A Multiscale Pyramid Transform for Graph Signals

Multiscale transforms designed to process analog and discrete-time signals and images cannot be directly applied to analyze high-dimensional data residing on the vertices of a weighted graph, as they do not capture the intrinsic geometric structure of the underlying graph data domain. In this paper, we adapt the Laplacian pyramid transform for signals on Euclidean domains so that it can be used to analyze high-dimensional data residing on the vertices of a weighted graph. Our approach is to study existing methods and develop new methods for the four fundamental operations of graph downsampling, graph reduction, and filtering and interpolation of signals on graphs. Equipped with appropriate notions of these operations, we leverage the basic multiscale constructs and intuitions from classical signal processing to generate a transform that yields both a multiresolution of graphs and an associated multiresolution of a graph signal on the underlying sequence of graphs.Comment: 16 pages, 13 figure

Diamond-based models for scientific visualization

Hierarchical spatial decompositions are a basic modeling tool in a variety of application domains including scientific visualization, finite element analysis and shape modeling and analysis. A popular class of such approaches is based on the regular simplex bisection operator, which bisects simplices (e.g. line segments, triangles, tetrahedra) along the midpoint of a predetermined edge. Regular simplex bisection produces adaptive simplicial meshes of high geometric quality, while simplifying the extraction of crack-free, or conforming, approximations to the original dataset. Efficient multiresolution representations for such models have been achieved in 2D and 3D by clustering sets of simplices sharing the same bisection edge into structures called diamonds. In this thesis, we introduce several diamond-based approaches for scientific visualization. We first formalize the notion of diamonds in arbitrary dimensions in terms of two related simplicial decompositions of hypercubes. This enables us to enumerate the vertices, simplices, parents and children of a diamond. In particular, we identify the number of simplices involved in conforming updates to be factorial in the dimension and group these into a linear number of subclusters of simplices that are generated simultaneously. The latter form the basis for a compact pointerless representation for conforming meshes generated by regular simplex bisection and for efficiently navigating the topological connectivity of these meshes. Secondly, we introduce the supercube as a high-level primitive on such nested meshes based on the atomic units within the underlying triangulation grid. We propose the use of supercubes to associate information with coherent subsets of the full hierarchy and demonstrate the effectiveness of such a representation for modeling multiresolution terrain and volumetric datasets. Next, we introduce Isodiamond Hierarchies, a general framework for spatial access structures on a hierarchy of diamonds that exploits the implicit hierarchical and geometric relationships of the diamond model. We use an isodiamond hierarchy to encode irregular updates to a multiresolution isosurface or interval volume in terms of regular updates to diamonds. Finally, we consider nested hypercubic meshes, such as quadtrees, octrees and their higher dimensional analogues, through the lens of diamond hierarchies. This allows us to determine the relationships involved in generating balanced hypercubic meshes and to propose a compact pointerless representation of such meshes. We also provide a local diamond-based triangulation algorithm to generate high-quality conforming simplicial meshes

Adaptive Scattered Data Fitting with Tensor Product Spline-Wavelets

The core of the work we present here is an algorithm that constructs a least squares approximation to a given set of unorganized points. The approximation is expressed as a linear combination of particular B-spline wavelets. It implies a multiresolution setting which constructs a hierarchy of approximations to the data with increasing level of detail, proceeding from coarsest to finest scales. It allows for an efficient selection of the degrees of freedom of the problem and avoids the introduction of an artificial uniform grid. In fact, an analysis of the data can be done at each of the scales of the hierarchy, which can be used to select adaptively a set of wavelets that can represent economically the characteristics of the cloud of points in the next level of detail. The data adaption of our method is twofold, as it takes into account both horizontal distribution and vertical irregularities of data. This strategy can lead to a striking reduction of the problem complexity. Furthermore, among the possible ways to achieve a multiscale formulation, the wavelet approach shows additional advantages, based on good conditioning properties and level-wise orthogonality. We exploit these features to enhance the efficiency of iterative solution methods for the system of normal equations of the problem. The combination of multiresolution adaptivity with the numerical properties of the wavelet basis gives rise to an algorithm well suited to cope with problems requiring fast solution methods. We illustrate this by means of numerical experiments that compare the performance of the method on various data sets working with different multi-resolution bases. Afterwards, we use the equivalence relation between wavelets and Besov spaces to formulate the problem of data fitting with regularization. We find that the multiscale formulation allows for a flexible and efficient treatment of some aspects of this problem. Moreover, we study the problem known as robust fitting, in which the data is assumed to be corrupted by wrong measurements or outliers. We compare classical methods based on re-weighting of residuals to our setting in which the wavelet representation of the data computed by our algorithm is used to locate the outliers. As a final application that couples two of the main applications of wavelets (data analysis and operator equations), we propose the use of this least squares data fitting method to evaluate the non-linear term in the wavelet-Galerkin formulation of non-linear PDE problems. At the end of this thesis we discuss efficient implementation issues, with a special interest in the interplay between solution methods and data structures

Multi-Modal Enhancement Techniques for Visibility Improvement of Digital Images

Image enhancement techniques for visibility improvement of 8-bit color digital images based on spatial domain, wavelet transform domain, and multiple image fusion approaches are investigated in this dissertation research. In the category of spatial domain approach, two enhancement algorithms are developed to deal with problems associated with images captured from scenes with high dynamic ranges. The first technique is based on an illuminance-reflectance (I-R) model of the scene irradiance. The dynamic range compression of the input image is achieved by a nonlinear transformation of the estimated illuminance based on a windowed inverse sigmoid transfer function. A single-scale neighborhood dependent contrast enhancement process is proposed to enhance the high frequency components of the illuminance, which compensates for the contrast degradation of the mid-tone frequency components caused by dynamic range compression. The intensity image obtained by integrating the enhanced illuminance and the extracted reflectance is then converted to a RGB color image through linear color restoration utilizing the color components of the original image. The second technique, named AINDANE, is a two step approach comprised of adaptive luminance enhancement and adaptive contrast enhancement. An image dependent nonlinear transfer function is designed for dynamic range compression and a multiscale image dependent neighborhood approach is developed for contrast enhancement. Real time processing of video streams is realized with the I-R model based technique due to its high speed processing capability while AINDANE produces higher quality enhanced images due to its multi-scale contrast enhancement property. Both the algorithms exhibit balanced luminance, contrast enhancement, higher robustness, and better color consistency when compared with conventional techniques. In the transform domain approach, wavelet transform based image denoising and contrast enhancement algorithms are developed. The denoising is treated as a maximum a posteriori (MAP) estimator problem; a Bivariate probability density function model is introduced to explore the interlevel dependency among the wavelet coefficients. In addition, an approximate solution to the MAP estimation problem is proposed to avoid the use of complex iterative computations to find a numerical solution. This relatively low complexity image denoising algorithm implemented with dual-tree complex wavelet transform (DT-CWT) produces high quality denoised images

Doctor of Philosophy

dissertationShape analysis is a well-established tool for processing surfaces. It is often a first step in performing tasks such as segmentation, symmetry detection, and finding correspondences between shapes. Shape analysis is traditionally employed on well-sampled surfaces where the geometry and topology is precisely known. When the form of the surface is that of a point cloud containing nonuniform sampling, noise, and incomplete measurements, traditional shape analysis methods perform poorly. Although one may first perform reconstruction on such a point cloud prior to performing shape analysis, if the geometry and topology is far from the true surface, then this can have an adverse impact on the subsequent analysis. Furthermore, for triangulated surfaces containing noise, thin sheets, and poorly shaped triangles, existing shape analysis methods can be highly unstable. This thesis explores methods of shape analysis applied directly to such defect-laden shapes. We first study the problem of surface reconstruction, in order to obtain a better understanding of the types of point clouds for which reconstruction methods contain difficulties. To this end, we have devised a benchmark for surface reconstruction, establishing a standard for measuring error in reconstruction. We then develop a new method for consistently orienting normals of such challenging point clouds by using a collection of harmonic functions, intrinsically defined on the point cloud. Next, we develop a new shape analysis tool which is tolerant to imperfections, by constructing distances directly on the point cloud defined as the likelihood of two points belonging to a mutually common medial ball, and apply this for segmentation and reconstruction. We extend this distance measure to define a diffusion process on the point cloud, tolerant to missing data, which is used for the purposes of matching incomplete shapes undergoing a nonrigid deformation. Lastly, we have developed an intrinsic method for multiresolution remeshing of a poor-quality triangulated surface via spectral bisection

State-of-the-Art and Trends in Scalable Video Compression with Wavelet Based Approaches

3noScalable Video Coding (SVC) differs form traditional single point approaches mainly because it allows to encode in a unique bit stream several working points corresponding to different quality, picture size and frame rate. This work describes the current state-of-the-art in SVC, focusing on wavelet based motion-compensated approaches (WSVC). It reviews individual components that have been designed to address the problem over the years and how such components are typically combined to achieve meaningful WSVC architectures. Coding schemes which mainly differ from the space-time order in which the wavelet transforms operate are here compared, discussing strengths and weaknesses of the resulting implementations. An evaluation of the achievable coding performances is provided considering the reference architectures studied and developed by ISO/MPEG in its exploration on WSVC. The paper also attempts to draw a list of major differences between wavelet based solutions and the SVC standard jointly targeted by ITU and ISO/MPEG. A major emphasis is devoted to a promising WSVC solution, named STP-tool, which presents architectural similarities with respect to the SVC standard. The paper ends drawing some evolution trends for WSVC systems and giving insights on video coding applications which could benefit by a wavelet based approach.partially_openpartially_openADAMI N; SIGNORONI. A; R. LEONARDIAdami, Nicola; Signoroni, Alberto; Leonardi, Riccard

Texture analysis and Its applications in biomedical imaging: a survey

Texture analysis describes a variety of image analysis techniques that quantify the variation in intensity and pattern. This paper provides an overview of several texture analysis approaches addressing the rationale supporting them, their advantages, drawbacks, and applications. This survey’s emphasis is in collecting and categorising over five decades of active research on texture analysis.Brief descriptions of different approaches are presented along with application examples. From a broad range of texture analysis applications, this survey’s final focus is on biomedical image analysis. An up-to-date list of biological tissues and organs in which disorders produce texture changes that may be used to spot disease onset and progression is provided. Finally, the role of texture analysis methods as biomarkers of disease is summarised.Manuscript received February 3, 2021; revised June 23, 2021; accepted September 21, 2021. Date of publication September 27, 2021; date of current version January 24, 2022. This work was supported in part by the Portuguese Foundation for Science and Technology (FCT) under Grants PTDC/EMD-EMD/28039/2017, UIDB/04950/2020, PestUID/NEU/04539/2019, and CENTRO-01-0145-FEDER-000016 and by FEDER-COMPETE under Grant POCI-01-0145-FEDER-028039. (Corresponding author: Rui Bernardes.)info:eu-repo/semantics/publishedVersio