Scientists working with large volumes of high-dimensional data, such as global climate patterns, stellar spectra, or human gene distributions, regularly confront the problem of dimensionality reduction: finding meaningful low-dimensional structures hidden in their high-dimensional observations. The human brain confronts the same problem in everyday perception, extracting from its high-dimensional sensory inputs-30,000 auditory nerve fibers or 10(6) optic nerve fibers-a manageably small number of perceptually relevant features. Here we describe an approach to solving dimensionality reduction problems that uses easily measured local metric information to learn the underlying global geometry of a data set. Unlike classical techniques such as principal component analysis (PCA) and multidimensional scaling (MDS), our approach is capable of discovering the nonlinear degrees of freedom that underlie complex natural observations, such as human handwriting or images of a face under different viewing conditions. In contrast to previous algorithms for nonlinear dimensionality reduction, ours efficiently computes a globally optimal solution, and, for an important class of data manifolds, is guaranteed to converge asymptotically to the true structure.

/pdf/a-global-geometric-framework-for-nonlinear-dimensionality-5brl02iqtj.pdf

A global geometric framework for nonlinear dimensionality reduction.

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.

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Eigenfaces vs. Fisherfaces: recognition using class specific linear projection

Is perception of the whole based on perception of its parts? There is psychological and physiological evidence for parts-based representations in the brain, and certain computational theories of object recognition rely on such representations. But little is known about how brains or computers might learn the parts of objects. Here we demonstrate an algorithm for non-negative matrix factorization that is able to learn parts of faces and semantic features of text. This is in contrast to other methods, such as principal components analysis and vector quantization, that learn holistic, not parts-based, representations. Non-negative matrix factorization is distinguished from the other methods by its use of non-negativity constraints. These constraints lead to a parts-based representation because they allow only additive, not subtractive, combinations. When non-negative matrix factorization is implemented as a neural network, parts-based representations emerge by virtue of two properties: the firing rates of neurons are never negative and synaptic strengths do not change sign.

Learning the parts of objects by non-negative matrix factorization

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 C1-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 with respect 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.

/pdf/robust-face-recognition-via-sparse-representation-4bfqv6y74o.pdf

Robust Face Recognition via Sparse Representation

Learning parts of objects by non-negative matrix factorization

In this work, we present a scalable least-squares solution for computing a seven degree-of-freedom similarity transform. Our method utilizes the generalized camera model to compute relative rotation, translation, and scale from four or more 2D-3D correspondences. In particular, structure and motion estimations from monocular cameras lack scale without specific calibration. As such, our methods have applications in loop closure in visual odometry and registering multiple structure from motion reconstructions where scale must be recovered. We formulate the generalized pose and scale problem as a minimization of a least squares cost function and solve this minimization without iterations or initialization. Additionally, we obtain all minima of the cost function. The order of the polynomial system that we solve is independent of the number of points, allowing our overall approach to scale favorably. We evaluate our method experimentally on synthetic and real datasets and demonstrate that our methods produce higher accuracy similarity transform solutions than existing methods.

http://vfragoso.com/pdfs/gdls_eccv_14.pdf

gDLS: A Scalable Solution to the Generalized Pose and Scale Problem

We present an approach to real-time tracking and mapping that supports any type of camera motion in 3D environments, that is, general (parallax-inducing) as well as rotation-only (degenerate) motions. Our approach effectively generalizes both a panorama mapping and tracking system and a keyframe-based Simultaneous Localization and Mapping (SLAM) system, behaving like one or the other depending on the camera movement. It seamlessly switches between the two and is thus able to track and map through arbitrary sequences of general and rotation-only camera movements.

/pdf/live-tracking-and-mapping-from-both-general-and-rotation-4o9pmagbcl.pdf

Live tracking and mapping from both general and rotation-only camera motion

Tracking hands in live video is a challenging task: the hand appearance can change too rapidly for appearance-based trackers to work, and color-based trackers (that do not rely on geometry) have to make limiting assumptions about the background color. This article shows the results of hand tracking with "Flocks of Features", a tracking method that combines motion cues and a learned foreground color distribution to achieve fast and robust 2D tracking of highly articulated objects. Many independent image artifacts are tracked from one frame to the next, adhering only to local constraints. This concept is borrowed from nature since these tracks mimic the flight of flocking birds - exhibiting local individualism and variability while maintaining a clustered entirety. Hand tracking has important applications for interaction with wearable computers, for intuitive manipulation of virtual objects, for detection of activity signatures, and much more. Tracking with Flocks of Features is not limited to hands - any articulated or appearance-changing object can benefit from this multi-cue tracking method.

/pdf/hand-tracking-with-flocks-of-features-4mltd9ax9y.pdf

Hand tracking with Flocks of Features

We propose a novel formulation for determining the absolute pose of a single or multi-camera system given a known vertical direction. The vertical direction may be easily obtained by detecting the vertical vanishing points with computer vision techniques, or with the aid of IMU sensor measurements from a smartphone. Our solver is general and able to compute absolute camera pose from two 2D-3D correspondences for single or multi-camera systems. We run several synthetic experiments that demonstrate our algorithm's improved robustness to image and IMU noise compared to the current state of the art. Additionally, we run an image localization experiment that demonstrates the accuracy of our algorithm in real-world scenarios. Finally, we show that our algorithm provides increased performance for real-time model-based tracking compared to solvers that do not utilize the vertical direction and show our algorithm in use with an augmented reality application running on a Google Tango tablet.

Efficient Computation of Absolute Pose for Gravity-Aware Augmented Reality

We propose an isometric self-organizing map (ISO-SOM) method for nonlinear dimensionality reduction, which integrates a self-organizing map model and an ISOMAP dimension reduction algorithm, organizing the high dimension data in a low dimension lattice structure. We apply the proposed method to the problem of appearance-based 3D hand posture estimation. As a learning stage, we use a realistic 3D hand model to generate data encoding the mapping between the hand pose space and the image feature space. The intrinsic dimension of such nonlinear mapping is learned by ISOSOM, which clusters the data into a lattice map. We perform 3D hand posture estimation on this map, showing that the ISOSOM algorithm performs better than traditional image retrieval algorithms for pose estimation. We also show that a 2.5D feature representation based on depth edges is clearly superior to intensity edge features commonly used in previous methods.

Matthew Turk

Papers

gDLS: A Scalable Solution to the Generalized Pose and Scale Problem

Live tracking and mapping from both general and rotation-only camera motion

Hand tracking with Flocks of Features

Efficient Computation of Absolute Pose for Gravity-Aware Augmented Reality

The Isometric Self-Organizing Map for 3D Hand Pose Estimation.