This paper addresses the problem of 3D face recognition using simultaneous sparse approximations on the sphere. The 3D face point clouds are first aligned with a novel and fully automated registration process. They are then represented as signals on the 2D sphere in order to preserve depth and geometry information. Next, we implement a dimensionality reduction process with simultaneous sparse approximations and subspace projection. It permits to represent each 3D face by only a few spherical functions that are able to capture the salient facial characteristics, and hence to preserve the discriminant facial information. We eventually perform recognition by effective matching in the reduced space, where Linear Discriminant Analysis can be further activated for improved recognition performance. The 3D face recognition algorithm is evaluated on the FRGC v.1.0 data set, where it is shown to outperform classical state-of-the-art solutions that work with depth images.
Many recognition systems use depth or range images that permit to formulate the 3D face recognition as a problem of dimensionality reduction for planar images, where each pixel value represents the distance from the sensor to the facial surface. Principal Component Analysis (PCA) and Eigenfaces'' can be used for dimensionality reduction @cite_15 , where the basis vectors are however typically holistic and of global support. PCA can be combined with Linear Discriminant Analysis (LDA) to form Fisherfaces'' with enhanced class separability properties @cite_26 . Alternatively, dimensionality reduction can be performed via variants of non-negative matrix factorization (NMF) algorithms @cite_17 @cite_4 @cite_14 that produce part-based decompositions of the depth images. Part-based decompositions based on non-negative sparse coding @cite_19 have recently been shown to provide improved recognition performance than NMF methods in face recognition @cite_27 . Recent methods have proposed to concentrate dimensionality reduction around facial landmarks like the nose tip @cite_8 or in multiple carefully chosen regions @cite_29 or to compute geodesic distances among the selected fiducial points @cite_25 . They however require a selection of the fiducial points or areas of interest that is often performed manually and prevents the implementation of fully automatic systems.
@cite_14 @cite_26 @cite_4 @cite_8 @cite_29 @cite_19 @cite_27 @cite_15 @cite_25 @cite_17
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Non-negative matrix factorization (NMF) is a recently developed technique for finding parts-based, linear representations of non-negative data. Although it has successfully been applied in several applications, it does not always result in parts-based representations. In this paper, we show how explicitly incorporating the notion of 'sparseness' improves the found decompositions. Additionally, we provide complete MATLAB code both for standard NMF and for our extension. Our hope is that this will further the application of these methods to solving novel data-analysis problems. We develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression. Taking a pattern classification approach, we consider each pixel in an image as a coordinate in a high-dimensional space. We take advantage of the observation that the images of a particular face, under varying illumination but fixed pose, lie in a 3D linear subspace of the high dimensional image space-if the face is a Lambertian surface without shadowing. However, since faces are not truly Lambertian surfaces and do indeed produce self-shadowing, images will deviate from this linear subspace. Rather than explicitly modeling this deviation, we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation. Our projection method is based on Fisher's linear discriminant and produces well separated classes in a low-dimensional subspace, even under severe variation in lighting and facial expressions. The eigenface technique, another method based on linearly projecting the image space to a low dimensional subspace, has similar computational requirements. Yet, extensive experimental results demonstrate that the proposed "Fisherface" method has error rates that are lower than those of the eigenface technique for tests on the Harvard and Yale face databases. A new variant ‘PMF’ of factor analysis is described. It is assumed that X is a matrix of observed data and σ is the known matrix of standard deviations of elements of X. Both X and σ are of dimensions n × m. The method solves the bilinear matrix problem X = GF + E where G is the unknown left hand factor matrix (scores) of dimensions n × p, F is the unknown right hand factor matrix (loadings) of dimensions p × m, and E is the matrix of residuals. The problem is solved in the weighted least squares sense: G and F are determined so that the Frobenius norm of E divided (element-by-element) by σ is minimized. Furthermore, the solution is constrained so that all the elements of G and F are required to be non-negative. It is shown that the solutions by PMF are usually different from any solutions produced by the customary factor analysis (FA, i.e. principal component analysis (PCA) followed by rotations). Usually PMF produces a better fit to the data than FA. Also, the result of PF is guaranteed to be non-negative, while the result of FA often cannot be rotated so that all negative entries would be eliminated. Different possible application areas of the new method are briefly discussed. In environmental data, the error estimates of data can be widely varying and non-negativity is often an essential feature of the underlying models. Thus it is concluded that PMF is better suited than FA or PCA in many environmental applications. Examples of successful applications of PMF are shown in companion papers. Neural networks in the visual system may be performing sparse coding of learnt local features that are qualitatively very similar to the receptive fields of simple cells in the primary visual cortex, V1. In conventional sparse coding, the data are described as a combination of elementary features involving both additive and subtractive components. However, the fact that features can ‘cancel each other out’ using subtraction is contrary to the intuitive notion of combining parts to form a whole. Thus, it has recently been argued forcefully for completely non-negative representations. This paper presents Non-Negative Sparse Coding (NNSC) applied to the learning of facial features for face recognition and a comparison is made with the other part-based techniques, Non-negative Matrix Factorization (NMF) and Local-Non-negative Matrix Factorization (LNMF). The NNSC approach has been tested on the Aleix–Robert (AR), the Face Recognition Technology (FERET), the Yale B, and the Cambridge ORL databases, respectively. In doing so, we have compared and evaluated the proposed NNSC face recognition technique under varying expressions, varying illumination, occlusion with sunglasses, occlusion with scarf, and varying pose. Tests were performed with different distance metrics such as the L1-metric, L2-metric, and Normalized Cross-Correlation (NCC). All these experiments involved a large range of basis dimensions. In general, NNSC was found to be the best approach of the three part-based methods, although it must be observed that the best distance measure was not consistent for the different experiments. We present a systematic procedure for selecting facial fiducial points associated with diverse structural characteristics of a human face. We identify such characteristics from the existing literature on anthropometric facial proportions. We also present three dimensional (3D) face recognition algorithms, which employ Euclidean geodesic distances between these anthropometric fiducial points as features along with linear discriminant analysis classifiers. Furthermore, we show that in our algorithms, when anthropometric distances are replaced by distances between arbitrary regularly spaced facial points, their performances decrease substantially. This demonstrates that incorporating domain specific knowledge about the structural diversity of human faces significantly improves the performance of 3D human face recognition algorithms. Non-negative matrix factorization (NMF) has previously been shown to be a useful decomposition for multivariate data. Two different multiplicative algorithms for NMF are analyzed. They differ only slightly in the multiplicative factor used in the update rules. One algorithm can be shown to minimize the conventional least squares error while the other minimizes the generalized Kullback-Leibler divergence. The monotonic convergence of both algorithms can be proven using an auxiliary function analogous to that used for proving convergence of the Expectation-Maximization algorithm. The algorithms can also be interpreted as diagonally rescaled gradient descent, where the rescaling factor is optimally chosen to ensure convergence.