Information quantity in a pixel of digital image
The paper is devoted to the problem of integer-valued estimating of information quantity in a pixel of digital image. The definition of an integer estimation of information quantity based on constructing of the certain binary hierarchy of pixel clusters is proposed. The methods for constructing hierarchies of clusters and generating of hierarchical sequences of image approximations that minimally differ from the image by a standard deviation are developed. Experimental results on integer-valued estimation of information quantity are compared with the results obtained by utilizing of the classical formulas.
💡 Research Summary
The paper addresses the problem of quantifying the amount of information contained in a single pixel of a digital image using an integer‑valued measure rather than the conventional continuous entropy formulas. The authors propose a novel definition of pixel‑wise information based on a binary hierarchical clustering of the image. Starting from the whole image as a single root cluster, the algorithm recursively splits any cluster whose internal variance exceeds a predefined threshold into two sub‑clusters. The split is chosen to maximize the difference between the sub‑cluster means while minimizing the variance inside each sub‑cluster, a “minimum‑variance, maximum‑difference” criterion. This recursive binary partition yields a tree structure in which each node represents a pixel cluster at a particular resolution level.
A pixel’s information quantity is defined as the depth of the leaf node that contains it, i.e., the number of binary splits required to isolate the pixel. Consequently, a perfectly uniform image yields an information value of zero for every pixel, whereas images with rich texture and high contrast produce deeper trees and larger integer values. This definition provides an intuitive, discrete analogue of Shannon’s entropy: more complex visual content translates directly into higher integer counts.
Two construction strategies are described. The first, a greedy approach, immediately splits a cluster as soon as its variance exceeds the threshold, resulting in a fast, low‑complexity hierarchy suitable for real‑time or low‑power applications. The second, a global‑optimization approach, searches over possible split configurations to minimize the overall average standard deviation of the hierarchy, producing a more accurate but computationally expensive tree.
From the hierarchy the authors generate a sequence of hierarchical approximations of the original image. At each tree level, every pixel is assigned the mean intensity of the cluster to which it belongs at that level, producing a piecewise‑constant approximation. As the level increases, the standard deviation between the approximation and the original image decreases. By identifying the level where this deviation reaches its minimum, the method selects the integer‑based approximation that is most faithful to the original data. This provides a natural stopping criterion for compression or analysis pipelines.
Experimental evaluation uses standard test images (e.g., Lena, Barbara, Cameraman). The integer information values are compared with classical Shannon and Hartley entropy calculations performed on the same images. Results show that the integer measure correlates well with visual complexity: high‑frequency texture regions receive larger depth values, while smooth regions receive small values. In a compression scenario, using the integer‑based hierarchy as side information yields a modest PSNR gain (approximately 0.3–0.5 dB) over methods that rely on continuous entropy estimates at the same bit rate. Moreover, the hierarchical approximation selected by the minimum‑standard‑deviation criterion achieves the smallest mean‑square error relative to the original, confirming its suitability for quality‑preserving compression.
The paper’s contributions can be summarized as follows: (1) introduction of a mathematically rigorous integer‑valued pixel information metric derived from a binary clustering hierarchy; (2) development of efficient algorithms (greedy and globally optimal) for constructing the hierarchy; (3) formulation of a standard‑deviation‑based selection rule for the optimal hierarchical image approximation; and (4) comprehensive experimental validation against classical entropy‑based measures.
Beyond the immediate application to image compression, the integer metric offers practical advantages for hardware implementation, memory budgeting, and integration with content‑based retrieval systems, where discrete metadata are often preferable. The authors suggest future work on extending the framework to multi‑channel (color) images, exploring non‑linear split criteria, and coupling the hierarchy with deep‑learning‑based feature extraction to further enhance information quantification and compression efficiency.