A Stochastic Model for Block Segmentation of Images Based on the Quadtree and the Bayes Code for It

Abstract

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.

Probability Distribution on Full Rooted Trees

“…Strictly speaking, our distribution has already been proposed independently in source coding [8], image processing [12], and machine learning [13], as mentioned above. The substantial novelty of our study is the extraction of the essence from the previous discussion, which depends on the applicational objects, and its representation as a clear mathematical theory.…”

“…Although Theorem 1 is also proved in [12,13], we extract the essential part of them as Lemma 1. In [10,11], a restricted case of Theorem 1 is proved, in which α v has a common value for all v ∈ I p .…”

“…, N} (v n end and v n end may different each other for n = n ). The remarkable previous studies using Corollary 2 are [8][9][10][11][12][13] (In [10,11], only (66) is used but (67) is not). In other words, they treat only the case under Condition 4.…”

“…Moreover, similar results obtained from rich real data analysis have been reported recently [10,11] (note that the prior form reported in [10,11] is extremely restricted and cannot be updated as a posterior, in contrast to that reported in [8,9]). In image processing, the authors of [12] were the first to regard the quadtree as a stochastic model, and its optimal estimation was derived under the Bayesian criteria. In machine learning, the authors of [13] redefined the decision tree as a stochastic generative model and improved most tree weighting methods (e.g., [5]).…”

“…However, these studies depend on specific data or generative models. This might have been the reason that more than 25 years passed before the first study [8] pertaining to text compression was applied to image processing [12] and machine learning [13].…”