Hypercomplex scaling and wavelet filters: their discovery and their application to colour vector image processing

The aim of this thesis is to extend existing work and find new discrete scaling and wavelet filters with quaternion coefficients and the first with Clifford Cl(1,1) and Cl(2,0) coefficients; and then to demonstrate the use of these filters by finding hypercomplex wavelet transforms of colour vector images. We solve certain symbolic matrix equations simultaneously to find our scaling filter coefficients and then use a numerical method involving paraunitary completion of the polyphase matrix to find related wavelet filter coefficients. We find that our symbolic solutions include full and ‘partial’ transposes of each other. Now complex numbers are isomorphic to each of the three two-dimensional subalgebras of the quaternions and some two-dimensional subalgebras of Cl(1,1) and Cl(2,0), Cl(1,0) being isomorphic to the rest: thus, we may use the values of the coefficients from complex and Cl(1,0) scaling and wavelet filters in the appropriate places of further quaternion, Cl(1,1) and Cl(2,0) scaling and wavelet filters. We use the cascade algorithm on all our filters and illustrate the resulting hypercomplex scaling and wavelet functions with plots of all possible projections onto two and three dimensions. We then use our filters to find hypercomplex wavelet transforms of some colour test images represented as arrays of pure hypercomplex numbers, ones with no scalars. To do this, for each one we place copies of one pair of filters down the leading diagonal of a zero matrix to produce a banded matrix. We pre-multiply a colour vector test image by this banded matrix and post-multiply by its conjugate transpose. This results in an array of full hypercomplex numbers. We then extract the approximation plus horizontal, vertical and diagonal detail images from the scalar (black and white) and vector (colour) parts of the result separately and illustrate them side by side, each arranged in the conventional format.