A Bayesian Approach to Manifold Topology Reconstruction

In this paper, we investigate the problem of statistical reconstruction of piecewise linear manifold topology. Given a noisy, probably undersampled point cloud from a one- or two-manifold, the algorithm reconstructs an approximated most likely mesh in a Bayesian sense from which the sample might have been taken. We incorporate statistical priors on the object geometry to improve the reconstruction quality if additional knowledge about the class of original shapes is available. The priors can be formulated analytically or learned from example geometry with known manifold tessellation. The statistical objective function is approximated by a linear programming / integer programming problem, for which a globally optimal solution is found. We apply the algorithm to a set of 2D and 3D reconstruction examples, demon-strating that a statistics-based manifold reconstruction is feasible, and still yields plausible results in situations where sampling conditions are violated

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

Reconstruction with velocities

Reconstruction is becoming a crucial procedure of galaxy clustering analysis for future spectroscopic redshift surveys to obtain subper cent level measurement of the baryon acoustic oscillation scale. Most reconstruction algorithms rely on an estimation of the displacement field from the observed galaxy distribution. However, the displacement reconstruction degrades near the survey boundary due to incomplete data and the boundary effects extend to ∼100 Mpc/h within the interior of the survey volume. We study the possibility of using radial velocities measured from the cosmic microwave background observation through the kinematic Sunyaev-Zeldovich effect to improve performance near the boundary. We find that the boundary effect can be reduced to ∼30 − 40 Mpc/h with the velocity information from Simons Observatory. This is especially helpful for dense low redshift surveys where the volume is relatively small and a large fraction of total volume is affected by the boundary

Euclidean Distance Matrices: Essential Theory, Algorithms and Applications

Euclidean distance matrices (EDM) are matrices of squared distances between points. The definition is deceivingly simple: thanks to their many useful properties they have found applications in psychometrics, crystallography, machine learning, wireless sensor networks, acoustics, and more. Despite the usefulness of EDMs, they seem to be insufficiently known in the signal processing community. Our goal is to rectify this mishap in a concise tutorial. We review the fundamental properties of EDMs, such as rank or (non)definiteness. We show how various EDM properties can be used to design algorithms for completing and denoising distance data. Along the way, we demonstrate applications to microphone position calibration, ultrasound tomography, room reconstruction from echoes and phase retrieval. By spelling out the essential algorithms, we hope to fast-track the readers in applying EDMs to their own problems. Matlab code for all the described algorithms, and to generate the figures in the paper, is available online. Finally, we suggest directions for further research.Comment: - 17 pages, 12 figures, to appear in IEEE Signal Processing Magazine - change of title in the last revisio

Sampling and Reconstruction of Shapes with Algebraic Boundaries

We present a sampling theory for a class of binary images with finite rate of innovation (FRI). Every image in our model is the restriction of \mathds{1}_{\{p\leq0\}} to the image plane, where \mathds{1} denotes the indicator function and $p$ is some real bivariate polynomial. This particularly means that the boundaries in the image form a subset of an algebraic curve with the implicit polynomial $p$. We show that the image parameters --i.e., the polynomial coefficients-- satisfy a set of linear annihilation equations with the coefficients being the image moments. The inherent sensitivity of the moments to noise makes the reconstruction process numerically unstable and narrows the choice of the sampling kernels to polynomial reproducing kernels. As a remedy to these problems, we replace conventional moments with more stable \emph{generalized moments} that are adjusted to the given sampling kernel. The benefits are threefold: (1) it relaxes the requirements on the sampling kernels, (2) produces annihilation equations that are robust at numerical precision, and (3) extends the results to images with unbounded boundaries. We further reduce the sensitivity of the reconstruction process to noise by taking into account the sign of the polynomial at certain points, and sequentially enforcing measurement consistency. We consider various numerical experiments to demonstrate the performance of our algorithm in reconstructing binary images, including low to moderate noise levels and a range of realistic sampling kernels.Comment: 12 pages, 14 figure