Perfect reconstruction with critically sampled filter banks and linear boundary conditions

This work is concerned with the (image) boundary conditions involved in processing a finite discrete-time signal with a critically sampled perfect reconstruction filter bank. It is desirable that the boundary conditions reduce edge effects and define a transformation into a space having the same dimensionality as the original signal. The complication that arises is in the computation of the inverse transform. Although it is straightforward to reconstruct the signal values that were not influenced by the boundary conditions, recovering those values on the boundaries is nontrivial. The solution of this problem is discussed for general linear boundary conditions. No symmetry assumptions are made on the boundary conditions or on the impulse responses of the analysis filters. A low-rank linear transform is derived that expresses the boundary values in terms of the transform coefficients, which in turn provides a method for inverting the subband decomposition. The application of the results in the case of two-channel orthonormal wavelet filters is discussed, and the effects of the filter support on the conditioning of the inverse problem are investigated