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 L1 regularizer. Unfortunately, the L1 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 L 1/2 sparsity constraint, which we name L 1/2 -NMF. The L 1/2 regularizer not only induces sparsity, but is also a better choice among Lq(0 < q < 1) regularizers. We propose an iterative estimation algorithm for L 1/2 -NMF, which provides sparser and more accurate results than those delivered using the L1 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.

Abstract-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.

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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.

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In this paper, we propose an approach to the problem of simultaneous shape and refractive index recovery from multispectral polarisation imagery captured from a single viewpoint. The focus of this paper is on dielectric surfaces which diffusely polarise light transmitted from the dielectric body into the air. The diffuse polarisation of the reflection process is modelled using a Transmitted Radiance Sinusoid curve and the Fresnel transmission theory. We provide a method of estimating the azimuth angle of surface normals from the spectral variation of the phase of polarisation. Moreover, to render the problem of simultaneous estimation of surface orientation and index of refraction well-posed, we enforce a generative model on the material dispersion equations for the index of refraction. This generative model, together with the Fresnel transmission ratio, permit the recovery of the index of refraction and the zenith angle simultaneously. We show results on shape recovery and rendering for real world and synthetic imagery.

Photometric stereo estimates the surface normal given a set of images acquired under different illumination conditions. To deal with diverse factors involved in the image formation process, recent photometric stereo methods demand a large number of images as input. We propose a method that can dramatically decrease the demands on the number of images by learning the most informative ones under different illumination conditions. To this end, we use a deep learning framework to automatically learn the critical illumination conditions required at input. Furthermore, we present an occlusion layer that can synthesize cast shadows, which effectively improves the estimation accuracy. We assess our method on challenging real-world conditions, where we outperform techniques elsewhere in the literature with a significantly reduced number of light conditions.

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.