Analysis and identification of multidimensional singularities using the continuous shearlet transform

Abstract In this chapter, we illustrate the properties of the continuous shearlet transform with respect to its ability to describe the set of singularities of multidimensional functions and distributions. This is of particular interest since singularities and other irregular structures typically carry the most essential information in multidimensional phenomena. Consider, for example, the edges of natural images or the moving fronts in the solutions of transport equations. In the following, we show that the continuous shearlet transform provides a precise geometrical characterization of the singularity sets of multidimensional functions and precisely characterizes the boundaries of 2D and 3D regions through its asymptotic decay at fine scales. These properties go far beyond the continuous wavelet transform and other classical methods, and set the groundwork for very competitive algorithms for edge detection and feature extraction of 2D and 3D data. Key words: analysis of singularities, continuous wavelet transform, shearlets, sparsity, wavefront set, wavelets

FACE, GENDER AND RACE CLASSIFICATION USING MULTI-REGULARIZED FEATURES LEARNING

This paper investigates a new approach for face, gender and race classification, called multi-regularized learning (MRL). This approach combines ideas from the recently proposed algorithms called multi-stage learning (MSL) and multi-task features learning (MTFL). In our approach, we first reduce the dimensionality of the training faces using PCA. Next, for a given a test (probe) face, we use MRL to exploit the relationships among multiple shared stages generated by changing the regularization parameter. Our approach results in convex optimization problem that controls the trade-off between the fidelity to the data (training) and the smoothness of the solution (probe). Our MRL algorithm is compared against different state-of-the-art methods on face recognition (FR), gender classification (GC) and race classification (RC) based on different experimental protocols with AR, LFW, FEI, Lab2 and Indian databases. Results show that our algorithm performs very competitively

Optimal restoration of noisy 3D X-ray data via shearlet

decomposition

The theory of reproducing systems on locally compact abelian groups

A reproducing system is a countable collection of functions {φj : j ∈ J } such that a general function f can be decomposed as f = ∑ j∈J cj (f ) φj , with some control on the analyzing coefficients cj (f ). Several such systems have been introduced very successfully in mathematics and its applications. We present a unified viewpoint to the study of reproducing systems on locally compact abelian groups G. This approach gives a novel characterization of the Parseval frame generators for a very general class of reproducing systems on L2(G). As an application of this result, we obtain a new characterization of Parseval frame generators for Gabor and affine systems on L2(G)

Advances in quantitative analysis of astrocytes using machine learning

Astrocytes, a subtype of glial cells, are starshaped cells that are involved in the homeostasis and blood flow control of the central nervous system (CNS). They are known to provide structural and functional support to neurons, including the regulation of neuronal activation through extracellular ion concentrations, the regulation of energy dynamics in the brain through the transfer of lactate to neurons, and the modulation of synaptic transmission via the release of neurotransmitters such as glutamate and adenosine triphosphate. In addition, astrocytes play a critical role in neuronal reconstruction after brain injury, including neurogenesis, synaptogenesis, angiogenesis, repair of the blood-brain barrier, and glial scar formation after traumatic brain injury (Zhou et al., 2020). The multifunctional role of astrocytes in the CNS with tasks requiring close contact with their targets is reflected by their morphological complexity, with processes and ramifications occurring over multiple scales where interactions are plastic and can change depending on the physiological conditions. Another major feature of astrocytes is reactive astrogliosis, a process occurring in response to traumatic brain injury, neurological diseases, or infection which involves substantial morphological alterations and is often accompanied by molecular, cytoskeletal, and functional changes that ultimately play a key role in the disease outcome (Schiweck et al., 2018). Because morphological changes in astrocytes correlate so significantly with brain injury and the development of pathologies of the CNS, there is a major interest in methods to reliably detect and accurately quantify such morphological alterations. We review below the recent progress in the quantitative analysis of images of astrocytes. We remark that, while our discussion is focused on astrocytes, the same methods discussed below can be applied to other types of complex glial cells.National Science Foundation grants Division of Mathmatical Scienc

Robust and stable region-of-interest tomographic reconstruction using a robust width prior

Region-of-interest computed tomography (ROI CT) aims at reconstructing a region within the field of view by using only ROI-focused projections. The solution of this inverse problem is challenging and methods of tomographic reconstruction that are designed to work with full projection data may perform poorly or fail when applied to this setting. In this work, we study the ROI CT problem in the presence of measurement noise and formulate the reconstruction problem by relaxing data fidelity and consistency requirements. Under the assumption of a robust width prior that provides a form of stability for data satisfying appropriate sparsity-inducing norms, we derive reconstruction performance guarantees and controllable error bounds. Based on this theoretical setting, we introduce a novel iterative reconstruction algorithm from ROI-focused projection data that is guaranteed to converge with controllable error while satisfying predetermined fidelity and consistency tolerances. Numerical tests on experimental data show that our algorithm for ROI CT is competitive with state-of-the-art methods especially when the ROI radius is small

Regularization with optimal space-time priors

We propose a variational regularization approach based on cylindrical shearlets to deal with dynamic imaging problems, with dynamic tomography as guiding example. The idea is that the mismatch term essentially integrates a sequence of separable, static problems, while the regularization term sees the non-stationary target as a spatio-temporal object. We motivate this approach by showing that cylindrical shearlets provide optimally sparse approximations for the class of cartoon-like videos, i.e., a class of functions useful to model spatio-temporal image sequences and videos, which we introduce extending the classic notion of cartoon-like images. To formulate our regularization model, we define cylindrical shearlet smoothness spaces, which is pivotal to obtain suitable embeddings in functional spaces. To complete our analysis, we prove that the proposed regularization strategy is well-defined, the solution of the minimisation problem exists and is unique (for $p > 1$). Furthermore, we provide convergence rates (in terms of the symmetric Bregman distance) under deterministic and random noise conditions, and within the context of statistical inverse learning. We numerically validate our theoretical results using both simulated and measured dynamic tomography data, showing that our approach leads to a practical and robust reconstruction strategy.Comment: 51 pages, 11 figure