A smooth, efficient representation of reflectance

Reflectance plays an important role in computer graphics. It describes the appearance of an object with two directional parameters. Reflectance is critical, because it determines the appearance of the object to be synthesized. Reflectance can be determined either by an analytical model, or by evaluating a fit to a measured reflectance data set. In general, analytical models are complex and computationally expensive to evaluate and it is difficult to control the parameters of the model to obtain a desired appearance. A popular method of fitting data is by using a basis function expansion. However, this method requires many basis functions to represent the strongly-peaked data and the result is computationally expensive. We propose a method to overcome this problem by using a modified N-dimensional multilevel B-spline approximation. Our method fits various reflectance data very well. Multilevel architecture makes it possible to control the accuracy of the fit. A higher level fit uses a denser control mesh and fits more accurately. In addition, the resulting fit is very smooth and efficient to evaluate. The time complexity of evaluation is a constant regardless of the fit level. A higher level fit requires more storage than a lower level fit. The storage might be a problem on memory intensive applications. To overcome this, we represent a data set with two fits, a diffuse fit and a specular fit and we can successfully compress the storage for finer fit without losing major performance from the original method. In addition, by utilizing minimal perfect hashing, we can retrieve the value of each control point efficiently from compressed table

Research issues in data modeling for scientific visualization

This article summarizes some topics of modeling as they impinge on the future development of scientific data visualization. The benefits from visualization techniques in analyzing data are well established, but to build on these pioneering efforts, one must recognize modeling as a distinct structural component in the larger context of visualization and problem-solving systems. Volume modeling is the entry way to this arena of future development, and model-based rendering describes how scientists will view the results. Important side developments such as multiresolution modeling and model-based segmentation will contribute structural capability to these systems. All of these components ultimately depend on the mathematical foundations of scattered data modeling and on model validation and standards to incorporate this modeling methodology into effective tools for scientific inquiry.Postprint (published version

A Fast Adaptive Wavelet Scheme in RBF Collocation for Nearly Singular Potential PDEs

We present a wavelet based adaptive scheme and investigate the efficiency of this scheme for solving nearly singular potential PDEs. Multiresolution wavelet analysis (MRWA) provides a firm mathematical foundation by projecting the solution of PDE onto a nested sequence of approximation spaces. The wavelet coefficients then were used as an estimation of the sensible regions for node adaptation. The proposed adaptation scheme requires negligible calculation time due to the existence of the fast DiscreteWavelet Transform (DWT). Certain aspects of the proposed adaptive scheme are discussed through numerical examples. It has been shown that the proposed adaptive scheme can detect the singularities both in the domain and near the boundaries. Moreover, the proposed adaptive scheme can be utilized for capturing the regions with high gradient both in the solution and its spatial derivatives. Due to the simplicity of the proposed method, it can be efficiently applied to large scale nearly singular engineering problems

Quantitative tools for seismic stratigraphy and lithology characterization

Seismological images represent maps of the earth's structure. Apparent bandwidth limitation of seismic data prevents successful estimation of transition sharpness by the multiscale wavelet transform. We discuss the application of two recently developed techniques for (non-linear) singularity analysis designed for bandwidth limited data, such as imaged seismic reflectivity. The first method is a generalization of Mallat's modulus maxima approach to a method capable of estimating coarse-grained local scaling/sharpness/Hölder regularity of edges/transitions from data residing at essentially one single scale. The method is based on a non-linear criterion predicting the (dis)appearance of local maxima as a function of the data's fractional integrations/differentiations. The second method is an extension of an atomic decomposition technique based on the greedy Matching Pursuit Algorithm. Instead of the ordinary Spline Wavelet Packet Basis, our method uses multiple Fractional Spline Wavelet Packet Bases, especially designed for seismic reflectivity data. The first method excels in pinpointing the location of the singularities (the stratigraphy). The second method improves the singularity characterization by providing information on the transition's location, magnitude, scale, order and direction (anti-/causal/symmetric). Moreover, the atomic decomposition entails data compression, denoising and deconvolution. The output of both methods produces a map of the earth's singularity structure. These maps can be overlayed with seismic data, thus providing us with a means to more precisely characterize the seismic reflectivity's litho-stratigraphical information content.Massachusetts Institute of Technology. Industry Consorti

Coronal Mass Ejection Detection using Wavelets, Curvelets and Ridgelets: Applications for Space Weather Monitoring

Coronal mass ejections (CMEs) are large-scale eruptions of plasma and magnetic feld that can produce adverse space weather at Earth and other locations in the Heliosphere. Due to the intrinsic multiscale nature of features in coronagraph images, wavelet and multiscale image processing techniques are well suited to enhancing the visibility of CMEs and supressing noise. However, wavelets are better suited to identifying point-like features, such as noise or background stars, than to enhancing the visibility of the curved form of a typical CME front. Higher order multiscale techniques, such as ridgelets and curvelets, were therefore explored to characterise the morphology (width, curvature) and kinematics (position, velocity, acceleration) of CMEs. Curvelets in particular were found to be well suited to characterising CME properties in a self-consistent manner. Curvelets are thus likely to be of benefit to autonomous monitoring of CME properties for space weather applications.Comment: Accepted for publication in Advances in Space Research (3 April 2010

Distance function wavelets - Part I: Helmholtz and convection-diffusion transforms and series

This report aims to present my research updates on distance function wavelets (DFW) based on the fundamental solutions and the general solutions of the Helmholtz, modified Helmholtz, and convection-diffusion equations, which include the isotropic Helmholtz-Fourier (HF) transform and series, the Helmholtz-Laplace (HL) transform, and the anisotropic convection-diffusion wavelets and ridgelets. The latter is set to handle discontinuous and track data problems. The edge effect of the HF series is addressed. Alternative existence conditions for the DFW transforms are proposed and discussed. To simplify and streamline the expression of the HF and HL transforms, a new dimension-dependent function notation is introduced. The HF series is also used to evaluate the analytical solutions of linear diffusion problems of arbitrary dimensionality and geometry. The weakness of this report is lacking of rigorous mathematical analysis due to the author's limited mathematical knowledge.Comment: Welcome any comments to [email protected]

Wavelet based Adaptive RBF Method for Nearly Singular Poisson-Type Problems on Irregular Domains

We present a wavelet based adaptive scheme and investigate the efficiency of this scheme for solving nearly singular potential PDEs over irregularly shaped domains. For a problem defined over Ω ∈ ℜd, the boundary of an irregularly shaped domain, Γ, is defined as a boundary curve that is a product of a Heaviside function along the normal direction and a piecewise continuous tangential curve. The link between the original wavelet based adaptive method presented in Libre, Emdadi, Kansa, Shekarchi, and Rahimian (2008, 2009) or LEKSR method and the generalized one is given through the use of simple Heaviside masking procedure. In addition level dependent thresholding were introduced to improve the efficiency and convergence rate of the solution. We will show how the generalized wavelet based adaptive method can be applied for detecting nearly singularities in Poisson type PDEs over irregular domains. The numerical examples have illustrated that the proposed method is powerful to analyze the Poisson type PDEs with rapid changes in gradients and nearly singularities