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.

    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.

    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.

    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.


  • 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.


  • 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.


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