The information rate-distortion-perception (RDP) function characterizes the three-way trade-off between description rate, average distortion, and perceptual quality measured by discrepancy between probability distributions and has been applied to emerging areas in communications empowered by generative modeling. We study several variants of the RDP functions through the lens of optimal transport to characterize their critical transitions. By transforming the information RDP function into a Wasserstein Barycenter problem, we identify the critical transitions when one of the constraints becomes inactive. Further, the non-strictly convexity brought by the perceptual constraint can be regularized by an entropy regularization term. We prove that the entropy regularized model converges to the original problem and propose an alternating iteration method based on the Sinkhorn algorithm to numerically solve the regularized optimization problem. In many practical scenarios, the computation of the Distortion-Rate-Perception (DRP) function offers a solution to minimize distortion and perceptual discrepancy under rate constraints. However, the interchange of the rate objective and the distortion constraint significantly amplifies the complexity. The proposed method effectively addresses this complexity, providing an efficient solution for DRP functions. Using our numerical method, we propose a reverse data hiding scheme that imperceptibly embeds a secret message into an image, ensuring perceptual fidelity and achieving a significant improvement in the perceptual quality of the stego image compared to traditional methods under the same embedding rate. Our theoretical results and numerical method lay an attractive foundation for steganographic communications with perceptual quality constraints.