9,716 research outputs found
Computation of Two-Dimensional Fourier Transforms for Noisy Band-Limited Signals
The computation of the two-dimensional Fourier transform by the sampling points creates an ill-posed problem. In this chapter, we will cover this problem for the band-limited signals in the noisy case. We will present a regularized algorithm based on the two-dimensional Shannon Sampling Theorem, the two-dimensional Fourier series, and the regularization method. First, we prove the convergence property of the regularized solution according to the maximum norm. Then an error estimation is given according to the L2-norm. The convergence property of the regularized Fourier series is given in theory, and some examples are given to compare the numerical results of the regularized Fourier series with the numerical results of the Fourier series
Scale-discretised ridgelet transform on the sphere
We revisit the spherical Radon transform, also called the Funk-Radon
transform, viewing it as an axisymmetric convolution on the sphere. Viewing the
spherical Radon transform in this manner leads to a straightforward derivation
of its spherical harmonic representation, from which we show the spherical
Radon transform can be inverted exactly for signals exhibiting antipodal
symmetry. We then construct a spherical ridgelet transform by composing the
spherical Radon and scale-discretised wavelet transforms on the sphere. The
resulting spherical ridgelet transform also admits exact inversion for
antipodal signals. The restriction to antipodal signals is expected since the
spherical Radon and ridgelet transforms themselves result in signals that
exhibit antipodal symmetry. Our ridgelet transform is defined natively on the
sphere, probes signal content globally along great circles, does not exhibit
blocking artefacts, supports spin signals and exhibits an exact and explicit
inverse transform. No alternative ridgelet construction on the sphere satisfies
all of these properties. Our implementation of the spherical Radon and ridgelet
transforms is made publicly available. Finally, we illustrate the effectiveness
of spherical ridgelets for diffusion magnetic resonance imaging of white matter
fibers in the brain.Comment: 5 pages, 4 figures, matches version accepted by EUSIPCO, code
available at http://www.s2let.or
S2LET: A code to perform fast wavelet analysis on the sphere
We describe S2LET, a fast and robust implementation of the scale-discretised
wavelet transform on the sphere. Wavelets are constructed through a tiling of
the harmonic line and can be used to probe spatially localised, scale-depended
features of signals on the sphere. The scale-discretised wavelet transform was
developed previously and reduces to the needlet transform in the axisymmetric
case. The reconstruction of a signal from its wavelets coefficients is made
exact here through the use of a sampling theorem on the sphere. Moreover, a
multiresolution algorithm is presented to capture all information of each
wavelet scale in the minimal number of samples on the sphere. In addition S2LET
supports the HEALPix pixelisation scheme, in which case the transform is not
exact but nevertheless achieves good numerical accuracy. The core routines of
S2LET are written in C and have interfaces in Matlab, IDL and Java. Real
signals can be written to and read from FITS files and plotted as Mollweide
projections. The S2LET code is made publicly available, is extensively
documented, and ships with several examples in the four languages supported. At
present the code is restricted to axisymmetric wavelets but will be extended to
directional, steerable wavelets in a future release.Comment: 8 pages, 6 figures, version accepted for publication in A&A. Code is
publicly available from http://www.s2let.or
The curvelet transform for image denoising
We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a` trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement
A recursive scheme for computing autocorrelation functions of decimated complex wavelet subbands
This paper deals with the problem of the exact computation of the autocorrelation function of a real or complex discrete wavelet subband of a signal, when the autocorrelation function (or Power Spectral Density, PSD) of the signal in the time domain (or spatial domain) is either known or estimated using a separate technique. The solution to this problem allows us to couple time domain noise estimation techniques to wavelet domain denoising algorithms, which is crucial for the development of blind wavelet-based denoising techniques. Specifically, we investigate the Dual-Tree complex wavelet transform (DT-CWT), which has a good directional selectivity in 2-D and 3-D, is approximately shift-invariant, and yields better denoising results than a discrete wavelet transform (DWT). The proposed scheme gives an analytical relationship between the PSD of the input signal/image and the PSD of each individual real/complex wavelet subband which is very useful for future developments. We also show that a more general technique, that relies on Monte-Carlo simulations, requires a large number of input samples for a reliable estimate, while the proposed technique does not suffer from this problem
Sparse image reconstruction on the sphere: implications of a new sampling theorem
We study the impact of sampling theorems on the fidelity of sparse image
reconstruction on the sphere. We discuss how a reduction in the number of
samples required to represent all information content of a band-limited signal
acts to improve the fidelity of sparse image reconstruction, through both the
dimensionality and sparsity of signals. To demonstrate this result we consider
a simple inpainting problem on the sphere and consider images sparse in the
magnitude of their gradient. We develop a framework for total variation (TV)
inpainting on the sphere, including fast methods to render the inpainting
problem computationally feasible at high-resolution. Recently a new sampling
theorem on the sphere was developed, reducing the required number of samples by
a factor of two for equiangular sampling schemes. Through numerical simulations
we verify the enhanced fidelity of sparse image reconstruction due to the more
efficient sampling of the sphere provided by the new sampling theorem.Comment: 11 pages, 5 figure
Slepian Spatial-Spectral Concentration on the Ball
We formulate and solve the Slepian spatial-spectral concentration problem on
the three-dimensional ball. Both the standard Fourier-Bessel and also the
Fourier-Laguerre spectral domains are considered since the latter exhibits a
number of practical advantages (spectral decoupling and exact computation). The
Slepian spatial and spectral concentration problems are formulated as
eigenvalue problems, the eigenfunctions of which form an orthogonal family of
concentrated functions. Equivalence between the spatial and spectral problems
is shown. The spherical Shannon number on the ball is derived, which acts as
the analog of the space-bandwidth product in the Euclidean setting, giving an
estimate of the number of concentrated eigenfunctions and thus the dimension of
the space of functions that can be concentrated in both the spatial and
spectral domains simultaneously. Various symmetries of the spatial region are
considered that reduce considerably the computational burden of recovering
eigenfunctions, either by decoupling the problem into smaller subproblems or by
affording analytic calculations. The family of concentrated eigenfunctions
forms a Slepian basis that can be used be represent concentrated signals
efficiently. We illustrate our results with numerical examples and show that
the Slepian basis indeeds permits a sparse representation of concentrated
signals.Comment: 33 pages, 10 figure
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