### Bayesian approach to current reconstruction

##### 2015-10-12

This is based on Ali Mohammad-Djafari’s ‘A Full Bayesian Approach for Inverse Problems.’

Our data is called $\phi^0\in\mathcal{R}^m$, and we have a parameterized relationship between $\phi^0$ and unknown parameters $g\in\mathcal{R}^n$, the local dipole current field. We assume $\phi^0 = M(z) g + \eta$ where $\eta$ is some Gaussian white noise of size $1/\sigma^2$ and $M\in\mathcal{R}^{m\times n}$ includes both the Biot-Savart law and the point spread function of the SQUID loop (parameterized by $z$). we can control the dimension $n$ of $g$, and if we make $n<m$ we can increase the noise-tolerance of the problem. Usually $n = m/a^2$ where $a$ is 2 or 3 for example.

Our hypothesis can be expressed probabilistically as: $\begin{equation} P(\phi | g, z, \sigma) = \frac{1}{(2\pi\sigma^2)^{m/2}}\exp\left(\frac{-1}{2\sigma^2}||\phi^0-Mg||^2 \right) \end{equation}$

This problem is in general referred to as blind deconvolution. Because it is ill-posed, we have have to include prior information to obtain a solution that is not totally corrupted by the experimental noise. A Gaussian prior on $g$ can be expressed as $\begin{equation} P(g|\mu) = \left(\frac{\mu^2}{2\pi}\right)^{n/2}\exp\left( -\frac{\mu}{2}||\Gamma g||^2\right) \end{equation}$

# Cramer-Rao bound

The minimum variance of a parameter $\xi$ of a measured image $I$ is:

Where $i$ indexes pixel measurements. For us this is $I_i = \phi_i = M_{ik}g_k$. (Derivation to come).