感知 Kalman 滤波器：完美感知质量约束下的在线状态估计
Perceptual Kalman Filters: Online State Estimation under a Perfect Perceptual-Quality Constraint
AI总结：
本文研究完美感知质量约束下的因果在线状态估计，提出未利用信息过程并推导感知滤波器递归公式，揭示其为保持时间一致性而有意忽略观测信息、导致 MSE 显著增加的根本困境。
摘要：
许多实际场景都需要从损坏或缺失的数据中重建时间信号。经典例子包括解码、跟踪、信号增强和去噪。由于重建后的信号最终由人类观看，因此希望获得令人类感知愉悦的重建结果。在数学上，当恢复信号的分布与自然信号的分布相同时，即达到完美感知质量；这一要求在静态估计场景（即一次性处理整个信号）中已得到大量研究。本文研究完美感知质量约束下的最优因果滤波问题，这是一项性质根本不同的任务。具体而言，我们分析通过线性含噪变换观测到的 Gaussian Markov 信号。在没有感知约束时，已知 Kalman 滤波器在该设置下按 MSE 意义是最优的。本文表明，加入完美感知质量约束（即时间一致性要求）会引入一个根本性困境：滤波器可能不得不“有意地”忽略观测所揭示的新信息，以便与其过去的决策保持一致。这通常会导致 MSE 显著增加（超过静态场景中遇到的增幅）。我们的分析超越了 Kalman 滤波器的经典新息过程，并引入了未利用信息过程这一新概念。利用该工具，我们给出了感知滤波器的递归公式，并在一个视频重建问题上展示了完美感知质量估计的定性影响。
英文摘要：
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.
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