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Many practical settings call for the reconstruction of temporal signals fromcorrupted or missing data. Classic examples include decoding, tracking, signalenhancement and denoising. Since the reconstructed signals are ultimatelyviewed by humans, it is desirable to achieve reconstructions that are pleasingto human perception. Mathematically, perfect perceptual-quality is achievedwhen the distribution of restored signals is the same as that of naturalsignals, a requirement which has been heavily researched in static estimationsettings (i.e. when a whole signal is processed at once). Here, we study theproblem of optimal causal filtering under a perfect perceptual-qualityconstraint, which is a task of fundamentally different nature. Specifically, weanalyze a Gaussian Markov signal observed through a linear noisytransformation. In the absence of perceptual constraints, the Kalman filter isknown to be optimal in the MSE sense for this setting. Here, we show thatadding the perfect perceptual quality constraint (i.e. the requirement oftemporal consistency), introduces a fundamental dilemma whereby the filter mayhave to "knowingly" ignore new information revealed by the observations inorder to conform to its past decisions. This often comes at the cost of asignificant increase in the MSE (beyond that encountered in static settings).Our analysis goes beyond the classic innovation process of the Kalman filter,and introduces the novel concept of an unutilized information process. Usingthis tool, we present a recursive formula for perceptual filters, anddemonstrate the qualitative effects of perfect perceptual-quality estimation ona video reconstruction problem.

Perceptual Kalman Filters: Online State Estimation under a Perfect Perceptual-Quality Constraint