In this paper, we investigate the rate-distortion-perception function (RDPF) of a source modeled by a Gaussian Process (GP) on a measure space $Ω$ under mean squared error (MSE) distortion and squared Wasserstein-2 perception metrics. First, we show that the optimal reconstruction process is itself a GP, characterized by a covariance operator sharing the same set of eigenvectors of the source covariance operator. Similarly to the classical rate-distortion function, this allows us to formulate the RDPF problem in terms of the Karhunen-Loève transform coefficients of the involved GPs. Leveraging the similarities with the finite-dimensional Gaussian RDPF, we formulate an analytical tight upper bound for the RDPF for GPs, which recovers the optimal solution in the "perfect realism" regime. Lastly, in the case where the source is a stationary GP and $Ω$ is the interval $[0, T]$ equipped with the Lebesgue measure, we derive an upper bound on the rate and the distortion for a fixed perceptual level and $T \to \infty$ as a function of the spectral density of the source process.
Rate-distortion-perception theory generalizes Shannon's rate-distortion theory by introducing a constraint on the perceptual quality of the output. The perception constraint complements the conventional distortion constraint and aims to enforce distribution-level consistencies. In this new theory, the information-theoretic limit is characterized by the rate-distortion-perception function. Although a coding theorem for the rate-distortion-perception function has recently been established, the fundamental nature of the optimal coding schemes remains unclear, especially regarding the role of rand…
Blau and Michaeli recently introduced a novel concept for inverse problems of signal processing, that is, the perception-distortion tradeoff. We introduce their tradeoff into the rate distortion theory of lossy source coding in information theory, and clarify the tradeoff among information rate, distortion and perception for general information sources.
Blau and Michaeli recently introduced a novel concept for inverse problems of signal processing, that is, the perception-distortion tradeoff. We introduce their tradeoff into the rate distortion theory of variable-length lossy source coding in information theory, and clarify the tradeoff among information rate, distortion and perception for general information sources. We also discuss the fixed-length coding with average distortion criterion that was missing in the previous letter.
The rate-distortion-perception function (RDPF; Blau and Michaeli, 2019) has emerged as a useful tool for thinking about realism and distortion of reconstructions in lossy compression. Unlike the rate-distortion function, however, it is unknown whether encoders and decoders exist that achieve the rate suggested by the RDPF. Building on results by Li and El Gamal (2018), we show that the RDPF can indeed be achieved using stochastic, variable-length codes. For this class of codes, we also prove that the RDPF lower-bounds the achievable rate
University of Toronto（多伦多大学）；McMaster University（麦克马斯特大学）
In the context of lossy compression, Blau & Michaeli (2019) adopt a mathematical notion of perceptual quality and define the information rate-distortion-perception function, generalizing the classical rate-distortion tradeoff. We consider the notion of universal representations in which one may fix an encoder and vary the decoder to achieve any point within a collection of distortion and perception constraints. We prove that the corresponding information-theoretic universal rate-distortion-perception function is operationally achievable in an approximate sense. Under MSE distortion, we show th…
University of Science and Technology of China（中国科学技术大学）
Signal degradation is ubiquitous and computational restoration of degraded signal has been investigated for many years. Recently, it is reported that the capability of signal restoration is fundamentally limited by the perception-distortion tradeoff, i.e. the distortion and the perceptual difference between the restored signal and the ideal `original' signal cannot be made both minimal simultaneously. Distortion corresponds to signal fidelity and perceptual difference corresponds to perceptual naturalness, both of which are important metrics in practice. Besides, there is another dimension wor…
School of Electrical and Computer Engineering Cornell University（康奈尔大学电气与计算机工程学院）；Cornell University（康奈尔大学）
A rate-distortion-perception (RDP) tradeoff has recently been proposed by Blau and Michaeli and also Matsumoto. Focusing on the case of perfect realism, which coincides with the problem of distribution-preserving lossy compression studied by Li et al., a coding theorem for the RDP tradeoff that allows for a specified amount of common randomness between the encoder and decoder is provided. The existing RDP tradeoff is recovered by allowing for the amount of common randomness to be infinite. The quadratic Gaussian case is examined in detail.
Asymptotic energy-distortion performance of zero-delay communication scenarios under additive white Gaussian noise is investigated. Using high-resolution analysis for quantizer design, the higher-order term in the logarithm of the distortion (termed the {\em energy-distortion dispersion}) is optimized while keeping the leading term (i.e., {\em energy-distortion exponent}) at its optimal value. For uniform and Gaussian sources, significant gains are observed compared to naïvely performed quantization, i.e., aimed at optimizing the source coding performance instead of the end-to-end distortion i…
We characterize the rate-distortion function for zero-mean stationary Gaussian sources under the MSE fidelity criterion and subject to the additional constraint that the distortion is uncorrelated to the input. The solution is given by two equations coupled through a single scalar parameter. This has a structure similar to the well known water-filling solution obtained without the uncorrelated distortion restriction. Our results fully characterize the unique statistics of the optimal distortion. We also show that, for all positive distortions, the minimum achievable rate subject to the uncorre…
Training-Free Rate-Distortion-Perception Traversal With Diffusion
Department of Information Engineering, The Chinese University of Hong Kong, Hong Kong（信息工程系，香港中文大学，香港）；College of Electronic and Information Engineering, Shenzhen University, Shenzhen（电子与信息工程学院，深圳大学，深圳）
Rate-distortion formulation is the information-theoretic approach to the study of signal encoding systems. Since a more general approach to model the nonstationarity exhibited by real-world signals is to use appropriately fitted time varying autoregressive (TVAR) models, we have investigated the rate-distortion function for the class of time varying nonstationary signals. In this study, we present formulations of the rate-distortion function for the Gaussian TVAR processes. The rate-distortion function can serve as an information-theoretic bound on the performance achievable by source encoding…
Alternating Minimization Schemes for Computing Rate-Distortion-Perception Functions with $f$-Divergence Perception Constraints
Communication Systems Department at EURECOM（EURECOM通信系统部门）；Andalusian Research Institute in Data Science and Computational Intelligence (DaSCI)（数据科学与计算智能安达卢西亚研究机构）；University of Granada（Granada大学）
Minimum energy required to achieve a distortion-noise profile, i.e., a function indicating the maximum allowed distortion value for each channel noise level. In this paper, the minimum energy required to achieve a distortion noise profile is studied for Gaussian sources which are transmitted robustly over Gaussian channels. We provide upper bound for the minimum energy behavior of the staircase profile using our proposed coding scheme. Conversely, utilizing a family of lower bounds originally derived for broadcast channels with power constraints, the minimum required energy is lower bounded fo…
The "water-filling" solution for the quadratic rate-distortion function of a stationary Gaussian source is given in terms of its power spectrum. This formula naturally lends itself to a frequency domain "test-channel" realization. We provide an alternative time-domain realization for the rate-distortion function, based on linear prediction. This solution has some interesting implications, including the optimality at all distortion levels of pre/post filtered vector-quantized differential pulse code modulation (DPCM), and a duality relationship with decision-feedback equalization (DFE) for inte…
We prove achievability of the recently characterized quadratic Gaussian rate-distortion function (RDF) subject to the constraint that the distortion is uncorrelated to the source. This result is based on shaped dithered lattice quantization in the limit as the lattice dimension tends to infinity and holds for all positive distortions. It turns out that this uncorrelated distortion RDF can be realized causally. This feature, which stands in contrast to Shannon's RDF, is illustrated by causal transform coding. Moreover, we prove that by using feedback noise shaping the uncorrelated distortion RD…
A single-letter lower bound on the sum rate of multiple description coding with tree-structured distortion constraints is established by generalizing Ozarow's celebrated converse argument through the introduction of auxiliary random variables that form a Markov tree. For the quadratic vector Gaussian case, this lower bound is shown to be achievable by an extended version of the El Gamal-Cover scheme, yielding a complete sum-rate characterization.
Lossy compression algorithms are typically designed and analyzed through the lens of Shannon's rate-distortion theory, where the goal is to achieve the lowest possible distortion (e.g., low MSE or high SSIM) at any given bit rate. However, in recent years, it has become increasingly accepted that "low distortion" is not a synonym for "high perceptual quality", and in fact optimization of one often comes at the expense of the other. In light of this understanding, it is natural to seek for a generalization of rate-distortion theory which takes perceptual quality into account. In this paper, we …
We provide a complete characterization of the achievable distortion region for the problem of sending a bivariate Gaussian source over bandwidth-matched Gaussian broadcast channels, where each receiver is interested in only one component of the source. This setting naturally generalizes the simple single Gaussian source bandwidth-matched broadcast problem for which the uncoded scheme is known to be optimal. We show that a hybrid scheme can achieve the optimum for the bivariate case, but neither an uncoded scheme alone nor a separation-based scheme alone is sufficient. We further show that in t…
We consider the joint source-channel coding problem of sending a Gaussian source on a K-user Gaussian broadcast channel with bandwidth mismatch. A new outer bound to the achievable distortion region is derived using the technique of introducing more than one additional auxiliary random variable, which was previously used to derive sum-rate lower bound for the symmetric Gaussian multiple description problem. By combining this outer bound with the achievability result based on source-channel separation, we provide approximate characterizations of the achievable distortion region within constant …
We establish a new extremal inequality, which is further leveraged to give a complete characterization of the rate region of the vector Gaussian CEO problem with the trace distortion constraint. The proof of this extremal inequality hinges on a careful analysis of the Karush-Kuhn-Tucker necessary conditions for the non-convex optimization problem associated with the Berger-Tung scheme, which enables us to integrate the perturbation argument by Wang and Chen with the distortion projection method by Rahman and Wagner.
The rate-distortion saddle-point problem considered by Lapidoth (1997) consists in finding the minimum rate to compress an arbitrary ergodic source when one is constrained to use a random Gaussian codebook and minimum (Euclidean) distance encoding is employed. We extend Lapidoth's analysis in several directions in this paper. Firstly, we consider refined asymptotics. In particular, when the source is stationary and memoryless, we establish the second-order, moderate, and large deviation asymptotics of the problem. Secondly, by "random Gaussian codebook", Lapidoth referred to a collection of ra…
We deal with zero-delay source coding of a vector Gaussian autoregressive (AR) source subject to an average mean squared error (MSE) fidelity criterion. Toward this end, we consider the nonanticipative rate distortion function (NRDF) which is a lower bound to the causal and zero-delay rate distortion function (RDF). We use the realization scheme with feedback proposed in [1] to model the corresponding optimal "test-channel" of the NRDF, when considering vector Gaussian AR(1) sources subject to an average MSE distortion. We give conditions on the vector Gaussian AR(1) source to ensure asymptoti…
In this paper, we study the rate distortion function of the i.i.d sequence of multiplications of a Bernoulli $p$ random variable and a gaussian random variable $\sim N(0,1)$. We use a new technique in the derivation of the lower bound in which we establish the duality between channel coding and lossy source coding in the strong sense. We improve the lower bound on the rate distortion function over the best known lower bound by $p\log_2\frac{1}{p}$ if distortion $D$ is small. This has some interesting implications on sparse signals where $p$ is small since the known gap between the lower and up…
We investigate the optimal performance of dense sensor networks by studying the joint source-channel coding problem. The overall goal of the sensor network is to take measurements from an underlying random process, code and transmit those measurement samples to a collector node in a cooperative multiple access channel with potential feedback, and reconstruct the entire random process at the collector node. We provide lower and upper bounds for the minimum achievable expected distortion when the underlying random process is Gaussian. When the Gaussian random process satisfies some general condi…
Recently, Tian et al. [1] considered joint source-channel coding of transmitting a Gaussian source over $K$-user Gaussian broadcast channel, and derived an outer bound on the admissible distortion region. In [1], they stated "due to its nonlinear form, it appears difficult to determine whether it is always looser than the trivial outer bound in all distortion regimes with bandwidth compression". However, in this correspondence we solve this problem and prove that for the bandwidth expansion case ($K\geq2$), this outer bound is strictly tighter than the trivial outer bound with each user being …
Minimum power required to achieve a distortion-noise profile, i.e., a function indicating the maximum allowed distortion value for each noise level, is studied for the transmission of Gaussian sources over Gaussian channels under a regime of bandwidth approaching zero. A simple but instrumental lower bound to the minimum required power for a given profile is presented. For an upper bound, a dirty-paper based coding scheme is proposed and its power-distortion tradeoff is analyzed. Finally, upper and lower bounds to the minimum power is analyzed and compared for specific distortion-noise profile…
In this paper, we consider a distributed remote source coding problem, where a sequence of observations of source vectors is available at the encoder. The problem is to specify the optimal rate for encoding the observations subject to a covariance matrix distortion constraint and in the presence of side information at the decoder. For this problem, we derive lower and upper bounds on the rate-distortion function (RDF) for the Gaussian case, which in general do not coincide. We then provide some cases, where the RDF can be derived exactly. We also show that previous results on specific instance…
This paper investigates the joint source-channel coding problem of sending two correlated memoryless sources with common part over a memoryless multiple access channel (MAC). An inner bound and two outer bounds on the achievable distortion region are derived. In particular, they respectively recover the existing bounds for several special cases, such as communication without common part, lossless communication, and noiseless communication. When specialized to quadratic Gaussian communication case, transmitting Gaussian sources with Gaussian common part over Gaussian MAC, the inner bound and ou…
We consider rate-distortion with two decoders, each with distinct side information. This problem is well understood when the side information at the decoders satisfies a certain degradedness condition. We consider cases in which this degradedness condition is violated but the source and the side information consist of jointly Gaussian vectors. We provide a hierarchy of four lower bounds on the optimal rate. These bounds are then used to determine the optimal rate for several classes of instances.
A Synonymous Variational Perspective on the Rate-Distortion-Perception Tradeoff
Key Laboratory of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications（信息与通信技术联合实验室，教育部，北京邮电大学）；State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications（网络与交换技术国家重点实验室，北京邮电大学）
