Multitransform/Multidimensional Signal Representation

A useful technique is presented to compactly represent multidimensional signals employing two or more transforms. This is performed by superimposing partial sets of basis functions of different transforms, which are in general, mutually non-orthogonal. Basically, the multidimensional signal is split into subsignals, each of which is represented by the dominant components of a transform, whose basis functions closely approximate the subsignal. The residual error, which is the difference between the original image and the reconstructed image is properly formulated. Adaptive algorithms are developed to minimize this error and to maximize the Signal to Noise ratio (SNR) of the reconstructed image for a fixed number of transform components. An optimization strategy is also proposed to select the dominant components from the various domains for adaptation. A simulation is presented to represent a synthetic image consisting of two sinusoids and two square waves. It is verified that it is possible to split the image spectrally into a narrowband and a broadband part, thus allowing the DCT to represent the sinusoids and the Walsh Hadamard transform to represent the square waves. This leads to excellent data compression