bspa
A Box Spline Subdivision Pyramid Algorithm
D. J. Hebert
Directory and file names begin with:   bspa
bibTex entry:
@article{KN:Heb98bsspa,
 author ="D. J. Hebert",
 title ="A Box Spline Subdivision Pyramid Algorithm",
 journal ="Applied Math Letters",
      volume ="12",
 date ="1999",
 pages ="57-62"
}
Abstract:
Using the refinement coefficients of the ZP-element box spline to form a lowpass filter mask, and a subdivision difference as a high pass filter, we scan  an image along a triangular Sierpinski path to create an efficient one-dimensional  pyramid algorithm for a two dimensional non-separablewavelet-like transform based  on the quincunx multiresolution analysis.

ciqaqm
Cyclic Interlaced Quadtree Algorithms for Quincunx Multiresolution
D. J. Hebert
Directory and file names begin with: ciqaqm
bibTex entry:
@article{KN:Heb98ciqaqm,
 author ="D. J. Hebert",
 title ="Cyclic Interlaced Quadtree Algorithms for Quincunx Multiresolution",
 journal ="Journal of Algorithms",
      volume ="27",
 pages ="97-128"
 year ="1998"
}
Abstract:
Recent advances in wavelet theory and in finite element computations draw attention to a well-known, simple and computationally efficient triangulation method.   We take a  new look at this triangulation, which is obtained by repeated symmetric bisection, starting with a half square. The cells   form the leaves of a binary tree and the nodes of a directed graph  consisting of a single simple cycle.  Computational speed  is facilitated by the binary and
quad-digit expression of triangle vertices, which reduce all vertex calculations to simple integer and logical operations.  The leaf cycle interlaces a pair of quadtrees whose  orientations differ by $\pi/4$.  Detailed analysis leads to algorithms which exploit the structure and computational efficiencies.

brwm
A Branching Random Walk Model for Diffusion-Reaction-Convection
D. J. Hebert
Directory and file names begin with: brwm
bibTex entry:
@article{KN:Heb97brwm,
 author ="D. J. Hebert",
 title= "A Branching Random Walk Model for Diffusion-Reaction-Convection",
 journal ="Advances in Mathematics",
 volume ="125",
 pages ="121-153",
 year ="1997"}
Abstract:
A discrete stochastic model is introduced for populations which are diffusing, interacting, and drifting on an integer lattice in a finite dimensional  space.  The model is used to construct a Markov process whose countable state space is  a set of location maps, assigning individuals to their positions.  The expected population sizes and population densities satisfy partial difference approximations to the nonlinear partial differential equations of diffusion-reaction-convection. Constructive convergence estimates  give convergence results for the random process and its ensemble average.

fwced
A Fast-wavelet Compass Edge Detector
D. J. Hebert and HyungJun Kim
Directory and file names begin with: fwced
Reseach Report: ICMA-95-196
Date of latest version: July, 1996
@article{KN:HebKim96fwced,
 author ="D. J. Hebert and HyungJun Kim",
 title ="A Fast Wavelet Compass Edge Detector",
 year ="1996",
 journal ="Proc. SPIE, Wavelet Applications in Signal and Image Processing",
 volume ="2825",
 year ="1996",
 pages ="432-442"
}
Abstract
As an approach to the wavelet detection of local scale and orientation in 2dimensional  images we make use of a well-known, computationally efficient triangulation of the  image domain and some of its lesser-known properties. We choose wavelets supported  by cartesian and quincunx lattice squares, antisymmetricabout the diagonal.  Computational algorithmsare based on properties of the triangulations such as  the following: The cells of the triangulation form the leaves of a binary tree  and the nodes ofa directed graph consisting of a simple cycle; the cells are  also identifiedwith blocks of interlaced quadtrees consisting of cartesian and  quincunx lattice points which also form the vertices of the cells. Pyramid algorithmsbased on hierarchical triangular scanning of the pixels and half-wavelet-supporting triangles provide efficient encoding and decodingbased on local triangle data and stacks.  As  examples we introduce a 2-dimensional Haar wavelet which detects diagonals of squares and we construct a triangular version of the  TS wavelet transform which has been recentlyproposed as an efficient approach to lossless and lossy image compression.We render an edge-enhanced image by reconstruction from  significant coefficients of edge-detecting wavelets.

ietw
Image Encoding with Triangulation Wavelets
D. J. Hebert and HyungJun Kim
Directory and file names begin with: ietw
Reseach Report: ICMA-95-xxx
Date of latest version: July, 1995
bibTex entry:
@article{KN:HebKim95ietw,
 author ="D. J. Hebert and HyungJun Kim",
 title ="Image Encoding with Triangulation Wavelets",
 journal ="Proc. SPIE, Wavelet Applications in Signal and Image Processing",
 volume ="2569",
 year ="1995",
 pages ="381-392"}
Abstract:
We demonstrate some wavelet-based image processing applications of a class of  simplicial grids  arising in finite element computations and computer graphics. The cells of a  triangular grid form the set of leaves of a binary tree and the nodes of a directed graph  consisting of a single cycle.  The leaf cycle of a uniform grid  forms a pattern for pixel image  scanning and for coherent computation of coefficients of  splines and wavelets.  A simple form
of image encoding  is accomplished with a one dimensional quadrature mirror filter whose  coefficients represent  an  expansion of the image in terms of two dimensional Haar wavelets with triangular support.  A combination the leaf cycle and an inherent quadtree structure allow  efficient neighbor finding, grid refinement, tree pruning and storage.   Pruning of the simplex tree yields  a partially compressed image which requires no decoding, but rather may be rendered as a shaded triangulation.  This structure and its  generalization to n-dimensions form a convenient setting for wavelet analysis and  computations based on simplicial grids.

slrtg
Symbolic Local Refinement of Tetrahedral Grids
D. J. Hebert
Directory and file names begin with: slrtg
Reseach Report: ICMA-xx-xxx
Date of latest version:
bibTex entry:
@article{KN:Heb94slrtg,
 author ="D. J. Hebert",
 title= "Symbolic Local Refinement of Tetrahedral Grids",
 journal ="J. Symbolic Computation",
 volume ="17",
 pages ="457-472",
 year ="1994"}
Abstract:
A recent local grid refinement algorithm for simplicial grids is shown to be suitable for  symbolic implementation in the 3-dimensional case. An addressing scheme  stores all the  geometric information about the tetrahedra in the refinement tree.  Location of vertices and  the addresses  of physically nearest neighbors are computed by decoding the symbols  of the simplex address. Bisection and face-compatible refinement of the simplex and its splitting  neighbors  are obtained by symbolic and logical operations on the leaves  of the tree.

ssrds
Simulations of Stochastic Reaction Diffusion Systems
D.J. Hebert
Directory and file names begin with: slrtg
Reseach Report: ICMA-xx-xxx
bibTex entry:
@article{KN:Heb92ssrds,
 author ="D. J. Hebert",
 title ="Simulations of Stochastic Reaction Diffusion Systems",
 journal ="Math. and Comput. in Simulation",
 volume ="34",
 pages ="411-432",
 year ="1992"}
Abstract
Direct simulation of a discrete stochastic model of reaction-diffusion provides a means of studying the fluctuations which occur when populations are finite.  This paper introduces the mathematical model along with the computational model which implements it,  and shows the relationship to some standard random methods for pde simulation.  Two examples are then given: First, the stochastic phase field model is introduced.  This attempt to study the influence of randomness in liquid-solid interface problems leads easily to meaningful extensions of the standard pde model. Second, some experiments with a model threshhold reaction  problem are described.  These illustrate many of the effects of the stochastic model which the pde idealization does not address. In particular, varying the population scale in the model leads to a variety of new observations and conjectures.