Joakim Gundersen

A new generation of wavelet shrinkage: adaptive strategies based on composition of Lorentz-type thresholding and Besov-type non-threshold shrinkage

This article is a systematic overview of compression, smoothing and denoising techniques based on shrinkage of wavelet coefficients, and proposes an advanced technique for generating enhanced composite wavelet shrinkage strategies.

Wavelet compression, data fitting and approximation based on adaptive composition of lorentz-type thresholding and besov-type non-threshold shrinkage

In this study we initiate the investigation of a new advanced technique, proposed in Section 6 of [3], for generating adaptive Besov–Lorentz composite wavelet shrinkage strategies We discuss some advantages of the Besov–Lorentz approach compared to firm thresholding.

Computing n -variate orthogonal discrete wavelet transforms on graphics processing units

In [4,5] an algorithm was proposed for isometric mapping between smooth n-variate m-dimensional vector fields and fractal curves and surfaces, by using orthonormal wavelet bases This algorithm matched only the orthonormal bases of scaling functions (the “V-spaces” of multiresolution analyses) In the present communication we shall consider a new algorithm which matches the orthonormal bases of wavelets (the “W-spaces” of multiresolution analyses) Being of Cantor diagonal type, it was applicable for both bounded and unbounded domains, but the complexity of its implementation was rather high In [3] we proposed a simpler algorithm for the case of boundary-corrected wavelet basis on a bounded hyper-rectangle In combination with the algorithm for the “V-spaces” from [4,5], the new algorithm provides the opportunity to compute multidimensional orthogonal discrete wavelet transform (DWT) in two ways – via the “classical” way for computing multidimensional wavelet transforms, and by using a commutative diagram of mappings of the bases, resulting in an equivalent computation on graphics processing units (GPUs) The orthonormality of the wavelet bases ensures that the direct and inverse transformations of the bases are mutually adjoint (transposed in the case of real entries) orthogonal matrices, which eases the computations of matrix inverses in the algorithm 1D and 2D orthogonal wavelet transforms have been first implemented for parallel computing on GPUs using C++ and OpenGL shading language around the year 2000; our new algorithm allows to extend general-purpose computing on GPUs (GPGPU) also to higher-dimensional wavelet transforms If used in combination with the Cantor diagonal type algorithm of [4,5] (the “V-space” basis matching) this algorithm can in principle be applied for computing DWT of n-variate vector fields defined on the whole ℝn However, if boundary-corrected wavelets are considered for vector-fields defined on a bounded hyper-rectangle in ℝn, then the present algorithm for GPU-based computation of n-variate orthogonal DWT can be enhanced with the new simple “V-space”-basis matching algorithm of [3] It is this version that we consider in detail in the present study.

Wavelet-based isometric conversion between dimension and resolution and some of its applications

In this paper we provide an overview about an orthonormal (multi) wavelet-based method for isometric immersion of smooth n-variate m-dimensional vector fields onto fractal curves and surfaces. This method was proposed in an earlier publication by two of the authors, with the purpose of extending the applicability of emerging GPU-programming to rich diversity of multidimensional problems. Here we propose (in Section 3) several directions for upgrading the method, with respective new applications.

On the Scientific Visualization of Complex‐Valued Functions of One Complex Variable

In [1] a method for colour‐map scientific visualization of parametric surfaces in 3D space was proposed. In [1] this method was applied for pixel‐based computation of the solution of geometrical intersection problems in 3D using the computer’s GPU (Graphics Processing Unit, i.e., graphics card) as a computational resource with parallel architecture. In the present work we represent the complex‐valued function w = f(z) of one complex argument as a surface in 3D (with parametrization x1 = φ(u,v), x2 = ψ(u,v), x3 = 0, where u = Rz, v = Sz, φ = Rw, ψ = Sw) and apply the above‐said method of [1]. The resulting approach is illustrated on several graphical examples.In its introduction the paper contains also a concise overview of other known approaches and techniques for scientific visualization of complex‐valued functions of a complex variable.Among some topics for future work, in the final section we discuss the possibility of extending the approach proposed here for visualization of graphs of vector functions...

WAVELET‐BASED LOSSLESS ONE‐AND TWO‐DIMENSIONAL REPRESENTATION OF MULTIDIMENSIONAL GEOMETRIC DATA

In the present communication we develop a complete representation of whole multidimensional manifolds, with the Cantor diagonal type of algorithm [1, 2] replaced by a new and simpler type of Cartesian‐indexing basis‐matching algorithm [3]. We provide graphical comparison between the results obtained via the Cantor diagonal algorithm and the Cartesian‐indexing algorithm. For this purpose, we test the algorithms on several different types of ‘benchmark’ multidimensional manifolds: Greens functions for linear PDEs, Cartesian products of 3‐dimensional manifolds, intersections of multidimensional manifolds. One new type of intersection problems which can be solved invoking the new representation is computing the intersections of multidimensional manifolds in parametric form (rather than only in implicit form, as earlier [3]).This work is based on research conducted within two consecutive Strategic Projects of the Norwegian Research Council: ‘GPGPU—Graphics Hardware as a High‐end Computational Resource’ (2004‐...

Vertex-based marching algorithms for finding multidimensional geometric intersections

This article is a survey of the current state of the art in vertex-based marching algorithms for solving systems of nonlinear equations and solving multidimensional intersection problems. It addresses also ongoing research and future work on the topic. Among the new topics discussed here for the first time is the problem of characterizing the type of singularities of piecewise affine manifolds, which are the numerical approximations to the solution manifolds, as generated by the most advanced of the considered vertex-based algorithms: the Marching-Simplex algorithm. Several approaches are proposed for solving this problem, all of which are related to modifications, extensions and generalizations of the Morse lemma in differential topology.