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CRC Press eBooks

Geometric Operations of Quantum Images

Zhixin Zhang, Xianhua Song, Ahmed A. Abd El‐Latif, Bassem Abd-El-Atty, Basma Abd El-Rahiem

Published May 15, 2026• 0 citations
Publisher

Abstract

To achieve a certain desired visual effect, the pixel positions of the input image are mapped to positions different from the original ones so as to change the display effect of the original image. This process is called the geometric transformation of the image. Geometric transformations mainly include geometric correction of the image and spatial transformations (scaling transformation, rotation, and affine transformation) [ 1 – 26 ] . This chapter proposes the geometric transformations of the QIRHSI quantum color image, including basic geometric transformations and scaling operations, and further studies the unitary operators of geometric transformations and the corresponding quantum circuits.

DOI
10.1201/9781003621423-5

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•Jan 1, 2010•107 citations
To achieve a certain desired visual effect, the pixel positions of the input image are mapped to positions different from the original ones so as to change the display effect of the original image. This process is called the geometric transformation of the image. Geometric transformations mainly include geometric correction of the image and spatial transformations (scaling transformation, rotation, and affine transformation) [ 1 – 26 ] . This chapter proposes the geometric transformations of the QIRHSI quantum color image, including basic geometric transformations and scaling operations, and further studies the unitary operators of geometric transformations and the corresponding quantum circuits.
The negative effects caused by geometric distortion can be removed to the maximum extent possible through appropriate geometric transformations, allowing us to focus on the image content itself in subsequent processing and recognition. Therefore, geometric transformations are often used as a pre-processing step for other image processing applications. In this paper, quantum algorithms are designed to implement geometric transformations, including two-point swapping, circular translation, flipping transformations and right-angle rotation, across a quantum image representation QIRHSI (Quantum Image Representation based on HSI color space) which is built on HSI (Hue-Saturation-Intensity) color space. The above geometric transformations are realized by quantum circuits composed of elementary quantum gates. By analyzing the complexity of the fundamental quantum gates needed for the above geometric transformations, it is found that the global transformations (circular translation, flip transformation and right-angle rotation) are lower than the local transformation (two-point swapping). The proposed geometric transforms are used to facilitate the applications of quantum images with low complexity and high efficiency.
Arithmetic operations on images are important primary transformations that can be applied to many complicated image operations. We mainly study the intensity-based arithmetic operations on the QIRHSI quantum image representation model to facilitate the development of quantum image processing. The quantum circuits and unitary matrices for four arithmetic operations including reverse, complement, addition, and subtraction are proposed. Finally, the complexity analysis of quantum networks is illustrated to show the effectiveness of the proposed arithmetic operations.
From the quantum images and their preparation methods described in Chapter 2, it can be seen that preparing a quantum image requires a large number of basic quantum gates. For example, to prepare a quantum color image Quantum Image Representation based on HSI (QIRHSI) with a size of https://www.w3.org/1998/Math/MathML" display="inline"> 2 8 × 2 8 pixels and an intensity value range of https://www.w3.org/1998/Math/MathML" display="inline"> 0 ∼ 2 8 − 1 , approximately the number of basic quantum gates required is 2 64 . Therefore, in this chapter, the resources for image preparation will be optimized. The principle of Boolean expression minimization will be used to simplify the quantum circuit in the quantum image QIRHSI. In essence, it is to replace the original preparation of the quantum image with a new unitary transformation that has fewer quantum logic gates.
Circuits to achieve geometric trans- formations including two-point swapping, flip, co- ordinate swapping, orthogonal rotations and their variants on N -sized quantum images are proposed based on the basic quantum gates; NOT, CNOT and Toffoli gates. The complexity of the circuits is O(log 2 N) for two-point swapping and O(logN) for flip, co-ordinate swapping and orthogonal rotations. The results indicate that local operations like two- point swapping are slower than global operations like flip, co-ordinate swapping, and orthogonal rotations in quantum image processing. This is in contrast to performing such operations in classical image process- ing where the local operations are faster. All the proposed transformations are confirmed by simula- tion on a classical computer. With their low com- plexity, these geometric transformations can be used as the major components to build circuits for other applications on quantum images.