Many practical settings call for the reconstruction of temporal signals from corrupted or missing data. Classic examples include decoding, tracking, signal enhancement and denoising. Since the reconstructed signals are ultimately viewed by humans, it is desirable to achieve reconstructions that are pleasing to human perception. Mathematically, perfect perceptual-quality is achieved when the distribution of restored signals is the same as that of natural signals, a requirement which has been heavily researched in static estimation settings (i.e. when a whole signal is processed at once). Here, we study the problem of optimal causal filtering under a perfect perceptual-quality constraint, which is a task of fundamentally different nature. Specifically, we analyze a Gaussian Markov signal observed through a linear noisy transformation. In the absence of perceptual constraints, the Kalman filter is known to be optimal in the MSE sense for this setting. Here, we show that adding the perfect perceptual quality constraint (i.e. the requirement of temporal consistency), introduces a fundamental dilemma whereby the filter may have to "knowingly" ignore new information revealed by the observations in order to conform to its past decisions. This often comes at the cost of a significant 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. Using this tool, we present a recursive formula for perceptual filters, and demonstrate the qualitative effects of perfect perceptual-quality estimation on a video reconstruction problem.

翻译：许多实际场景需要从受损或缺失数据中重建时变信号。经典例子包括解码、跟踪、信号增强和去噪。由于重建信号最终由人类观测，因此获得符合人类感知的重建结果至关重要。从数学角度看，当重建信号的分布与自然信号分布相同时实现完美感知质量——这一要求在静态估计场景（即一次性处理完整信号）中已被广泛研究。本文研究完美感知质量约束下的最优因果滤波问题，该任务具有根本不同的性质。具体而言，我们分析通过线性含噪变换观测到的高斯马尔可夫信号。在无感知约束时，卡尔曼滤波器在此场景下在均方误差意义上已知为最优。研究发现，添加完美感知质量约束（即时间一致性要求）会引入根本性困境：滤波器可能需要“故意”忽略观测揭示的新信息以符合其先前决策。这通常以均方误差显著增加（超过静态场景）为代价。我们的分析超越传统卡尔曼滤波器的创新过程，引入未利用信息过程这一新概念。基于该工具，我们提出感知滤波器的递归公式，并通过视频重建问题展示完美感知质量估计的定性效果。