83 research outputs found
Graph Spectral Image Processing
Recent advent of graph signal processing (GSP) has spurred intensive studies
of signals that live naturally on irregular data kernels described by graphs
(e.g., social networks, wireless sensor networks). Though a digital image
contains pixels that reside on a regularly sampled 2D grid, if one can design
an appropriate underlying graph connecting pixels with weights that reflect the
image structure, then one can interpret the image (or image patch) as a signal
on a graph, and apply GSP tools for processing and analysis of the signal in
graph spectral domain. In this article, we overview recent graph spectral
techniques in GSP specifically for image / video processing. The topics covered
include image compression, image restoration, image filtering and image
segmentation
Semiclassical bounds for spectra of biharmonic operators
We provide complementary semiclassical bounds for the Riesz means of
the eigenvalues of various biharmonic operators, with a second term in the
expected power of . The method we discuss makes use of the averaged
variational principle (AVP), and yields two-sided bounds for individual
eigenvalues, which are semiclassically sharp. The AVP also yields comparisons
with Riesz means of different operators, in particular Laplacians
Time integration and steady-state continuation for 2d lubrication equations
Lubrication equations allow to describe many structurin processes of thin
liquid films. We develop and apply numerical tools suitable for their analysis
employing a dynamical systems approach. In particular, we present a time
integration algorithm based on exponential propagation and an algorithm for
steady-state continuation. In both algorithms a Cayley transform is employed to
overcome numerical problems resulting from scale separation in space and time.
An adaptive time-step allows to study the dynamics close to hetero- or
homoclinic connections. The developed framework is employed on the one hand to
analyse different phases of the dewetting of a liquid film on a horizontal
homogeneous substrate. On the other hand, we consider the depinning of drops
pinned by a wettability defect. Time-stepping and path-following are used in
both cases to analyse steady-state solutions and their bifurcations as well as
dynamic processes on short and long time-scales. Both examples are treated for
two- and three-dimensional physical settings and prove that the developed
algorithms are reliable and efficient for 1d and 2d lubrication equations,
respectively.Comment: 33 pages, 16 figure
Interactive Medical Image Registration With Multigrid Methods and Bounded Biharmonic Functions
Interactive image registration is important in some medical applications since automatic image registration is often slow and sometimes error-prone. We consider interactive registration methods that incorporate user-specified local transforms around control handles. The deformation between handles is interpolated by some smooth functions, minimizing some variational energies. Besides smoothness, we expect the impact of a control handle to be local. Therefore we choose bounded biharmonic weight functions to blend local transforms, a cutting-edge technique in computer graphics. However, medical images are usually huge, and this technique takes a lot of time that makes itself impracticable for interactive image registration.
To expedite this process, we use a multigrid active set method to solve bounded biharmonic functions (BBF). The multigrid approach is for two scenarios, refining the active set from coarse to fine resolutions, and solving the linear systems constrained by working active sets. We\u27ve implemented both weighted Jacobi method and successive over-relaxation (SOR) in the multigrid solver. Since the problem has box constraints, we cannot directly use regular updates in Jacobi and SOR methods. Instead, we choose a descent step size and clamp the update to satisfy the box constraints. We explore the ways to choose step sizes and discuss their relation to the spectral radii of the iteration matrices. The relaxation factors, which are closely related to step sizes, are estimated by analyzing the eigenvalues of the bilaplacian matrices. We give a proof about the termination of our algorithm and provide some theoretical error bounds.
Another minor problem we address is to register big images on GPU with limited memory. We\u27ve implemented an image registration algorithm with virtual image slices on GPU. An image slice is treated similarly to a page in virtual memory. We execute a wavefront of subtasks together to reduce the number of data transfers.
Our main contribution is a fast multigrid method for interactive medical image registration that uses bounded biharmonic functions to blend local transforms. We report a novel multigrid approach to refine active set quickly and use clamped updates based on weighted Jacobi and SOR. This multigrid method can be used to efficiently solve other quadratic programs that have active sets distributed over continuous regions
The Bilaplacian with Robin Boundary Conditions
We introduce Robin boundary conditions for biharmonic operators, which are a model for elastically supported plates and are closely related to the study of spaces of traces of Sobolev functions. We study the dependence of the operator, its eigenvalues, and eigenfunctions on the Robin parameters. We show in particular that when the parameters go to plus infinity the Robin problem converges to other biharmonic problems, and we obtain estimates on the rate of divergence when the parameters go to minus infinity. We also analyze the dependence of the operator on smooth perturbations of the domain, computing the shape derivatives of the eigenvalues and giving a characterization for critical domains under volume and perimeter constraints. We include a number of open problems arising in the context of our results
On the behavior of clamped plates under large compression
We determine the asymptotic behavior of eigenvalues of clamped plates under large compression by relating this problem to eigenvalues of the Laplacian with Robin boundary conditions. Using the method of fundamental solutions, we then carry out a numerical study of the extremal domains for the first eigenvalue, from which we see that these depend on the value of the compression, and start developing a boundary structure as this parameter is increased. The corresponding number of nodal domains of the first eigenfunction of the extremal domain also increases with the compression.This work was partially supported by the Funda ̧c ̃ao para a Ciˆencia e a Tecnologia(Portugal) through the program “Investigador FCT” with reference IF/00177/2013 and the projectExtremal spectral quantities and related problems(PTDC/MAT-CAL/4334/2014).info:eu-repo/semantics/publishedVersio
- …