Why Shape Coding? Asymptotic Analysis of the Entropy Rate for Digital Images

TL;DR

This paper establishes the theoretical limits of shape-based image compression, proving asymptotic optimality under certain conditions and validating these findings with the MNIST database.

Contribution

It introduces a shape coding framework that achieves the entropy rate for image sources and confirms the shape-pixel ratio condition in practical datasets.

Findings

Shape coding can asymptotically reach the entropy rate for image sources.

The shape-pixel ratio condition $O({1 ig/ {\log t}})$ is validated on MNIST.

Shape coding is near-optimal for lossless image compression.

Abstract

This paper focuses on the ultimate limit theory of image compression. It proves that for an image source, there exists a coding method with shapes that can achieve the entropy rate under a certain condition where the shape-pixel ratio in the encoder/decoder is $O({1 \over {\log t}})$. Based on the new finding, an image coding framework with shapes is proposed and proved to be asymptotically optimal for stationary and ergodic processes. Moreover, the condition $O({1 \over {\log t}})$ of shape-pixel ratio in the encoder/decoder has been confirmed in the image database MNIST, which illustrates the soft compression with shape coding is a near-optimal scheme for lossless compression of images.