  There is no method to quantify the spatial complexity within colon polyps.
This paper describes a spatial transformation that translates the tissue
architecture within a polyp, or a normal colon lining, into a complex sinusoid
wave composed of discrete points. This sinusoid wave can then undergo the Fast
Fourier Transform to obtain a spectrum of frequencies that represents the
sinusoid wave. This spectrum can then serve as a signature of the spatial
complexity [an index] within the polyp. By overlaying vertical lines that
radiate from the bottom middle [like a fold-out fan] of an image of a polyp
stained by hematoxylin and eosin, the image is segmented into sectors. Each
vertical line also forms an angle with the horizontal axis of the image,
ranging from 0 degrees to 180 degrees rising counter clockwise. Each vertical
line will intersect with various features of the polyp [border of lumens,
border of epithelial lining]. Each of these intersections is a point that can
be characterized by its distance from the origin [this distance is also a
magnitude of that point]. Thus, each intersection between radial line and polyp
feature can be mapped by polar coordinates [radius length, angle measure]. By
summing the distance of all points along the same radial line, each radial line
that divides the image becomes one value. Plotting these values [y variable]
against the angle of each radial line from the horizontal axis [x variable]
results in a sinusoid wave consisting of discrete points. This method is
referred to as the Linearized Compressed Polar Coordinates [LCPC] Transform.
The LCPC transform, in conjunction with the Fast Fourier Transform, can reduce
the complexity of visually hidden histological grades in colon polyps into
categories of similar wave frequencies [each histological grade has a signature
consisting of a handful of frequencies].
