Lung Motion

During breathing the thoracic cavity is expanded by contraction of the diaphragm and outer intercostal muscles. This causes changes in the intrapleural pressure which acts as a force upon the lung surface: Lung expands, and (since the pleu-ral cavity is filled with a fluid) during this process the visceral pleura is sliding frictionlessly down the internal surface of the thoracic cavity.uniform negative pressure is applied to a lung surface representing some initial state of breathing (e.g. state of end-expiration). This causes the lung to expand. We limit expansion by a geometry representing the lung shape at a final state of breathing (e.g. end-inhalation); see the fig. for illustration. If there occurs a contact between the two geometries, the contact is modeled to be frictionless. For simplistic purposes lung tissue is assumed to be an isotropic linear elastic and homogeneous medium.

Singular Value Decomposition

M=1,000,000，N=500,000

div definition

Let x, y, z be a system of Cartesian coordinates on a 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors.

The divergence of a continuously differentiable vector field F = F_x i + F_y j + F_z k is defined to be the scalar-valued function:

Although expressed in terms of coordinates, the result is invariant under orthogonal transformations, as the physical interpretation suggests.

The common notation for the divergence ∇·Fis a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product: take the components of ∇ (see del), apply them to the components of F, and sum the results. As a result, this is considered an abuse of notation.

PDEs Solutions

Scott A. Sarra, Adaptive radial basis function methods for time dependent partial differential equations, Applied Numerical Mathematics Volume 54, Issue 1, June 2005, Pages 79-94.

Alfio Borzi, Austria, Introduction to multigrid methods, http://www.uni-graz.at/imawww/borzi/index.html.

Reading Schedul September 15,2008

1. Huzefa Neemuchwala, Alfred Hero and Paul Carson, Image matching using alpha-entropy measures and entropic graphs.Signal Processing. Volume 85, Issue 2, February 2005, Pages 277-296.
2. Clustering with Bregman Divergences, Journal of Machine Learning Research 6 (2005) 1705–1749.
3. A variational framework for integrating segmentation and registration
through active contours, Medical Image Analysis 7 (2003) 171–185.

Work like them

It's very good to do research like them, add the research paper stack to the schedul.

http://groups.csail.mit.edu/vision/golland/group_meeting/doku.php

Whole Body Medical Image Registration

It seems that whole body medical image registrtion is solved already as the researchers claimed in the publications. MI and B-spline are used in my test, the LBFGS optimization is employed in the process. CT-CT image can be aligned well, but when the algorithm is used to solve PET-CT image registration, the result is not good enough, even can not be accepted. I can not draw a conclusion the classical method can not solve these kind of problems because this is a primary test. The gobal rigid and local norigid registration may be combined together for the problem. Another issue we should pay attention is the images volume are 3D and the multi-resolution stretegy can improve the computation speed.

A New Class of Quasi Newton Method

The main idea of the following papers is to approximate the Hessian Matrix by a algebra method. They've used it for MLP training, the dimension of the problem upto 2034. From the theroy and their application, this optimzation method can be used for medical image
registration. But unfortunately, the experimental results are not satisfied to me. I used the Limited-memory BFGS (L-BFGS) method[Nocedal80] and the method provided by these paper. libLBFGS is employed for implementing L-BFGS and the method.
I also compare three different line search method for both algorithms. Strong wolfe, standard wolfe and backtrack line search methods are used in my implementation.

C. Di Fiore - S. Fanelli - P. Zellini
"On the best least square fit to a matrix and its applications"
(Algebra and Algebraic Topology, Nova Science Publishers, Inc., 73-109 , 2007)

C. Di Fiore - S. Fanelli - F. Lepore - P. Zellini
"Matrix algebras in quasi-newton methods for unconstrained minimization"
(Numerische Mathematik, 94 , 479-500 , 2003)

A. Bortoletti - C. Di Fiore - S. Fanelli - P. Zellini
"A new class of quasi-newtonian methods for optimal learning in MLP-networks"
(IEEE Transactions on Neural Networks, 14 , 2 , 263-273 , 2003)