Maximally Localized Radial Profiles for Tight Steerable Wavelet Frames

A crucial component of steerable wavelets is the radial profile of the generating function in the frequency domain. In this paper, we present an infinite-dimensional optimization scheme that helps us find the optimal profile for a given criterion over the space of tight frames. We consider two classes of criteria that measure the localization of the wavelet. The first class specifies the spatial localization of the wavelet profile, and the second that of the resulting wavelet coefficients. From these metrics and the proposed algorithm, we construct tight wavelet frames that are optimally localized and provide their analytical expression. In particular, one of the considered criterion helps us finding back the popular Simoncelli wavelet profile. Finally, the investigation of local orientation estimation, image reconstruction from detected contours in the wavelet domain, and denoising indicate that optimizing wavelet localization improves the performance of steerable wavelets, since our new wavelets outperform the traditional ones.

Wavelet-Based Reconstruction for Magnetic Resonance Imaging

Matthieu Guerquin-Kern

Magnetic resonance imaging (MRI) scanners produce raw measurements that are unfit to direct interpretation, unless an algorithmic step, called reconstruction, is introduced. Up to the last decade, this reconstruction was performed by algorithms of moderate complexity. This worked because substantial efforts were devoted to adjust the MRI hardware to suit the algorithmic component. More recently, new techniques have reversed this trend by putting more emphasis on the algorithms and alleviating the constraints on the hardware. Whereas many new methods suffer from a marked increase in computational complexity, this thesis focuses on the development of reconstruction algorithms that are faster and simpler than state-of-the-art solutions, while preserving their quality. First, we present the physical principles that underlie the acquisition of MRI data and motivate the classical linear model. Based on this continuous equation, we derive efficient implementations of a discrete model. Standard and state-of-the-art reconstruction algorithms are reviewed and presented in a general framework where reconstruction is regarded as an optimization problem that can naturally integrate regularization. Next, we propose novel simulation tools for the validation of reconstruction methods. Those tools model the sensitivity of the receiving coil, which allows for the simulation of parallel MRI experiments. To honor the continuous nature of the underlying physics, we suggest the use of analytical phantoms. Unlike rasterized simulations, our phantoms do not introduce aliasing artifacts. Instead, they allow us to study how rasterization itself impacts the quality of reconstruction. To achieve this goal, we were able to work out closed-form solutions for the Fourier transform of parametric regions that can realistically reproduce anatomical features. Our results show that the inverse-crime situation impairs significantly the assessment of the performance of reconstruction methods, particularly, the non-linear ones. Finally, we investigate the design of algorithms that achieve reconstruction with a sparsity constraint expressed in a wavelet domain. Based on the latest developments in large-scale convex optimization, we derive an acceleration strategy that can be tailored to the MRI setup and provide theoretical evidence of its benefit. We develop it into a practical method that combines the advantages of speed and quality. Applied on challenging reconstruction problems, with simulated and in-vivo data, we significantly reduce the reconstruction time over state-of-the-art techniques without compromising quality.

EPFL2012

Wavelet-Based Reconstruction for Magnetic Resonance Imaging

Matthieu Guerquin-Kern

EPFL2012