Correlation Length Controllable Priors

From Theory of Measurements Wiki

1 Abstract
2 Introduction
3 One Dimensional Prior
- 3.1 First Prior
- 3.2 Second Prior
4 See Also

Abstract

Caption

In this paper, we consider how to make statistical inversion with smoothness priors, which have a finite and controllable correlation length and amplitude.

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Introduction

In this paper, we consider stochastic regularization and linear inverse problems with additive noise. In general linear inverse problems can be given with the following equation.

$m=Ax+\varepsilon$

We assume, that the measurements $m$ are known and spatially fixed. We further assume, that the theory $A\,$ is known and the statistical properties of the noise process $\varepsilon$ are exactly known.

If the operator $A\,$ is ill-posed in the sense of Hadamard [2], one has to use some a priori information, in order to solve the linear inverse problem. The information in statistical inverse theory is given in the form of a stochastic process. If we assume, that the process is stationary and Gaussian process, then the process is fully defined with the mean and covariance

$\textrm{mean}(x_t) = \mathbf{E}(x_t)$

$\textrm{cov}(x_t,x_{t'}) = \mathbf{E}((x_t-\mathbf{E}(x_t))^T(x_{t'}-\mathbf{E}(x_{t'})))$

One could for example give the prior process as zero mean and with the covariance

$\textrm{cov} (x_t,x_{t'}) = \alpha \exp(-\frac{1}{l} |t-t'|), \quad \alpha>0,l > 0$

When making inversion in practice, one has to discretize the covariance and invert it. If we used prior process given in equation \ref{tayskovarianssi}, the covariance matrix would a full matrix. Inverting full matrices costs always lots of computing power and memory. Due to the finiteness of computing power and memory, one is limited to rather low-dimensional inverse problems, if one uses prior processes, which have full covariance structures. Because of this, one rather uses for example smoothness priors. However, when using such priors, the question is how to control the correlation structure and how to take account of the grid. In this paper, we propose two prior distributions in discrete form, which have a controllable correlation amplitude and length. Moreover, the prior distributions are also essentially independent of the discretization of the unknown.

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One Dimensional Prior

The Green's priors suggested in [4], offer a way of making statistical inversion in a way, which is essentially independent of the chosen discretization of the unknown $x$ . This means basically, that Green's priors fix the correlation length in some natural units, such as centimeters. However, if we want to use Green's priors and also change the correlation length and amplitude of correlation in a unified way, we end up in problems. This has been overcome for example with fixing the correlation length in some approximative way [1].

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First Prior

We propose in this paper, that with the priors introduced in this paper, we can control the correlation length in a unified way. The first correlation length controlling prior is given by the following two equations.

$x_i = \epsilon_i , \quad \epsilon_i \sim N(0,\alpha cl/h),$ $x_i-x_{i-1} = \epsilon_i, \quad \epsilon_i \sim N(0,\alpha h/(cl))$

We call the parameter $\alpha\,$ the amplitude of the correlation and $l\,$ the length of the correlation. The discretization of the unknown is given by the discretization step $h\,$ . The parameter $c\,$ is a scaling parameter defined by $c=t_{final}-t_{init}\,$ . The purpose of the scaling parameter is to change the time axis from real time axis $[t_{init},t_{final}]\,$ to $[0,1]\,$ .

The usage of this kind of a prior distribution, differs from previous practices in the sense, that we are using different kinds of smoothness priors at the same time. Moreover, the correlation length and grid has been taken also into account. Because, this prior is based on the Green's priors and because of the definition of Green's priors [4], we call also this prior a Green's prior.

Traditionally this kind of a correlation control prior is given as follows.

$x_i = \epsilon_i , \quad \epsilon_i \sim N(0,\sigma_0^2/h),$ $x_i-x_{i-1} = \epsilon_i, \quad\epsilon_i \sim N(0,\sigma_1^2h)$

This kind of an approach suffers from a proper explanation of the prior. It can be easily seen, that the amplitude of the prior and the length of the prior can be given as follows.

$α = σ 0 σ 1$

$l = \frac{1}{|t_{final}-t_{init}|}\frac{\sigma_0}{\sigma_1}$

$c = | t f i n a l - t i n i t |$

By giving the regularization parameters $\sigma_0^2\,$ and $\sigma_1^2\,$ in this way, offers us also a clear interpretation of the regularization parameters within the formalism of stochastic processes.

In figure \ref{malli1} the covariance matrix is depicted as a surface plot. In figure \ref{ekat} we have plotted the first row of the covariance and varied the parameters $l\,$ and $\alpha\,$ . By numerical computations, we have verified, that

$\int \textrm{cov}(x_t,x_{t'})dt \approx \sum_i \textrm{cov}(x_i,x_j) \Delta t= \alpha cl, \quad j \textrm{ fixed}.$

Moreover, we have also verified numerically, that

$\iint \textrm{cov}(x_t,x_{t'})dtdt' \approx \sum_j\sum_i \textrm{cov}(x_i,x_j)\Delta t\Delta t' = \alpha cl.$

Images of First Order Prior Samples

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Second Prior

A second kind of a one-dimensional correlation controllable prior can be given by the following two equations.

$x_i = \epsilon_i , \quad \epsilon_i \sim N(0,\alpha cl/h)$ $x_{i+1}-2x_i+x_{i-1}=\epsilon_i,\quad\epsilon_i\sim N(0, \alpha h^3/(cl)^3)$

Traditionally this prior has been given as follows.

$x_i = \epsilon_i , \quad \epsilon_i \sim N(0,\sigma_0^2/h)$ $x_{i+1}-2x_i+x_{i-1}=\epsilon_i,\quad\epsilon_i\sim N(0, \sigma_2^2h^3)$

The amplitude and length of the prior can be given now as follows.

$\alpha = \sqrt{\sigma_0^3\sigma_2}$

$l = \sqrt{\frac{\sigma_0}{\sigma_2}}$

[1] S R Arridge, J P Kaipio, V Kolehmainen, M Schweiger, E Somersalo, T Tarvainen and M Vauhkonen, Approximation errors and model reduction in optical diffusion tomography, Inverse Problems 22 (2006) 175-195.

[2] J Hadamard, Le Probléme de Cauchy et les Equations aux Dérivées Partielles Linéaires Hyperboliques, Herman et Cie, Paris, 1932.

[3] J Kaipio and E Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Series 160, Springer-Verlag 2004.

[4] S Lasanen and L Roininen, Statistical Inversion with Green's Priors, 5th Int. Conf. Inv. Prob. Eng. Proc., Cambridge, 2005.

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Correlation Length Controllable Priors

From Theory of Measurements Wiki

Contents

Abstract

Introduction

One Dimensional Prior

First Prior

Second Prior

See Also

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