# PUBLICATIONS

FFTs on the Rotation Group
Earlier work by
Driscoll and Healy
has produced an efficient
$O(B^2\log^2 B)$
algorithm for computing the Fourier transform of band-limited
functions on the 2-sphere. In this paper, we discuss an implementation
of an $O(B^4)$ algorithm for the numerical computation of Fourier transforms of
functions defined on the rotation group, $SO(3)$. This compares with the
direct $O(B^6)$ approach. The algorithm we implemented is
based on the ``Separation of Variables'' technique, e.g. as presented
by Maslen and Rockmore. In conjunction with the techniques
developed by Driscoll and Healy, the $SO(3)$ algorithm we implemented may be made
truly fast, $O(B^3\log^2 B)$.
Basic results will be
presented establishing the algorithm's numerical stability,
and examples of applications will be presented.

- P. J. Kostelec and D. N. Rockmore,
"FFTs on the Rotation Group,"
*Santa Fe Institute Working Papers Series*
Paper #03-11-060, 2003,
pdf.

(January 2008) An updated and revised version of this preprint will appear in
*The Journal of Fourier Analysis and Applications*.

**Note:**
C code implementing the algorithms described in this
paper may be found here.

Computational Harmonic Analysis for Tensor Fields on the Two-Sphere
In this paper we describe algorithms for the numerical
computation of Fourier transforms of tensor fields on the two-sphere.
These algorithms reduce the computation of an expansion on tensor
spherical harmonics to expansions in scalar spherical harmonics, and
hence can take advantage of recent improvements in the
efficiency of computation of scalar spherical harmonic transforms.

- P. Kostelec, D. Maslen, D. Rockmore and D. Healy, Jr.,
"Computational Harmonic Analysis for Tensor Fields on the Two-Sphere".
*J. Computational Physics* **162** (2000), pp. 514 - 535; in
.pdf (around 310 kb),
.ps (around 385 kb),
.ps.Z , (around 180 kb) and
.ps.gz (around 130 kb) formats.

**Note:** Further information regarding fast Fourier analysis on groups
may be found
here.

FFTs for the 2-Sphere - Improvements and Variations
Earlier work by
Driscoll and Healy has produced an efficient
algorithm for computing the Fourier transform of band-limited functions on
the 2-sphere. In this paper we present a reformulation and variation of the
original algorithm which results in a greatly improved inverse transform, and
consequent improved convolution algorithm for such functions. All require
at most $O(N\log^2 N)$ operations where $N$ is the number of
sample points. We also address implementation considerations and give
heuristics for allowing reliable and computationally efficient floating point
implementations of slightly modified algorithms.
These claims are supported by
extensive numerical experiments from our implementation in C on DEC, HP
and SGI platforms. These results indicate that variations of the algorithm
are both reliable and efficient for a large range of useful problem sizes.
Performance appears to be architecture-dependent.
The paper concludes with a brief discussion of a few potential
applications.

- D. Healy Jr., D. Rockmore, P. Kostelec and S. Moore,
"FFTs for the 2-Sphere - Improvements and Variations,"
in
.pdf (around 1 meg),
.ps.Z , (around 1.4 megs) and
.ps.gz (around 880 kb) formats.
An updated and revised version of this preprint appears in
*The Journal of Fourier Analysis and Applications*
**9:4** (2003), pp. 341 - 385.

**Note:** Further information regarding fast Fourier analysis on groups
may be found
here. Also, C code implementing the algorithms described in this
paper may be found
here.

Multiresolution Elastic Image Registration
We have developed a multiscale algorithm for elastic (or molded) alignment
of images. There is a wide array of medical applications of elastic
(as opposed to strictly rigid) alignment: Subtraction of previous images
from current ones to identify changes is perhaps the most obvious.
We present preliminary results of this molding technique on
a variety of images, and conclude with some
closing remarks about this and future directions and goals
of this work.

- P. Kostelec, J. Weaver and D. Healy Jr.,
"Multiresolution Elastic Image Registration".
*Medical Physics* **25(9)** (1998), pp. 1593--1604; in
.ps.Z , (around 2.3 megs) and
.ps.gz (around 2 megs) formats.

The Use of Hamilton's Principle to Derive
Time--Advance Algorithms for Ordinary Differential Equations in Time
Hamilton's principle is applied to derive a class of numerical algorithms for
systems of ordinary differential equations when the equations are derivable from
a Lagrangian. This is an important extension into the time domain of an earlier
use of Hamilton's principle to derive algorithms for the spatial operators in
Maxwell's equations. In that work, given a set of expansion functions for spatial
dependencies, the Vlasov--Maxwell equations were replaced by a system of ordinary
differential equations in time, but the question of solving the ordinary
differential equations was not addressed. Advantageous properties of the new
time-advance algorithms have been identified analytically and by numerical
comparison with other methods, such as Runge--Kutta and symplectic algorithms. This
approach to time advance can be extended to include partial differential equations
and the Vlasov--Maxwell equations. An interesting issue that could be studied
is whether a collisionless plasma simulation completely based on Hamilton's
principle can be used to obtain a convergent computation of average properties,
such as the electric energy, even when the underlying particle motion is
characterized by sensitive dependence on initial conditions.

- H. R. Lewis and P. Kostelec,
"The Use of Hamilton's Principle to Derive
Time--Advance Algorithms for Ordinary Differential Equations in Time".
*Computer Physics Communications* **96** (1996), pp. 129--151.

Interpolating Uniquely with only a Finite Class of
Polynomials
This paper draws from theorems in transcendental number theory to answer
questions about interpolation with a finite class of multivariable polynomials.
In particular we describe sets of data points at which a unique polynomial
from a particular class gives us a good approximation of the output.

- A. DeStefano, P. Kostelec and D. Wallace,
"Uniquely with only a Finite Class of
Polynomials".
*Proceedings of the International Symposium MTNS--89* **2** (1990),
pp. 497--505.

Return to
my homepage.