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Bangmod International Journal of Mathematical and Computational Science

QUANTUM RENYI ENTROPY WITH APPLICATION IN IMAGE PROCESSING

Wachirapong Jirakitpuwapat, Poom Kumam, Tanapat Deesuwan, Sompong Dhompongsa, Printaporn Sanguansuttigul

Published Dec 31, 2020• 0 citations
OpenAlex

Abstract

Thresholding is a basic method for image segmentation. First, we plot a histogram of image to a quantum state. The quantum state represents pixel intensities, and the density matrix represents their probability distribution. Secondly, we plot a histogram of the segment image to a quantum state. The quantum state represents pixel intensities of classes. Finally, we obtain the optimal thresholding by quantum Renyi entropy which explains physical interpretation in quantum language.

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Thresholding is a basic method for image segmentation. First, we plot a histogram of image to a quantum state. The quantum state represents pixel intensities, and the density matrix represents their probability distribution. Secondly, we plot a histogram of the segment image to a quantum state. The quantum state represents pixel intensities of classes. Finally, we obtain the optimal thresholding by quantum Renyi entropy which explains physical interpretation in quantum language.
Aiming to implement image segmentation precisely and efficiently, we exploit new ways to encode images and achieve the optimal thresholding on quantum state space. Firstly, the state vector and density matrix are adopted for the representation of pixel intensities and their probability distribution, respectively. Then, the method based on global quantum entropy maximization (GQEM) is proposed, which has an equivalent object function to Otsu's, but gives a more explicit physical interpretation of image thresholding in the language of quantum mechanics. To reduce the time consumption for searching for optimal thresholds, the method of quantum lossy-encoding-based entropy maximization (QLEEM) is presented, in which the eigenvalues of density matrices can give direct clues for thresholding, and then, the process of optimal searching can be avoided. Meanwhile, the QLEEM algorithm achieves two additional effects: (1) the upper bound of the thresholding level can be implicitly determined according to the eigenvalues; and (2) the proposed approaches ensure that the local information in images is retained as much as possible, and simultaneously, the inter-class separability is maximized in the segmented images. Both of them contribute to the structural characteristics of images, which the human visual system is highly adapted to extract. Experimental results show that the proposed methods are able to achieve a competitive quality of thresholding and the fastest computation speed compared with the state-of-the-art methods.
In the rapidly advancing field of image analysis and processing, accurately segmenting images into meaningful regions remains a critical challenge. Drawing from recent advancements in quantum computing and information theory, our research introduces an innovative approach to image segmentation. This work presents a novel multilevel segmentation method that utilizes a two-dimensional quantum image representation, offering a more sophisticated and efficient technique for image thresholding. In this framework, the image's 2D histogram is treated as a quantum system, with quantum Rényi entropy used to quantify the information contained within the image. To enhance segmentation quality, we first improve the contrast of the images by applying a new contrast enhancement algorithm before performing the segmentation. The resulting entropy-based fitness function is then optimized using Differential Evolution (DE) and Particle Swarm Optimization (PSO) algorithms to determine the optimal thresholding values. A comprehensive comparative analysis is conducted between the proposed quantum method and traditional classical approaches, evaluated on a set of benchmark images using nine metrics, including the Wilcoxon test for statistical significance. Experimental results demonstrate the effectiveness of the PSO optimizer, the superiority of the two-dimensional quantum image representation.
We propose a new quantum watermarking scheme based on threshold selection using informational entropy of quantum image. The core idea of this scheme is to embed information into object and background of cover image in different ways. First, a threshold method adopting the quantum informational entropy is employed to determine a threshold value. The threshold value can then be further used for segmenting the cover image to a binary image, which is an authentication key for embedding and extraction information. By a careful analysis of the quantum circuits of the scheme, that is, translating into the basic gate sequences which show the low complexity of the scheme. One of the simulation-based experimental results is entropy difference which measures the similarity of two images by calculating the difference in quantum image informational entropy between watermarked image and cover image. Furthermore, the analyses of peak signal-to-noise ratio, histogram and capacity of the scheme are also provided.
The classical image segmentation algorithm based on local adaptive threshold can effectively segment images with uneven illumination, but with the increase of the image data, the real-time problem gradually emerges. In this paper, a quantum segmentation algorithm based on local adaptive threshold for NEQR image is proposed, which can use quantum mechanism to simultaneously compute local thresholds for all pixels in a gray-scale image and quickly segment the image into a binary image. In addition, several quantum circuit units, including median calculation, quantum binarization, etc. are designed in detail, and then a complete quantum circuit is designed to segment NEQR images by using fewer qubits and quantum gates. For a [Formula: see text] image with [Formula: see text] gray-scale levels, the complexity of our algorithm can be reduced to O[Formula: see text], which is an exponential speedup compared to the classic counterparts. Finally, the experiment is conducted on IBM Q to show the feasibility of our algorithm in the noisy intermediate-scale quantum (NISQ) era.
Image segmentation is a fundamental operation in computer vision, used to partition an image into meaningful regions according to various attributes or spatial relationships among pixels. This paper presents a novel quantum circuit for two-threshold image segmentation implemented in the Novel Enhanced Quantum Representation (NEQR) format. Threshold-based segmentation is particularly attractive for quantum implementation due to its conceptual simplicity and inherent parallelism, but existing approaches suffer from a critical correctness issue: quantum RESET operations are applied to auxiliary qubits that may become entangled with the image data register during the segmentation process. We demonstrate, both theoretically and experimentally, that such RESET operations implicitly perform a measurement that collapses the superposition of the entangled quantum state, leading to non-reproducible segmentation results across repeated executions. To address this limitation, we propose a seven-step quantum segmentation pipeline that explicitly preserves entanglement integrity by allocating dedicated auxiliary qubits for each comparison stage, thereby eliminating unsafe qubit reuse. The proposed circuit encodes pixel intensities using the NEQR representation, applies upper and lower threshold comparators in sequence, and integrates both comparison outcomes to classify pixels into three intensity regions. Experiments conducted on a Qiskit StatevectorSimulator demonstrate that the proposed design produces stable and reproducible segmentation results across all test images, in contrast to the inconsistent behaviour observed in prior work. For images requiring 33 qubits, simulations consumed up to 128 GB of RAM with a total execution time of approximately 27 h on an Intel® Xeon® Gold 6240 CPU @ 2.60 GHz. The proposed approach establishes a robust foundation for quantum image segmentation on future fault-tolerant hardware.