A Matricial Algorithm for Polynomial Refinement

In order to have a multiresolution analysis, the scaling function must be refinable. That is, it must be the linear combination of 2-dilation, $\mathbb{Z}$-translates of itself. Refinable functions used in connection with wavelets are typically compactly supported. In 2002, David Larson posed the question in his REU site, "Are all polynomials (of a single variable) finitely refinable?" That summer the author proved that the answer indeed was true using basic linear algebra. The result was presented in a number of talks but had not been typed up until now. The purpose of this short note is to record that particular proof

Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis

Recently, compressed sensing techniques in combination with both wavelet and directional representation systems have been very effectively applied to the problem of image inpainting. However, a mathematical analysis of these techniques which reveals the underlying geometrical content is completely missing. In this paper, we provide the first comprehensive analysis in the continuum domain utilizing the novel concept of clustered sparsity, which besides leading to asymptotic error bounds also makes the superior behavior of directional representation systems over wavelets precise. First, we propose an abstract model for problems of data recovery and derive error bounds for two different recovery schemes, namely l_1 minimization and thresholding. Second, we set up a particular microlocal model for an image governed by edges inspired by seismic data as well as a particular mask to model the missing data, namely a linear singularity masked by a horizontal strip. Applying the abstract estimate in the case of wavelets and of shearlets we prove that -- provided the size of the missing part is asymptotically to the size of the analyzing functions -- asymptotically precise inpainting can be obtained for this model. Finally, we show that shearlets can fill strictly larger gaps than wavelets in this model.Comment: 49 pages, 9 Figure

Compatibility of matrix-assisted laser desorption/ionization- mass spectrometry imaging with porous fingerprint development techniques

This thesis encompasses work done to aid in the incorporation of matrix-assisted laser desorption/ionization - mass spectrometry imaging (MALDI-MSI) into the forensic workflow. MALDI-MSI is a valuable technique in that chemical information, in addition to the location of the compounds, can be obtained. In the first chapter, an introduction to the technique and its applications and benefits for the field of forensics are presented. The following chapter details a set of experiments comparing MALDI-MSI with porous surface fingerprint development techniques, specifically ninhydrin and iodine-fuming. It was determined that MALDI-MSI is compatible with both techniques and able to reveal chemical information from a fingerprint. The MS image quality was also compared, and iodine-fumed fingerprints showed analogous images to the non-developed fingerprint, while ninhydrin-developed fingerprints seldom did. With iodine-fumed fingerprints maintaining ridge detail, further information can be determined, such as the deconvolution of the fingerprint and the surface surrounding the fingerprint. The final chapter concludes with an outlook on future directions of research related to this work