Permutation filters: a class of nonlinear filters based on set permutations

In this paper we introduce and analyze a new class of non{linear lters which have their roots in permutation theory. We show that a large body of non{linear lters proposed to date constitute a proper subset of Permutation Filters (P Filters). In particular, rank{order lters, weighted rank{order lters, and stack lters embody limited permutation transformations of a set. Indeed, by using the full potential of a permutation group transformation we can design very eecient estimation algorithms. Permutation groups inherently utilize both rank{order and temporal{order information; thus, the estimation of non{stationary processes in Gaussian/non{Gaussian environments with frequency selection can be eeec-tively addressed. An adaptive design algorithm which minimizes the mean absolute error criterion is described as well as a more exible adaptive algorithm which attains the optimal permutation lter under a deterministic least normed error criterion. Simulation results are presented to illustrate the performance of permutation lters in comparison with other widely used lters. Permission to publish this abstract separately is granted. 2 Filter learning curves for the case with tone interference and (0:05; 2:0; 15:0) distributed additive contaminated Gaussian noise. (The L 1 norm was used for the LNE optimizations. 4 Observed and lter estimate signals for the case with tone interference and (0:05, 2:0; 15:0) distributed additive contaminated Gaussian noise: (a) observed, (b) linear FIR, (c) combination, (d) stack, (e) permutation, (f) reduced set permutation. : : 30 5 Power spectral densities of the observed and lter estimate signals for the case with tone interference and (0:05; 2:0; 15:0) distributed additive contaminated Gaussian noise: (a) observed, (b) linear FIR, (c) combination, (d) stack, (e) permutation, (f) reduced set 6 Top: Video sequence estimate errors for 33 window permutation and stack lters. The lters were trained two ways, using all 120 frames, and only the rst frame. Bottom: The diierences between estimate errors based on training the lters with all 120 frames and only the 2 The output distributions for permutation lters operating on i.i.d. input variables with common distribution (). The output distribution is given as a polynomial of for all possible values of K 1 ; K 2 , and K 3. The number of unique lters for each set of K 1 ; K 2 , and K 3 values as listed as K , and the output mean and variance are given for zero mean unit variance bi{exponentially distributed input samples. : : : 22 3 …