Fast space-varying convolution in stray light reduction, fast matrix vector multiplication using the sparse matrix transform, and activation detection in fMRI data analysis.

摘要

In this dissertation, I will address three interesting problems in the area of image processing: fast space-varying convolution in stray light reduction, fast matrix vector multiplication using the sparse matrix transform, and activation detection in functional Magnetic Resonance Imaging (fMRI) data analysis.In the first topic, we study the problem of space-varying convolution which often arises in the modeling or restoration of images captured by optical imaging systems. Specifically, in the application of stray light reduction, where the stray light point spread function varies across the field of view, accurate restoration requires the use of space-varying convolution. While space-invariant convolution can be efficiently implemented with the Fast Fourier Transform (FFT), space-varying convolution requires direct implementation of the convolution operation, which can be very computationally expensive when the convolution kernel is large.In this work, we developed two general approaches for the efficient implementation of space-varying convolution through the use of piecewise isoplanatic approximation and matrix source coding techniques. In the piecewise isoplanatic approximation approach, we partition the image into isoplanatic patches based on vector quantization, and use a piecewise isoplanatic model to approximate the fully space-varying model. Then the space-varying convolution can be efficiently computed by using FFT to compute space-invariant convolution of each patch and adding all pieces together. In the matrix source coding approach, we dramatically reduce computation by approximately factoring the dense space-varying convolution operator into a product of sparse transforms. This approach leads to a trade-off between the accuracy and speed of the operation. The experimental results show that our algorithms can achieve a dramatic reduction in computation while achieving high accuracy.In the second topic, we aim at developing a fast algorithm for computing matrix-vector products which are often required to solve problems in linear systems. If the matrix size is P x P and is dense, then the complexity for computing this matrix-vector product is generally O(P2). So as the dimension P goes up, the computation and storage required can increase dramatically, in some cases making the operation infeasible to compute. When the matrix is Toeplitz, the product can be efficiently computed in O(P log P) by using the fast Fourier transform (FFT). However, in the general problems the matrix is not Toeplitz, which makes the computation difficult.In this work, we adopt the concept of the recently introduced sparse matrix transform (SMT) and propose an algorithm to approximately decompose the matrix into the product of a series of sparse matrices and a diagonal matrix. In fact, we show that this decomposition forms an approximate singular value decomposition of a matrix. With our decomposition, the matrix-vector product can be computed with order O(P). Our approach is analogous to the way in which an FFT can be used for fast convolution but importantly, the proposed SMT decomposition can be applied to matrices that represent space or time-varying operations.We test the effectiveness of our algorithm on a space-varying convolution example. The particular application we consider is stray-light reduction in digital cameras, which requires the convolution of each new image with a large space-varying stray light point spread function (PSF). We demonstrate that our proposed method can dramatically reduce the computation required to accurately compute the required convolution of an image with the space-varying PSF.In the third topic, we study the problem of fMRI activation detection, which is to detect which region of the brain is activated when the subject is presented with a specific stimulus. FMRI data is subject to severe noise and some extent of blur. So recovering the signal and successfully identifying the activated voxels are challenging inverse problems.We propose a new model for event-related fMRI which explicitly incorporates the spatial correlation introduced by the scanner, and develop a new set of tools for activation detection. We propose simple, efficient algorithms to estimate model parameters. We develop an activation detection algorithm which consists of two parts: image restoration and maximum a posteriori (MAP) estimation. During the image restoration stage, a total-variation (TV) based approach is employed to restore each data slice, for each time index. At the MAP estimation stage, we estimate parameters of a parametric hemodynamic response function (HRF) model for each pixel from the restored data. We employ the generalized Gaussian Markov random field (GGMRF) model to enforce spatial regularity when we compute the MAP estimate of the HRF parameters. We then threshold the amplitude parameter map to obtain the final activation map. Through comparison with the widely used general linear model method, in synthetic and real data experiments, we demonstrate the promise and advantage of our algorithm.