Research

Research Interests

Numerical Analysis, Numerical Optimization, Statistical Learning, Inverse Problems, Image Processing, Dynamic Sampling
My main research is in computational methods for inverse problems in image processing applications. I have worked on methods from numerical optimization and statistical learning, and I have worked on many different applications of inverse problem as well.

Hybrid Projection Methods for Large-scale Inverse Problems

To solve large-scale ill-posed inverse problems, a good choice of the prior is critical to compute a reasonable solution. Current approach using Gaussian prior is required to fix or setup a prior covariance matrix in advance before using iterative projection methods to solve the corresponding regularized problem.

\[ \min\limits_{\textbf{s}} \dfrac{1}{2}\|\textbf{As} - \textbf{d} \|_{\textbf{R}^{-1}}^2 + \dfrac{\lambda^2}{2} \| \textbf{s}-\boldsymbol{\mu} \|_{\textbf{Q}^{-1}}^2 \]

where: \(\textbf{d}\in\mathbb{R}^m\) is observed data, \(\textbf{A}\in\mathbb{R}^{m\times n}\) models the forward process, \(\textbf{s}\in\mathbb{R}^n\) is the desired parameters, \(\boldsymbol{\epsilon}\sim{\cal N}(\textbf{0},\textbf{R})\), and the prior knowledge \(\textbf{s}\sim{\cal N}(\boldsymbol{\mu},\lambda^{-2}\textbf{Q})\).

I have studied various approaches either pre-determine the covariance matrix \(\textbf{Q}\), estimate \(\textbf{Q}\) from training set, or mixing two different covariance matrices. I have used Golub-Kahan bidiagonalization to solve efficiently the large-scale inverse problem.

Computational Tools for Inversion and Uncertainty Estimation

For large-scale experiments (e.g., long scans with high frequency) that have many uncertainties, reconstructing real-time signal of interest from noisy observation is a computationally challenging nonlinear inverse problem. From separable nonlinear setting, we focus on both estimating input and blur parameter for forward model.

\[ \min\limits_{\textbf{x}\textbf{y}} \dfrac{\sigma^2}{2}\| \textbf{A}(\textbf{y})\textbf{x} - \textbf{b} \|_2^2 + \dfrac{\lambda^2}{2}\|\textbf{x}-\boldsymbol{\mu}\|^2_{\textbf{Q}^{-1}} +\dfrac{\alpha^2}{2}\|\textbf{y}-\textbf{s}\|^2_{\textbf{R}^{-1}} \]

where: \(\textbf{b}\) is observation, \(\textbf{x}\sim{\cal N}(\boldsymbol{\mu},\lambda^{-2}\textbf{Q})\) is the desired parameter, \(\textbf{y}\sim{\cal N}(\textbf{s},\alpha^{-2}\textbf{R})\) is parameter of forward model \(\textbf{A}\), and noise \(\boldsymbol{\epsilon}\sim{\cal N}(\textbf{0},\sigma^{-2}\textbf{I})\).