Wednesday, December 19, 2007

Compressed Sensing Thoughts, Diffusion maps methods for Brain Imaging and for Computer Generated Graphics

Muthu Muthukrishnan is wondering aloud about the substring matching problem and how it can be used to find the reconstruction of a signal made of subsignals that are sparse using the compressed sensing framework.

I also found items of interest and relevant to dimensionality reduction applied to different domains. The first one is an article on Recovering cerebral white matter structures with Spectral Clustering of Diffusion MRI Data by Demian Wassermann, Maxime Descoteaux and Rachid Deriche.
The abstract reads:

White matter fiber clustering allows to get insight about anatomical structures in order to generate atlases, perform clear visualizations and compute statistics across subjects, all important and current neuroimaging problems. In this work, we present a Diffusion Maps clustering method applied to diffusion MRI in order to cluster and segment complex white matter fiber bundles. It is well-known that Diffusion Tensor Imaging (DTI) is restricted in complex fiber regions with crossings and this is why recent High Angular Resolution Diffusion Imaging (HARDI) such has Q-Ball Imaging (QBI) have been introduced to overcome these limitations. QBI reconstructs the diffusion orientation distribution function (ODF), a spherical function that has its maxima agreeing with the underlying fiber populations. In this paper, we introduce the usage of the Diffusion Maps technique and show how it can be used to directly cluster set of fiber tracts, that could be obtained through a streamline tractography for instance, and how it can also help in segmenting fields of ODF images, obtained through a linear and regularized ODF estimation algorithm based on a spherical harmonics representation of the Q-Ball data. We first show the advantage of using Diffusion Maps clustering over classical methods such as N-Cuts and Laplacian Eigenmaps in both cases. In particular, our Diffusion Maps requires a smaller number of hypothesis from the input data, reduces the number of artifacts in fiber tract clustering and ODF image segmentation and automatically exhibits the number of clusters in both cases by using an adaptive scale-space parameter. We also show that our ODF Diffusion Maps clustering can reproduce published results using the diffusion tensor (DT) clustering with N-Cuts on simple synthetic images without crossings. On more complex data with crossings, we show that our ODF-based method succeeds to separate fiber bundles and crossing regions whereas the DT-based methods generate artifacts and exhibit wrong number of clusters. Finally, we illustrate the potential of our approach on a real brain dataset where we successfully segment well-known fiber bundles.
It is pretty long but worth reading.

Also, Gabriel Peyre makes available his course note and some matlab programs to help out on the topic of Data Processing Using Manifold Methods. This course seems specific to the use of dimensionality reduction technique like diffusion maps to do some manipulation of 3-D graphics. The presentation of his course reads:

This course is an introduction to the computational theory of manifolds. Manifold models arise in various area of mathematics, image processing, data mining or computer science. Surfaces of arbitrary dimension can be used to model non-linear datasets that one encounters in modern data processing. Numerical methods allow to exploit this geometric non-linear prior in order to extract relevant information from the data. These methods include in particular local differential computations (related to the Laplacian operator and its variants) and global distance methods (related to geodesic computations). In this course, you will learn how to perform differential and geodesic computations on images, volumes, surfaces and high dimensional graphs.

The course includes a set of Matlab experiments. These experiments give an overview of various tasks in computer vision, image processing, learning theory and mesh processing. This includes computation of shortest paths, Voronoi segmentations, geodesic Delaunay triangulations, surface flattening, dimensionality reduction and mesh processing.

One can notice that the flattening of meshes has a parallel with methods like the MVU, except that in the MVU case, additional interdistance constraints have to be met.

Then again in CG, people are ok with "it's good enough" :-)


Unknown said...

The flattening algorithm that is used for texture mapping and remeshing does not rely on laplacian eigenmaps (or diffusion wavelets). It only requires the solution of a sparse linear system (no eigenvector extraction).

And it is not just enough, it is perfect, since Tutte theorem guaranties that it is 1:1 :)

Igor said...


It just goes to show that I should probably take your course :)
I don't recall seeing an explanation for it on your slides, is there a reference where I can find an explanation/implementation of it ?