## Estimation and inference in problems from imaging and biophysics

##### Date

2018

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University of Delaware

##### Abstract

Many physical processes and phenomena have solutions containing sums of exponential functions. These exponential functions generally describe the decay of processes and signals they emit. In order to better understand them, one needs to determine the parameters that underly these processes, given measured data. The estimation and inference of these parameters from data is an inverse problem, and is usually ill-posed in that multiple sets of parameters may generate the same, or similar data. ☐ In this dissertation, we study exponential analysis in three different applications. The first application is inferring the transition rates of a birth-death process (BDP) from its extinction time (ET) distribution. We first investigated the analytical solution with noise-free data as the sum of exponential functions, and then solved small BDP problems from exact ETs. Then with maximum sites as additional information, we proposed a new numerical scheme to infer the transition rates from a BDP of length $N + 1$, in the context of protein folding with atomic force microscopy (AFM) data. This method focuses on the coefficients of the characteristic polynomial of the underlying ODE, and establishes recurrence relations between them. Transition rates are recovered sequentially with initial errors propagating exponentially in the BDP. ☐ The second problem arises in nuclear magnetic resonance (NMR) and medical imaging. In an effort to determine the type of tissue in NMR experiments, we need to apply the inverse Laplace transform (ILT) on NMR signals which are functions of two relaxation times $T_1$ and $T_2$. Since ILT is ill-conditioned, we use Tikhonov regularization to recover the distribution of relaxation times. The inversion can be done either in a single step by resizing the solution matrix as a large vector with one parameter, or in two steps by sequential inversion of $T_1$ and $T_2$ that involves two parameters. We show that the one-parameter approach performs well, and adding extra parameters does not improve the result. ☐ The third problem is to fit a two-term exponential probability distribution given measured data. When the exponents are close to each other, many classic methods fail. We propose the moment constraint method, which is revised from the modified Prony method, that takes into account moments of data, regularization, and expectation-maximization (EM) techniques to overcome the difficulties. This method outperforms many other methods, such as maximum-likelihood, when two exponents are very close. The moment constraint method is also applied to four-term exponential fitting problems whose exponents can be separated into two groups by magnitude. It breaks down to two-term exponential fitting subproblems, after a preprocessing step involving Tikhonov regularization and EM sorting, and yields results with reasonable accuracy.

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##### Keywords

Pure sciences, Applied sciences, Birth-death process, Exponential analysis, Inference