Inverse Problems

To reconstruct images or signals from seemingly insufficient or noisy data, we exploit mathematical properties of the solution to make the most of the available data. Noteworthy is the new field of compressed sensing, which exploits sparsity (or compressibility) to recover signals very well (even perfectly) from far less data than previously imagined. Decoding highly corrupted bit streams, image or 3-D object reconstruction from very few radiographs, and network tomography are just a few of the many possible applications.

• Exact Reconstructions from Surprisingly Little Data
• Abel Inversion Using Total-Variation Regularization

Geometric Measure Theory

Geometric analysis provides insights into the precise nature of sets, measures and functions. These can be exploited for creation of novel algorithms for the analysis and modeling of data. Our current work in this area includes shape statistics, exact minimizers for image analysis functionals, image signatures for robust metrics, shape priors for fine tuning image segmentation, and null-Riemannian geometry for selective metrics.

• L1TV Computes the flat norm for boundaries
• Total Variation Regularization for Image Denoising: Geometric Theory
• Some Properties of Minimizers for the L1-TV Functional

Derivative-Free and Mixed-Variable Optimization

Optimization problems seek some optimal set of variables that serve as a description of some situation, data, or object. Optimality is measured by a so-called objective function on the variables that is designed to achieve a minimum value for the best possible description. Many algorithms use derivative information on the objective function to achieve efficient convergence to the optimum. We develop and use optimization methods that do not use derivatives. These methods are important for several reasons. Derivatives may be unavailable as is the case for black box and computer code objectives. Derivatives may be unreliable due to stochastic noise. Derivatives may be difficult to compute such as for objectives defined only in complex non-cartesian geometries. Derivatives may be undefined such as for variables that are discrete (integer, date, grid location, etc.), categorical (color, material, table entry, etc.), or mixed.

• Sparse Radiographic Tomography and System Identification Imaging from Single View, Multiple Time Sample Density Plots
• Equality Constraints, Riemannian Manifolds and Direct Search Methods
• Direct Search Algorithms over Riemannian Manifolds

Metrics

Many data science challenges require rigorously defensible comparisons between similar data types. Examples include object recognition, image reconstruction, validation of simulations using experiments, and quantification of uncertainty in many settings. Understanding how to build desirable properties into the metrics used to quantify these comparisons often requires substantial mathematical insight.

• Classification Modulo Invariance with Application to Face Recognition
• Defensible Metrics and Merit Functions

Regularization and Dimension Reduction

In any data driven problem such as image denoising, statistical inference from data, or validation of scientific predictions we must use prior information and the simplicity implicit in the problem domain to turn data into conclusions. An example is the use of the knowledge of band-limiting to turn a measured time series back into the continuous signal from which the measurements came using only the measurements. Another example is the use of knowledge that image information is contained in the edges to suggest the use of total variation (TV) regularization in denoising applications. A final example is the exploitation of the fact that high dimensional data often actually lies on low dimensional submanifolds in the high-dimensional data spaces.

• A Numerically Implementable Close-to-Isometric Embedding Algorithm
• Nonconvex Regularization for Inverse Problems

Dynamical Systems

The continuation of invariant manifolds is a common task in numerical dynamical systems. A particularly interesting application is the continuation of tori that are invariant under a smooth flow. One class of methods, based on invariance equations known as "orthogonality conditions," are robust and stable under many circumstances. We investigate ways to discretize these topological conditions and solve the resultant linear algebraic systems. This picture illustrates an invariant torus born out of a Neimark-Sacker bifurcation in the forced van der Pol oscillator. The image shows the distortion of the torus as it progresses toward breakdown. The dotted line is a repelling periodic orbit, calculated using the same numerical method as the torus. See the torus tour below for more images.

• Torus tour (allow popups from ddma.lanl.gov to view)
• A Geometrical Method for the Approximation of Invariant Tori

Signal Processing

Low dimensional time series arise in many ways. For example, many engineering systems are tracked and controlled using scalar time series from transducers measuring pressure, temperature, pH, concentrations of chemical components, speeds and velocities at various physical locations, etc. Tasks such as detection of an event signature in a flood of noise, prediction of future signal behavior, signal compression, or inference of the signal between measurements (interpolation) all require insight into the structure of the structure of these times series and the domain from which they come.

Though the bulk of our work deals with higher dimensional signals (images, for example) we have expertise and an interest in low dimensional signal processing. Examples of past projects include detection of aliasing in measured signals and validation of simulations through metrics designed for the comparison of scalar time series.

• Detection of Aliasing in Persistant Signals
• Finding the Time of Arrival of a Signal