Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Antje Vollrath
Technische Universität Braunschweig
A new algorithm for fast Fourier transforms on the rotation group
We will discuss an approximate fast algorithm to calculate the discrete Fourier transform on the rotation
group SO(3). The algorithms to compute such transforms are based on evaluating the so-called Wigner-D
functions Dmn ℓ that yield an orthogonal basis of L2 (SO(3)). Using these basis functions, our method needs
O(L3 logL + Q) arithmetic operations for a degree-L transform at Q nodes free of choice, with the desired
accuracy, instead of O(L3Q) in a naive approach.
This acceleration, is achieved by exploiting the tensor product character of the Wigner-D functions. With
this decomposition arises a set of orthogonal polynomials closely related to Jacobi polynomials - the
Wigner-d functions.
The talk will focus on a new efficient method to calculate a particular linear transformation that allows us
to replace Wigner-d functions of arbitrary orders with those of low orders and eventually with Chebyshev
polynomials. Based on the differential equations, whose solutions are the Wigner-d functions, we show
that the linear mapping for these conversions appears as the eigenvector matrix of certain semiseparable
matrices. This enables us to employ a known divide-and-conquer algorithm for symmetric semiseparable
eigenproblems together with the fast multipole method to calculate the desired change of basis.
Finally by replacing the Chebyshev expansions by expansions of complex exponentials we can employ
the well-analysed nonequispaced fast Fourier transform algorithm for the computations.
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