Barycentric coordinates for triangles are commonly used in computer graphics, geometric modelling, and other computational sciences for various purposes, because they provide a convenient way to linearly interpolate data that is given at the corners of a triangle. The concept of barycentric coordinates can also be extended in several ways to convex polygons with more than three vertices, but most of these constructions break down when used in the non-convex setting. One choice that is not limited to convex configurations are the mean value coordinates and we show that they are well-defined for arbitrary planar polygons without self-intersections. Besides many other important properties, these coordinate functions are smooth and allow an efficient and robust implementation. They are particularly useful for interpolating data that is given at the vertices of the polygons and we present several examples of their application to common problems in computer graphics and geometric modelling.

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A warping is a function that deforms images by mapping between image domains. The choice of function is formulated statistically as maximum penalised likelihood, where the likelihood measures the similarity between images after warping and the penalty is a measure of distortion of a warping. The paper addresses two issues simultaneously, of how tochoose the warping function and how to assess the alignment. A new, Fouriervon Mises image model is identified, with phase differences between Fourier-transformed images having von Mises distributions. Also, new, null-set distortion criteria are proposed, with each criterion uniquely minimised by a particular set of polynomial functions. A conjugate gradient algorithm is used to estimate the warping function, which isnumerically approximated by a piecewise bilinear function. The method is motivated by, and used to solve, three applied problems: to register a remotely-sensed image with a map, to align microscope images obtained using different optics, and to discriminate between species of fish from photographic images.

Abstract We present a new approach for aligning families of 2D gels. Instead of choosing one of the gels as reference and performing pairwise alignment, we construct an ideal gel that is representative for the entire family and obtain a set of piecewise affine transformations that optimally align each gel of the family to the ideal gel. The coefficients defining the transformations as well as the ideal landmarks are obtained as the solution of a large-scale quadratic programming problem that can be solved efficiently by interiorpoint methods. 1 Proteomics and 2-D PAGE Proteome analysis involves the separation, visualization and analysis of complex mixtures containing as many as several thousand proteins obtained from whole cells, tissues or organisms. Two-dimensional polyacrylamide gel electrophoresis (2D-PAGE), first introduced by O’Farrell [9] and Klose [8] in 1975, remains a core technology for separating complex protein mixtures in the majority of proteome projects. The main goal of protein separation methods is to detect differentially expressed proteins across treatment groups. However, a major bottleneck toward that goal is the misalignment of gels due to warping and thus confounding biological variation with non-biologically relevant distortions. This paper provides a computationally feasible gel alignment methods based on powerful optimization techniques such as interior point methods. With 2D-PAGE technique, proteins are separated orthogonally according to their charge and size. The separated proteins are then stained so that they are readily detectable, and the gels are digitally scanned into a database for storage. Gel images are

Abstract. Associating speci c gene activity with speci c functional locations in the brain anatomy results in a greater understanding of the role of the gene's products. To perform such an association for the over 20,000 or so genes in the mammalian genome, reliable automated methods that characterize the distribution of gene expression in relation to a standard anatomical model are required. In this work, we propose a new automatic method that results in the segmentation of gene expression images into distinct anatomical regions in which the expression can be quanti ed and compared with other images. Our method utilizes models of shape of training images, texture differentiation at region boundaries, and features of anatomical landmarks to deform a subdivision mesh-based atlas to t gene expression images. The subdivision mesh provides a common coordinate system for internal brain data through which gene expression patterns can be compared across images. The automated large-scale annotation will help scientists interpret gene expression patterns at cellular resolution more ef ciently. 1

Building and studying 3D representations of anatomical structures, such as the brain, plays an important role in modern biology and medical science. While 3D imaging methods such as MRI and CT have been widely applied, 2D imaging methods such as optical microscopy typically generate images with much higher resolution. In this thesis I describe how to construct a high-resolution 3D atlas of the mouse brain from 2D microscopic images. In particular, I focus on using computer graphics techniques, such as image registration to correcting tissue distortions and polygonal modeling to build surfaces representing the partitioning of anatomical regions. The methods are being applied to construct a high-quality polygonal atlas of an adult mouse brain from 350 histological tissue sections. While the resulting brain atlas will contribute to a larger project of building a spatial database of gene expressions over the mouse brain, the methods described in this thesis are well suited for the general purpose of building polygonal atlases of arbitrary anatomical structures from tissue sections. Acknowledgements I would like to express my greatest thanks to my advisor, Dr. Joe Warren, whose broad knowledge and dynamic thinking guided me throughout the process of research and thesis writing. I would like to thank Dr. Ron Goldman for his valuable sugges-tions on both the contents of the thesis and my presentation skills. I would also like to thank Dr. Richard Baraniuk and Dr. Lydia Kavraki, who have spent their valuable time reading this work and attending the defense. I would like to thank my collaborators at Baylor College of Medicine and University of Houston: Dr. Carson, Dr. Chiu, Dr. Gregor Eichele Dr. Thaller, Dr. Kakadiaris and Musodiq Bello. Without their help I could not have finished this work. I would like to give my special thanks to my wife Ming whose patient love enabled me to complete this work. ii

Abstract The class of tensor product cubic splines is considered in an optimization process for two-dimensional polyacrylamide gel electrophoresis (2D-PAGE) image alignment problems arising in proteomics studies. Numerical results are presented along with comparisons with previous results based on piecewise affine and bilinear transformations. 1 1

Image warping plays an extremely important role in face matching and recognition and there-fore has wide applications and great significance in computer vision, biomedical imaging and homeland security. One popular warping algorithm developed by Alex Rosen, known as Alex-Warp has the drawback that it only allows one pair of landmarks being specified at one time. The consequence of this is that the destination landmarks are displaced in the subsequent warp-ing. We develop an image warping algorithm using thin-plate splines that allows multi-landmarks being specified and warped simultaneously. The space transformation is the solution of minimizing a thin-plate deformation energy. QR decomposition techniques are used to stabi-lize the solution. We apply the new warping algorithm to various face matching situations.

Abstract A new method for aligning families of two-dimensional polyacrylamide gel electrophoresis (2D-PAGE) images arising in proteomics studies is presented. Forward piecewise bilinear transformations are used to determine an ideal gel and to obtain an initial alignment of the family of gels to this ideal gel. Both the ideal landmarks and the coefficients defining the transformations are obtained by solving a quadratic programming problem. The alignment is then improved by using inverse transformations on finer grids. Numerical results for a family of 123 gels are reported.

Introduction A common method of analysing images is to model them as deformed versions of a "standard image," called a template. The method is often applied to medical images and can be used to analyse shape variation, particularly that due to some pathology. Our methodology was developed with reference to a particular data set of human femora arising from a paleopathological study of osteoarthritis of the knee, but it is intended to be of much more general applicability. AMS (1991) subject classification. Primary 62H35; secondary 42C40. Key words and phrases. Arthritis, deformed template, image analysis, iterated conditional mode, mixture distribution, paleopathology, penalized least squares, shape statistics, wavelets. 150 T.R. DOWNIE AND B.W. SILVERMAN It is usual in statistical work to model the deformation in terms of its expansion in