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Theory

We propose to estimate local scaling functions and spatially variable temporal shifts by modifying plane-wave destruction (Fomel, 2002) to include scaling. A scaling function is incorporated in the description of high-order plane-wave destructors. These filters are described in the

-transform notation as

$\begin{displaymath} C(p,Z_1,Z_2) = B(p,Z_1^{-1}) - Z_2B(p,Z_1) \end{displaymath}$

(1)

where is the local slope (corresponding to a time shift), and are local shifts in time and space, respectively, and is a polynomial filter. We modify this formulation to incorporate a scaling function as follows:

$\begin{displaymath} C(a,p,Z_1,Z_4) = B(p,Z_1^{-1})-aZ_4B(p,Z_1) \end{displaymath}$

(2)

where

is a scaling coefficient and

represents a shift between images. In the matrix-vector notation, equation (2) is expressed as

$\begin{displaymath} \mathbf{C}(\mathbf{a},\mathbf{p})\mathbf{d} = \mathbf{B}_l(\... ...{d} -\mbox{diag}(\mathbf{a})\mathbf{B}_r(\mathbf{p})\mathbf{d} \end{displaymath}$

(3)

where $\mathbf{B}$ and $\mathbf{C}$ denote the convolution operator with the filters and , respectively, and $\mathbf{d}$ is the time-lapse data. and denote the right and left hand side of the polynomial filter in equation (2). Our goal for the warped and scaled monitor image is to match the base image, therefore

$\begin{displaymath} \mathbf{C}(\mathbf{a},\mathbf{p})\mathbf{d}\approx 0 . \end{displaymath}$

(4)

The depedence of $\mathbf{C}$ on $\mathbf{a}$ is linear, however $\mathbf{p}$ enters in a nonlinear way (Chen et al., 2013a). We separate this problem into a linear and nonlinear part and use the variable projection technique (Golub and Pereyra, 1973; Kaufman, 1975).

We describe the algorithm below.

Set initial values as $\mathbf{p} = \mathbf{0}$ and $\mathbf{a} = \mathbf{1}$ .
Hold the shift constant and compute the scaling weight $\mathbf{a}$ by the smooth division of the right and left side of the plane-wave destruction filter $\mathbf{C}$ in equation(3):

$\begin{displaymath} \mathbf{a} = \left<\frac{\mathbf{B}_r(\mathbf{p})\mathbf{d}}{\mathbf{B}_l(\mathbf{p})\mathbf{d}}\right> \end{displaymath}$ (5)
Scale the monitor image with the estimated weight.
Hold the scale $\mathbf{a}$ constant and compute the shift $\mathbf{p}$ using slope estimation by accelerated plane-wave destruction.
Shift the monitor image using the estimated slope.
Iterate until convergence (return to step 2).

This algorithm efficiently shifts and scales monitor images to match the base image. The estimated shifts and scaling weights are constrained to be smooth using shaping regularization (Fomel, 2007).

Next: Synthetic example Up: Phillips & Fomel: Amplitude-adjusted Previous: Introduction

2022-08-08