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Formulation of the Problem

Let us assume that we were given a set of $n$ images $\mathbf{I}_{1},\mathbf{I}_{2},..,\mathbf{I}_{n}$. Each of these images is two-dimensional - in the simpler case at least. An image is $N$ pixels nigh and $M$ pixels wide. Each perturbation (displacement) $\Delta$ has a direction $\overrightarrow{\Lambda}$ and an intensity (vector length) associated with it. Let us define the total displacement to be $\Delta_{total}$ and the average $\Delta_{average}$ accordingly. Then,

$$\Delta_{total}=\begin{array}{c} n\\ \sum\\ k=1\end{array}\... ...ay}\begin{array}{c} M\\ \sum\\ i=1\end{array}\Delta_{k_{i,j}}$$ (1)

and

$$\Delta_{average}=\frac{\Delta_{total}}{nMN}.$$ (2)

Having formulated it in this way, a more proper average should treat the displacements in each image separately and not aggregate displacements in each of the $n$ images.

$$\mathbf{I}'_{k_{i,j}}=\begin{array}{c} n\\ \sum\\ k=1\end{... ...\\ \sum\\ i=1\end{array}\mathbf{I_{k_{i,j}}}+\Delta_{k_{i,j}}$$ (3)

only reflects on how the new set of images are generated. Rather than calculating a sum, it forms a set of matrices (in a vector-wise assignment).

Some displacement $\Delta_{k_{i,j}}$ has been applied to each of the pixels, $\mathbf{I_{k_{i,j}}},$ in each of of the images in the set. We seek a way of selecting $\Delta$ in a way which obeys certain rules. The goal is the obtain a stack whose members are the images $\mathbf{I}_{1},\mathbf{I}_{2},..,\mathbf{I}_{n}$ and where each member of the stack has an increasing amount of displacement applied. The perturbation method needs to sensibly pick values for $\Delta_{k_{i,j}}$ so that a clear relationship among stack member should emerge.