Polynomial mapped bases: theory and applications

In this paper, we collect the basic theory and the most important applications of a novel technique that has shown to be suitable for scattered data interpolation, quadrature, bio-imaging reconstruction. The method relies on polynomial mapped bases allowing, for instance, to incorporate data or function discontinuities in a suitable mapping function. The new technique substantially mitigates the Runge's and Gibbs effects

Treating the Gibbs phenomenon in barycentric rational interpolation and approximation via the S-Gibbs algorithm

In this work, we extend the so-called mapped bases or fake nodes approach to the barycentric rational interpolation of Floater-Hormann and to AAA approximants. More precisely, we focus on the reconstruction of discontinuous functions by the S-Gibbs algorithm introduced in [12]. Numerical tests show that it yields an accurate approximation of discontinuous functions

Image registration and visualization of in situ gene expression images.

In the age of high-throughput molecular biology techniques, scientists have incorporated the methodology of in-situ hybridization to map spatial patterns of gene expression. In order to compare expression patterns within a common tissue structure, these images need to be registered or organized into a common coordinate system for alignment to a reference or atlas images. We use three different image registration methodologies (manual; correlation based; mutual information based) to determine the common coordinate system for the reference and in-situ hybridization images. All three methodologies are incorporated into a Matlab tool to visualize the results in a user friendly way and save them for future work. Our results suggest that the user-defined landmark method is best when considering images from different modalities; automated landmark detection is best when the images are expected to have a high degree of consistency; and the mutual information methodology is useful when the images are from the same modality

More properties of (β,γ)-Chebyshev functions and points

Recently, (β,γ)-Chebyshev functions, as well as the corresponding zeros, have been introduced as a generalization of classical Chebyshev polynomials of the first kind and related roots. They consist of a family of orthogonal functions on a subset of [−1,1], which indeed satisfies a three-term recurrence formula. In this paper we present further properties, which are proven to comply with various results about classical orthogonal polynomials. In addition, we prove a conjecture concerning the Lebesgue constant's behavior related to the roots of (β,γ)-Chebyshev functions in the corresponding orthogonality interval

Oversampling errors in multimodal medical imaging are due to the Gibbs effect

To analyse multimodal 3-dimensional medical images, interpolation is required for resampling which - unavoidably - introduces an interpolation error. In this work we consider three segmented 3-dimensional images resampled with three different neuroimaging software tools for comparing undersampling and oversampling strategies and to identify where the oversampling error lies. The results indicate that undersampling to the lowest image size is advantageous in terms of mean value per segment errors and that the oversampling error is larger where the gradient is steeper, showing a Gibbs effect

Digital resampling and timing recovery in QAM systems

Digital resampling is a process that converts a digital signal from one sampling rate to another. This process is performed by means of interpolating between the input samples to produce output samples at an output sampling rate. The digital interpolation process is accomplished with an interpolation filter. The problem of resampling digital signals at an output sampling rate that is incommensurate with the input sampling rate is the first topic of this thesis. This problem is often encountered in practice, for example in multiplexing video signals from different sources for the purpose of distribution. There are basically two approaches to resample the signals. Both approaches are thoroughly described and practical circuits for hardware implementation are provided. A comparison of the two circuits shows that one circuit requires a division to compute the new sampling times. This time scaling operation adds complexity to the implementation with no performance advantage over the other circuit, and makes the 'division free' circuit the preferred one for resampling. The second topic of this thesis is performance analysis of interpolation filters for Quadrature Amplitude Modulation (QAM) signals in the context of timing recovery. The performance criterion of interest is Modulation Error Ratio (MER), which is considered to be a very useful indicator of the quality of modulated signals in QAM systems. The methodology of digital resampling in hardware is employed to describe timing recovery circuits and propose an approach to evaluate the performance of interpolation filters. A MER performance analysis circuit is then devised. The circuit is simulated with MATLAB/Simulink as well as implemented in Field Programmable Gate Array (FPGA). Excellent agreement between results obtained from simulation and hardware implementation proves the validity of the methodology and practical application of the research works

More properties of (β,γ)(\beta,\gamma)-Chebyshev functions and points

Recently, $(\beta,\gamma)$-Chebyshev functions, as well as the corresponding zeros, have been introduced as a generalization of classical Chebyshev polynomials of the first kind and related roots. They consist of a family of orthogonal functions on a subset of $[-1,1]$, which indeed satisfies a three-term recurrence formula. In this paper we present further properties, which are proven to comply with various results about classical orthogonal polynomials. In addition, we prove a conjecture concerning the Lebesgue constant's behavior related to the roots of $(\beta,\gamma)$-Chebyshev functions in the corresponding orthogonality interval

Efficient Deformable Shape Correspondence via Kernel Matching

We present a method to match three dimensional shapes under non-isometric deformations, topology changes and partiality. We formulate the problem as matching between a set of pair-wise and point-wise descriptors, imposing a continuity prior on the mapping, and propose a projected descent optimization procedure inspired by difference of convex functions (DC) programming. Surprisingly, in spite of the highly non-convex nature of the resulting quadratic assignment problem, our method converges to a semantically meaningful and continuous mapping in most of our experiments, and scales well. We provide preliminary theoretical analysis and several interpretations of the method.Comment: Accepted for oral presentation at 3DV 2017, including supplementary materia

Experimental land observing data system feasibility study

An end-to-end data system to support a Shuttle-based Multispectral Linear Array (MLA) mission in the mid-1980's was defined. The experimental Land Observing System (ELOS) is discussed. A ground system that exploits extensive assets from the LANDSAT-D Program to effectively meet the objectives of the ELOS Mission was defined. The goal of 10 meter pixel precision, the variety of data acquisition capabilities, and the use of Shuttle are key to the mission requirements, Ground mission management functions are met through the use of GSFC's Multi-Satellite Operations Control Center (MSOCC). The MLA Image Generation Facility (MIGF) combines major hardware elements from the Applications Development Data System (ADDS) facility and LANDSAT Assessment System (LAS) with a special purpose MLA interface unit. LANDSAT-D image processing techniques, adapted to MLA characteristics, form the basis for the use of existing software and the definition of new software required