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

Implications for compressed sensing of a new sampling theorem on the sphere

A sampling theorem on the sphere has been developed recently, requiring half as many samples as alternative equiangular sampling theorems on the sphere. A reduction by a factor of two in the number of samples required to represent a band-limited signal on the sphere exactly has important implications for compressed sensing, both in terms of the dimensionality and sparsity of signals. We illustrate the impact of this property with an inpainting problem on the sphere, where we show the superior reconstruction performance when adopting the new sampling theorem compared to the alternative.Comment: 1 page, 2 figures, Signal Processing with Adaptive Sparse Structured Representations (SPARS) 201

How Much Confidence Do We Have in a MRI Tractography Experiment?

When performing a tractography experiment it is essential to know whether a reconstructed tract results from the diffusion signal itself or from some random effect or noise. In this study, we introduce a way to associate to every connection a confidence level. The reason why the latter greatly varies with the length of the tract is analyzed. We use this method to filter out the connections likely to be the result of noise and show the effect on the connectivity of the human visual system

Geometric Moments in Scale-Spaces

In this paper we present a generalization of geometric moments in scale-spaces derived from the general heat diffusion equation, with a particular interest for th

A Variational Model for Object Segmentation Using Boundary Information, Statistical Shape Prior and the Mumford-Shah Functional

In this paper, we propose a variational model to segment an object belonging to a given scale space using the active contour method, a geometric shape prior and the Mumford-Shah functional. We define an energy functional composed by three complementary terms. The first one detects object boundaries from image gradients. The second term constrains the active contour to get a shape compatible with a statistical shape model of the shape of interest. And the third part drives globally the shape prior and the active contour towards a homogeneous intensity region. The segmentation of the object of interest is given by the minimum of our energy functional. This minimum is computed with the calculus of variations and the gradient descent method that provide a system of evolution equations solved with the well-known level set method. We also prove the existence of this minimum in the space of functions with bounded variation. Applications of the proposed model are presented on synthetic and medical images

Affine invariant Matching Pursuit-based shape representation and recognition using scale-space

In this paper, we propose an analytical low-level representation of images, obtained by a decomposition process, here the matching pursuit (MP) algorithm, as a new way of describing objects through a general continuous description using an affine invariant dictionary of basis functions. This description is used to recognize objects in images. In the learning phase, a template object is decomposed, and the extracted subset of basis functions, called meta-atom, gives the description of our object. We then extend naturally this description into the linear scale-space using the definition of our basis functions, and thus bringing a more general representation of our object. We use this enhanced description as a predefined dictionary of the object to conduct an MP-based shape recognition (MPSR) task into the linear scale-space. The introduction of the scale-space approach improves the robustness of our method, and permits to avoid local minima problems encountered when minimizing a non-convex energy function. We show results for the detection of complex synthetic shapes, as well as natural (aerial and medical) images

Rotation and scale invariant shape representation and recognition using Matching Pursuit

Using a low-level representation of images, like matching pursuit, we introduce a new way of describing objects through a general description using a translation, rotation, and isotropic scale invariant dictionary of basis functions. We then use this description as a predefined dictionary of the object to conduct a shape recognition task. We show some promising results for the detection with simple shapes