Decon in the log domain with variable gain

Code modifications required by regularization

Consider regularization of the form $0\approx u_\tau - u_{-\tau}$ . In matrix form this is $\mathbf{0} \approx \mathbf{r}_m = \mathbf{J} \mathbf{u}$ where the matrix $\mathbf{J}$ is defined below with six vector components in the ordering required by the fast Fourier transform program.

$$\mathbf{0} \ \approx\ \left[ \begin{array}{c} r_m(1)\\ r_m(... ...u(1)\\ u(2)\\ u(3)\\ u(4)\\ u(5)\\ u(6) \end{array}\right]$$ (30)

Note that $\mathbf{J}^\ast =\mathbf{J}$ . The gradient search direction is

$$\Delta\mathbf{u} \ =\ \frac {\partial\,\mathbf{r}_m^\ast \mathbf... ...\, {\mathbf{W} \mathbf{r}_m} \ \ = \ \ \mathbf{J}^\ast \mathbf{W} \mathbf{r}_m$$ (31)

where $\mathbf{W}$ is a diagonal matrix of weights. Again for six components, the diagonal contains $(1,w_1,w_2,0,w_2,w_1)$ .

Here are the modifications needed to incorporate $\ell_2$ regularization on $u_\tau$ :

$${\rm argmin}_{\bold u}\ \sum_{t,x} H(q_t) + \frac{\epsilon}{2} \, \mathbf{u}^\ast \mathbf{J}^\ast \mathbf{W} \mathbf{J} \mathbf{u}$$ (32)

$$\Delta u_t = {\rm as\ before }\ + \epsilon \, \mathbf{J}^\ast \mathbf{W} \mathbf{r}_m$$ (33)

$$\Delta r_t = {\rm as\ before}$$ (34)

$$\alpha = -\ \frac{(\sum_{x,t} \Delta q_t H_t' ) \ +\ \epsilon\,\ (\bold r... ...elta q_t)^2 H_t'' ) \ +\ \epsilon\,(\Delta \bold r_m \cdot \Delta \bold r_m) }$$ (35)

In a least squares problem we compute a step size $\alpha$ as minus a ratio $\bold r \cdot \Delta \bold r$ over $\Delta \bold r \cdot \Delta \bold r$ . Adding a least squares regularization to any convex fitting problem we simply add $\epsilon\, (\bold r_m \cdot \Delta \bold r_m)$ to the numerator and $\epsilon \,(\Delta \bold r_m \cdot \Delta \bold r_m)$ to the denominator.

Actually, another regularization is desireable. We should also request $u_\tau$ to be small for large anticausal lags, lags more negative than the range we are considering for the antisymmetry regularization. This might be handled by truncating the gradient rather than as a regularization.

A third regularization can be added to weaken $u_\tau$ at its large positive lags in circumstances where we feel we have insufficient data to estimate trace-long filters.

Decon in the log domain with variable gain