TY - JOUR

T1 - A stochastic model for block segmentation of images based on the quadtree and the bayes code for it†

AU - Nakahara, Yuta

AU - Matsushima, Toshiyasu

N1 - Funding Information:
Funding: This work was supported in part by JSPS KAKENHI Grant Numbers JP17K06446 and JP19K04914.
Publisher Copyright:
© 2021 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2021/8

Y1 - 2021/8

N2 - In information theory, lossless compression of general data is based on an explicit assumption of a stochastic generative model on target data. However, in lossless image compression, researchers have mainly focused on the coding procedure that outputs the coded sequence from the input image, and the assumption of the stochastic generative model is implicit. In these studies, there is a difficulty in discussing the difference between the expected code length and the entropy of the stochastic generative model. We solve this difficulty for a class of images, in which they have non-stationarity among segments. In this paper, we propose a novel stochastic generative model of images by redefining the implicit stochastic generative model in a previous coding procedure. Our model is based on the quadtree so that it effectively represents the variable block size segmentation of images. Then, we construct the Bayes code optimal for the proposed stochastic generative model. It requires the summation of all possible quadtrees weighted by their posterior. In general, its computational cost increases exponentially for the image size. However, we introduce an efficient algorithm to calculate it in the polynomial order of the image size without loss of optimality. As a result, the derived algorithm has a better average coding rate than that of JBIG.

AB - In information theory, lossless compression of general data is based on an explicit assumption of a stochastic generative model on target data. However, in lossless image compression, researchers have mainly focused on the coding procedure that outputs the coded sequence from the input image, and the assumption of the stochastic generative model is implicit. In these studies, there is a difficulty in discussing the difference between the expected code length and the entropy of the stochastic generative model. We solve this difficulty for a class of images, in which they have non-stationarity among segments. In this paper, we propose a novel stochastic generative model of images by redefining the implicit stochastic generative model in a previous coding procedure. Our model is based on the quadtree so that it effectively represents the variable block size segmentation of images. Then, we construct the Bayes code optimal for the proposed stochastic generative model. It requires the summation of all possible quadtrees weighted by their posterior. In general, its computational cost increases exponentially for the image size. However, we introduce an efficient algorithm to calculate it in the polynomial order of the image size without loss of optimality. As a result, the derived algorithm has a better average coding rate than that of JBIG.

KW - Bayes code

KW - Lossless image compression

KW - Quadtree

KW - Stochastic generative model

UR - http://www.scopus.com/inward/record.url?scp=85111675377&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85111675377&partnerID=8YFLogxK

U2 - 10.3390/e23080991

DO - 10.3390/e23080991

M3 - Article

AN - SCOPUS:85111675377

VL - 23

JO - Entropy

JF - Entropy

SN - 1099-4300

IS - 8

M1 - 991

ER -