Let us assume that we were given a set of images . Each of these images is two-dimensional - in the simpler case at least. An image is pixels nigh and pixels wide. Each perturbation (displacement) has a direction and an intensity (vector length) associated with it. Let us define the total displacement to be and the average accordingly. Then,
(1) |
and
(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 images.
(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 has been applied to each of the pixels, in each of of the images in the set. We seek a way of selecting in a way which obeys certain rules. The goal is the obtain a stack whose members are the images and where each member of the stack has an increasing amount of displacement applied. The perturbation method needs to sensibly pick values for so that a clear relationship among stack member should emerge.