Piecewise rigid curve deformation via a Finsler steepest descent

This paper introduces a novel steepest descent flow in Banach spaces. This extends previous works on generalized gradient descent, notably the work of Charpiat et al., to the setting of Finsler metrics. Such a generalized gradient allows one to take into account a prior on deformations (e.g., piecewise rigid) in order to favor some specific evolutions. We define a Finsler gradient descent method to minimize a functional defined on a Banach space and we prove a convergence theorem for such a method. In particular, we show that the use of non-Hilbertian norms on Banach spaces is useful to study non-convex optimization problems where the geometry of the space might play a crucial role to avoid poor local minima. We show some applications to the curve matching problem. In particular, we characterize piecewise rigid deformations on the space of curves and we study several models to perform piecewise rigid evolution of curves

Tubular Surface Evolution for Segmentation of the Cingulum Bundle From DW-MRI

Presented at the 2nd MICCAI Workshop on Mathematical Foundations of Computational Anatomy: Geometrical and Statistical Methods for Biological Shape Variability Modeling, September 6th, 2008, Kimmel Center, New York, USA.This work provides a framework for modeling and extracting the Cingulum Bundle (CB) from Diffusion-Weighted Imagery (DW-MRI) of the brain. The CB is a tube-like structure in the brain that is of potentially of tremendous importance to clinicians since it may be helpful in diagnosing Schizophrenia. This structure consists of a collection of fibers in the brain that have locally similar diffusion patterns, but vary globally. Standard region-based segmentation techniques adapted to DW-MRI are not suitable here because the diffusion pattern of the CB cannot be described by a global set of simple statistics. Active surface models extended to DW-MRI are not suitable since they allow for arbitrary deformations that give rise to unlikely shapes, which do not respect the tubular geometry of the CB. In this work, we explicitly model the CB as a tube-like surface and construct a general class of energies defined on tube-like surfaces. An example energy of our framework is optimized by a tube that encloses a region that has locally similar diffusion patterns, which differ from the diffusion patterns immediately outside. Modeling the CB as a tube-like surface is a natural shape prior. Since a tube is characterized by a center-line and a radius function, the method is reduced to a 4D (center-line plus radius) curve evolution that is computationally much less costly than an arbitrary surface evolution. The method also provides the center-line of CB, which is potentially of clinical significance

Shape Calculus for Shape Energies in Image Processing

Many image processing problems are naturally expressed as energy minimization or shape optimization problems, in which the free variable is a shape, such as a curve in 2d or a surface in 3d. Examples are image segmentation, multiview stereo reconstruction, geometric interpolation from data point clouds. To obtain the solution of such a problem, one usually resorts to an iterative approach, a gradient descent algorithm, which updates a candidate shape gradually deforming it into the optimal shape. Computing the gradient descent updates requires the knowledge of the first variation of the shape energy, or rather the first shape derivative. In addition to the first shape derivative, one can also utilize the second shape derivative and develop a Newton-type method with faster convergence. Unfortunately, the knowledge of shape derivatives for shape energies in image processing is patchy. The second shape derivatives are known for only two of the energies in the image processing literature and many results for the first shape derivative are limiting, in the sense that they are either for curves on planes, or developed for a specific representation of the shape or for a very specific functional form in the shape energy. In this work, these limitations are overcome and the first and second shape derivatives are computed for large classes of shape energies that are representative of the energies found in image processing. Many of the formulas we obtain are new and some generalize previous existing results. These results are valid for general surfaces in any number of dimensions. This work is intended to serve as a cookbook for researchers who deal with shape energies for various applications in image processing and need to develop algorithms to compute the shapes minimizing these energies

Molecular Emission in Dense Massive Clumps from the Star-Forming Regions S231-S235

The article deals with observations of star-forming regions S231-S235 in 'quasi-thermal' lines of ammonia (NH$_3$), cyanoacetylene (HC$_3$N) and maser lines of methanol (CH$_3$OH) and water vapor (H$_2$O). S231-S235 regions is situated in the giant molecular cloud G174+2.5. We selected all massive molecular clumps in G174+2.5 using archive CO data. For the each clump we determined mass, size and CO column density. After that we performed observations of these clumps. We report about first detections of NH$_3$ and HC$_3$N lines toward the molecular clumps WB89 673 and WB89 668. This means that high-density gas is present there. Physical parameters of molecular gas in the clumps were estimated using the data on ammonia emission. We found that the gas temperature and the hydrogen number density are in the ranges 16-30 K and 2.8-7.2$\times10^3$ cm$^{-3}$, respectively. The shock-tracing line of CH$_3$OH molecule at 36.2 GHz is newly detected toward WB89 673.Comment: 16 pages, 4 figure

UV line diagnostics of accretion disk winds in cataclysmic variables

The IUE data base is used to analyze the UV line shapes of cataclysmic variables RW Sex, RW Tri, and V Sge. Observed lines are compared to synthetic line profiles computed using a model of rotating bi-conical winds from accretion disks. The wind model calculates the wind ionization structure self-consistently including photoionization from the disk and boundary layer and treats 3-D line radiation transfer in the Sobolev approximation. It is found that winds from accretion disks provide a good fit for reasonable parameters to the observed UV lines which include the P Cygni profiles for low inclination systems and pure emission at large inclination. Disk winds are preferable to spherical winds which originate on the white dwarf because they (1) require a much lower ratio of mass loss rate to accretion rate and are therefore more plausible energetically, (2) provide a natural source for a bi-conical distribution of mass outflow which produces strong scattering far above the disk leading to P Cygni profiles for low inclination systems, and pure line emission profiles at high inclination with the absence of eclipses in UV lines, and (3) produce rotation broadened pure emission lines at high inclination