The supplementary material to the NIPS version of this paper [4] contains a critical error, which was discovered several days before the conference. Unfortunately, it was too late to withdraw the paper from the proceedings. Fortunately, since that time, a correct analysis of the proposed convex programming relaxation has been developed by Emmanuel Candes of Stanford University. That analysis is reported in a joint paper, Robust Principal Component Analysis? by Emmanuel Candes, Xiaodong Li, Yi Ma and John Wright,

Many classic and contemporary face recognition algorithms work well on public data sets, but degrade sharply when they are used in a real recognition system. This is mostly due to the difficulty of simultaneously handling variations in illumination, image misalignment, and occlusion in the test image. We consider a scenario where the training images are well controlled, and test images are only loosely controlled. We propose a conceptually simple face recognition system that achieves a high degree of robustness and stability to illumination variation, image misalignment, and partial occlusion. The system uses tools from sparse representation to align a test face image to a set of frontal training images. The region of attraction of our alignment algorithm is computed empirically for public face datasets such as Multi-PIE. We demonstrate how to capture a set of training images with enough illumination variation that they span test images taken under uncontrolled illumination. In order to evaluate how our algorithms work under practical testing conditions, we have implemented a complete face recognition system, including a projector-based training acquisition system. Our system can efficiently and effectively recognize faces under a variety of realistic conditions, using only frontal images under the proposed illuminations as training.

Abstract—We consider the problem of recovering a lowrank matrix when some of its entries, whose locations are not known a priori, are corrupted by errors of arbitrarily large magnitude. It has recently been shown that this problem can be solved efficiently and effectively by a convex program named Principal Component Pursuit (PCP), provided that the fraction of corrupted entries and the rank of the matrix are both sufficiently small. In this paper, we extend that result to show that the same convex program, with a slightly improved weighting parameter, exactly recovers the low-rank matrix even if “almost all ” of its entries are arbitrarily corrupted, provided the signs of the errors are random. We corroborate our result with simulations on randomly generated matrices and errors. I.

Abstract—Compressive sensing provides a framework for recovering sparse signals of length N from M ≪ N measurements. If the measurements contain noise bounded by ɛ, then standard algorithms recover sparse signals with error at most Cɛ. However, these algorithms perform suboptimally when the measurement noise is also sparse. This can occur in practice due to shot noise, malfunctioning hardware, transmission errors, or narrowband interference. We demonstrate that a simple algorithm, which we dub Justice Pursuit (JP), can achieve exact recovery from measurements corrupted with sparse noise. The algorithm handles unbounded errors, has no input parameters, and is easily implemented via standard recovery techniques. I.

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Inspired by syndrome source coding using linear error-correcting codes, we explore a new form of measurement matrix for compressed sensing. The proposed matrix is constructed in the systematic form [A I], where A is a randomly generated submatrix with elements distributed according to i.i.d. Gaussian, and I is the identity matrix. In the noiseless setting, this systematic construction retains similar property as the conventional Gaussian ensemble achieves. However, in the noisy setting with gross errors of arbitrary magnitude, where Gaussian ensemble fails catastrophically, systematic construction displays strong stability. In this paper, we prove its stable reconstruction property. We further show its ℓ1-norm sparsity recovery property by proving its restricted isometry property (RIP). We also demonstrate how the systematic matrix can be used to design a family of lossy-to-lossless compressed sensing schemes where the number of measurements trades off the reconstruction distortions. 1

Partially occluded faces are common in many applications of face recognition. While algorithms based on sparse representation have demonstrated promising results, they achieve their best performance on occlusions that are not spatially correlated (i.e. random pixel corruption). We show that such sparsity-based algorithms can be significantly improved by harnessing prior knowledge about the pixel error distribution. We show how a Markov Random Field model for spatial continuity of the occlusion can be integrated into the computation of a sparse representation of the test image with respect to the training images. Our algorithm efficiently and reliably identifies the corrupted regions and excludes them from the sparse representation. Extensive experiments on both laboratory and real-world datasets show that our algorithm tolerates much larger fractions and varieties of occlusion than current state-of-the-art algorithms. 1.

This paper develops new theory and algorithms to recover signals that are approximately sparse in some general (i.e., basis, frame, over-complete, or incomplete) dictionary but corrupted by a combination of measurement noise and interference having a sparse representation in a second general dictionary. Particular applications covered by our framework include the restoration of signals impaired by impulse noise, narrowband interference, or saturation, as well as image in-painting, super-resolution, and signal separation. We develop efficient recovery algorithms and deterministic conditions that guarantee stable restoration and separation. Two application examples demonstrate the efficacy of our approach.

An application of compressive sensing (CS) theory in imagebased robust face recognition is considered. Most contemporary face recognition systems suffer from limited abilities to handle image nuisances such as illumination, facial disguise, and pose misalignment. Motivated by CS, the problem has been recently cast in a sparse representation framework: The sparsest linear combination of a query image is sought using all prior training images as an overcomplete dictionary, and the dominant sparse coefficients reveal the identity of the query image. The ability to perform dense error correction directly in the image space also provides an intriguing solution to compensate pixel corruption and improve the recognition accuracy exceeding most existing solutions. Furthermore, a local iterative process can be applied to solve for an image transformation applied to the face region when the query image is misaligned. Finally, we discuss the state of the art in fast ℓ1-minimization to improve the speed of the robust face recognition system. The paper also provides useful guidelines to practitioners working in similar fields, such as acoustic/speech recognition. Index Terms: face recognition, compressive sensing, ℓ1minimization 1.

Inspired by syndrome source coding using linear error-correcting codes, we explore a new form of measurement matrix for compressed sensing. The proposed matrix is constructed in the systematic form [A I], where A is a randomly generated submatrix with elements distributed according to i.i.d. Gaussian, and I is the identity matrix. In the noiseless setting, this systematic construction retains similar property as the conventional Gaussian ensemble achieves. However, in the noisy setting with gross errors of arbitrary magnitude, where Gaussian ensemble fails catastrophically, systematic construction displays strong stability. In this paper, we prove its stable reconstruction property. We further show its ℓ1-norm sparsity recovery property by proving its restricted isometry property (RIP). We also demonstrate how the systematic matrix can be used to design a family of lossy-to-lossless compressed sensing schemes where the number of measurements trades off the reconstruction distortions. 1