{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T11:12:08Z","timestamp":1762081928403,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,27]],"date-time":"2022-12-27T00:00:00Z","timestamp":1672099200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Research and Development Program of China","award":["2021.YFA1000500(4)"],"award-info":[{"award-number":["2021.YFA1000500(4)"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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\/logt). 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\/logt) 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.<\/jats:p>","DOI":"10.3390\/e25010048","type":"journal-article","created":{"date-parts":[[2022,12,28]],"date-time":"2022-12-28T05:42:48Z","timestamp":1672206168000},"page":"48","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Why Shape Coding? Asymptotic Analysis of the Entropy Rate for Digital Images"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8168-6870","authenticated-orcid":false,"given":"Gangtao","family":"Xin","sequence":"first","affiliation":[{"name":"Department of Electronic Engineering, Tsinghua University, Beijing 100084, China"},{"name":"Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing 100084, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0658-6079","authenticated-orcid":false,"given":"Pingyi","family":"Fan","sequence":"additional","affiliation":[{"name":"Department of Electronic Engineering, Tsinghua University, Beijing 100084, China"},{"name":"Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing 100084, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2519-6401","authenticated-orcid":false,"given":"Khaled B.","family":"Letaief","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, Hong Kong University of Science and Technology (HKUST), Hong Kong"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,27]]},"reference":[{"key":"ref_1","first-page":"4194","article-title":"Learning End-to-End Lossy Image Compression: A Benchmark","volume":"44","author":"Hu","year":"2022","journal-title":"IEEE Trans. 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