Abstract — We consider the problem of automatically recognizing human faces from frontal views with varying expression and illumination, as well as occlusion and disguise. We cast the recognition problem as one of classifying among multiple linear regression models, and argue that new theory from sparse signal representation offers the key to addressing this problem. Based on a sparse representation computed by ℓ 1-minimization, we propose a general classification algorithm for (image-based) object recognition. This new framework provides new insights into two crucial issues in face recognition: feature extraction and robustness to occlusion. For feature extraction, we show that if sparsity in the recognition problem is properly harnessed, the choice of features is no longer critical. What is critical, however, is whether the number of features is sufficiently large and whether the sparse representation is correctly computed. Unconventional features such as downsampled images and random projections perform just as well as conventional features such as Eigenfaces and Laplacianfaces, as long as the dimension of the feature space surpasses certain threshold, predicted by the theory of sparse representation. This framework can handle errors due to occlusion and corruption uniformly, by exploiting the fact that these errors are often sparse w.r.t. to the standard (pixel) basis. The theory of sparse representation helps predict how much occlusion the recognition algorithm can handle and how to choose the training images to maximize robustness to occlusion. We conduct extensive experiments on publicly available databases to verify the efficacy of the proposed algorithm, and corroborate the above claims.

Humans are visual animals, and imaging sensors that extend our reach – cameras – have improved dramatically in recent times thanks to the introduction of CCD and CMOS digital technology. Consumer digital cameras in the mega-pixel range are now ubiquitous thanks to the happy coincidence that the semiconductor material of choice for large-scale electronics integration (silicon) also happens to readily convert photons at visual wavelengths into electrons. On the contrary, imaging at wavelengths where silicon is blind is considerably more complicated, bulky, and expensive. Thus, for comparable resolution, a $500 digital camera for the visible becomes a $50,000 camera for the infrared. In this paper, we present a new approach to building simpler, smaller, and cheaper digital cameras that can operate efficiently across a much broader spectral range than conventional silicon-based cameras. Our approach fuses a new camera architecture based on a digital mi-cromirror device (DMD – see Sidebar: Spatial Light Modulators) with the new mathematical theory and algorithms of compressive sampling (CS – see Sidebar: Compressive Sampling in a Nutshell). CS combines sampling and compression into a single nonadaptive linear measurement process [1–4]. Rather than measuring pixel samples of the scene under view, we measure inner products

The theory of compressive sensing (CS) enables the reconstruction of a sparse or compressible image or signal from a small set of linear, non-adaptive (even random) projections. However, in many applications, including object and target recognition, we are ultimately interested in making a decision about an image rather than computing a reconstruction. We propose here a framework for compressive classification that operates directly on the compressive measurements without first reconstructing the image. We dub the resulting dimensionally reduced matched filter the smashed filter. The first part of the theory maps traditional maximum likelihood hypothesis testing into the compressive domain; we find that the number of measurements required for a given classification performance level does not depend on the sparsity or compressibility of the images but only on the noise level. The second part of the theory applies the generalized maximum likelihood method to deal with unknown transformations such as the translation, scale, or viewing angle of a target object. We exploit the fact the set of transformed images forms a low-dimensional, nonlinear manifold in the high-dimensional image space. We find that the number of measurements required for a given classification performance level grows linearly in the dimensionality of the manifold but only logarithmically in the number of pixels/samples and image classes. Using both simulations and measurements from a new single-pixel compressive camera, we demonstrate the effectiveness of the smashed filter for target classification using very few measurements.

Compressed sensing is an emerging signal acquisition technique that enables signals to be sampled well below the Nyquist rate, given that the signal has a sparse representation in an orthonormal basis. In fact, sparsity in an orthonormal basis is only one possible signal model that allows for sampling strategies below the Nyquist rate. In this paper we consider a more general signal model and assume signals that live on or close to the union of linear subspaces of low dimension. We present sampling theorems for this model that are in the same spirit as the Nyquist-Shannon sampling theorem in that they connect the number of required samples to certain model parameters. Contrary to the Nyquist-Shannon sampling theorem, which gives a necessary and sufficient condition for the number of required samples as well as a simple linear algorithm for signal reconstruction, the model studied here is more complex. We therefore concentrate on two aspects of the signal model, the existence of one to one maps to lower dimensional observation spaces and the smoothness of the inverse map. We show that almost all linear maps are one to one when the observation space is at least of the same dimension as the largest dimension of the convex hull of the union of any two subspaces in the model. However, we also show that in order for the inverse map to have certain smoothness properties such as a given finite Lipschitz constant, the required observation dimension necessarily depends logarithmically

The recently introduced theory of compressive sensing enables the recovery of sparse or compressible signals from a small set of nonadaptive, linear measurements. If properly chosen, the number of measurements can be much smaller than the number of Nyquist-rate samples. Interestingly, it has been shown that random projections are a near-optimal measurement scheme. This has inspired the design of hardware systems that directly implement random measurement protocols. However, despite the intense focus of the community on signal recovery, many (if not most) signal processing problems do not require full signal recovery. In this paper, we take some first steps in the direction of solving inference problems—such as detection, classification, or estimation—and filtering problems using only compressive measurements and without ever reconstructing the signals involved. We provide theoretical bounds along with experimental results.

The recently introduced theory of compressed sensing enables the reconstruction of sparse or compressible signals from a small set of nonadaptive, linear measurements. If properly chosen, the number of measurements can be much smaller than the number of Nyquist rate samples. Interestingly, it has been shown that random projections are a satisfactory measurement scheme. This has inspired the design of physical systems that directly implement similar measurement schemes. However, despite the intense focus on the reconstruction of signals, many (if not most) signal processing problems do not require a full reconstruction of the signal – we are often interested only in solving some sort of detection problem or in the estimation of some function of the data. In this report, we show that the compressed sensing framework is useful for a wide range of statistical inference tasks. In particular, we demonstrate how to solve a variety of signal detection and estimation problems given the measurements without ever reconstructing the signals themselves. We provide theoretical bounds along with experimental results. 1

In this paper we show that dimensionality reduction (i.e., Johnson-Lindenstrauss lemma) preserves not only the distances between static points, but also between moving points, and more generally between low-dimensional flats, polynomial curves, curves with low winding degree, and polynomial surfaces. We also show that surfaces with bounded doubling dimension can be embedded into low dimension with small additive error. Finally, we show that for points with polynomial motion, the radius of the smallest enclosing ball can be preserved under dimensionality reduction. 1

In this paper, we examine the role of feature selection in face recognition from the perspective of sparse representation. We cast the recognition problem as finding a sparse representation of the test image features w.r.t. the training set. The sparse representation can be accurately and efficiently computed by ℓ 1-minimization. The proposed simple algorithm generalizes conventional face recognition classifiers such as nearest neighbors and nearest subspaces. Using face recognition under varying illumination and expression as an example, we show that if sparsity in the recognition problem is properly harnessed, the choice of features is no longer critical. What is critical, however, is whether the number of features is sufficient and whether the sparse representation is correctly found. We conduct extensive experiments to validate the significance of imposing sparsity using the Extended Yale B database and the AR database. Our thorough evaluation shows that, using conventional features such as Eigenfaces and facial parts, the proposed algorithm achieves much higher recognition accuracy on face images with variation in either illumination or expression. Furthermore, other unconventional features such as severely down-sampled images and randomly projected features perform almost equally well with the increase of feature dimensions. The differences in performance between different features become insignificant as the feature-space dimension is sufficiently large.

This article proposes a new class of models for natural signals and images. The set of patches extracted from the data to analyze is constrained to be close to a low dimensional manifold. This manifold structure is detailed for various ensembles suitable for natural signals, images and textures modeling. These manifolds provide a low-dimensional parameterization of the local geometry of these datasets. These manifold models can be used to regularize inverse problems in signal and image processing. The restored signal is represented as a smooth curve or surface traced on the manifold that matches the forward measurements. A manifold pursuit algorithm computes iteratively a solution of the manifold regularization problem. Numerical simulations on inpainting and compressive sensing inversion show that manifolds models bring an improvement for the recovery of data with geometrical features. Key words: signal processing, image modeling, texture, manifold. PACS: code, code Capturing the complex geometry of signals and images is at the core of recent advances in sound and natural image processing. Edges and texture patterns create complex non-local interactions. This paper studies these geometries for several sounds, images and textures models. The set of local patches in the dataset is modeled using smooth manifolds. These local features trace a continuous curve (resp. surface) on the manifold, which is a prior that can be used to solve inverse problems.

We propose a framework for exploiting dimension-reducing random projections in detection and classification problems. Our approach is based on the generalized likelihood ratio test; in the case of image classification, it exploits the fact that a set of images of a fixed scene under varying articulation parameters forms a low-dimensional, nonlinear manifold. Exploiting recent results showing that random projections stably embed a smooth manifold in a lower-dimensional space, we develop the multiscale smashed filter as a compressive analog of the familiar matched filter classifier. In a practical target classification problem using a single-pixel camera that directly acquires compressive image projections, we achieve high classification rates using many fewer measurements than the dimensionality of the images.