Image restoration using sparse approximations of spatially varying blur operators in the wavelet domain

6 pagesInternational audienceRestoration of images degraded by spatially varying blurs is an issue of increasing importance in the context of photography, satellite or microscopy imaging. One of the main difficulty to solve this problem comes from the huge dimensions of the blur matrix. It prevents the use of naive approaches for performing matrix-vector multiplications. In this paper, we propose to approximate the blur operator by a matrix sparse in the wavelet domain. We justify this approach from a mathematical point of view and investigate the approximation quality numerically. We finish by showing that the sparsity pattern of the matrix can be pre-defined, which is central in tasks such as blind deconvolution

Modulation Invariant Bilinear T(1) Theorem

We prove a T(1) theorem for bilinear singular integral operators (trilinear forms) with a one-dimensional modulation symmetry

Using diffusion distances for flexible molecular shape comparison

<p>Abstract</p> <p>Background</p> <p>Many molecules are flexible and undergo significant shape deformation as part of their function, and yet most existing molecular shape comparison (MSC) methods treat them as rigid bodies, which may lead to incorrect shape recognition.</p> <p>Results</p> <p>In this paper, we present a new shape descriptor, named Diffusion Distance Shape Descriptor (DDSD), for comparing 3D shapes of flexible molecules. The diffusion distance in our work is considered as an average length of paths connecting two landmark points on the molecular shape in a sense of inner distances. The diffusion distance is robust to flexible shape deformation, in particular to topological changes, and it reflects well the molecular structure and deformation without explicit decomposition. Our DDSD is stored as a histogram which is a probability distribution of diffusion distances between all sample point pairs on the molecular surface. Finally, the problem of flexible MSC is reduced to comparison of DDSD histograms.</p> <p>Conclusions</p> <p>We illustrate that DDSD is insensitive to shape deformation of flexible molecules and more effective at capturing molecular structures than traditional shape descriptors. The presented algorithm is robust and does not require any prior knowledge of the flexible regions.</p

Smoothing of commutators for a Hörmander class of bilinear pseudodifferential operators

Commutators of bilinear pseudodifferential operators with symbols in the H\"ormander class BS_{1, 0}^1 and multiplication by Lipschitz functions are shown to be bilinear Calder\'on-Zygmund operators. A connection with a notion of compactness in the bilinear setting for the iteration of the commutators is also made.Comment: 16 page

A Guide to Localized Frames and Applications to Galerkin-like Representations of Operators

This chapter offers a detailed survey on intrinsically localized frames and the corresponding matrix representation of operators. We re-investigate the properties of localized frames and the associated Banach spaces in full detail. We investigate the representation of operators using localized frames in a Galerkin-type scheme. We show how the boundedness and the invertibility of matrices and operators are linked and give some sufficient and necessary conditions for the boundedness of operators between the associated Banach spaces.Comment: 32 page

Wavelet-based identification of DNA focal genomic aberrations from single nucleotide polymorphism arrays

<p>Abstract</p> <p>Background</p> <p>Copy number aberrations (CNAs) are an important molecular signature in cancer initiation, development, and progression. However, these aberrations span a wide range of chromosomes, making it hard to distinguish cancer related genes from other genes that are not closely related to cancer but are located in broadly aberrant regions. With the current availability of high-resolution data sets such as single nucleotide polymorphism (SNP) microarrays, it has become an important issue to develop a computational method to detect driving genes related to cancer development located in the focal regions of CNAs.</p> <p>Results</p> <p>In this study, we introduce a novel method referred to as the wavelet-based identification of focal genomic aberrations (WIFA). The use of the wavelet analysis, because it is a multi-resolution approach, makes it possible to effectively identify focal genomic aberrations in broadly aberrant regions. The proposed method integrates multiple cancer samples so that it enables the detection of the consistent aberrations across multiple samples. We then apply this method to glioblastoma multiforme and lung cancer data sets from the SNP microarray platform. Through this process, we confirm the ability to detect previously known cancer related genes from both cancer types with high accuracy. Also, the application of this approach to a lung cancer data set identifies focal amplification regions that contain known oncogenes, though these regions are not reported using a recent CNAs detecting algorithm GISTIC: SMAD7 (chr18q21.1) and FGF10 (chr5p12).</p> <p>Conclusions</p> <p>Our results suggest that WIFA can be used to reveal cancer related genes in various cancer data sets.</p