Linear Colour Image Processing in Hypercomplex Algebra guided by Genetic Algorithms

Linear vector colour image processing (LVCIP) has been researched for about two decades in the field of pattern recognition and computer vision, but no sound mathematical framework has been suggested. In this thesis, some general mathematical frameworks are investigated for LVCIP based on canonical hypercomplex convolution mask and introduced new linear colour vector image filters. Using linear filtering, multiple complex geometrical operations are embedded into multiple convolutions and reduced into a canonic convolution based on linear quaternion system (LQS) since LQS is a canonic form of a general linear quaternion function of the first degree. The proposed mathematical frameworks are specific for linear filtering since the convolution operations are carried out in projective 4D RGB colour space instead of Euclidean 3D RGB colour space as is conventionally done in quaternion-based CIP. In this new approach, colour image pixel values are mapped as projective vectors in form of full quaternions using homogeneous coordinates instead of Euclidean vectors as pure quaternions using Cartesian coordinates. The use of projective geometry provides a much richer set of geometric operations in the colour space compared to the Euclidean geometry. In the proposed framework, each part of a quaternion-valued mask (scalar and imaginary parts) can be encoded for applying different specific geometrical operations. Initially, a genetic algorithm (GA) is used as a guiding tool for finding primitive designs of such filters, since the designing of such filters is proved to be difficult manually. Results from GA run without constraints are inspected manually to deduce a specific pattern of mask coefficients which are applied in a subsequent GA computation to generate a better set of coefficients with a specific structure. Then the better set of coefficients found by GA are used to deduce simple ad-hoc designs of such filters manually with complete parameters specification.