In this paper, we derive a "convolution theorem" suitable for the Hirschman optimal transform (HOT), a unitary transform derived from a discrete-time, discrete-frequency version of the entropy-based uncertainty measure first described by Hirschman. We use the result to develop transform domain adaptive filters. First, we show how our method can be used to implement a fast block-LMS adaptive filter that we call the HOT block-LMS adaptive filter. This filter requires slightly less than half of the computations that are required in an FFT-based block-LMS adaptive filter. We also develop another transformbased adaptive filter algorithm that uses a sliding window instead of a block of data. The HOT version of these sliding algorithms is also significantly computationally more efficient (by N , where N is the filter order) than the sliding DFT version. Because our work is at an early stage, we develop simulations that explore basic convergence characteristics.

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In this paper we propose a novel image restoration method that effectively combines a particle filter with wavelet shrinkage to achieve robust performance against inhomogeneous noise mixtures. Specifically, the particle filter acts to suppress outlier-rich components of the noise while, in a subsequent step, the wavelet domain shrinkage attenuates any remaining, less heavily tailed noise components. We present late breaking preliminary examples demonstrating excellent rejection of salt-and-pepper like Cauchy noise mixed with additive white Gaussian noise (AWGN). Although limited in scope, these preliminary results suggest that the combination of particle filters with more traditional restoration techniques is a powerful approach that can provide a new dimension of flexibility for addressing noise mixtures involving difficult nonlinear and non-Gaussian components.

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