THIS IS A CONTINUATION OF N00014-14-1-0683 Imaging science algorithms based on finite and infinite dimensional Hamilton-Jacobi equations
Abstract
Statement of Work Many image or signal acquisition and processing problems are formulated as an estimation problem in an infinite dimensional framework or in a finite dimensional framework. Standard approaches to solve these problems consist in using variational methods and/or Bayesian approaches, either in a finite or infinite dimensional formalism. These imaging science estimation tasks are a growing field at the confluence of applied mathematics and computer science. They require sound theoretical grounds (i) to provide fine analysis, optimal performances, fast algorithms and (ii) to improve the image processing chain (i.e., from the acquisition of the data up to the image or signal analysis). The work we shall propose consists in three complementary tasks that we briefly describe now: Task 1: "Hamilton-Jacobi equations and finite dimensional variational methods". Recently, Darbon showed a connection between certain finite dimensional convex imaging problems and Hamilton-Jacobi (H-J) equations. Task 1 proposes to extend this connection to cope with 1) 1- homogeneous data fidelity terms (like the l1+TV model), 2) image decomposition models (like Meyer s cartoon+texture model), 3) inverse scale space based image restoration models. These results will allow us to improves image restoration methods over standard non-adaptive methods. We also propose to investigate this H-J formulation to provide new efficient numerical algorithms. Task 2: "Stochastic and Bayesian point of views". Task 2 proposes to explore potential connections between viscous H-J equations and sctochastic and Bayesian models. First, we would like to better understand the connections between viscous H-J equations and Bayesian estimators. We intend to develop a mathematical theory that aims to better understand the behavior of Bayesian restoration methods for image processing tasks. Second, we will address questions related to the formulation of Bayesian estimators in terms of Feynman-Kac formulas and path integral formulations. Finally, we expect to be able to highlight deep connections between large deviation theory and H-J equations. Task 3: "New algorithms for super resolution and computational photography based on infinite dimensional H-J equations". Task 3 proposes to bring the H-J formalism to treat the data acquisition step in addition to the data processing. In order to complete this task we need to use an infinite dimensional version of H-J equations to cope with the continuous observed world. The obtained framework will allow us to improve the imaging chain (from the data acquisition up to the image or signal analysis). The point of view given by a new infinite dimensional H-J formula will be applied to 1) the super resolution of sparse signals and 2) to the computational photography paradigm. This work will allow us to have a deep understanding of the feasibility and difficulty of these two problems. In addition, it will allow us to develop fast new numerical algorithms. Timeline: Year 1: The task 1, entitled "Hamilton-Jacobi and finite dimensional variational methods", with some preliminary work on Task 2 and 3 could be developed the first year (July 2014-June 2015). Year 2: The task 2, entitled "Stochastic and Bayesian point of views", with some further work on task 3 could be developed the second year (July 2015-June 2016). Year 3: The task 3, entitled "New algorithms for super resolution and computational photography based on infinite dimensional H-J equations", could be developed the third year (July 2016-October 2017).
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jun 10, 2016
- Source ID
- N000141612157
Entities
People
- Stanley Osher
Organizations
- Office of Naval Research
- United States Navy
- University of California, Los Angeles