Decon in the log domain with variable gain

TAKING THE STEP

We adopt the convention that components of a vector $\mathbf{ u}$ range over the values of $(u_t)$ , likewise for other vectors. Given the gradient direction $\Delta\mathbf{ u}$ we need to know the residual change $\Delta\mathbf{r}$ and a distance $\alpha$ to go: $\alpha\Delta\mathbf{r}$ and $\alpha\Delta\mathbf{u}$ .

A two-term example demonstrates a required linearization.

$$e^{\alpha\Delta U} = e^{\alpha(\Delta u_1 Z + \Delta u_2 Z^2)}$$ (18)

$$e^{\alpha\Delta U} = 1 + \alpha (\Delta u_1 Z +\Delta u_2 Z^2) + \alpha^2(\cdots)$$ (19)

$${\rm FT}^{-1}\ e^{\alpha\Delta U} = (1 , \alpha \Delta u_1 , \alpha\Delta u_2 ) + \alpha^2(\cdots)$$ (20)

$${\rm FT}^{-1}\ e^{\alpha\Delta U} = (1 , \alpha \Delta \mathbf{ u} ) + \alpha^2(\cdots)$$ (21)

With that background, neglecting $\alpha^2$ , and knowing the gradient $\Delta\mathbf{ u}$ , let us work out the forward operator to find $\Delta \mathbf{q}$ . Let ``$\ast$ '' denote convolution.

$$\mathbf{r} + \alpha \Delta \mathbf{r} = {\rm FT}^{-1}( D e^{ U +\alpha \Delta U})$$ (22)

$$= {\rm FT}^{-1} (De^ U e^{\alpha \Delta U})$$ (23)

$$= {\rm FT}^{-1} (De^ U ) \ast {\rm FT}^{-1} ( e^{\alpha \Delta U})$$ (24)

$$= \mathbf{r} \ast ( 1 , \alpha \Delta \mathbf{u})$$ (25)

$$= \mathbf{r} + \alpha \, \mathbf{r} \ast \Delta \mathbf{u}$$ (26)

$$\Delta \mathbf{ r} = \mathbf{ r} \ \ast\ \Delta \mathbf{u}$$ (27)

$$\Delta q_t = g_t\ \Delta r_t$$ (28)

It is pleasing that $\Delta\mathbf{r}$ is proportional to $\mathbf{r}$ . This might mean we can deal with a wide dynamic range within $r_t$ . The convolution, a physical process, occurs in the physical domain which is only later gained to the statistical domain $q_t$ . Naturally, the convolution may be done as a product in the frequency domain.

To minimize $H(\mathbf{q}+\alpha\Delta\mathbf{q})$ express it as a Taylor series approximation to quadratic order. Minimizing yields

$$\alpha = -\ \sum_t \Delta q_t H_t' \ \ /\ \ \sum_t (\Delta q_t)^2 H_t''$$ (29)

Update $q_t = q_t + \alpha \Delta q_t$ and $U=U+\alpha \Delta U$ , optionally (Newton method) iterate (because the locations of the many Taylor series have changed slightly with the change in $\mathbf{q}$ ).

Decon in the log domain with variable gain