Translating the Architectural Complexity of the Colon or Polyp into a Sinusoid Wave for Classification via the Fast Fourier Transform

Background: 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 (i.e. an index) within the polyp. Method: By overlaying vertical lines that radiate from the bottom-middle (like a fold-out fan) of an image of a polyp stained by hematoxylin & eosin, the image is segmented into sectors. Each vertical line also forms an angle with the horizontal axis of the image, ranging from 0° to 180° rising counter-clockwise. Each vertical line will intersect with various features of the polyp (i.e. border of lumens, border of epithelial lining). Each of these intersections is a point that can be characterized by it’s 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 (yvariable) 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. Discussion: 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 (i.e. each histological grade has a signature consisting of a “handful” of frequencies). Step 1 – Map each border (or feature of interest) using polar coordinates. Step 2 – Compress magnitude of all points on a line into one value for each line. Step 3 – Apply Fast Fourier Transform Graphical Abstract Applications • Detect subtle changes in architecture, or cellular components, of colon lining or polyps. • Classify the complexity of tissue architecture as simple collection of wave frequencies. • Identify refined polyp grading system to predict clinical outcome or response to therapy of new polyp subtypes. Linearized Compressed Polar Coordinate (LCPC) Transform David H. Nguyen. “Translating the Architectural Complexity of the Colon or Polyp into a Sinusoid Wave for Classification via the Fast Fourier Transform.” January 2018, ArXiv. Figure 1 – Definition of Method: Linearized Compressed Polar Coordinates 90° 45° 135° 0° 180° 90° 45° 135°