(Contact djh@pitt.edu for information)

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.