Total Variation (TV) and related extensions have been popular in image restoration due to their robust performance and wide applicability. While the original formulation is still relevant after two decades of extensive research, its extensions that combine derivatives of first and second orders are now being explored for better performance, with examples being Combined Order TV (COTV) and Total Generalized Variation (TGV). As an improvement over such multi-order convex formulations, we propose a novel non-convex regularization functional which adaptively combines Hessian-Schatten (HS) norm and first order TV (TV1) functionals with spatially varying weight. This adaptive weight itself is controlled by another regularization term; the total cost becomes the sum of this adaptively weighted HS-TV1 term, the regularization term for the adaptive weight, and the data-fitting term. The reconstruction is obtained by jointly minimizing w.r.t. the required image and the adaptive weight. We construct a block coordinate descent method for this minimization with proof of convergence, which alternates between minimization w.r.t. the required image and the adaptive weights. We derive exact computational formula for minimization w.r.t. the adaptive weight, and construct an ADMM algorithm for minimization w.r.t. to the required image. We compare the proposed method with existing regularization methods, and a recently proposed Deep GAN method using image recovery examples including MRI reconstruction and microscopy deconvolution.

Semiconductor quantum dots (QDs) are new fluorescent probes with great promise for ultrasensitive biological imaging. When detected at the single-molecule level, QD-tagged molecules can be observed and tracked in the membrane of live cells over unprecedented durations. The motion of these individual molecules, recorded in sequences of fluorescence images, can reveal aspects of the dynamics of cellular processes that remain hidden in conventional ensemble imaging. Due to QD complex optical properties, such as fluorescence intermittency, the quantitative analysis of these sequences is, however, challenging and requires advanced algorithms. We present here a novel approach, which, instead of a frame by frame analysis, is based on perceptual grouping in a spatiotemporal volume. By applying a detection process based on an image fluorescence model, we first obtain an unstructured set of points. Individual molecular trajectories are then considered as minimal paths in a Riemannian metric derived from the fluorescence image stack. These paths are computed with a variant of the fast marching method and few parameters are required. We demonstrate the ability of our algorithm to track intermittent objects both in sequences of synthetic data and in experimental measurements obtained with individual QD-tagged receptors in the membrane of live neurons. While developed for tracking QDs, this method can, however, be used with any fluorescent probes.