4,439 research outputs found
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
Ship Wake Detection in SAR Images via Sparse Regularization
In order to analyse synthetic aperture radar (SAR) images of the sea surface,
ship wake detection is essential for extracting information on the wake
generating vessels. One possibility is to assume a linear model for wakes, in
which case detection approaches are based on transforms such as Radon and
Hough. These express the bright (dark) lines as peak (trough) points in the
transform domain. In this paper, ship wake detection is posed as an inverse
problem, which the associated cost function including a sparsity enforcing
penalty, i.e. the generalized minimax concave (GMC) function. Despite being a
non-convex regularizer, the GMC penalty enforces the overall cost function to
be convex. The proposed solution is based on a Bayesian formulation, whereby
the point estimates are recovered using maximum a posteriori (MAP) estimation.
To quantify the performance of the proposed method, various types of SAR images
are used, corresponding to TerraSAR-X, COSMO-SkyMed, Sentinel-1, and ALOS2. The
performance of various priors in solving the proposed inverse problem is first
studied by investigating the GMC along with the L1, Lp, nuclear and total
variation (TV) norms. We show that the GMC achieves the best results and we
subsequently study the merits of the corresponding method in comparison to two
state-of-the-art approaches for ship wake detection. The results show that our
proposed technique offers the best performance by achieving 80% success rate.Comment: 18 page
Thermoacoustic tomography with detectors on an open curve: an efficient reconstruction algorithm
Practical applications of thermoacoustic tomography require numerical
inversion of the spherical mean Radon transform with the centers of integration
spheres occupying an open surface. Solution of this problem is needed (both in
2-D and 3-D) because frequently the region of interest cannot be completely
surrounded by the detectors, as it happens, for example, in breast imaging. We
present an efficient numerical algorithm for solving this problem in 2-D
(similar methods are applicable in the 3-D case). Our method is based on the
numerical approximation of plane waves by certain single layer potentials
related to the acquisition geometry. After the densities of these potentials
have been precomputed, each subsequent image reconstruction has the complexity
of the regular filtration backprojection algorithm for the classical Radon
transform. The peformance of the method is demonstrated in several numerical
examples: one can see that the algorithm produces very accurate reconstructions
if the data are accurate and sufficiently well sampled, on the other hand, it
is sufficiently stable with respect to noise in the data
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