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Approximation Theory XV: San Antonio 2016
These proceedings are based on papers presented at the international conference Approximation Theory XV, which was held May 22\u201325, 2016 in San Antonio, Texas. The conference was the fifteenth in a series of meetings in Approximation Theory held at various locations in the United States, and was attended by 146 participants. The book contains longer survey papers by some of the invited speakers covering topics such as compressive sensing, isogeometric analysis, and scaling limits of polynomials and entire functions of exponential type.
The book also includes papers on a variety of current topics in Approximation Theory drawn from areas such as advances in kernel approximation with applications, approximation theory and algebraic geometry, multivariate splines for applications, practical function approximation, approximation of PDEs, wavelets and framelets with applications, approximation theory in signal processing, compressive sensing, rational interpolation, spline approximation in isogeometric analysis, approximation of fractional differential equations, numerical integration formulas, and trigonometric polynomial approximation
Compressed sensing MRI using masked DCT and DFT measurements
This paper presents modification of the TwIST algorithm for Compressive
Sensing MRI images reconstruction. Compressive Sensing is new approach in
signal processing whose basic idea is recovering signal form small set of
available samples. The application of the Compressive Sensing in biomedical
imaging has found great importance. It allows significant lowering of the
acquisition time, and therefore, save the patient from the negative impact of
the MR apparatus. TwIST is commonly used algorithm for 2D signals
reconstruction using Compressive Sensing principle. It is based on the Total
Variation minimization. Standard version of the TwIST uses masked 2D Discrete
Fourier Transform coefficients as Compressive Sensing measurements. In this
paper, different masks and different transformation domains for coefficients
selection are tested. Certain percent of the measurements is used from the
mask, as well as small number of coefficients outside the mask. Comparative
analysis using 2D DFT and 2D DCT coefficients, with different mask shapes is
performed. The theory is proved with experimental results
One-Bit Compressive Sensing of Dictionary-Sparse Signals
One-bit compressive sensing has extended the scope of sparse recovery by showing that sparse signals can be accurately reconstructed even when their linear measurements are subject to the extreme quantization scenario of binary samples—only the sign of each linear measurement is maintained. Existing results in one-bit compressive sensing rely on the assumption that the signals of interest are sparse in some fixed orthonormal basis. However, in most practical applications, signals are sparse with respect to an overcomplete dictionary, rather than a basis. There has already been a surge of activity to obtain recovery guarantees under such a generalized sparsity model in the classical compressive sensing setting. Here, we extend the one-bit framework to this important model, providing a unified theory of one-bit compressive sensing under dictionary sparsity. Specifically, we analyze several different algorithms—based on convex programming and on hard thresholding—and show that, under natural assumptions on the sensing matrix (satisfied by Gaussian matrices), these algorithms can efficiently recover analysis-dictionary-sparse signals in the one-bit model
Differential Phase-contrast Interior Tomography
Differential phase contrast interior tomography allows for reconstruction of
a refractive index distribution over a region of interest (ROI) for
visualization and analysis of internal structures inside a large biological
specimen. In this imaging mode, x-ray beams target the ROI with a narrow beam
aperture, offering more imaging flexibility at less ionizing radiation.
Inspired by recently developed compressive sensing theory, in numerical
analysis framework, we prove that exact interior reconstruction can be achieved
on an ROI via the total variation minimization from truncated differential
projection data through the ROI, assuming a piecewise constant distribution of
the refractive index in the ROI. Then, we develop an iterative algorithm for
the interior reconstruction and perform numerical simulation experiments to
demonstrate the feasibility of our proposed approach
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