Georgia Institute of Technology College of Sciences

Using the technique of intertwining multiresolution analysis piecewise linear, continuous, orthogonal, wavelets on a regular hexagon are constructed. We will review the technique of intertwining multiresolution analysis in the one variable case then indicate the modifications necessary for the two variable construction. This is work with George Donovan and Doug Hardin.

Medical imaging is the application of mathematical and engineering models to create images of the human body for clinical purposes or medical science by using a medical device. One of the main objectives of medical imaging research is to find the boundary of the region of the interest. The procedure to find the boundary of the region of the interest is called a segmentation. The purpose of this talk is to present a variational region based algorithm that is able to deal with spatial perturbations of the image intensity directly. Image segmentation is obtained by using a Gamma-Convergence approximation for a multi-scale piecewise smooth model. This model overcomes the limitations of global region models while avoiding the high sensitivity of local approaches. The proposed model is implemented efficiently using recursive Gaussian convolutions. The model is applied to magnetic resonance (MR) images where image quality depends highly on the acquisition protocol. Numerical experiments on 2-dimensional human liver MR images show that our model compares favorably to existing methods.This work is done in collaborated with Mikael Rousson and Chenyang Xu.

Biomimetic synthetic cilia can be effectively utilized for regulating microscale transport processes at interfaces. Using computer simulations, we examine how polymeric cilia can be harnessed to control the motion of microscopic particles suspended in a viscous fluid. The cilia are modeled as deformable, elastic filaments and our simulations capture the complex fluid-structure interactions among these filaments, channel walls and surrounding solution. We show that non-motile cilia that are tilted with respect to the surface can hydrodynamically direct solid particles towards channel walls, thereby, inducing their rapid deposition. When synthetic cilia are actuated by a sinusoidal force that is applied at the free ends, the beating cilia can either drive particles downwards toward the substrate or expelled particles into the fluid above the actuated cilial layer. This dynamic behavior can be regulated by changing the driving frequency. The findings uncover new routes for controlling the deposition of microscopic particles in microfluidic devices.

Image restoration has been an active research topic in imageprocessing and computer vision. There are vast of literature, mostof which rely on the regularization, or prior information of theunderlying image. In this work, we examine three types of methodsranging from local, nonlocal to global with various applications.A classical approach for local regularization term is achieved bymanipulating the derivatives. We adopt the idea in the localpatch-based sparse representation to present a deblurringalgorithm. The key observation is that the sparse coefficientsthat encode a given image with respect to an over-complete basisare the same that encode a blurred version of the image withrespect to a modified basis. Following an``analysis-by-synthesis'' approach, an explicit generative modelis used to compute a sparse representation of the blurred image,and its coefficients are used to combine elements of the originalbasis to yield a restored image.We follows the framework that generates the neighborhood filtersto an variational formulation for general image reconstructionproblems. Specifically, two extensions regarding to the weightcomputation are investigated. One is to exploit the recurrence ofstructures at different locations, orientations and scales in animage. While previous methods based on ``nonlocal filtering'' identify corresponding patches only up to translations, we consider more general similarity transformation.The second algorithm utilizes a preprocessed data as input for theweight computation. The requirements for preprocessing are (1) fastand (2) containing sharp edges. We get superior results in theapplications of image deconvolution and tomographic reconstruction.A Global approach is explored in a particular scenario, that is,taking a burst of photographs under low light conditions with ahand-held camera. Since each image of the burst is sharp but noisy,our goal is to efficiently denoise these multiple images. Theproposed algorithm is a complex chain involving accurateregistration, video equalization, noise estimation and the use ofstate-of-the-art denoising methods. Yet, we show that this complexchain may become risk free thanks to a key feature: the noise modelcan be estimated accurately from the image burst.