Vector-valued Data Research Articles

Radial basis function based level set interpolation and evolution for deformable modelling

We present a study in level set representation and evolution using radial basis functions (RBFs) for active contour and active surface models. It builds on recent works by others who introduced RBFs into level sets for structural topology optimisation. Here, we introduce the concept into deformable models and present a new level set formulation able to handle more complex topological changes, in particular perturbation away from the evolving front. In the conventional level set technique, the initial active contour/surface is implicitly represented by a signed distance function and periodically re-initialised to maintain numerical stability. We interpolate the initial distance function using RBFs on a much coarser grid, which provides great potential in modelling in high dimensional space. Its deformation is considered as an updating of the RBF interpolants, an ordinary differential equation (ODE) problem, instead of a partial differential equation (PDE) problem, and hence it becomes much easier to solve. Re-initialisation is found no longer necessary, in contrast to conventional finite difference method (FDM) based level set approaches. The proposed level set updating scheme is efficient and does not suffer from self-flattening while evolving, hence it avoids large numerical errors. Further, more complex topological changes are readily achievable and the initial contour or surface can be placed arbitrarily in the image. These properties are extensively demonstrated on both synthetic and real 2D and 3D data. We also present a novel active contour model, implemented with this level set scheme, based on multiscale learning and fusion of image primitives from vector-valued data, e.g. colour images, without channel separation or decomposition.

Affine invariant non-linear anisotropic diffusion smoothing strategy for vector-valued images

The Gaussian noise introduced into the diffusion tensor images (DTIs) can bring serious impacts on tensor calculation and fibre tracking. To decrease the effects of the Gaussian noise, many Euclidean invariant gradient (EIG) based anisotropic diffusion denoising methods have been presented. In this paper, the effects of the Gaussian noise on calculated tensors were analysed and an affine invariant gradient (AIG) based non-linear anisotropic smoothing strategy was presented. The AIG based smoothing strategy is the development of the affine invariant non-linear anisotropic diffusion (AINAD) restoration model, introduced by Guillermo Sapiro, and adopted to restore vector-valued data. To evaluate the efficiency of the AINAD model in accounting for the Gaussian noise introduced into the vector-valued data, the peak to peak signal-to-noise ratio (PSNR) and signal-to-mean squared error ratio (SMSE) metrics are used. The experiment results acquired from the synthetic and real data prove the good performance of the presented filter.

AIG based nonlinear anisotropic smoothing strategy for vector-valued images

The effects of the Rician noise on the calculated tensors are analyzed and an affine invariant gradient (AIG) based nonlinear anisotropic smoothing strategy is presented. The AIG based smoothing strategy is a development of the affine invariant nonlinear anisotropic diffusion (AINAD) restoration model, introduced by Guillermo Sapiro, and adopted to restore vector-valued data. To evaluate the efficiency of the presented AINAD model in accounting for the Rician noise introduced into the vector-valued data, the peak-to-peak signal-to-noise ratio (PSNR), signal-to-mean squared error ratio (SMSE) and Beta(parameter that stands for edge preservation) metrics are used. The experiment results acquired from the synthetic and real data prove the good performance of the presented filter.

Convex Hodge Decomposition and Regularization of Image Flows

The total variation (TV) measure is a key concept in the field of variational image analysis. In this paper, we focus on vector-valued data and derive from the Hodge decomposition of image flows a definition of TV regularization for vector-valued data that extends the standard componentwise definition in a natural way. We show that our approach leads to a convex decomposition of arbitrary vector fields, providing a richer decomposition into piecewise harmonic fields rather than piecewise constant ones, and motion texture. Furthermore, our regularizer provides a measure for motion boundaries of piecewise harmonic image flows in the same way, as the TV measure does for contours of scalar-valued piecewise constant images.

A Review of Recent Advances in Surface Defect Detection using Texture analysis Techniques

In this paper, we systematically review recent advances in surface inspection using computer vision andimage processing techniques, particularly those based on texture analysis methods. The aim is to reviewthe state-of-the-art techniques for the purposes of visual inspection and decision making schemes that areable to discriminate the features extracted from normal and defective regions. This field is so vast that itis impossible to cover all the aspects of visual inspection. This paper focuses on a particular but importantsubset which generally treats visual surface inspection as texture analysis problems. Other topics related tovisual inspection such as imaging system and data acquisition are out of the scope of this survey.The surface defects are loosely separated into two types. One is local textural irregularities which is themain concern for most visual surface inspection applications. The other is global deviation of colour and/ortexture, where local pattern or texture does not exhibit abnormalities. We refer this type of defects as shadeor tonality problem. The second type of defects have been largely neglected until recently, particularly whencolour imaging system has been widely used in visual inspection and where chromatic consistency plays animportant role in quality control. The emphasis of this survey though is still on detecting local abnormalities,given the fact that majority of the reported works are dealing with the first type of defects.The techniques used to inspect textural abnormalities are discussed in four categories, statistical approaches,structural approaches, filter based methods, and model based approaches, with a comprehensivelist of references to some recent works. Due to rising demand and practice of colour texture analysis inapplication to visual inspection, those works that are dealing with colour texture analysis are discussedseparately. It is also worth noting that processing vector-valued data has its unique challenges, which conventionalsurface inspection methods have often ignored or do not encounter.We also compare classification approaches with novelty detection approaches at the decision makingstage. Classification approaches often require supervised training and usually provide better performancethan novelty detection based approaches where training is only carried out on defect-free samples. However,novelty detection is relatively easier to adapt and is particularly desirable when training samples areincomplete

Simple integrative preprocessing preserves what is shared in data sources

BackgroundBioinformatics data analysis toolbox needs general-purpose, fast and easily interpretable preprocessing tools that perform data integration during exploratory data analysis. Our focus is on vector-valued data sources, each consisting of measurements of the same entity but on different variables, and on tasks where source-specific variation is considered noisy or not interesting. Principal components analysis of all sources combined together is an obvious choice if it is not important to distinguish between data source-specific and shared variation. Canonical Correlation Analysis (CCA) focuses on mutual dependencies and discards source-specific "noise" but it produces a separate set of components for each source.ResultsIt turns out that components given by CCA can be combined easily to produce a linear and hence fast and easily interpretable feature extraction method. The method fuses together several sources, such that the properties they share are preserved. Source-specific variation is discarded as uninteresting. We give the details and implement them in a software tool. The method is demonstrated on gene expression measurements in three case studies: classification of cell cycle regulated genes in yeast, identification of differentially expressed genes in leukemia, and defining stress response in yeast. The software package is available at .ConclusionWe introduced a method for the task of data fusion for exploratory data analysis, when statistical dependencies between the sources and not within a source are interesting. The method uses canonical correlation analysis in a new way for dimensionality reduction, and inherits its good properties of being simple, fast, and easily interpretable as a linear projection.

Recovery Algorithms for Vector-Valued Data with Joint Sparsity Constraints

Vector-valued data appearing in concrete applications often possess sparse expansions with respect to a preassigned frame for each vector component individually. Additionally, different components may also exhibit common sparsity patterns. Recently, there were introduced sparsity measures that take into account such joint sparsity patterns, promoting coupling of nonvanishing components. These measures are typically constructed as weighted $\ell_1$ norms of componentwise $\ell_q$ norms of frame coefficients. We show how to compute solutions of linear inverse problems with such joint sparsity regularization constraints by fast thresholded Landweber algorithms. Next we discuss the adaptive choice of suitable weights appearing in the definition of sparsity measures. The weights are interpreted as indicators of the sparsity pattern and are iteratively updated after each new application of the thresholded Landweber algorithm. The resulting two-step algorithm is interpreted as a double-minimization scheme for a suitable target functional. We show its $\ell_2$-norm convergence. An implementable version of the algorithm is also formulated, and its norm convergence is proven. Numerical experiments in color image restoration are presented.

Iterative thresholding algorithms

This article provides a variational formulation for hard and firm thresholding. A related functional can be used to regularize inverse problems by sparsity constraints. We show that a damped hard or firm thresholded Landweber iteration converges to its minimizer. This provides an alternative to an algorithm recently studied by the authors. We prove stability of minimizers with respect to the parameters of the functional by means of Γ-convergence. All investigations are done in the general setting of vector-valued (multi-channel) data.

Orthogonal wavelet frames and vector-valued wavelet transforms

Motivated by the notion of orthogonal frames, we describe sufficient conditions for the construction of orthogonal MRA wavelet frames in L 2 ( R ) from a suitable scaling function. These constructions naturally lead to filter banks in ℓ 2 ( Z ) with similar orthogonality relations and, through these filter banks, the orthogonal wavelet frames give rise to a vector-valued discrete wavelet transform (VDWT). The novelty of these constructions lies in their potential for use with vector-valued data, where the VDWT seeks to exploit correlation between channels. Extensions to higher dimensions are natural and the constructions corresponding to the bidimensional case are presented along with preliminary results of numerical experiments in which the VDWT is applied to color image data.

Nonlinear structure tensors

In this article, we introduce nonlinear versions of the popular structure tensor, also known as second moment matrix. These nonlinear structure tensors replace the Gaussian smoothing of the classical structure tensor by discontinuity-preserving nonlinear diffusions. While nonlinear diffusion is a well-established tool for scalar and vector-valued data, it has not often been used for tensor images so far. Two types of nonlinear diffusion processes for tensor data are studied: an isotropic one with a scalar-valued diffusivity, and its anisotropic counterpart with a diffusion tensor. We prove that these schemes preserve the positive semidefiniteness of a matrix field and are, therefore, appropriate for smoothing structure tensor fields. The use of diffusivity functions of total variation (TV) type allows us to construct nonlinear structure tensors without specifying additional parameters compared to the conventional structure tensor. The performance of nonlinear structure tensors is demonstrated in three fields where the classic structure tensor is frequently used: orientation estimation, optic flow computation, and corner detection. In all these cases, the nonlinear structure tensors demonstrate their superiority over the classical linear one. Our experiments also show that for corner detection based on nonlinear structure tensors, anisotropic nonlinear tensors give the most precise localisation.

Vector-valued image regularization with PDEs: a common framework for different applications

In this paper, we focus on techniques for vector-valued image regularization, based on variational methods and PDEs. Starting from the study of PDE-based formalisms previously proposed in the literature for the regularization of scalar and vector-valued data, we propose a unifying expression that gathers the majority of these previous frameworks into a single generic anisotropic diffusion equation. On one hand, the resulting expression provides a simple interpretation of the regularization process in terms of local filtering with spatially adaptive Gaussian kernels. On the other hand, it naturally disassembles any regularization scheme into the smoothing process itself and the underlying geometry that drives the smoothing. Thus, we can easily specialize our generic expression into different regularization PDEs that fulfill desired smoothing behaviors, depending on the considered application: image restoration, inpainting, magnification, flow visualization, etc. Specific numerical schemes are also proposed, allowing us to implement our regularization framework with accuracy by taking the local filtering properties of the proposed equations into account. Finally, we illustrate the wide range of applications handled by our selected anisotropic diffusion equations with application results on color images.

Variational Problems and Partial Differential Equations on Implicit Surfaces

A novel framework for solving variational problems and partial differential equations for scalar and vector-valued data defined on surfaces is introduced in this paper. The key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed Cartesian coordinate system using this new embedding function. The equations are then both intrinsic to the surface and defined in the embedding space. This approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces, as is commonly done in the literature. We describe the framework and present examples in computer graphics and image processing applications, including texture synthesis, flow field visualization, and image and vector field intrinsic regularization for data defined on 3D surfaces.

On the level lines and geometry of vector-valued images

In this letter, we extend the concept of level lines of scalar images to vector-valued data. Consistent with the scalar case, we define the level-lines of vector-valued images as the integral curves of the directions of minimal vectorial change. This direction, and the magnitude of the change, are computed using classical Riemannian geometry. As an example of the use of this new concept, we show how to visualize the basic geometry of vector-valued images with a scalar image.

Color Snakes

A framework for object segmentation in vector-valued images is presented in this paper. The first scheme proposed is based on geometric active contours moving toward the objects to be detected in the vector-valued image. Object boundaries are obtained as geodesics or minimal weighted-distance curves, where the metric is given by a definition of edges in vector-valued data. The curve flow corresponding to the proposed active contours holds formal existence, uniqueness, stability, and correctness results. The scheme automatically handles changes in the deforming curve topology. The technique is applicable, for example, to color and texture images as well as multiscale representations. We then present an extension of these vector active contours, proposing a possible image flow for vector-valued image segmentation. The algorithm is based on moving each one of the image level sets according to the proposed vector active contours. This extension also shows the relation between active contours and a number of partial-differential-equations-based image processing algorithms as anisotropic diffusion and shock filters.

The compass (star) identity for vector-valued rational interpolants

The compass identity (Wynn's five point star identity) for Pade approximants connects neighbouring elements called N, S, E, W and C in the Pade table. Its form has been extended to the cases of rational interpolation of ordinary (scalar) data and interpolation of vector-valued data. In this paper, full specifications of the associated five point identity for the scalar denominator polynomials and the vector numerator polynomials of the vector-valued rational interpolants on real data points are given, as well as the related generalisations of Frobenius' identities. Unique minimal forms of the polynomials constituting the interpolants and results about unattainable points correspond closely to their counterparts in the scalar case.

Probabilistic models in cluster analysis

This paper discusses cluster analysis in a probabilistic and inferential framework as opposed to more exploratory, heuristic or algorithmic approaches. It presents a broad survey on probabilistic models for partition-type, hierarchical and tree-like clustering structures and points to the relevant literature. It is shown how suitable clustering criteria or grouping methods may be derived from these models in the case of vector-valued data, dissimilarity matrices and similarity relations. In particular, we discuss hypothesis testing for homogeneity or for a grouping structure, the asymptotic distribution of test statistics, the use of random graph theory and combinatorial methods for simulating random dendrograms. Our presentation of hierarchies includes, e.g., Markovian branching processes and phylogenetic inference based on molecular sequence data.

Termination and continuity of greedy growing for tree-structured vector quantizers

Tree-structured vector quantizers (TSVQ) provide a computationally efficient, variable-rate method of compressing vector-valued data. In applications, the problem of designing a TSVQ from empirical training data is critical. Greedy growing algorithms are a common and effective approach to the design problem. They are recursive procedures that produce a TSVQ one node at a time by optimizing a simple splitting criterion at each step. While unsupervised greedy growing algorithms are well-understood from an experimental point of view, there has been little theory to support their use, or to examine their behavior on large training sets. The authors present a rigorous analysis of a greedy growing algorithm proposed by Riskin (1990), Riskin and Gray (1991), and Balakrishnan (1991). The first part of the paper is a description of the algorithm and an examination of its asymptotic behavior as it applies to a fixed, absolutely continuous distribution. The second part of the paper establishes the structural consistency of the algorithm with respect to a convergent sequence of distributions. As an application, the authors obtain results concerning the large-sample empirical behavior of the algorithm when it is applied to stationary ergodic training vectors.

Discriminant Analysis Using a Bicriteria Linear Program

In the past decade, there has been some interest shown in the use of linear programs to solve discriminant analysis problems. In this paper we present and clarify an alternative to the statistical approach for discriminating between two (or more) groups of vector-valued data in which an order relation is given. By the nature of this problem, a bicriteria linear program can be used instead of a statistical approach and hence no statistical assumptions need to be imposed on the problem.

Vector Correlation: Review, Exposition, and Geographic Application

Abstract Correlation provides one of the most useful and widely employed techniques of data analysis in geography and in the larger scientific community. Most correlation coefficients describe relations among scalar data. Vector-valued data have received less attention despite their obvious geographic importance: vectors can describe motions across the landscape and relations between locations. After a brief review of the meaning and past applications of vector correlation, we present a vector correlation measure for comparing two-dimensional vectors. This vector correlation describes the goodness-of-fit of a relationship between two sets of vectors that includes translation, scaling, and either rotational or reflectional dependency. Linear regression parameters additionally emerge from the computation. We illustrate the properties of the vector correlation using wind velocity data.

Vector-valued, rational interpolants III

In 1963, Wynn proposed a method for rational interpolation of vector-valued quantities given on a set of distinct interpolation points. He used continued fractions, and generalized inverses for the reciprocals of vector-valued quantities. In this paper, we present an axiomatic approach to vector-valued rational interpolation. Uniquely defined interpolants are constructed for vector-valued data so that the components of the resulting vector-valued rational interpolant share a common denominator polynomial. An explicit determinantal formula is given for the denominator polynomial for the cases of (i) vector-valued rational interpolation on distinct real or complex points and (ii) vector-valued Pade approximation. We derive the connection with thee-algorithm of Wynn and Claessens, and we establish a five-term recurrence relation for the denominator polynomials.