RESEARCH PROJECTS

• Project 1:, deals with quantifying (numerically) the neurological and topological differences and similarities between pairs of (MRI, fMRI, PET, CT) brain scans. We were able to design metrics on the space of Fractal/Wavelet Transforms of signals, that help us make quantitative distinctions between equivalent medical images, using their transforms.
  
• If we wish to compare two images and identify corresponding anatomical features (or regions of activation, for functional data) we need to use a "warping" technique to deform one of the images to an image similar to the second one. This brings up the question of "What kind of a deformation should we use?". In Project 2, we constructed a mathematical model (based on Fractal and Wavelet Analyses) that helps classifying warps and warping techniques.
  
• Segmentation of medical images is the topic of Project 3. Using the discrete dynamical system induced by our fractal transform we designed a segmentation algorithm. The two major goals in brain image segmentation are: Determining the regions of high concentration of White Matter, Grey Matter and CSF (Cerebral Spinal Fluid); and Reducing the data complexity and dimensionality.
  
• Our models, and our metrics, turn out also to be useful for image magnification. In Project 4, we compared the current state-of-the-art (bilinear) Interpolation techniques for image zoom in, to the novel Fractal magnification algorithms. We were able to show that our model outperforms the interpolation method in some aspects. Blowing up images using their fractal transforms reveals more details (at lower resolution) and avoids the smearing and blurring effects of the interpolation.
  
• Fractal-like transformations could be used for automatic pattern recognition and feature extraction. Project 5 deals with a simple application of such techniques. We are able to show that a decent pattern recognition algorithm could be used for image registration and alignment - a very useful tool for image comparison.
  
• In Project 6, we develop a new technique for determining the statistically significant metabolic variations in single/multi-subject human brain functional studies. The new method, called Sub-Volume Thresholding (SVT), models the difference images as "locally" stationary Gaussian random fields. Thus adding more flexibility to the commonly used "globally" stationary random approaches. Our model naturally encounters a class of continuous functions we showed induces a family of permissible covariance matrices (valid covariograms). Using the SVT technique we are trying to identify local perfusions and differences in groups of; left vs right hand motor studies; amnesia vs memory-retrieval deficit AD (Alzheimer's disease) patients; and groups of hallucinations vs delusion patients.
  
• My work in the Optimization project includes developing, implementing and testing algorithms for solving min/max, linear/non-linear problems/systems/inequalities. Using Subdivision Traversing and other topological algorithms we introduce a class of simple, fast and robust algorithms for function optimization. The casting problem serves as a motivation in this project. When casting an air-plain wing, for example, there are a number of input variables (like: Temperature, Pressure, Flow velocity, alloy proportions, etc.) and a list of output characteristics (like: Strength, Number of voids, etc.). The problem is to increase the strength of the wing, decrease the number of bubbles (voids) etc, without actually knowing the function connecting the two types of variables. Currently, this problem is approached by some sort of uniform (or random) selection of test points (input variables), conducting an experiment and observing the output. We have designed an algorithm, that solves an optimization problem to optimize the search for the "right" input based on the previously obtained functional values at prior test points.