Inverse polynomial reconstruction method in DCT domain

The discrete cosine transform (DCT) offers superior energy compaction properties for a large class of functions and has been employed as a standard tool in many signal and image processing applications. However, it suffers from spurious behavior in the vicinity of edge discontinuities in piecewise smooth signals. To leverage the sparse representation provided by the DCT, in this article, we derive a framework for the inverse polynomial reconstruction in the DCT expansion. It yields the expansion of a piecewise smooth signal in terms of polynomial coefficients, obtained from the DCT representation of the same signal. Taking advantage of this framework, we show that it is feasible to recover piecewise smooth signals from a relatively small number of DCT coefficients with high accuracy. Furthermore, automatic methods based on minimum description length principle and cross-validation are devised to select the polynomial orders, as a requirement of the inverse polynomial reconstruction method in practical applications. The developed framework can considerably enhance the performance of the DCT in sparse representation of piecewise smooth signals. Numerical results show that denoising and image approximation algorithms based on the proposed framework indicate significant improvements over wavelet counterparts for this class of signals.