## 1 Introduction
The perceptual constraint slightly increases the bitrate required in theory, suggesting an increase in traffic in communication networks. **(b)** Information rate–distortion–perception frontier for a Gaussian source with MSE distortion and 2-Wasserstein perceptual constraint. **(c)** Rate–distortion performance evaluation of various practical image compression algorithms using the PSNR distortion metric. The evaluation is conducted with the CompressAI package [14], covering a range of methods from the standard JPEG to SoTA learning-based compression techniques such as [15]. cally combines both distortion-oriented MSE and perception-oriented cross-entropy losses, weighted to balance distortion and perception constraints (see Equation (21)). For downstream tasks such as image synthesis, a key challenge is to balance the faithfulness to user inputs and the realism of the generated outputs [22]. Within the RDP framework, the perceptual quality of reconstruction is assessed by the similarity between the distributions of the input and the reconstruction, with the encoder and decoder potentially aided by generative models. This alignment naturally supports the objectives of generative modeling. Thus, in contemporary intelligent compression and communication systems, the quality of reconstruction should be evaluated based on both the perceptions of end users, whether human or machine, and the average distortions. The perception axis plays a pivotal role in both source coding and communication systems in general. In information theory, the source coding problem aims to minimize the amount of information needed to transmit a message by eliminating redundancy, thereby achieving effective compression. To assess the performance, a certain level of reconstruction error is permissible. For instance, when compressing and subsequently recovering a bit string using a decoder, the Hamming distance can be used to quantify the number of errors, thereby providing a measure of distortion. Conventional RD theory, originally developed by [PERSON], has effectively characterized the minimal rate required to satisfy distortion constraints. For stationary sources, it is well established that deterministic encoders and decoders can achieve asymptotically optimal RD performance [23]. However, the introduction of perceptual constraints suggests that deterministic coding may no longer be optimal [24]. Randomization has been demonstrated to be essential for achieving realism constraints through stochastic coding [25; 26; 27]. The requirement for the output to conform to a specified distribution is known as the realism constraint [28; 29; 30; 31]. The zero-distortion constraint corresponds to the well-known zero-error source coding problem [23], which also implies perfect realism. However, perfect realism, which requires the distributions of the source and reconstruction to be identical, does not necessarily imply zero distortion. Nevertheless, perceptual constraints restrict the reconstruction process, thereby increasing the required rate. For example, consider a pangram--a sentence using each letter of the English alphabet with balanced letter counts. If a decoder is required to reconstruct a sentence while adhering to a specific letter distribution, this scenario exemplifies the perfect realism condition. A \"perfect pangram\" is extremely rare due to the overuse of common letters and the scarcity of rare ones; for instance, the classic pangram \"The quick brown fox jumps over a lazy dog\" contains repeated Es and Os. This suggests that perceptual constraints are closely related to semantics and can be highly restrictive, thereby carrying significant amounts of information. In practical communication systems, the idealized assumptions underlying the separation theorem, upon which modern communication systems are founded, are rarely met.