Two-dimensional filter bank design for optimal reconstruction using limited subband information

In this correspondence, we propose design techniques for analysis and synthesis filters of 2-D perfect reconstruction filter banks (PRFB's) that perform optimal reconstruction when a reduced number of subband signals is used. Based on the minimization of the squared error between the original signal and some low-resolution representation of it, the 2-D filters are optimally adjusted to the statistics of the input images so that most of the signal's energy is concentrated in the first few subband components. This property makes the optimal PRFB's efficient for image compression and pattern representations at lower resolutions for classification purposes. By extending recently introduced ideas from frequency domain principal component analysis to two dimensions, we present results for general 2-D discrete nonstationary and stationary second-order processes, showing that the optimal filters are nonseparable. Particular attention is paid to separable random fields, proving that only the first and last filters of the optimal PRFB are separable in this case. Simulation results that illustrate the theoretical achievements are presented.