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

The application of compressive sampling to radio astronomy I: Deconvolution

Compressive sampling is a new paradigm for sampling, based on sparseness of signals or signal representations. It is much less restrictive than Nyquist-Shannon sampling theory and thus explains and systematises the widespread experience that methods such as the H\"ogbom CLEAN can violate the Nyquist-Shannon sampling requirements. In this paper, a CS-based deconvolution method for extended sources is introduced. This method can reconstruct both point sources and extended sources (using the isotropic undecimated wavelet transform as a basis function for the reconstruction step). We compare this CS-based deconvolution method with two CLEAN-based deconvolution methods: the H\"ogbom CLEAN and the multiscale CLEAN. This new method shows the best performance in deconvolving extended sources for both uniform and natural weighting of the sampled visibilities. Both visual and numerical results of the comparison are provided.Comment: Published by A&A, Matlab code can be found: http://code.google.com/p/csra/download

Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions

We develop a robust uncertainty principle for finite signals in C^N which states that for almost all subsets T,W of {0,...,N-1} such that |T|+|W| ~ (log N)^(-1/2) N, there is no sigal f supported on T whose discrete Fourier transform is supported on W. In fact, we can make the above uncertainty principle quantitative in the sense that if f is supported on T, then only a small percentage of the energy (less than half, say) of its Fourier transform is concentrated on W. As an application of this robust uncertainty principle (QRUP), we consider the problem of decomposing a signal into a sparse superposition of spikes and complex sinusoids. We show that if a generic signal f has a decomposition using spike and frequency locations in T and W respectively, and obeying |T| + |W| <= C (\log N)^{-1/2} N, then this is the unique sparsest possible decomposition (all other decompositions have more non-zero terms). In addition, if |T| + |W| <= C (\log N)^{-1} N, then this sparsest decomposition can be found by solving a convex optimization problem.Comment: 25 pages, 9 figure

Decoding by Linear Programming

This paper considers the classical error correcting problem which is frequently discussed in coding theory. We wish to recover an input vector $f \in \R^n$ from corrupted measurements $y = A f + e$. Here, $A$ is an $m$ by $n$ (coding) matrix and $e$ is an arbitrary and unknown vector of errors. Is it possible to recover $f$ exactly from the data $y$? We prove that under suitable conditions on the coding matrix $A$, the input $f$ is the unique solution to the $\ell_1$-minimization problem ($\|x\|_{\ell_1} := \sum_i |x_i|$) $$\min_{g \in \R^n} \| y - Ag \|_{\ell_1}$$ provided that the support of the vector of errors is not too large, $\|e\|_{\ell_0} := |\{i : e_i \neq 0\}| \le \rho \cdot m$ for some $\rho > 0$. In short, $f$ can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; $f$ is recovered exactly even in situations where a significant fraction of the output is corrupted.Comment: 22 pages, 4 figures, submitte

The Dantzig selector: Statistical estimation when pp is much larger than nn

In many important statistical applications, the number of variables or parameters $p$ is much larger than the number of observations $n$. Suppose then that we have observations $y=X\beta+z$, where $\beta\in\mathbf{R}^p$ is a parameter vector of interest, $X$ is a data matrix with possibly far fewer rows than columns, $n\ll p$, and the $z_i$'s are i.i.d. $N(0,\sigma^2)$. Is it possible to estimate $\beta$ reliably based on the noisy data $y$? To estimate $\beta$, we introduce a new estimator--we call it the Dantzig selector--which is a solution to the $\ell_1$-regularization problem \min_{\tilde{\b eta}\in\mathbf{R}^p}\|\tilde{\beta}\|_{\ell_1}\quad subject to\quad \|X^*r\|_{\ell_{\infty}}\leq(1+t^{-1})\sqrt{2\log p}\cdot\sigma, where $r$ is the residual vector $y-X\tilde{\beta}$ and $t$ is a positive scalar. We show that if $X$ obeys a uniform uncertainty principle (with unit-normed columns) and if the true parameter vector $\beta$ is sufficiently sparse (which here roughly guarantees that the model is identifiable), then with very large probability, $$\|\hat{\beta}-\beta\|_{\ell_2}^2\le C^2\cdot2\log p\cdot \Biggl(\sigma^2+\sum_i\min(\beta_i^2,\sigma^2)\Biggr).$$ Our results are nonasymptotic and we give values for the constant $C$. Even though $n$ may be much smaller than $p$, our estimator achieves a loss within a logarithmic factor of the ideal mean squared error one would achieve with an oracle which would supply perfect information about which coordinates are nonzero, and which were above the noise level. In multivariate regression and from a model selection viewpoint, our result says that it is possible nearly to select the best subset of variables by solving a very simple convex program, which, in fact, can easily be recast as a convenient linear program (LP).Comment: This paper discussed in: [arXiv:0803.3124], [arXiv:0803.3126], [arXiv:0803.3127], [arXiv:0803.3130], [arXiv:0803.3134], [arXiv:0803.3135]. Rejoinder in [arXiv:0803.3136]. Published in at http://dx.doi.org/10.1214/009053606000001523 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Solving Quadratic Equations via PhaseLift when There Are About As Many Equations As Unknowns

This note shows that we can recover a complex vector x in C^n exactly from on the order of n quadratic equations of the form ||^2 = b_i, i = 1, ..., m, by using a semidefinite program known as PhaseLift. This improves upon earlier bounds in [3], which required the number of equations to be at least on the order of n log n. We also demonstrate optimal recovery results from noisy quadratic measurements; these results are much sharper than previously known results.Comment: 6 page

A probabilistic and RIPless theory of compressed sensing

This paper introduces a simple and very general theory of compressive sensing. In this theory, the sensing mechanism simply selects sensing vectors independently at random from a probability distribution F; it includes all models - e.g. Gaussian, frequency measurements - discussed in the literature, but also provides a framework for new measurement strategies as well. We prove that if the probability distribution F obeys a simple incoherence property and an isotropy property, one can faithfully recover approximately sparse signals from a minimal number of noisy measurements. The novelty is that our recovery results do not require the restricted isometry property (RIP) - they make use of a much weaker notion - or a random model for the signal. As an example, the paper shows that a signal with s nonzero entries can be faithfully recovered from about s log n Fourier coefficients that are contaminated with noise.Comment: 36 page