We address the problem of separating source signals of fMRI images. This problem is similar to the Blind Source Separation (BSS) under the assumption that the sources are independent over time.
Abstract
We address the problem of separating source signals of fMRI images. This problem is similar to the Blind Source Separation (BSS) under the assumption that the sources are independent over time when using the BOLD (Blood Oxygenation Level Dependent) contrast mechanism. It was recently shown that sparse representation of mixed sources improves dramatically the quality of the separation and its complexity. In this project we demonstrate and analyze geometrical techniques and the Infomax algorithm on sparse representation of the mixing sources. We also compare some sparse criteria and sparse representation techniques, all of which are demonstrated in simulation and experiments.
Blind Source Separation
The blind source separation problem is to extract the original signals from a set of their linear mixtures, where the mixing matrix is unknown and without having any preliminary knowledge about the original sources.
The classic formulation of this problem (in a matrix form)
where A is the mixing matrix, Si are the original signals and Xi are the mixed output signals.
The basic approach
Motivation for sparsness:
The sparseness of signals usually assures that the coefficients of different signals will not exist at the same indexes.
Sparse Criteria
We saw the importance of sparse representation of signals and images; it is left to determine an automatic criterion that will indicate weather the representation achieved is sparse and later to choose the best sparse representation.
L0 norm – counts the variables that differ from a certaim threshold.
L1, L2 norm – is an empiric criterion based on the L1 & L2 norms.
Shannon entropy – can be regard as the smallest number of bits needed to represent the signal. The connection of entropy and sparseness derives from images with a large number of bits equal to zero will have small entropy since we expect to receive more zeros than other gray level values.
Image Separation Algorithms
Algorithms used for retrieving the mixing matrix:
Geometric algorithms for retreiving the mixing matrix (2 mixtures & 2 sources)
1. Angular histogram – representing the points in a scatter plot of the 2 mixtures in a histogram, the maximums suggest the most likly angles.
2. Hypersphere Projection (clustering) – Projecting on half a unit sphere and retreiving the centers of mass of these clusters using FCM (Fuzzy C-means Clustering) function.
3. ICA – infomax – ICA is a statistical technique to recover independent sources given only the sensor observations which are a linear mixture of independent sources. ICA reduces higher-order statistical dependencies, attempting to make the signals as independent as possible. The Bell-Sejnowski Infomax algorithm is based on maximizing the mutual information between the inputs and outputs of a neural network.
The first step in the above algorithms is to find a good sparse representation for these images. This part is crucial. Natural images and synthetic images will probably have different transformations that will lead to their sparsest representation, thus it is very important to choose the right transformation.
Sparse transformation that were examined are:
- The first and second numerical derivatives
- Applying the above on the whole mixed image, blocks, overlapping blocks
- Wavelet decomposition using the db wavelet
Conclusions
Sparseness Criteria
- The best results were obtained by using either Q(L1,L2) with 25% of its best blocks or choosing 25% of all blocks of all criteria. These approache gave the highest and most stable SNR results
Sparse Representation
- ICA – Infomax – the best sparse represntation was the second order numerical derivative (Laplacian) generally sparse represntation had little impact on this algorithm
- Angular Histogram – the only represenation that gave good results were the first derivative and the WP decompostion
- Hypersphere Clustering – the best sparse represntation was the second order numerical derivative (Laplacian) apllied on overlapping blocks and the first derivative
The Reconstruction Algorithm
The ICA infomax algorithm:
- The best of the three we checked, It has the highest SNR average and it is stable. A very important feature it has is its independency on the mixing matrix, which can be very crucial for the geometric algorithms
- Can handle problem of high dimension
- More stable than the geometric algorithms and its average performance usually outperforms the best results given by the geometric
The Geometric Algorithms:
- The hyper-sphere algorithm doesn’t give good results for systems with more than two sources. On the other hand the angular histogram seems to have the same problem
- Both require another information about the mixing matrix (like normalized columns) which in real life is not practical
- They are unstable
Real fMRI data example
The strongest 9 sources that were separated from the real fMRI data with 3 artificial sources with energy ratio of 90%
Tools
The Project was programmed in Matlab 6, on a PC platform.
The main Matlab tools used were the Wavelet toolbox, image processing, clustering and PCA tools.
The ICA toolbox, developed by Scott Makeig of the Salk Institute, was used to implement Infomax separation.
Acknowledgment
I would like to thank my project supervisors, Michael & Alex Bronstein and Dr. Michael Zibulevsky, for their guidance help and patience. I would also like to thank Johanan Erez and the rest of the Image Science lab staff.
I’m also grateful to the Ollendorff Minerva Center Fund for supporting this project.