Total Variation Prior

2015-11-17

In another post I computed the negative-loglikelihood of an image given current dipole fields $g$ with a quadratic prior probability on $g$. The quadratic priors penalize sharp edges too much, causing the reconstruction to suffer near line currents. To remedy this I switch to a total variation (TV) regularization, which imposes sparse gradients on the image:

$\begin{eqnarray} p(g) &=& C\exp\left(-\lambda TV(g)\right) \\ \text{where } TV(g) &=& \int | \nabla g |^2 d^2x \end{eqnarray}$

This prior favors images with constant patches, where only the boundaries contribute to $TV(g)$ and reduce the probability. Since we’re mostly interested in the currents, which are the derivatives of $g$, we will instead impose a sparsity of the second derivatives or equivalently a sparsity of the second derivatives of $g$:

$\begin{eqnarray} p(g) &=& C\exp(-\lambda TV_2(g)) \\ \text{where } TV_2 &=& \sum_i \sqrt{(D_h^2g_i)^2+(D_v^2g_i)^2 } \end{eqnarray}$

for horizontal ($x$) derivatives $D_h$ and vertical ($y$) derivatives $D_v$. Then, as in Gradients of NLL we define

$\mathcal{L} = \frac{1}{2\sigma^2}\left(|| {(\phi^0 - \phi_{ext}) - Mg}||^2 + 2\sigma^2\mu TV_2(g+g_{ext})\right)$

Colin Clement

Total Variation Prior

2015-11-17