We propose a new definition of the total variation (TV) norm for vector-valued functions that can be applied to restore color and other vector-valued images. The new TV norm has the desirable properties of 1) not penalizing discontinuities (edges) in the image, 2) being rotationally invariant in the image space, and 3) reducing to the usual TV norm in the scalar case. Some numerical experiments on denoising simple color images in red-green-blue (RGB) color space are presented.

We analyze theoretically and with numerical simulations the phenomenon of super-resolution in time-reversal acoustics. A signal that is recorded and then re-transmitted by an array of transducers, propagates back though the medium and refocuses approximately on the source that emitted it. In a homogeneous medium, the refocusing resolution of the time-reversed signal is limited by diffraction. When the medium has random inhomogeneities the resolution of the refocused signal can in some circumstances beat the diffraction limit. This is super-resolution.We give a theoretical treatment of this phenomenon and present numerical simulations which confirm the theory.

We describe some numerical techniques for the Total Variation image restoration method, namely a primaldual linearization for the Euler-Lagrange equations and some preconditioning issues. We also highlight extension of this technique to color images, blind deconvolution and the staircasing effect.

Spleen tyrosine kinase (Syk) is an attractive drug target in autoimmune, inflammatory, and oncology disease indications. The most advanced Syk inhibitor, R406, 1 (or its prodrug form fostamatinib, 2), has shown efficacy in multiple therapeutic indications, but its clinical progress has been hampered by dose-limiting adverse effects that have been attributed, at least in part, to the off-target activities of 1. It is expected that a more selective Syk inhibitor would provide a greater therapeutic window. Herein we report the discovery and optimization of a novel series of imidazo[1,2-a]pyrazine Syk inhibitors. This work culminated in the identification of GS-9973, 68, a highly selective and orally efficacious Syk inhibitor which is currently undergoing clinical evaluation for autoimmune and oncology indications.

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