Multidimensional, paraunitary principal component filter banks

In this correspondence, the one-dimensional (1-D) principal component filter banks (PCFB's) derived by Tsatsatsanis and Giannakis (1995) are generalized to higher dimensions. As presented by Tsatsatsanis and Giannakis, PCFB's minimize the mean-squared error (MSE) when only Q out of P subbands are retained. Previously, two-dimensional (2-D) PCFB's were proposed by Tirakis et al. (1995). The work by Tirakis et al. was limited to 2-D signals and separable resampling operators. The formulation presented here is general in that it can easily accommodate signals of arbitrary (yet finite) dimension and nonseparable sampling. A major result presented in this paper is that in addition to minimizing MSE when reconstructing from Q out of p subbands, the PCFB's result in maximizing theoretical coding gain (TCG) thereby performing optimally in a energy compaction sense.