Lexing Ying

Fast Discrete Curvelet Transforms

This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform in two and three dimensions. The first digital transformation is based on unequally spaced fast Fourier transforms, while the second is based on the wrapping of specially selected Fourier samples. The two implementations essentially differ by the choice of spatial grid used to translate curvelets at each scale and angle. Both digital transformations return a table of digital curvelet coefficients indexed by a scale parameter, an orientation parameter, and a spatial location parameter. And both implementations are fast in the sense that they run in O(n^2 log n) flops for n by n Cartesian arrays; in addition, they are also invertible, with rapid inversion algorithms of about the same complexity. Our digital transformations improve upon earlier implementations—based upon the first generation of curvelets—in the sense that they are conceptually simpler, faster, and far less redundant. The software CurveLab, which implements both transforms presented in this paper, is available at http://www.curvelet.org.

Texture and shape synthesis on surfaces

We present a novel method for texture synthesis on surfaces from examples. We consider a very general type of textures, including color, transparency and displacements. Our method synthesizes the texture directly on the surface, rather than synthesizing a texture image and then mapping it to the surface. The synthesized textures have the same qualitative visual appearance as the example texture, and cover the surfaces without the distortion or seams of conventional texture-mapping. We describe two synthesis methods, based on the work of Wei and Levoy and Ashikhmin; our techniques produce similar results, but directly on surfaces.

3D discrete curvelet transform

In this paper, we present the first 3D discrete curvelet transform. This transform is an extension to the 2D transform described in Candes et al..1 The resulting curvelet frame preserves the important properties, such as parabolic scaling, tightness and sparse representation for singularities of codimension one. We describe three different implementations: in-core, out-of-core and MPI-based parallel implementations. Numerical results verify the desired properties of the 3D curvelets and demonstrate the efficiency of our implementations.

A simple manifold-based construction of surfaces of arbitrary smoothness

We present a smooth surface construction based on the manifold approach of Grimm and Hughes. We demonstrate how this approach can relatively easily produce a number of desirable properties which are hard to achieve simultaneously with polynomial patches, subdivision or variational surfaces. Our surfaces are C∞-continuous with explicit nonsingular C∞ parameterizations, high-order flexible at control vertices, depend linearly on control points, have fixed-size local support for basis functions, and have good visual quality.

Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers

This paper introduces a new sweeping preconditioner for the iterative solution of the variable coefficient Helmholtz equation in two and three dimensions. The algorithms follow the general structur ...

Fast Directional Multilevel Algorithms for Oscillatory Kernels

This paper introduces a new directional multilevel algorithm for solving

A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains

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SelInv---An Algorithm for Selected Inversion of a Sparse Symmetric Matrix

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Fast Computation of Fourier Integral Operators

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Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework I: Total energy calculation

-point problems with highly oscillatory kernels. These systems often result from the boundary integral formulations of scattering problems and are difficult due to the oscillatory nature of the kernel and the non-uniformity of the particle distribution. We address the problem by first proving that the interaction between a ball of radius