Antonio Robles-Kelly

Hyperspectral Unmixing via

Hyperspectral unmixing is a crucial preprocessing step for material classification and recognition. In the last decade, nonnegative matrix factorization (NMF) and its extensions have been intensively studied to unmix hyperspectral imagery and recover the material end-members. As an important constraint for NMF, sparsity has been modeled making use of the <i>L</i>₁ regularizer. Unfortunately, the <i>L</i>₁ regularizer cannot enforce further sparsity when the full additivity constraint of material abundances is used, hence limiting the practical efficacy of NMF methods in hyperspectral unmixing. In this paper, we extend the NMF method by incorporating the <i>L</i>_1/2 sparsity constraint, which we name <i>L</i>_1/2 -NMF. The <i>L</i>_1/2 regularizer not only induces sparsity but is also a better choice among <i>Lq</i>(0 <; <i>q</i> <; 1) regularizers. We propose an iterative estimation algorithm for <i>L</i>_1/2-NMF, which provides sparser and more accurate results than those delivered using the <i>L</i>₁ norm. We illustrate the utility of our method on synthetic and real hyperspectral data and compare our results to those yielded by other state-of-the-art methods.

L_{1/2}

This paper is concerned with computing graph edit distance. One of the criticisms that can be leveled at existing methods for computing graph edit distance is that they lack some of the formality and rigor of the computation of string edit distance. Hence, our aim is to convert graphs to string sequences so that string matching techniques can be used. To do this, we use a graph spectral seriation method to convert the adjacency matrix into a string or sequence order. We show how the serial ordering can be established using the leading eigenvector of the graph adjacency matrix. We pose the problem of graph-matching as a maximum a posteriori probability (MAP) alignment of the seriation sequences for pairs of graphs. This treatment leads to an expression in which the edit cost is the negative logarithm of the a posteriori sequence alignment probability. We compute the edit distance by finding the sequence of string edit operations which minimizes the cost of the path traversing the edit lattice. The edit costs are determined by the components of the leading eigenvectors of the adjacency matrix and by the edge densities of the graphs being matched. We demonstrate the utility of the edit distance on a number of graph clustering problems.

Sparsity-Constrained Nonnegative Matrix Factorization

Multiple instance learning (MIL) is a paradigm in supervised learning that deals with the classification of collections of instances called bags. Each bag contains a number of instances from which features are extracted. The complexity of MIL is largely dependent on the number of instances in the training data set. Since we are usually confronted with a large instance space even for moderately sized real-world data sets applications, it is important to design efficient instance selection techniques to speed up the training process without compromising the performance. In this paper, we address the issue of instance selection in MIL. We propose MILIS, a novel MIL algorithm based on adaptive instance selection. We do this in an alternating optimization framework by intertwining the steps of instance selection and classifier learning in an iterative manner which is guaranteed to converge. Initial instance selection is achieved by a simple yet effective kernel density estimator on the negative instances. Experimental results demonstrate the utility and efficiency of the proposed approach as compared to the state of the art.

Graph edit distance from spectral seriation

In this paper, we make use of the relationship between the Laplace-Beltrami operator and the graph Laplacian, for the purposes of embedding a graph onto a Riemannian manifold. To embark on this study, we review some of the basics of Riemannian geometry and explain the relationship between the Laplace-Beltrami operator and the graph Laplacian. Using the properties of Jacobi fields, we show how to compute an edge-weight matrix in which the elements reflect the sectional curvatures associated with the geodesic paths on the manifold between nodes. For the particular case of a constant sectional curvature surface, we use the Kruskal coordinates to compute edge weights that are proportional to the geodesic distance between points. We use the resulting edge-weight matrix to embed the nodes of the graph onto a Riemannian manifold. To do this, we develop a method that can be used to perform double centring on the Laplacian matrix computed from the edge-weights. The embedding coordinates are given by the eigenvectors of the centred Laplacian. With the set of embedding coordinates at hand, a number of graph manipulation tasks can be performed. In this paper, we are primarily interested in graph-matching. We recast the graph-matching problem as that of aligning pairs of manifolds subject to a geometric transformation. We show that this transformation is Pro-crustean in nature. We illustrate the utility of the method on image matching using the COIL database.

MILIS: Multiple Instance Learning with Instance Selection

In this paper, we address the problem of combining linear support vector machines (SVMs) for classification of large-scale nonlinear datasets. The motivation is to exploit both the efficiency of linear SVMs (LSVMs) in learning and prediction and the power of nonlinear SVMs in classification. To this end, we develop a LSVM mixture model that exploits a divide-and-conquer strategy by partitioning the feature space into subregions of linearly separable datapoints and learning a LSVM for each of these regions. We do this implicitly by deriving a generative model over the joint data and label distributions. Consequently, we can impose priors on the mixing coefficients and do implicit model selection in a top-down manner during the parameter estimation process. This guarantees the sparsity of the learned model. Experimental results show that the proposed method can achieve the efficiency of LSVMs in the prediction phase while still providing a classification performance comparable to nonlinear SVMs.

A Riemannian approach to graph embedding

This paper describes a new robust regular polygon detector. The regular polygon transform is posed as a mixture of regular polygons in a five dimensional space. Given the edge structure of an image, we derive the a posteriori probability for a mixture of regular polygons, and thus the probability density function for the appearance of a mixture of regular polygons. Likely regular polygons can be isolated quickly by discretising and collapsing the search space into three dimensions. The remaining dimensions may be efficiently recovered subsequently using maximum likelihood at the locations of the most likely polygons in the subspace. This leads to an efficient algorithm. Also the a posteriori formulation facilitates inclusion of additional a priori information leading to real-time application to road sign detection. The use of gradient information also reduces noise compared to existing approaches such as the generalised Hough transform. Results are presented for images with noise to show stability. The detector is also applied to two separate applications: real-time road sign detection for on-line driver assistance; and feature detection, recovering stable features in rectilinear environments.

Mixing Linear SVMs for Nonlinear Classification

This paper shows how the eigenstructure of the adjacency matrix can be used for the purposes of robust graph matching. We commence from the observation that the leading eigenvector of a transition probability matrix is the steady state of the associated Markov chain. When the transition matrix is the normalized adjacency matrix of a graph, then the leading eigenvector gives the sequence of nodes of the steady state random walk on the graph. We use this property to convert the nodes in a graph into a string where the node-order is given by the sequence of nodes visited in the random walk. We match graphs represented in this way, by finding the sequence of string edit operations which minimize edit distance.

Regular polygon detection

In this paper, we address the problem of photometric invariance in multispectral imaging making use of an optimisation approach based upon the dichromatic model. In this manner, we cast the problem of recovering the spectra of the illuminant, the surface reflectance and the shading and specular factors in a structural optimisation setting. Making use of the additional information provided by multispectral imaging and the structure of image patches, we recover the dichromatic parameters of the scene. To do this, we formulate a target cost function combining the dichromatic error and the smoothness priors for the surfaces under study. The dichromatic parameters are recovered through minimising this cost function in a coordinate descent manner. The algorithm is quite general in nature, admitting the enforcement of smoothness constraints and extending in a straightforward manner to trichromatic settings. Moreover, the objective function is convex with respect to the subset of variables to be optimised in each alternating step of the minimisation strategy. This gives rise to an optimal closed-form solution for each of the iterations in our algorithm. We illustrate the effectiveness of our method for purposes of illuminant spectrum recovery, skin recognition, material clustering and specularity removal. We also compare our results to a number of alternatives.

STRING EDIT DISTANCE, RANDOM WALKS AND GRAPH MATCHING

In this paper, we present a method for object of interest detection. This method is statistical in nature and hinges in a model which combines salient features using a mixture of linear support vector machines. It exploits a divide-and-conquer strategy by partitioning the feature space into sub-regions of linearly separable data-points. This yields a structured learning approach where we learn a linear support vector machine for each region, the mixture weights, and the combination parameters for each of the salient features at hand. Thus, the method learns the combination of salient features such that a mixture of classifiers can be used to recover objects of interest in the image. We illustrate the utility of the method by applying our algorithm to the MSRA Salient Object Database.

A Solution of the Dichromatic Model for Multispectral Photometric Invariance

This paper describes work aimed at the unsupervised learning of shape-classes from shock trees. We commence by considering how to compute the edit distance between weighted trees. We show how to transform the tree edit distance problem into a series of maximum weight clique problems, and show how to use relaxation labeling to find an approximate solution. This allows us to compute a set of pairwise distances between graph-structures. We show how the edit distances can be used to compute a matrix of pairwise affinities using χ2 statistics. We present a maximum likelihood method for clustering the graphs by iteratively updating the elements of the affinity matrix. This involves interleaved steps for updating the affinity matrix using an eigendecomposition method and updating the cluster membership indicators. We illustrate the new tree clustering framework on shock-graphs extracted from the silhouettes of 2D shapes.