Low-dimensional Linear Subspace Research Articles

Closed subschemes of quadrics with large groups of linear automorphisms

Let Qn ⊂ P n+1 , n ≥ 2, be an n-dimensional smooth quadric hypersurface. Fix P ∈ Qn. Set Gn := Aut(Qn) and Gn(P ): ={f ∈: f (P )= P }. We first study low dimensional linear subspaces of H 0 (P n+1 , OPn+1(k)). Then we study the zero-dimensional schemes Z ⊂ Qn such that f (Z )= Z for all f ∈ Gn(P ) and length(Z) is small.

Face Recognition Using 3-D Models: Pose and Illumination

Unconstrained illumination and pose variation lead to significant variation in the photographs of faces and constitute a major hurdle preventing the widespread use of face recognition systems. The challenge is to generalize from a limited number of images of an individual to a broad range of conditions. Recently, advances in modeling the effects of illumination and pose have been accomplished using three-dimensional (3-D) shape information coupled with reflectance models. Notable developments in understanding the effects of illumination include the nonexistence of illumination invariants, a characterization of the set of images of objects in fixed pose under variable illumination (the illumination cone), and the introduction of spherical harmonics and low-dimensional linear subspaces for modeling illumination. To generalize to novel conditions, either multiple images must be available to reconstruct 3-D shape or, if only a single image is accessible, prior information about the 3-D shape and appearance of faces in general must be used. The 3-D Morphable Model was introduced as a generative model to predict the appearances of an individual while using a statistical prior on shape and texture allowing its parameters to be estimated from single image. Based on these new understandings, face recognition algorithms have been developed to address the joint challenges of pose and lighting. In this paper, we review these developments and provide a brief survey of the resulting face recognition algorithms and their performance

Face recognition from a single training image under arbitrary unknown lighting using spherical harmonics

In this paper, we propose two novel methods for face recognition under arbitrary unknown lighting by using spherical harmonics illumination representation, which require only one training image per subject and no 3D shape information. Our methods are based on the recent result which demonstrated that the set of images of a convex Lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low-dimensional linear subspace. We provide two methods to estimate the spherical harmonic basis images spanning this space from just one image. Our first method builds the statistical model based on a collection of 2D basis images. We demonstrate that, by using the learned statistics, we can estimate the spherical harmonic basis images from just one image taken under arbitrary illumination conditions if there is no pose variation. Compared to the first method, the second method builds the statistical models directly in 3D spaces by combining the spherical harmonic illumination representation and a 3D morphable model of human faces to recover basis images from images across both poses and illuminations. After estimating the basis images, we use the same recognition scheme for both methods: we recognize the face for which there exists a weighted combination of basis images that is the closest to the test face image. We provide a series of experiments that achieve high recognition rates, under a wide range of illumination conditions, including multiple sources of illumination. Our methods achieve comparable levels of accuracy with methods that have much more onerous training data requirements. Comparison of the two methods is also provided.

Visual tracking and recognition using probabilistic appearance manifolds

This paper presents an algorithm for modeling, tracking, and recognizing human faces in video sequences within one integrated framework. Conventional video-based face recognition systems have usually been embodied with two independent components: the tracking and recognition modules. In contrast, our algorithm emphasizes an algorithmic architecture that tightly couples these two components within a single framework. This is accomplished through a novel appearance model which is utilized simultaneously by both modules, even with their disparate requirements and functions. The complex nonlinear appearance manifold of each registered person is partitioned into a collection of submanifolds where each models the face appearances of the person in nearby poses. The submanifold is approximated by a low-dimensional linear subspace computed by principal component analysis using images sampled from training video sequences. The connectivity between the submanifolds is modeled as transition probabilities between pairs of submanifolds, and these are learned directly from training video sequences. The integrated task of tracking and recognition is formulated as a maximum a posteriori estimation problem. Within our framework, the tracking and recognition modules are complementary to each other, and the capability and performance of one are enhanced by the other. Our approach contrasts sharply with more rigid conventional approaches in which these two modules work independently and in sequence. We report on a number of experiments and results that demonstrate the robustness, effectiveness, and stability of our algorithm.

Segmented Linear Subspaces for Illumination-Robust Face Recognition

All images of a convex Lambertian surface captured with a fixed pose under varying illumination are known to lie in a convex cone in the image space that is called the illumination cone. Since this cone model is too complex to be built in practice, researchers have attempted to approximate it with simpler models. In this paper, we propose a segmented linear subspace model to approximate the cone. Our idea of segmentation is based on the fact that the success of low dimensional linear subspace approximations of the illumination cone increases if the directions of the surface normals get close to each other. Hence, we propose to cluster the image pixels according to their surface normal directions and to approximate the cone with a linear subspace for each of these clusters separately. We perform statistical performance evaluation experiments to compare our system to other popular systems and demonstrate that the performance increase we obtain is statistically significant.

Lambertian reflectance and linear subspaces

We prove that the set of all Lambertian reflectance functions (the mapping from surface normals to intensities) obtained with arbitrary distant light sources lies close to a 9D linear subspace. This implies that, in general, the set of images of a convex Lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low-dimensional linear subspace, explaining prior empirical results. We also provide a simple analytic characterization of this linear space. We obtain these results by representing lighting using spherical harmonics and describing the effects of Lambertian materials as the analog of a convolution. These results allow us to construct algorithms for object recognition based on linear methods as well as algorithms that use convex optimization to enforce nonnegative lighting functions. We also show a simple way to enforce nonnegative lighting when the images of an object lie near a 4D linear space. We apply these algorithms to perform face recognition by finding the 3D model that best matches a 2D query image.

Multi-Frame Correspondence Estimation Using Subspace Constraints

When a rigid scene is imaged by a moving camera, the set of all displacements of all points across multiple frames often resides in a low-dimensional linear subspace. Linear subspace constraints have been used successfully in the past for recovering 3D structure and 3D motion information from multiple frames (e.g., by using the factorization method of Tomasi and Kanade (1992, International Journal of Computer Vision, 9:137–154)). These methods assume that the 2D correspondences have been precomputed. However, correspondence estimation is a fundamental problem in motion analysis. In this paper we show how the multi-frame subspace constraints can be used for constraining the 2D correspondence estimation process itself. We show that the multi-frame subspace constraints are valid not only for affine cameras, but also for a variety of imaging models, scene models, and motion models. The multi-frame subspace constraints are first translated from constraints on correspondences to constraints directly on image measurements (e.g., image brightness quantities). These brightness-based subspace constraints are then used for estimating the correspondences, by requiring that all corresponding points across all video frames reside in the appropriate low-dimensional linear subspace. The multi-frame subspace constraints are geometrically meaningful, and are {not} violated at depth discontinuities, nor when the camera-motion changes abruptly. These constraints can therefore replace {heuristic} constraints commonly used in optical-flow estimation, such as spatial or temporal smoothness.

From few to many: illumination cone models for face recognition under variable lighting and pose

We present a generative appearance-based method for recognizing human faces under variation in lighting and viewpoint. Our method exploits the fact that the set of images of an object in fixed pose, but under all possible illumination conditions, is a convex cone in the space of images. Using a small number of training images of each face taken with different lighting directions, the shape and albedo of the face can be reconstructed. In turn, this reconstruction serves as a generative model that can be used to render (or synthesize) images of the face under novel poses and illumination conditions. The pose space is then sampled and, for each pose, the corresponding illumination cone is approximated by a low-dimensional linear subspace whose basis vectors are estimated using the generative model. Our recognition algorithm assigns to a test image the identity of the closest approximated illumination cone. Test results show that the method performs almost without error, except on the most extreme lighting directions.

Multi-frame estimation of planar motion

Traditional plane alignment techniques are typically performed between pairs of frames. We present a method for extending existing two-frame planar motion estimation techniques into a simultaneous multi-frame estimation, by exploiting multi-frame subspace constraints of planar surfaces. The paper has three main contributions: 1) we show that when the camera calibration does not change, the collection of all parametric image motions of a planar surface in the scene across multiple frames is embedded in a low dimensional linear subspace; 2) we show that the relative image motion of multiple planar surfaces across multiple frames is embedded in a yet lower dimensional linear subspace, even with varying camera calibration; and 3) we show how these multi-frame constraints can be incorporated into simultaneous multi-frame estimation of planar motion, without explicitly recovering any 3D information, or camera calibration. The resulting multi-frame estimation process is more constrained than the individual two-frame estimations, leading to more accurate alignment, even when applied to small image regions.

Bandlimited extrapolation using time-bandwidth dimension

The problem of extrapolating discrete-index bandlimited signals from a finite number of samples is addressed in this paper. The algorithm presented in this paper exploits the fact that the set of bandlimited signals that are also essentially time-limited is approximated well by a low-dimensional linear subspace. This fact, which is well known for one-dimensional (1-D) signals with contiguous passbands and time-concentration intervals, is established for a more general class of multidimensional (m-D) signals with discontiguous passbands and discontiguous time-concentration regions. A criterion is presented for determining the dimension of the approximating subspace and the minimax optimal subspace itself based on knowledge of the passband, time-concentration regions, energy concentration factor, and bounds on the tolerable extrapolation error. The extrapolation is constrained to lie in this subspace, and parameters characterizing the extrapolation are obtained from the data by solving a linear system of equations. For certain sampling patterns, the system is ill conditioned, and a second rank reduction is needed to reduce the deleterious effects of observation noise and modeling error. A novel criterion for rank selection based on known bounds on noise power and modeling error is presented. The effectiveness of the new algorithm and the rank selection criterion are demonstrated by means of computer simulations.