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Statistical characterizations of complex network structures can be obtained from both the Ihara Zeta function (in terms of prime cycle frequencies) and the partition function from statistical mechanics. However, these two representations are usually regarded as separate tools for network analysis, without exploiting the potential synergies between them. In this paper, we establish a link between the Ihara Zeta function from algebraic graph theory and the partition function from statistical mechanics, and exploit this relationship to obtain a deeper structural characterisation of network structure. Specifically, the relationship allows us to explore the connection between the microscopic structure and the macroscopic characterisation of a network. We derive thermodynamic quantities describing the network, such as entropy, and show how these are related to the frequencies of prime cycles of various lengths. In particular, the n-th order partial derivative of the Ihara Zeta function can be used to compute the number of prime cycles in a network, which in turn is related to the partition function of Bose–Einstein statistics. The corresponding derived entropy allows us to explore a phase transition in the network structure with critical points at high and low-temperature limits. Numerical experiments and empirical data are presented to evaluate the qualitative and quantitative performance of the resulting structural network characterisations. [ABSTRACT FROM AUTHOR]

Low-rank decomposition (LRD) is an effective scheme to explore the affinity among superpixels in the image and video segmentation. However, the superpixel feature collected based on colour, shape, and texture may be rough, incompatible, and even conflicting if multiple features extracted in various manners are vectored and stacked straight together. It poses poor correlation, inconsistence on intra-category superpixels, and similarities on inter-category superpixels. This paper proposes a correlation-enhanced superpixel for video segmentation in the framework of LRD. Our algorithm mainly consists of two steps, feature analysis to establish the initial affinity among superpixels, followed by construction of a correlation-enhanced superpixel. This work is very helpful to perform LRD effectively and find the affinity accurately and quickly. Experiments conducted on datasets validate the proposed method. Comparisons with the state-of-the-art algorithms show higher speed and more precise in video segmentation. [ABSTRACT FROM AUTHOR]

Measuring the intrinsic deformability of arbitrary small-scale subdivision of a shape is an interesting meanwhile valuable research topic. Such measurement can be directly utilised as a reliable criteria to partition shape into small components and then assist in shape modelling and description. Compared to global modelling, through constructing subdivision-based complex shape description, the accuracy and flexibility of shape representation can be significantly improved. In this study, the authors propose a line segment advection (LSA)-based vertex-level three-dimensional shape deformability measuring method. It can highlight the deformability characteristics of each shape part in any scale and size. The measurement is realised mainly based on the advection of line segments connecting neighbouring shape mesh vertices. For 3D shapes, since the line segment of triangular mesh facet directly reflects the minimal neighbourhood relationships and mesh microstructure, its advection can capture the finest details of shape deformability. Then, after transferring that information into neighbouring vertices, a vertex-level shape deformability measurement can be acquired. Besides, to demonstrate the value of the proposed measuring method to shape partitioning and piecewise shape modelling, a straightforward shape partitioning method is introduced as well. Extensive experiments on three publicly available databases are conducted to verify the effectiveness of proposed methods. [ABSTRACT FROM AUTHOR]

The estimation of mutual information between graphs has been an elusive problem until the formulation of graph matching in terms of manifold alignment. Then, graphs are mapped to multi-dimensional sets of points through structure preserving embeddings. Point-wise alignment algorithms can be exploited in this context to re-cast graph matching in terms of point matching. Methods based on bypass entropy estimation must be deployed to render the estimation of mutual information computationally tractable. In this paper the novel contribution is to show how manifold alignment can be combined with copula-based entropy estimators to efficiently estimate the mutual information between graphs. We compare the empirical copula with an Archimedean copula (the independent one) in terms of retrieval/recall after graph comparison. Our experiments show that mutual information built in both choices improves significantly state-of-the art divergences. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two fast subgraph kernels based on a depth-based representation of graph-structure. Both methods gauge depth information through a family of K -layer expansion subgraphs rooted at a vertex [1] . The first method commences by computing a centroid-based complexity trace for each graph, using a depth-based representation rooted at the centroid vertex that has minimum shortest path length variance to the remaining vertices [2] . This subgraph kernel is computed by measuring the Jensen–Shannon divergence between centroid-based complexity entropy traces. The second method, on the other hand, computes a depth-based representation around each vertex in turn. The corresponding subgraph kernel is computed using isomorphisms tests to compare the depth-based representation rooted at each vertex in turn. For graphs with n vertices, the time complexities for the two new kernels are O ( n 2 ) and O ( n 3 ) , in contrast to O ( n 6 ) for the classic Gärtner graph kernel [3] . Key to achieving this efficiency is that we compute the required Shannon entropy of the random walk for our kernels with O ( n 2 ) operations. This computational strategy enables our subgraph kernels to easily scale up to graphs of reasonably large sizes and thus overcome the size limits arising in state-of-the-art graph kernels. Experiments on standard bioinformatics and computer vision graph datasets demonstrate the effectiveness and efficiency of our new subgraph kernels. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the heat kernel embedding as a route to graph representation. The heat kernel of the graph encapsulates information concerning the distribution of path lengths and, hence, node affinities on the graph; and is found by exponentiating the Laplacian eigen-system over time. A Young-Householder decomposition is performed on the heat kernel to obtain the matrix of the embedded coordinates for the nodes of the graph. With the embeddings at hand, we establish a graph characterization based on differential geometry by computing sets of curvatures associated with the graph edges and triangular faces. A sectional curvature computed from the difference between geodesic and Euclidean distances between nodes is associated with the edges of the graph. Furthermore, we use the Gauss-Bonnet theorem to compute the Gaussian curvatures associated with triangular faces of the graph. [ABSTRACT FROM AUTHOR]

Many computer vision and pattern recognition problems may be posed as the analysis of a set of dissimilarities between objects. For many types of data, these dissimilarities are not euclidean (i.e., they do not represent the distances between points in a euclidean space), and therefore cannot be isometrically embedded in a euclidean space. Examples include shape-dissimilarities, graph distances and mesh geodesic distances. In this paper, we provide a means of embedding such non-euclidean data onto surfaces of constant curvature. We aim to embed the data on a space whose radius of curvature is determined by the dissimilarity data. The space can be either of positive curvature (spherical) or of negative curvature (hyperbolic). We give an efficient method for solving the spherical and hyperbolic embedding problems on symmetric dissimilarity data. Our approach gives the radius of curvature and a method for approximating the objects as points on a hyperspherical manifold without optimisation. For objects which do not reside exactly on the manifold, we develop a optimisation-based procedure for approximate embedding on a hyperspherical manifold. We use the exponential map between the manifold and its local tangent space to solve the optimisation problem locally in the euclidean tangent space. This process is efficient enough to allow us to embed data sets of several thousand objects. We apply our method to a variety of data including time warping functions, shape similarities, graph similarity and gesture similarity data. In each case the embedding maintains the local structure of the data while placing the points in a metric space. [ABSTRACT FROM AUTHOR]

Pairwise dissimilarity representations are frequently used as an alternative to feature vectors in pattern recognition. One of the problems encountered in the analysis of such data is that the dissimilarities are rarely Euclidean, while statistical learning algorithms often rely on Euclidean dissimilarities. Such non-Euclidean dissimilarities are often corrected or a consistent Euclidean geometry is imposed on them via embedding. This paper commences by reviewing the available algorithms for analysing non-Euclidean dissimilarity data. The novel contribution is to show how the Ricci flow can be used to embed and rectify non-Euclidean dissimilarity data. According to our representation, the data is distributed over a manifold consisting of patches. Each patch has a locally uniform curvature, and this curvature is iteratively modified by the Ricci flow. The raw dissimilarities are the geodesic distances on the manifold. Rectified Euclidean dissimilarities are obtained using the Ricci flow to flatten the curved manifold by modifying the individual patch curvatures. We use two algorithmic components to implement this idea. Firstly, we apply the Ricci flow independently to a set of surface patches that cover the manifold. Second, we use curvature regularisation to impose consistency on the curvatures of the arrangement of different surface patches. We perform experiments on three real world datasets, and use these to determine the importance of the different algorithmic components, i.e. Ricci flow and curvature regularisation. We conclude that curvature regularisation is an essential step needed to control the stability of the piecewise arrangement of patches under the Ricci flow. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we aim to characterize graphs in terms of a structural measure of complexity. Our idea is to decompose a graph into layered substructures of increasing size, and then to measure the information content of these substructures. To locate dominant substructures within a graph, we commence by identifying a centroid vertex which has the minimum shortest path length variance to the remaining vertices. For each graph a family of centroid expansion subgraphs is derived from the centroid vertex in order to capture dominant structural characteristics of the graph. Since the centroid vertex is identified through a global analysis of the shortest path length distribution, the expansion subgraphs provide a fine representation of a graph structure. We then show how to characterize graphs using depth-based complexity traces. Here we explore two different strategies. The first strategy is to measure how the entropies on the centroid expansion subgraphs vary with the increasing size of the subgraphs. The second strategy is to measure how the entropy differences vary with the increasing size of the subgraphs. We perform graph classification in the principal component space of the complexity trace vectors. Experiments on graph datasets abstracted from some bioinformatics and computer vision databases demonstrate the effectiveness and efficiency of the proposed graph complexity traces. Our methods are competitive to state of the art methods. [Copyright &y& Elsevier]

Abstract: In this paper we show how to use blind source separation to estimate shape from polarised images from a fixed viewing and illumination direction. Our method does not require prior knowledge of the polariser angles nor the refractive index. We use weighted singular vector decomposition (SVD) to estimate the unknown parameters, where the weights are dictated by the physics of polarisation. In a two-step process, we firstly use least square fitting to obtain the polariser angles, and then estimate the refractive index using a mutual information criterion function. We show that the proposed method is capable of calculating robust shape information compared with alternative approaches relying on the same input information. The proposed method can be applied when using uncalibrated polarisation filters. It can also be used effectively when the subject moves during image capture. [Copyright &y& Elsevier]

Abstract: In many data analysis tasks, one is often confronted with the problem of selecting features from very high dimensional data. The feature selection problem is essentially a combinatorial optimization problem which is computationally expensive. To overcome this problem it is frequently assumed that features either independently influence the class variable or do so only involving pairwise feature interaction. To overcome this problem, we draw on recent work on hyper-graph clustering to select the most informative feature subset (mIFS) from a set of objects using high-order (rather than pairwise) similarities. There are two novel ingredients. First, we use a new information theoretic criterion referred to as the multidimensional interaction information (MII) to measure the significance of different feature combinations with respect to the class labels. Secondly, we use hypergraph clustering to select the most informative feature subset (mIFS), which has both low redundancy and strong discriminating power. The advantage of MII is that it incorporates third or higher order feature interactions. Hypergraph clustering, which extracts the most informative features. The size of the most informative feature subset (mIFS) is determined automatically. Experimental results demonstrate the effectiveness of our feature selection method on a number of standard data-sets. [Copyright &y& Elsevier]

Most existing feature selection methods focus on ranking individual features based on a utility criterion, and select the optimal feature set in a greedy manner. However, the feature combinations found in this way do not give optimal classification performance, since they neglect the correlations among features. In an attempt to overcome this problem, we develop a novel feature selection technique using the spectral data transformation and by using ℓ1-norm regularized models for subset selection. Specifically, we propose a new two-step spectral regression technique for unsupervised feature selection. In the first step, we use kernel entropy component analysis (kECA) to transform the data into a lower-dimensional space so as to improve class separation. Second, we use ℓ1-norm regularization to select the features that best align with the data embedding resulting from kECA. The advantage of kECA is that dimensionality reducing data transformation maximally preserves entropy estimates for the input data whilst also best preserving the cluster structure of the data. Using ℓ1-norm regularization, we cast feature discriminant analysis into a regression framework which accommodates the correlations among features. As a result, we can evaluate joint feature combinations, rather than being confined to consider them individually. Experimental results demonstrate the effectiveness of our feature selection method on a number of standard face datasets. [ABSTRACT FROM AUTHOR]

Abstract: This paper provides a review of recent work on the analysis of graphs, focussing in depth and comparing work by the Bern and York groups. The paper also offers directions for future investigation. [Copyright &y& Elsevier]

In this paper we use the Birkhoff-von Neumann decomposition of the diffusion kernel to compute a polytopal measure of graph complexity. We decompose the diffusion kernel into a series of weighted Birkhoff combinations and compute the entropy associated with the weighting proportions (polytopal complexity). The maximum entropy Birkhoff combination can be expressed in terms of matrix permanents. This allows us to introduce a phase-transition principle that links our definition of polytopal complexity to the heat flowing through the network at a given diffusion time. The result is an efficiently computed complexity measure, which we refer to as flow complexity. Moreover, the flow complexity measure allows us to analyze graphs and networks in terms of the thermodynamic depth. We compare our method with three alternative methods described in the literature (Estrada's heterogeneity index, the Laplacian energy, and the von Neumann entropy). Our study is based on 217 protein-protein interaction (PPI) networks including histidine kinases from several species of bacteria. We find a correlation between structural complexity and phylogeny (more evolved species have statistically more complex PPIs). Although our methods outperform the alternatives, we find similarities with Estrada's heterogeneity index in terms of network size independence and predictive power. [ABSTRACT FROM AUTHOR]

Abstract: The Ihara zeta function has proved to be a powerful tool in the analysis of graph structures. It is determined by the prime cycles of a finite graph and can be characterized in terms of a quasi characteristic polynomial of the adjacency matrix T of the oriented line graph associated to G. The coefficients of this polynomial, referred to as Ihara coefficients, have been used to characterize graphs in a permutation-invariant manner, and allow for an efficient evaluation of the Ihara zeta function. In this paper we present a novel method for computing the Ihara coefficients. We first establish a characterization of the Ihara coefficients in terms of complete Bell polynomials and, by exploiting a recursive relation for the latter, we show how the Ihara coefficients can be efficiently computed in , provided that the eigenvalues of T are known. [Copyright &y& Elsevier]

This paper addresses the problem of robust template tracking in image sequences. Our work falls within the discriminative framework in which the observations at each frame yield direct probabilistic predictions of the state of the target. Our primary contribution is that we explicitly address the problem that the prediction accuracy for different observations varies, and in some cases, can be very low. To this end, we couple the predictor to a probabilistic classifier which, when trained, can determine the probability that a new observation can accurately predict the state of the target (that is, determine the "relevance" or "reliability" of the observation in question). In the particle filtering framework, we derive a recursive scheme for maintaining an approximation of the posterior probability of the state in which multiple observations can be used and their predictions moderated by their corresponding relevance. In this way, the predictions of the "relevant" observations are emphasized, while the predictions of the "irrelevant" observations are suppressed. We apply the algorithm to the problem of 2D template tracking and demonstrate that the proposed scheme outperforms classical methods for discriminative tracking both in the case of motions which are large in magnitude and also for partial occlusions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the use of heat kernels as a means of embedding the individual nodes of a graph in a vector space. The reason for turning to the heat kernel is that it encapsulates information concerning the distribution of path lengths and hence node affinities on the graph. The heat kernel of the graph is found by exponentiating the Laplacian eigensystem over time. In this paper, we explore how graphs can be characterized in a geometric manner using embeddings into a vector space obtained from the heat kernel. We explore two different embedding strategies. The first of these is a direct method in which the matrix of embedding co-ordinates is obtained by performing a Young–Householder decomposition on the heat kernel. The second method is indirect and involves performing a low-distortion embedding by applying multidimensional scaling to the geodesic distances between nodes. We show how the required geodesic distances can be computed using parametrix expansion of the heat kernel. Once the nodes of the graph are embedded using one of the two alternative methods, we can characterize them in a geometric manner using the distribution of the node co-ordinates. We investigate several alternative methods of characterization, including spatial moments for the embedded points, the Laplacian spectrum for the Euclidean distance matrix and scalar curvatures computed from the difference in geodesic and Euclidean distances. We experiment with the resulting algorithms on the COIL database. [Copyright &y& Elsevier]

Abstract: This paper develops new geometrical filtering and edge detection algorithms for processing non-Euclidean image data. We view image data as residing on a Riemannian manifold, and we work with a representation based on the exponential map for this manifold together with the Riemannian weighted mean of image data. We show how the weighted mean can be efficiently computed using Newton''s method, which converges faster than the gradient descent method described elsewhere in the literature. Based on geodesic distances and the exponential map, we extend the classical median filter and the Perona–Malik anisotropic diffusion technique to smooth non-Euclidean image data. We then propose an anisotropic Gaussian kernel for image filtering, and we also show how both the median filter and the anisotropic Gaussian filter can be combined to develop a new edge preserving filter, which is effective at removing both Gaussian noise and impulse noise. By using the intrinsic metric of the feature manifold, we also generalise Di Zenzo''s structure tensor to non-Euclidean images for edge detection. We demonstrate the applications of our Riemannian filtering and edge detection algorithms both on directional and tensor-valued images. [Copyright &y& Elsevier]

In this paper we show how to estimate facial surface reflectance properties (a slice of the BRDF and the albedo) in conjunction with the facial shape from a single image. The key idea underpinning our approach is to iteratively interleave the two processes of estimating reflectance properties based on the current shape estimate and updating the shape estimate based on the current estimate of the reflectance function. For frontally illuminated faces, the reflectance properties can be described by a function of one variable which we estimate by fitting a curve to the scattered and noisy reflectance samples provided by the input image and estimated shape. For non-frontal illumination, we fit a smooth surface to the scattered 2D reflectance samples. We make use of a novel statistical face shape constraint which we term ‘model-based integrability’ which we use to regularise the shape estimation. We show that the method is capable of recovering accurate shape and reflectance information from single grayscale or colour images using both synthetic and real world imagery. We use the estimated reflectance measurements to render synthetic images of the face in varying poses. To synthesise images under novel illumination, we show how to fit a parametric model of reflectance to the estimated reflectance function. [ABSTRACT FROM AUTHOR]

Abstract: Graph structures have been proved important in high level-vision since they can be used to represent structural and relational arrangements of objects in a scene. One of the problems that arises in the analysis of structural abstractions of objects is graph clustering. In this paper, we explore how permutation invariants computed from the trace of the heat kernel can be used to characterize graphs for the purposes of measuring similarity and clustering. The heat kernel is the solution of the heat equation and is a compact representation of the path-length distribution on a graph. The trace of the heat kernel is given by the sum of the Laplacian eigenvalues exponentiated with time. We explore three different approaches to characterizing the heat kernel trace as a function of time. Our first characterization is based on the zeta function, which from the Mellin transform is the moment generating function of the heat kernel trace. Our second characterization is unary and is found by computing the derivative of the zeta function at the origin. The third characterization is derived from the heat content, i.e. the sum of the elements of the heat kernel. We show how the heat content can be expanded as a power series in time, and the coefficients of the series can be computed using the Laplacian spectrum. We explore the use of these characterizations as a means of representing graph structure for the purposes of clustering, and compare them with the use of the Laplacian spectrum. Experiments with the synthetic and real-world databases reveal that each of the three proposed invariants is effective and outperforms the traditional Laplacian spectrum. Moreover, the heat-content invariants appear to consistently give the best results in both synthetic sensitivity studies and on real-world object recognition problems. [Copyright &y& Elsevier]

Abstract: Standard particle filtering technique have previously been applied to the problem of fiber tracking by Brun et al. [Brun, A., Bjornemo, M., Kikinis, R., Westin, C.F., 2002. White matter tractography using sequential importance sampling. In: Proceedings of the ISMRM Annual Meeting, p. 1131] and Bjornemo et al. [Bjornemo, M., Brun, A., Kikinis, R., Westin, C.F., 2002. Regularized stochastic white matter tractography using diffusion tensor MRI, In: Proc. MICCAI, pp. 435–442]. However, these previous attempts have not utilised the full power of the technique, and as a result the fiber paths were tracked in a goal directed way. In this paper, we provide an advanced technique by presenting a fast and novel probabilistic method for white matter fiber tracking in diffusion weighted MRI (DWI), which takes advantage of the weighting and resampling mechanism of particle filtering. We formulate fiber tracking using a non-linear state space model which captures both smoothness regularity of the fibers and the uncertainties in the local fiber orientations due to noise and partial volume effects. Global fiber tracking is then posed as a problem of particle filtering. To model the posterior distribution, we classify voxels of the white matter as either prolate or oblate tensors. We then construct the orientation distributions for prolate and oblate tensors separately. Finally, the importance density function for particle filtering is modeled using the von Mises–Fisher distribution on a unit sphere. Fast and efficient sampling is achieved using Ulrich–Wood’s simulation algorithm. Given a seed point, the method is able to rapidly locate the globally optimal fiber and also provides a probability map for potential connections. The proposed method is validated and compared to alternative methods both on synthetic data and real-world brain MRI datasets. [Copyright &y& Elsevier]

Abstract: In this paper we develop a new formulation of probabilistic relaxation labelling using the theory of diffusion processes on graphs. Our aim is to tackle the problem of labelling objects consistently and unambiguously using information concerning label consistency and initial label probabilities. We abstract this problem using a support graph with each graph node an object-label assignment. Initial object-label probabilities then evolve across the graph under the governance of the Fokker–Planck equation in terms of an infinitesimal generator matrix computed from the edge weights of the support graph. In this way we effectively kernelise probabilistic relaxation. Encouraging results are obtained in applying the new relaxation process in the applications of scene labelling, data classification, and feature correspondence matching. [Copyright &y& Elsevier]

Abstract: A new method for smoothing both gray-scale and color images is presented that relies on the heat diffusion equation on a graph. We represent the image pixel lattice using a weighted undirected graph. The edge weights of the graph are determined by the Gaussian weighted distances between local neighboring windows. We then compute the associated Laplacian matrix (the degree matrix minus the adjacency matrix). Anisotropic diffusion across this weighted graph-structure with time is captured by the heat equation, and the solution, i.e. the heat kernel, is found by exponentiating the Laplacian eigensystem with time. Image smoothing is accomplished by convolving the heat kernel with the image, and its numerical implementation is realized by using the Krylov subspace technique. The method has the effect of smoothing within regions, but does not blur region boundaries. We also demonstrate the relationship between our method, standard diffusion-based PDEs, Fourier domain signal processing and spectral clustering. Experiments and comparisons on standard images illustrate the effectiveness of the method. [Copyright &y& Elsevier]

Abstract: In this paper, we present a new 3D triangular mesh parameterization method that is computationally efficient and yields minimized distance errors. The method has four steps. Firstly, multidimensional scaling (MDS) is used to flatten each submesh consisting of one vertex and its direct neighbours on the 3D triangular mesh. Secondly, an optimal method is used to compute the linear reconstructing weights of each vertex with respect to its neighbours. Thirdly, a spectral decomposition method is used to obtain initial 2D parameterization coordinates. Fourthly, the initial coordinates are rotated and scaled to minimize the distance errors. It is demonstrated that this method can be used for texture mapping. Analyses and examples show the effectiveness of this parameterization method compared with alternatives. [Copyright &y& Elsevier]

This paper presents a novel method for 3D surface reconstruction that uses polarization and shading information from two views. The method relies on the polarization data acquired using a standard digital camera and a linear polarizer. Fresnel theory is used to process the raw images and to obtain initial estimates of surface normals, assuming that the reflection type is diffuse. Based on this idea, the paper presents two novel contributions to the problem of surface reconstruction. The first is a technique to enhance the surface normal estimates by incorporating shading information into the method. This is done using robust statistics to estimate how the measured pixel brightnesses depend on the surface orientation. This gives an estimate of the object material reflectance function, which is used to refine the estimates of the surface normals. The second contribution is to use the refined estimates to establish correspondence between two views of an object. To do this, surface patches are extracted from each view, which are then aligned by minimising an energy functional based on the surface normal estimates and local topographic properties. The optimum alignment parameters for different patch pairs are then used to establish stereo correspondence. This process results in an unambiguous field of surface normals, which can be integrated to recover the surface depth. Our technique is most suited to smooth nonmetallic surfaces. It complements existing stereo algorithms since it does not require salient surface features to obtain correspondences. An extensive set of experiments, yielding reconstructed objects and reflectance functions, are presented and compared to ground truth. [ABSTRACT FROM AUTHOR]

This paper exploits the properties of the commute time between nodes of a graph for the purposes of clustering and embedding and explores its applications to image segmentation and multibody motion tracking. Our starting point is the lazy random walk on the graph, which is determined by the heat kernel of the graph and can be computed from the spectrum of the graph Laplacian. We characterize the random walk using the commute time (that is, the expected time taken for a random walk to travel between two nodes and return) and show how this quantity may be computed from the Laplacian spectrum using the discrete Green's function. Our motivation is that the commute time can be anticipated to be a more robust measure of the proximity of data than the raw proximity matrix. In this paper, we explore two applications of the commute time. The first is to develop a method for image segmentation using the eigenvector corresponding to the smallest eigenvalue of the commute time matrix. We show that our commute time segmentation method has the property of enhancing the intragroup coherence while weakening intergroup coherence and is superior to the normalized cut. The second application is to develop a robust multibody motion tracking method using an embedding based on the commute time. Our embedding procedure preserves commute time and is closely akin to kernel PCA, the Laplacian eigenmap, and the diffusion map. We illustrate the results on both synthetic image sequences and real-world video sequences and compare our results with several alternative methods. [ABSTRACT FROM AUTHOR]

Abstract: This paper exploits the properties of the commute time for the purposes of graph simplification and matching. Our starting point is the lazy random walk on the graph, which is determined by the heat kernel of the graph and can be computed from the spectrum of the graph Laplacian. We characterise the random walk using the commute time between nodes, and show how this quantity may be computed from the Laplacian spectrum using the discrete Green''s function. In this paper, we explore two different, but essentially dual, simplified graph representations delivered by the commute time. The first representation decomposes graphs into concentric layers. To do this we augment the graph with an auxiliary node which acts as a heat source. We use the pattern of commute times from this node to decompose the graph into a sequence of layers. Our second representation is based on the minimum spanning tree of the commute time matrix. The spanning trees located using commute time prove to be stable to structural variations. We match the graphs by applying a tree-matching method to the spanning trees. We experiment with the method on synthetic and real-world image data, where it proves to be effective. [Copyright &y& Elsevier]

Abstract: This paper focusses on how reflectance models based on scattering theory and reported in the physics literature can be used for making estimates of surface roughness parameters using reflectance measurements obtained with a digital camera. We commence by reviewing the Beckmann–Kirchhoff (B–K) scatter theory, and the recent modification to it by Vernold and Harvey. We show how this model can be used to estimate surface roughness parameters for dielectric surfaces using pixel brightness measurements. Using the roughness parameter measurements we compare the model with reflectance measurements from the CUReT database. This comparison shows that the Vernold–Harvey modification of the B–K model gives a better fit to data than the Oren–Nayar model for certain types of rough surface. [Copyright &y& Elsevier]

Abstract: In this paper we address the problem of how to learn a structural prototype that can be used to represent the variations present in a set of trees. The prototype serves as a pattern space representation for the set of trees. To do this we construct a super-tree to span the union of the set of trees. This is a chicken and egg problem, since before the structure can be estimated correspondences between the nodes of the super-tree and the nodes of the sample tree must be to hand. We demonstrate how to simultaneously estimate the structure of the super-tree and recover the required correspondences by minimizing the sum of the tree edit-distances over pairs of trees, subject to edge consistency constraints. Each node of the super-tree corresponds to a dimension of the pattern space, and for each tree we construct a pattern vector in which the elements of the weights corresponding to each of the dimensions of the super-tree. We perform pattern analysis on the set of trees by performing principal components analysis on the vectors. The method is illustrated on a shape analysis problem involving shock-trees extracted from the skeletons of 2D objects. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

This paper offers two new directions to shape-from-shading, namely the use of the heat equation to smooth the field of surface normals and the recovery of surface height using a low-dimensional embedding. Turning our attention to the first of these contributions, we pose the problem of surface normal recovery as that of solving the steady state heat equation subject to the hard constraint that Lambert's law is satisfied. We perform our analysis on a plane perpendicular to the light source direction, where the z component of the surface normal is equal to the normalized image brightness. The x — y or azimuthal component of the surface normal is found by computing the gradient of a scalar field that evolves with time subject to the heat equation. We solve the heat equation for the scalar potential and, hence, recover the azimuthal component of the surface normal from the average image brightness, making use of a simple finite difference method. The second contribution is to pose the problem of recovering the surface height function as that of embedding the field of surface normals on a manifold so as to preserve the pattern of surface height differences and the lattice footprint of the surface normals. We experiment with the resulting method on a variety of real-world image data, where it produces qualitatively good reconstructed surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we show how a statistical model of facial shape can be embedded within a shape-from-shading algorithm. We describe how facial shape can be captured using a statistical model of variations in surface normal direction. To construct this model, we make use of the azimuthal equidistant projection to map the distribution of surface normals from the polar representation on a unit sphere to Cartesian points on a local tangent plane. The distribution of surface normal directions is captured using the covariance matrix for the projected point positions. The eigenvectors of the covariance matrix define the modes of shape-variation in the fields of transformed surface normals. We show how this model can be trained using surface normal data acquired from range images and how to fit the model to intensity images of faces using constraints on the surface normal direction provided by Lambert's law. We demonstrate that the combination of a global statistical constraint and local irradiance constraint yields an efficient and accurate approach to facial shape recovery and is capable of recovering fine local surface details. We assess the accuracy of the technique on a variety of images with ground truth and real-world images. [ABSTRACT FROM AUTHOR]

This paper describes how a facial albedo map can be recovered from a single image using a statistical model that captures variations in surface normal direction. We fit the model to intensity data using constraints on the surface normal direction provided by Lambert's law and then use the differences between observed and reconstructed image brightness to estimate the albedo. We show that this process is stable under varying illumination. We then show how eigenfaces trained on albedo maps may provide a better representation for illumination insensitive recognition than those trained on raw image intensity. [ABSTRACT FROM AUTHOR]

This paper poses the problem of tree-clustering as that of fitting a mixture of tree unions to a set of sample trees. The tree-unions are structures from which the individual data samples belonging to a cluster can be obtained by edit operations. The distribution of observed tree nodes in each cluster sample is assumed to be governed by a Bernoulli distribution. The clustering method is designed to operate when the correspondences between nodes are unknown and must be inferred as part of the learning process. We adopt a minimum description length approach to the problem of fitting the mixture model to data. We make maximum-likelihood estimates of the Bernoulli parameters. The tree-unions and the mixing proportions are sought so as to minimize the description length criterion. This is the sum of the negative logarithm of the Bernoulli distribution, and a message-length criterion that encodes both the complexity of the union-trees and the number of mixture components. We locate node correspondences by minimizing the edit distance with the current tree unions, and show that the edit distance is linked to the description length criterion. The method can be applied to both unweighted and weighted trees. We illustrate the utility of the resulting algorithm on the problem of classifying 2D shapes using a shock graph representation. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates spectral approaches to the problem of point pattern matching. We make two contributions. First, we consider rigid point-set alignment. Here we show how kernel principal components analysis (kernel PCA) can be effectively used for solving the rigid point correspondence matching problem when the point-sets are subject to outliers and random position jitter. Specifically, we show how the point- proximity matrix can be kernelised, and spectral correspondence matching transformed into one of kernel PCA. Second, we turn our attention to the matching of articulated point-sets. Here we show label consistency constraints can be incorporated into definition of the point proximity matrix. The new methods are compared to those of Shapiro and Brady and Scott and Longuet-Higgins, together with multidimensional scaling. We provide experiments on both synthetic data and real world data. [Copyright &y& Elsevier]

The Hamilton-Jacobi approach has proven to be a powerful and elegant method for extracting the skeleton of two-dimensional (2-D) shapes. The approach is based on the observation that the normalized flux associated with the inward evolution of the object boundary at nonskeletal points tends to zero as the size of the integration area tends to zero, while the flux is negative at the locations of skeletal points. Nonetheless, the error in calculating the flux on the image lattice is both limited by the pixel resolution and also proportional to the curvature of the boundary evolution front and, hence, unbounded near endpoints. This makes the exact location of endpoints difficult and renders the performance of the skeleton extraction algorithm dependent on a threshold parameter. This problem can be overcome by using interpolation techniques to calculate the flux with subpixel precision. However, here, we develop a method for 2-D skeleton extraction that circumvents the problem by eliminating the curvature contribution to the error. This is done by taking into account variations of density due to boundary curvature. This yields a skeletonization algorithm that gives both better localization and less susceptibility to boundary noise and parameter choice than the Hamilton-Jacobi method. [ABSTRACT FROM AUTHOR]

Abstract: Although inexact graph-matching is a problem of potentially exponential complexity, the problem may be simplified by decomposing the graphs to be matched into smaller subgraphs. If this is done, then the process may cast into a hierarchical framework and hence rendered suitable for parallel computation. In this paper we describe a spectral method which can be used to partition graphs into non-overlapping subgraphs. In particular, we demonstrate how the Fiedler-vector of the Laplacian matrix can be used to decompose graphs into non-overlapping neighbourhoods that can be used for the purposes of both matching and clustering. [Copyright &y& Elsevier]

Abstract: It is well known that many surfaces exhibit reflectance that is not well modelled by Lambert''s law. This is the case not only for surfaces that are rough or shiny, but also those that are matte and composed of materials that are particle suspensions. As a result, standard Lambertian shape from shading methods cannot be applied directly to the analysis of rough and shiny surfaces. In order to overcome this difficulty, in this paper, we consider how to reconstruct the Lambertian component for rough and shiny surfaces when the object is illuminated in the viewing direction (retroreflection). To do this we make use of the diffuse reflectance models described by Oren and Nayar, and by Wolff. Our experiments with synthetic and real-world data reveal the effectiveness of the correction method, leading to improved surface normal and height recovery. [Copyright &y& Elsevier]

Abstract: In this paper we describe a new shape-from-shading method. We show how the parallel transport of surface normals can be used to impose curvature consistency and also to iteratively update surface normal directions so as to improve the brightness error. We commence by showing how to make local estimates of the Hessian matrix from surface normal information. With the local Hessian matrix to hand, we develop an “EM-like” algorithm for updating the surface normal directions. At each image location, parallel transport is applied to the neighbouring surface normals to generate a sample of local surface orientation predictions. From this sample, a local weighted estimate of the image brightness is made. The transported surface normal which gives the brightness prediction which is closest to this value is selected as the revised estimate of surface orientation. The revised surface normals obtained in this way may in turn be used to re-estimate the Hessian matrix, and the process iterated until stability is reached. We experiment with the method on a variety of real world and synthetic data. Here we explore the properties of the fields of surface normals and the height data delivered by the method. [Copyright &y& Elsevier]

Abstract: This paper describes a graph-spectral method for 3D surface integration. The algorithm takes as its input a 2D field of surface normal estimates, delivered, for instance, by a shape-from-shading or shape-from-texture procedure. We commence by using the surface normals to obtain an affinity weight matrix whose elements are related to the surface curvature. The weight matrix is used to compute a row-normalized transition probability matrix, and we pose the recovery of the integration path as that of finding the steady-state random walk for the Markov chain defined by this matrix. The steady-state random walk is given by the leading eigenvector of the original affinity weight matrix. By threading the surface normals together along the path specified by the magnitude order of the components of the leading eigenvector we perform surface integration. The height increments along the path are simply related to the traversed path length and the slope of the local tangent plane. The method is evaluated on needle-maps delivered by a shape-from-shading algorithm applied to real-world data and also on synthetic data. The method is compared with the local geometric height reconstruction method of Bors, Hancock and Wilson, and the global methods of Horn and Brooks and Frankot and Chellappa. [Copyright &y& Elsevier]

Graph structures have proven computationally cumbersome for pattern analysis. The reason for this is that, before graphs can be converted to pattern vectors, correspondences must be established between the nodes of structures which are potentially of different size. To overcome this problem, in this paper, we turn to the spectral decomposition of the Laplacian matrix. We show how the elements of the spectral matrix for the Laplacian can be used to construct symmetric polynomials that are permutation invariants. The coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner. We extend this representation to graphs in which there are unary attributes on the nodes and binary attributes on the edges by using the spectral decomposition of a Hermitian property matrix that can be viewed as a complex analogue of the Laplacian. To embed the graphs in a pattern space, we explore whether the vectors of invariants can be embedded in a low-dimensional space using a number of alternative strategies, including principal components analysis (PCA), multidimensional scaling (MDS), and locality preserving projection (LPP). Experimentally, we demonstrate that the embeddings result in well-defined graph clusters. Our experiments with the spectral representation involve both synthetic and real-world data. The experiments with synthetic data demonstrate that the distances between spectral feature vectors can be used to discriminate between graphs on the basis of their structure. The real-world experiments show that the method can be used to locate clusters of graphs. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we explore how graph-spectral methods can be used to develop a new shape-from-shading algorithm. We characterize the field of surface normals using a weight matrix whose elements are computed from the sectional curvature between different image locations and penalize large changes in surface normal direction. Modeling the blocks of the weight matrix as distinct surface patches, we use a graph seriation method to find a surface integration path that maximizes the sum of curvature-dependent weights and that can be used for the purposes of height reconstruction. To smooth the reconstructed surface, we fit quadrics to the height data for each patch. The smoothed surface normal directions are updated ensuring compliance with Lambert's law. The processes of height recovery and surface normal adjustment are interleaved and iterated until a stable surface is obtained. We provide results on synthetic and real-world imagery. [ABSTRACT FROM AUTHOR]

This paper presents an iterative spectral framework for pairwise clustering and perceptual grouping. Our model is expressed in terms of two sets of parameters. Firstly, there are cluster memberships which represent the affinity of objects to clusters. Secondly, there is a matrix of link weights for pairs of tokens. We adopt a model in which these two sets of variables are governed by a Bernoulli model. We show how the likelihood function resulting from this model may be maximised with respect to both the elements of link-weight matrix and the cluster membership variables. We establish the link between the maximisation of the log-likelihood function and the eigenvectors of the link-weight matrix. This leads us to an algorithm in which we iteratively update the link-weight matrix by repeatedly refining its modal structure. Each iteration of the algorithm is a three-step process. First, we compute a link-weight matrix for each cluster by taking the outer-product of the vectors of current cluster-membership indicators for that cluster. Second, we extract the leading eigenvector from each modal link-weight matrix. Third, we compute a revised link weight matrix by taking the sum of the outer products of the leading eigenvectors of the modal link-weight matrices. [Copyright &y& Elsevier]

The modal correspondence method of Shapiro and Brady aims to match point-sets by comparing the eigenvectors of a pairwise point proximity matrix. Although elegant by means of its matrix representation, the method is notoriously susceptible to differences in the relational structure of the point-sets under consideration. In this paper, we demonstrate how the method can be rendered robust to structural differences by adopting a hierarchical approach. To do this, we place the modal matching problem in a probabilistic setting in which the correspondences between pairwise clusters can be used to constrain the individual point correspondences. We demonstrate the utility of the method on a number of synthetic and real-world point-pattern matching problems. [ABSTRACT FROM AUTHOR]

This paper demonstrates how the EM algorithm can be used for learning and matching mixtures of point distribution models. We make two contributions. First, we show how shape-classes can be learned in an unsupervised manner. We present a fast procedure for training point distribution models using the EM algorithm. Rather than estimating the class means and covariance matrices needed to construct the PDM, the method iteratively refines the eigenvectors of the covariance matrix using a gradient ascent technique. Second, we show how recognition by alignment can be realised by fitting a mixture of linear shape deformations. We evaluate the method on the problem of learning the class-structure and recognising Arabic characters. [Copyright &y& Elsevier]

This paper develops a maximum a posteriori (MAP) probability estimation framework for shape-from-shading (SFS) from synthetic aperture radar (SAR) images. The aim is to use this method to reconstruct surface topography from a single radar image of relatively complex terrain. Our MAP framework makes explicit how the recovery of local surface orientation depends on the whereabouts of terrain edge features and the available radar reflectance information. To apply the resulting process to real world radar data, we require probabilistic models for the appearance of terrain features and the relationship between the orientation of surface normals and the radar reflectance. We show that the SAR data can be modeled using a Rayleigh-Bessel distribution and use this distribution to develop a maximum likelihood algorithm for detecting and labeling terrain edge features. Moreover, we show how robust statistics can be used to estimate the characteristic parameters of this distribution. We also develop an empirical model for the SAR reflectance function. Using the reflectance model, we perform Lambertian correction so that a conventional SFS algorithm can be applied to the radar data. The initial surface normal direction is constrained to point in the direction of the nearest ridge or ravine feature. Each surface normal must fall within a conical envelope whose axis is in the direction of the radar illuminant. The extent of the envelope depends on the corrected radar reflectance and the variance of the radar signal statistics. We explore various ways of smoothing the field of surface normals using robust statistics. Finally, we show how to reconstruct the terrain surface from the smoothed field of surface normal vectors. The proposed algorithm is applied to various SAR data sets containing relatively complex terrain structure. [ABSTRACT FROM AUTHOR]

Analyzes the recurrent correlation associative memory (RCAM) model of Chiueh and Goodman. Pattern recall in the RCAM; Pattern model; Measuring recall performance of the excitation function; Conclusion.

This paper presents a new method for computing the tree edit distance problem with uniform edit cost. We commence by showing that any tree obtained with a sequence of cut operations is a subtree of the transitive closure of the original tree, we show that the necessary condition for any subtree to be a solution can be reduced to a clique problem in a derived structure. Using this idea we transform the problem of computing tree edit distance into a series of maximum weight clique problems. We, then use relaxation labeling to find an approximation to the tree edit distance. [Copyright &y& Elsevier]