Hypercomplex Auto-And-Cross-Correlation of Color Images

Autocorrelation and cross-correlation have been defined and utilized in signal and image processing for many years, but not for color or vector images. In this poster we present for the first time a definition of correlation applicable to color images, based on quaternions or hypercomplex numbers. We have devised a visualization of the result using the polar form of a quaternion in which color denotes quaternion eigenaxis and phase, and a grayscale image represents the modulus. 3. Color image representation A color image in rgb color space may be represented using hypercomplex numbers by encoding the red, green and blue components of the rgb value as a pure quaternion: f (x, y) = ir(x, y) + jg(x, y) + kb(x, y) where r(x, y) is the red component of the color image and similarly for the green and blue components. The reason for choosing this representation is that the rgb values represent a 3-space vector (a point in rgb space), as does the pure quaternion. 6. Results – autocorrelation of natural images Autocorrelation of the ‘Lena’ image (128 × 128 pixels). Left to right: original image, modulus, phase, eigenaxis. Autocorrelation of the ‘Boat’ image (128 × 128 pixels). Left to right: original image, modulus, phase, eigenaxis. (Original images from the USC-SIPI image database.) 8. Results – cross-correlation Cross-correlation of two checkerboard images. Top row: original images; bottom row; left to right: modulus; phase; eigenaxis.