Design of optimum multi-dimensional energy compaction filters

We discuss the design of optimum signal-adapted multi-dimensional energy compaction filters. As in the one-dimensional (1-D) case, the energy compaction problem is linear in the auto-correlation coefficients of the compaction filter, which must also satisfy the multi-dimensional (m-D) equivalent of the Nyquist-(M) condition. If a minimum-phase spectral factor exists the optimum compaction filter is recovered using the m-D Discrete Hilbert Transform (DHT). If a minimum phase spectral factor does not exist, an iterative algorithm based on multi-objective goal attainment is proposed. The Nyquist-M condition is enforced while simultaneously forcing the autocorrelation coefficients of the compaction filter to be as close as possible to the coefficients of the product filter and the compaction gain of the optimum compaction filter to be close to the compaction gain produced by using the optimum product filter.