Volume 1, Issue 1, January 2011 - DROPS - Schloss Dagstuhl

Stephan Dahlke, Michael Elad, Yonina Eldar, Gitta Kutyniok, and Gerd Teschke 115
3 J. Zujovic and O. G. Guleryuz, Complexity Regularized Pattern Matching, Proc. IEEE
Int’l Conf. on Image Proc. (ICIP2009), Cairo, Egypt, Nov. 2009.
4 Rubinstein, R., Zibulevsky, M. and Elad, M., Double Sparsity: Learning Sparse Dictionaries
for Sparse Signal Approximation, IEEE Transactions on Signal Processing 58(3) (2010).
5 E. Chou and O. G. Guleryuz, A Majorization-minimization algorithm for sparse factorization
and related applications, (in preparation).
6 O. G. Guleryuz, Nonlinear Approximation Based Image Recovery Using Adaptive Sparse
Reconstructions and Iterated Denoising: Parts I and II, IEEE Transactions on Image
Processing, (2006).
3.6 Stable discretizations of the hyperbolic cross fast Fourier transform
Stefan Kunis (Universität Osnabrück, DE)
License Creative Commons BY-NC-ND 3.0 Unported license
© Stefan Kunis
Joint work of Kämmerer, Lutz; Kunis, Stefan; Potts, Daniel
A straightforward discretization of problems in d spatial dimensions with 2 n , n ∈ N, grid
points in each coordinate leads to an exponential growth 2 dn in the number of degrees of
freedom.
We restrict the frequency domain to the hyperbolic cross
�
H d n =
j∈N d 0 ,�j�1=n
(−2 j1−1 , 2 j1−1 ] × . . . × (−2 jd−1 , 2 jd−1 ] ∩ Z d ,
and ask for the fast approximate evaluation of the trigonometric polynomial
f(x) = �
ˆfk e 2πikx , (1)
k∈H d n
at nodes xℓ ∈ T d , ℓ = 1, . . . , M.
We note that the reduced problem size is cd2 n n d−1 ≤ |H d n| ≤ Cd2 n n d−1 and a classical
result states the computation of (1) for all sparse grid nodes takes at most Cd2 n n d floating
point operations.
This has been generalized for arbitrary spatial sampling nodes and both algorithms are
available in the Matlab toolbox nhcfft.
◮ Theorem 1. [1] The computation of (1) at all nodes xℓ ∈ T d , ℓ = 1, . . . , |H d n|, takes at
most Cd2 n n 2d−2 (| log ε| + log n) d , where ε > 0 denotes the target accuracy.
More recently, we analyzed the numerical stability of these sampling sets and in sharp
contrast to the ordinary FFT which is unitary, we found the following negative result.
◮ Theorem 2. [2] The computation of (1) at the sparse grid has condition number
cd2 n
2 n 2d−3
2 ≤ κ ≤ Cd2 n
2 n 2d−2 .
Although random sampling offers a stable spatial discretization with high probability if
M ≥ C|H d n| log |H d n|, the fast algorithm [1] relies on an oversampled sparse grid and thus
suffers from the same instability.
Ongoing work [3] considers lattices as spatial discretization for the hyperbolic cross fast
Fourier transform. These turn out to have quite large cardinality asymptotically but offer
perfect stability and outperform known algorithms by at least one order of magnitude with
respect to CPU timings for moderate problem sizes.
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