing, etc., [18, 19]. Iterative soft thresholding algorithm (ISTA) [20, 21] defines a generative model for the observed multivariate data, in which the data variables are assumed to be linear mixtures of some known latent variables. These latent variables are assumed to be non-Gaussian and mutually independent sources of the observed data. In this paper, we propose a novel depth estimation method, based on DE and Improved ISTA (IISTA) for reconstructing the 3D face model of a human face from a sequence of 2D face images. The significant advantage of the proposed hybrid algorithm is the provision of a general frame work to incorporate prior information so as to make a more accurate and reliable estimation. In our proposed algorithm, the rotation and translation process for converting a frontal-view face image to a non-frontal-view face is formulated as a optimization problem by referring to the shape alignment approach in [5]. During the past decades, the CANDIDE 3-D face model [23] have been used for 3-D face representation and recognition, mainly because of its simplicity and public availability. The model is a parameterized face mask specifically developed for the modelbase coding of human faces with expressions. The third version of Candide model, Candide-3, is composed of 113 vertices and 168 triangular surfaces as shown in Fig. 1(a). Each vertex is represented by its 3D Coordinates. Considering the depth values [z-coordinates] of the Candide
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Fig. 1: (a) Candide model (b) Bosphorus database image features numbered model as one input along with the frontal view coordinates forms initial input 3D face structure to the optimization problem. The work in this paper is summarized as follows: First, Differential Evolution Optimization is used to estimate the head pose of the subject i.e., Rotation Matrix. Second Improved Iterative Soft Thresholding algorithm is used to estimate the depth values of important feature points from the 2D face image. Experimentation is carried on 3D Bosphorus Database and the results demonstrate the feasibility and efficiency of the proposed method. The remainder part of the paper is organized as follows. In Section 2, we present our proposed hybrid optimization algorithm, experimental results and related discussions are given in Section 3 and concluding remarks are presented in Section 4.
2
Depth Estimation using Hybrid Optimization
In this paper a hybrid optimization algorithm is proposed to recover the pose and 3D structure of the human face based on 2D images by projecting its 3D model on to the 2D plane, i.e. 2D to 3D problem. We assume that n shape feature points represented by the (xi , yi )i=1:n coordinates shown in Fig. 1(b) are marked accurately. (Mxi , Myi , Mzi ) represents the i-th feature point of a frontal-view 3D face model M . Mxi and Myi are measured from the image being adapted, while Mzi is initially set at the default values of the CANDIDE model Zc . (qxi , qyi ) is a i-th feature point of a non-frontal-view 2D face q. The rotation matrix R for q is given as follows: cos φ sin φ 0 cos ψ 0 − sin ψ 1 0 0 r11 r12 r13 0 0 cos θ sin θ = r21 r22 r23 R= (1) − sin φ cos φ 0 1 0 0 0 1 sin ψ 0 cos ψ 0 − sin θ cos θ r31 r32 r33 Where the pose parameters φ, ψ, and θ are the rotation angles around the x, y, and z axes, respectively. Then the rotation and translation process for mapping the frontal-view face image to the non-frontal-view face image can be given by similarity transform: Mx1 Mx2 Mx3 · · · Mxn qx1 qx2 · · · qxn r11 r12 r13 tx1 tx2 · · · txn =k· (2) My1 My2 My3 · · · Myn + qy1
qy2
···
qyn
r21
r22
r23
Mz1
Mz2
Mz3
···
Mzn
ty1
ty2
···
tyn
Where k is the scale factor and (tx , ty ) are the translations along x and y axes. In matrix form equation above can be written as follows: q = k · R2×3 · M + T
(3)
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Initializing of Parameter Vectors
Difference Vectors Based Mutation
Crossover/ Recombination
Selection
Fig. 2: Differential evolution algorithm stages Where q is a 2 × n matrix such that each column represents the (x, y) coordinates (qxi , qyi )T of one feature point, M is a matrix such that each column represents (x, y, z) the coordinates (Mxi , Myi , Mzi )T of one feature point, and T is a 2 × n matrix such that all columns are (txi , tyi )T . In terms of the shape-alignment approach in [5], the translation term T can be eliminated if both q and M are centered at the origin, i.e., q ← q − q¯ ¯ M ←M −M q = k · R2×3 · M
(4)
Denote A = k · R2×3 , then Eq. (4) can be written as q =A·M
(5)
In the Eq. (5), the Rotation Matrix A and the depth values (Mz ) in M = (Mx , My , Mz ) are to be estimated. The estimation is carried in two stages. In the first stage, the rotation matrix is estimated using Differential Evolution incorporating the initial depths with Candide-3 depth values. The initial Candide depths are corresponding to general face structure, therefore, in the second stage, the depth values of important features are estimated using our proposed IISTA.
2.1
Hybrid optimization algorithm
2.1.1
Differential evolution optimization
Differential Evolution algorithm is a recently proposed real parameter optimization algorithm [24], and its functioning is similar to the Genetic Algorithm and works through a simple cycle of stages shown in Fig. 2. Differential Evolution like the method of Genetic Algorithm generates the initial population (poses) that are randomly generated and evenly distributed to form the initial population. To evolve new individuals that will be part of the next generation are created by combining members of the current population. Every individual acts as a parent and is associated to a donor vector. In the basic version of DE, the donor vector Di for the i-th parent (Xi ) is generated, by combining three random and distinct population members (Xa ), (Xb ) and (Xc ) as follows: ∀i ∈ n : Di = Xa + F · (Xb − Xc ) (6) Where i, a, b, c are distinct. Where (Xb ) and (Xc ) are randomly chosen and (Xa ) is chosen either randomly or as one of the best members of the population, F (Scale Factor) is a real valued parameter that strongly influences DE performance and typically lies in the interval [0.4,1]. Other mutation strategies have been applied to DE, experimenting with different base vectors and different number of vectors for perturbation. After mutation a trial vector Ti,j is generated by choosing between the donor vector and the previous generation for each element (j) according
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Table 1: Six strategies of DE Mutation Scheme
abbreviation
DE\rand\1
DE S1
DE\local-to-best\1
DE S2
DE\best\1 with jitter
DE S3
DE\rand\1 with per-vector-dither
DE S4
DE\rand\1 with per-generation-dither
DE S5
DE\rand\1 either-or-algorithm
DE S6
to the crossover rate CR which lies in the interval [0,1], for each element in the vector we choose either the corresponding element form the previous generation vector or from the donor vector such that ∀i, j : if (ramdom < CR || j = jrand )thenTi,j = Di,j otherwiseTi,j = Xi,j
(7)
Where jrand is randomly chosen for each iteration through i and ensures that no Ti is exactly the same as the corresponding Xi . Then the trial vectors fitness is evaluated, and for each member of the new generation, Xi , we choose the better performing of the previous generation, Xi , or the trial vector, Ti . The six strategies of DE proposed by storn and price with binomial crossover are listed in Table 1. In the first stage the optimal pose parameters in rotation matrix R are estimated using Differential Evolution Optimization with CANDIDE depths as initial values. If the pose of 3D face model and the depths of the feature points fit the non-frontal-view face image q, the following distance will be a minimum. d = kq − A · M k22
(8)
The fitness of each candidate in the population is measured based on Eq. (8). In the second stage given the estimated pose parameters i.e., matrix A, the depth values Mzi are estimated using Improved Iterative Soft Thresholding algorithm (IISTA). 2.1.2
Improved Iterative Soft Thresholding Algorithm
Differential Evolution optimization is used to estimate the matrix A for q with initial depth values of Candide model in the first iteration, the Candide face structure is an reference of general face structure only. Therefore, to accurately estimate the depth values in M , we propose Improved Iterative Soft Thresholding Algorithm to recover the source signals. This algorithm in general form appeared in optimization literature as ISTA and used for the purpose of wavelet-based 1D signal restoration [20, 22]. In this work we are extending the general 1D ISTA algorithm to 2D ISTA i.e., Improved Iterative Soft Thresholding Algorithm and is derived based on concepts of majorization-minimization and on l1 -norm regularization leading to soft-thresholding for the estimation of 3D structure. The orthographic projection model from 3D to 2D is given in Eq. (5) and we are interested in estimating the depth values in M , and A is the matrix representing the observation process (rotation, translation and scaling) and is estimated in stage 1 using DE. The estimation of M from q given A can be viewed as a linear inverse problem. A standard approach to solve linear inverse problems is to define a suitable objective function J(M ) and to find the
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depth values minimizing J(M ). The objective function J(M ) in general expressed as sum of two terms: J(M ) = D(q, A · M ) + λ · R(M ) (9) where D(q, A · M ) measures the discrepancy between q and A · M , R(M ) is a regularization term. The parameter λ is called the regularization parameter and is used to adjust the trade-off between the two terms; λ should be a positive value. For D(q, A · M ) we will use the mean square error, namely D(q, A · M ) = kq − A · M k22 (10) The notation kV k22 represents the sum of squares of the vector V , kV k22 = v12 + v22 + · · · + vn2 . Minimizing this D(q, A · M ) will give a signal M which is the estimated 3D model corresponding to q according to the square error criterion. We could try to minimize D(q, A · M ) by setting M = A−1 · q; however, A may not be invertible. Even if A were invertible, it may be very illconditioned in which case this solution may not reflect the 3D model i.e., the estimated depth values may vary much from actual depth values of face under consideration. The role of the regularization term is exactly to address this problem. The regularizer R(M ) should be chosen so as to penalize undesirable/unwanted behavior in M . As the signal M is a non-Gaussian signal, we choose l1 -norm of M as the regularization term, which is defined as
Mx1 Mx2 Mx3
Mxn
(11) kM k1 =
My1 + My2 + My3 + · · · + Myn
Mz1 Mz2 Mz3
Mzn Hence, the approach is to estimate M , from q by minimizing the objective function J(M ) = kq − A · M k22 + λ · kM k22
(12)
This is called an l1 -norm regularized linear inverse problem. To minimize the Eq. (12) to estimate the M we employ Majorization-minimization approach. 2.1.3
Majorization minimization
Majorization-minimization (MM) replaces the difficult minimization problem in Eq. (12) by a sequence of easier minimization problems. The MM approach generates a sequence of Vectors [Mxk , Myk , Mzk ]Tk=0,1,2,3,4,··· . which converge to desired solution. The MM can be described as follows. Suppose we have a vector M k , a guess for the minimum of J(M ). Based on M k , we would like to find a new M k+1 which further decrease i.e., we want to find M k+1 such that J(M k+1 < J(M k )). The MM approach asks us first to choose a new function which majorizes J(M ) and, second, that we minimize the new function to get M k+1 . MM puts some requirements on this new function, call it G(M ). We should choose G(M ) such that G(M ) ≥ J(M ) for all M . In addition G(M ) should equal J(M ) at M k . We find M k+1 by minimizing G(M ). The function G(M ) will be different at each iteration. So we denote it Gk (M ). To summarize, the majorization-minimization algorithm for the minimization of a function J(M ) is given by the following iteration: Step 1 Set k = 0. Initialize M 0 .
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Step 2 Choose Gk (M ) such that. [a] Gk (M ) ≥ J(M ). [b] Gk (M k ) = J(M k ). Step 3 Set M k+1 as the minimize of Gk (M ). Step 4 Set k = k + 1 and go to Step 2. Applying the majorization and minimization procedure with l1 -norm for estimating the 3D face structure results in iterative update equation i.e., IISTA and is given below: λ 1 (13) M k+1 = sof t(M k + AT (q − A · M k ), ) α 2α where α ≥ maxeig(AT A). The soft threshold-ed rule is the non-linear function defined as
sof t(x, T ) =
3
x + T when x 6 −T
(14)
0 when |x| 6 T x − T when x > T
Experimental Results on Bosphorus Database
The first 30 subjects from the Bosphorus database [25] were used in the experiments. Note that images with unseen feature points cannot be selected as training images, as the corresponding depth values cannot be estimated. As a result, only five non-frontal-view face images, PR D, PR SD, PR SU, PR U and YR R10 can be used to train the model in the experiments and are shown in Fig. 3 for one subject in the database. In the experiments, the pose parameters φ, ψ, and θ are initially set to be zeros. The scale parameter k is set to be 1. We can obtain one set of depth values for the facial-feature points when each non-frontal-view face image is combined with its corresponding frontal-view face image for the hybrid optimization. We have used DE six strategies with Cr=0.2, F=0.4 and refined during experimentation with initial population of 30 for the first stage and IISTA with λ = 0.0001 chosen based on empirical study. All the simulations were conducted using MATLAB running on an ordinary computer. Further, we have measured the similarity of the estimated depths with true depths using Pearson linear correlation [26] coefficient, a metric commonly used for similarity index and are given in Table. 2. Fig. 4 & 5
Fig. 3: (a) Front (b) PR D
(c) PR SD
(d) PR SU
(e) PR U
(f) YR R10
shows the extensive experimental comparison results of the correlation coefficients of the estimated depth values and Candide depths, estimated depths and true depth values and True depth values and candide depth values with one non-frontal-view face image PR D, PR SD, PR SU, PR U & YR R10 and corresponding frontal view using DE-S1-IISTA & DE-S2-IISTA for the 31 subjects of the Bosphorus database.
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Table 2: Pearson correlation coefficients of estimated depth values and true depth values of five subjects of bosphorus database
S.No
PR D
PR SD
PR SU
PR U
YR R10
Average±Std
Subject1
0.89547
0.891205
0.886641
0.883051
0.896535
0.8905±0.005744
Subject2
0.88663
0.876436
0.880269
0.877998
0.896683
0.8836±0.008279
Subject6
0.86981
0.862595
0.858267
0.857091
0.867473
0.8630±0.005564
Subject8
0.91441
0.915564
0.918433
0.917968
0.919148
0.9171±0.002019
Subject12
0.90739
0.895185
0.894074
0.905831
0.899861
0.9004±0.006038
1
1
1
1
0.9
0.9
0.9
0.9
1
c(Mz,Mzb) c(Mzb,Mzc)
0.5
c(Mz,Mzc)
0.7
c(Mz,Mzb) c(Mzb,Mzc)
0.6
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c(Mz,Mzb) c(Mzb,Mzc)
0.6
0.5
0.8 Correlation Coefficients
c(Mz,Mzc)
0.6
0.8
Correlation Coefficients
0.7
Correlation Coefficients
Correlation Coefficients
Correlation Coefficients
0.9
0.8
0.8 c(Mz,Mzc) 0.7
c(Mz,Mzb) c(Mzb,Mzc)
0.6
0.7 0.6 c(Mz,Mzc)
0.5
c(Mz,Mzb) 0.4
c(Mzb,Mzc)
0.3 0.2
0.5
0.1 0.4
0
5
10
15 20 Subject Numbers
25
30
35
0.4
0
5
10
15 20 Subject Numbers
25
30
35
0.4
0
5
10
15 20 Subject Numbers
25
30
35
0.4
0
5
10
15 20 Subject Numbers
25
30
35
0
0
5
10
15 20 Subject Numbers
25
30
35
Fig. 4: Pearson correlation coefficients of estimated and true depths for 31 subjects using PR D, PR SD, PR SU, PR U & YR R10 as training sample using DE-S1 + IISTA
3.1
Comparison with similar Depth Estimation algorithms
The proposed approach belongs to the class of SFM, as a comparison we present here the Mean µ and Standard Deviation σ of the computed Pearson correlation coefficients for the estimated and true depths on the 3D Bosphorus Database with similar algorithms, rank-3 factorization method [14] (denoted as Rank-three), the expectation maximization algorithm [14] (denoted Exp-max), Trajectory-space representation [27] (denoted Traj-space), Depth Estimation based on genetic algorithm [5] (denoted SM), in Table 3. From the Table 3 it can be observed that the proposed hybrid optimization algorithm is having highest µ and least σ in comparison to the similar stateof-the-art general depth estimation algorithms. Our algorithm is specifically designed to recover the sparse 3D structure for offline applications.
4
Conclusion
In this paper, Hybrid Optimization algorithm is proposed to estimate the 3D face structure from 2D images. The Hybrid Optimization scheme is comprised of Differential Evolution and followed by our proposed IISTA. Further, our algorithm requires only one frontal and one non-frontal view face image to estimate the 3D face structure. The experimental results verify that the proposed hybrid optimization scheme can improve the 3D face reconstruction accuracy compared to similar depth estimation algorithms. Experimental results on 3D Bosphorus database have demonstrated the feasibility and efficiency of the proposed method. In future work, we will further improve the estimation accuracy of the 3D face model.
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1
1
1
1
0.9
0.9
0.9
0.9
7111
1
c(Mz,Mzb) c(Mzb,Mzc)
0.6
0.5
0.8 c(Mz,Mzc) c(Mz,Mzb)
0.7
c(Mzb,Mzc) 0.6
0.5
0.8
0.7
c(Mz,Mzc) c(Mz,Mzb)
0.6
c(Mzb,Mzc)
0.5
0.8 Correlation Coefficients
c(Mz,Mzc)
0.7
Correlation Coefficients
0.8
Correlation Coefficients
Correlation Coefficients
Correlation Coefficients
0.9
0.8
0.7
c(Mz,Mzc) c(Mz,Mzb) c(Mzb,Mzc)
0.6
0.7
c(Mz,Mzc)
0.6
c(Mz,Mzb)
0.5
c(Mzb,Mzc)
0.4 0.3 0.2
0.5
0.1 0.4
0
5
10
15 20 Subject Numbers
25
30
35
0.4
0
5
10
15 20 Subject Numbers
25
30
0.4
35
0
5
10
15 20 Subject Numbers
25
30
0.4
35
0
5
10
15 20 Subject Numbers
25
30
35
0
0
5
10
15 20 Subject Numbers
25
30
35
Fig. 5: Pearson correlation coefficients of estimated and true depths for 31 subjects using PR D, PR SD, PR SU, PR U & YR R10 as training sample using DE-S2 + IISTA
Table 3: Mean and standard deviation of pearson linear correlation coefficients for different SFM algorithms
Algorithm
µ
σ
DE-S1-IISTA
0.9171
0.0020
DE-S2-IISTA
0.9172
0.0025
Rank-Three
0.9096
0.0327
Exp-Max
0.8424
0.1658
Traj-Space
0.6440
0.2384
SM
0.6198
0.2608
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