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arXiv:2507.06642

Quantum algorithm for edge detection in digital grayscale images

Mohit Rohida, Alok Shukla, Prakash Vedula

Published Jul 9, 2025• 0 citations
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Abstract

In this work, we propose a novel quantum algorithm for edge detection in digital grayscale images, based on the sequency-ordered Walsh-Hadamard transform. The proposed method significantly improves upon existing quantum techniques for edge detection by using a quantum algorithm for the sequency-ordered Walsh-Hadamard transform, achieving a circuit depth of $\mathcal{O}(n)$ (where $n$ is the number of qubits). This represents a notable enhancement over the Quantum Fourier Transform (QFT), which has a circuit depth of $\mathcal{O}(n^{2})$. Furthermore, our approach for edge detection has a computational cost (both gate complexity and quantum circuit depth) of $\mathcal{O}(\log_{2}(N_{1}N_{2}))$ for an image of size $N_{1}\times N_{2}$, offering a considerable improvement over the Quantum Hadamard Edge Detection (QHED) algorithm, which incurs a cost of $\mathcal{O}(\text{poly}(\log_{2}(N_{1}N_{2})))$. By integrating a quantum high-pass filter with the sequency-ordered Walsh-Hadamard transform, the algorithm effectively extracts edge information from images. Computational examples are provided to demonstrate the efficacy of the proposed algorithm which provides a better performance in comparison to QHED.

DOI
10.48550/arxiv.2507.06642

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In this work, we propose a novel quantum algorithm for edge detection in digital grayscale images, based on the sequency-ordered Walsh-Hadamard transform. The proposed method significantly improves upon existing quantum techniques for edge detection by using a quantum algorithm for the sequency-ordered Walsh-Hadamard transform, achieving a circuit depth of O(n)\mathcal{O}(n)O(n) (where nnn is the number of qubits). This represents a notable enhancement over the Quantum Fourier Transform (QFT), which has a circuit depth of O(n2)\mathcal{O}(n^{2})O(n2). Furthermore, our approach for edge detection has a computational cost (both gate complexity and quantum circuit depth) of O(log⁡2(N1N2))\mathcal{O}(\log_{2}(N_{1}N_{2}))O(log2​(N1​N2​)) for an image of size N1×N2N_{1}\times N_{2}N1​×N2​, offering a considerable improvement over the Quantum Hadamard Edge Detection (QHED) algorithm, which incurs a cost of O(poly(log⁡2(N1N2)))\mathcal{O}(\text{poly}(\log_{2}(N_{1}N_{2})))O(poly(log2​(N1​N2​))). By integrating a quantum high-pass filter with the sequency-ordered Walsh-Hadamard transform, the algorithm effectively extracts edge information from images. Computational examples are provided to demonstrate the efficacy of the proposed algorithm which provides a better performance in comparison to QHED.
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Quantum edge detection is one of the important part of quantum image processing. In this paper, a quantum edge detection algorithm is designed for the quantum representation of multi-wavelength image (QRMW) model. The algorithm includes all stages of filtering, enhancement and detection. The proposed algorithm is also designed to apply any filtering operation to QRMW images, not only for a particular filtering operation. The proposed algorithm aims to solve the problems that quantum edge detection algorithms in the literature have processing only for a particular operator and noise reduction. Moreover, the algorithm aims to perform operations more efficiently by using less resources. Low-pass filter (LPF) smoothing operators are applied in the filtering stage for the noise reduction problem. In order to apply all filtering operations to the image, arithmetic operators that can operate with all signed integers are used in the algorithm. The operators Sobel, Prewitt and Scharr in the enhancement stage and the gradient method in the detection stage are used for both verification of the proposed algorithm and comparisons with the existing algorithms. A method with quantitative outcomes is shown to evaluate the performance of the edge detection algorithms. Analysis of the simulations performed on sample images with different operators. The circuit complexity of the algorithm is presented and the comparisons are made with the existing studies. The superiority of the proposed algorithm and its flexibility to be used in other studies are clearly demonstrated by analysis.
Image processing and detection are natural extensions of detecting the edges, curves, and areas of sudden changes in brightness or, more specifically, missing pixels. Edge detection is a set of mathematical algorithms. Change detection is a term used to describe the process of identifying discrete changes in single-digit data. The dimension. The ability to detect and remove image edges is fundamental to image processing and computer and machine vision. One way to detect and remove features from images is to use edge detection. In this work, we first provide an overview of research in this area, and then list some of the concerns that have been raised regarding rapid development and its achievement. We see “quantum image classification and recognition” as the ideal application to showcase the potential of quantum technology. Quantum Probabilistic Image Encoding (QPIE:QHED) was used in our study to transform classical data into quantum states. It is necessary to apply quantum algorithms to classical problems, a method for quantum Hadamard edge detection, and also uses them to process quantum image data, often produced by QPIE, using QHED, a modern edge detection technique. Quite simply, QHED is a technology for precise edge detection in images that uses quantum computing.
This work introduces a novel quantum algorithm for gradient-based edge detection that operates entirely within the quantum circuit model. Grayscale images are encoded using the Novel Enhanced Quantum Representation (NEQR), allowing exact arithmetic on pixel intensities. Directional gradients are computed by generating superpositions of neighboring pixels via cyclic shift operations and performing subtraction with an exact quantum arithmetic circuit. To refine accuracy, we introduce a direction-aware shifting mechanism that aligns edges with the darker side of intensity transitions. Our novel Quantum Partitioning Algorithm enables efficient in-place thresholding of edge candidates. This work exhibits polynomial-time improvements and optimizes the ancilla count compared to previous NEQR-based quantum edge detection algorithms. These results demonstrate a resource-efficient and fully quantum approach to edge detection, highlighting a practical quantum advantage in image processing.
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