786 research outputs found
FPS-SFT: A Multi-dimensional Sparse Fourier Transform Based on the Fourier Projection-slice Theorem
We propose a multi-dimensional (M-D) sparse Fourier transform inspired by the
idea of the Fourier projection-slice theorem, called FPS-SFT. FPS-SFT extracts
samples along lines (1-dimensional slices from an M-D data cube), which are
parameterized by random slopes and offsets. The discrete Fourier transform
(DFT) along those lines represents projections of M-D DFT of the M-D data onto
those lines. The M-D sinusoids that are contained in the signal can be
reconstructed from the DFT along lines with a low sample and computational
complexity provided that the signal is sparse in the frequency domain and the
lines are appropriately designed. The performance of FPS-SFT is demonstrated
both theoretically and numerically. A sparse image reconstruction application
is illustrated, which shows the capability of the FPS-SFT in solving practical
problems
Shearlets and Optimally Sparse Approximations
Multivariate functions are typically governed by anisotropic features such as
edges in images or shock fronts in solutions of transport-dominated equations.
One major goal both for the purpose of compression as well as for an efficient
analysis is the provision of optimally sparse approximations of such functions.
Recently, cartoon-like images were introduced in 2D and 3D as a suitable model
class, and approximation properties were measured by considering the decay rate
of the error of the best -term approximation. Shearlet systems are to
date the only representation system, which provide optimally sparse
approximations of this model class in 2D as well as 3D. Even more, in contrast
to all other directional representation systems, a theory for compactly
supported shearlet frames was derived which moreover also satisfy this
optimality benchmark. This chapter shall serve as an introduction to and a
survey about sparse approximations of cartoon-like images by band-limited and
also compactly supported shearlet frames as well as a reference for the
state-of-the-art of this research field.Comment: in "Shearlets: Multiscale Analysis for Multivariate Data",
Birkh\"auser-Springe
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