Figure 1 - available via license: Creative Commons Attribution 4.0 International

Content may be subject to copyright.

Source publication

A ﬂexible representation of quantum images (FRQI) was proposed to facilitate the extension of classical (non-quantum)-like image processing applications to the quantum computing domain. The representation encodes a quantum image in the form of a normalized state, which captures information about colours and their corresponding positions in the imag...

## Contexts in source publication

**Context 1**

... commonly used quantum gates, such as NOT, Hadamard and CNOTgates in Figure 1, are often employed to break down the complex transform in the quantum circuit model into simpler ones. The gate that applies on k qubits is generally denoted as a 2 k × 2 k unitary matrix, in addition, the number of qubits in the input of the gate has to be equal to the output end. The mission of the final step in quantum simulation is to convert the quantum information into the classical form, which is realized by analyzing the probability distributions of the readouts from the quantum measurement. To distinguish the probabilistic classical bit from a qubit, a double-line wire is adopted, as depicted in Figure 2. The probability p of a measurement result r occurring when state ψ is measured is ψ | M r † M r | ψ . The state of the system after measurement, | ψ , ...

**Context 2**

... commonly used quantum gates, such as NOT, Hadamard and CNOTgates in Figure 1, are often employed to break down the complex transform in the quantum circuit model into simpler ones. The gate that applies on k qubits is generally denoted as a 2 k × 2 k unitary matrix, in addition, the number of qubits in the input of the gate has to be equal to the output end. ...

**Context 3**

... quantum image |I(n) X mc has all of its colors coming from the original image |I(n) mc by shifting the θ angle on the R, G, B or α channel. The quantum circuits of U X (U R , U G , U B and U α ) are C 2 R y (2θ) gates and are shown in Figure 9, and the C 2 R y (2θ) can be constructed from elementary gates (controlled rotation and CNOT gates), as shown in Figure 10. . C 2 R y (2θ) can be constructed from basic gates, CR y (θ), CR y (−θ) and CNOT gates. ...

**Context 4**

... quantum image |I(n) Y mc is obtained from the original image |I(n) mc by applying the CS Y operator, and specifically, quantum circuits of U Y (U RG , U RB and U GB ) are shown in Figure 11. From Figure 12, at most, three quantum basic gates are utilized to build the CS operator. ...

**Context 5**

... quantum image |I(n) Y mc is obtained from the original image |I(n) mc by applying the CS Y operator, and specifically, quantum circuits of U Y (U RG , U RB and U GB ) are shown in Figure 11. From Figure 12, at most, three quantum basic gates are utilized to build the CS operator. The general quantum circuit of U Y operations: (a) U RG ; (b) U RB ; and (c) U GB . ...

**Context 6**

... that images A and B are two same-sized MCQI quantum images with four components (R, G, B and α), where A is the image to be blended and B is the background image. In order to encode the two images in the MCQI quantum states, one ancilla qubit is used to accompany with the MCQI qubits (three color qubits and 2n position qubits for 2 n × 2 n image), as shown in Figure 13, where C i A and C i B are the color states of images A and B, respectively, which are defined as: ...

**Context 8**

... quantum circuit of U AB is shown in Figure 11. Figure 13. Input images (A and B) and the circuit structure to realize α blending. ...

**Context 9**

... seen in Figure 14, the size of a strip in the representation captures the input state of the strip comprising 2 m quantum images. Each image in the strip is an FRQI state, while the combination of such states in the strip is best represented as a multiple FRQI or simply the mFRQI state. ...

**Context 10**

... representation of strip, which was introduced in Section 4.1, facilitates smooth comparison of multiple quantum images of equal size. The scheme to compare quantum images in parallel consists of three steps, as detailed in Figure 15. These steps are outlined in this subsection. ...

**Context 11**

... proposal of the strip comprising 2 m images as defined in Definition 1 provides us a crucial condition to make it possible, because the operation on the strip wires can transform the information in every image simultaneously. The generalized circuit structure of comparing 2 m − 1 pairs of quantum images in parallel is presented in Figure 16. By applying a Hadamard operation on the r − th strip wire in the circuit, s r , the mathematical expressions between the two images being compared are realized. ...

**Context 12**

... this subsection, the more complicated cases of quantum image comparison are discussed, such as comparing arbitrary pairs of images, comparing sub-blocks from two images in a strip. The circuit structure for realizing such processes is presented in Figure 17. ...

**Context 13**

... operation G I is needed when the sub-blocks being compared are at different positions from the two images. The state in the circuit after applying the Hadamard gate on the r − th strip wire is transformed into H r G s |S G I , as shown in Figure 17. The similarity between the sub-blocks from two FRQI quantum images encoded in a strip is: ...

**Context 14**

... purpose of this experiment is to realize the comparison between two sub-blocks from two arbitrary images in a strip. As shown in Figure 18 The corresponding circuit structure to compare them is presented in Figure 19. There are four steps to accomplish this comparison: ...

**Context 15**

... purpose of this experiment is to realize the comparison between two sub-blocks from two arbitrary images in a strip. As shown in Figure 18 The corresponding circuit structure to compare them is presented in Figure 19. There are four steps to accomplish this comparison: ...

**Context 16**

... seen in Figure 21, the size of a Z-strip in the representation captures the input state comprising 2 m+1 quantum images. The Z-axis differentiates the strip that is located on the left and the right position. ...

**Context 17**

... the quantum CD, quantum player and movie reader, a framework as shown in Figure 31 is presented in order to represent and produce movies on quantum computers. The movie enhancement of the movie reader is considered for the purpose of enhancing the content of each frame before final display to the audience. ...

**Context 19**

... when restricted within the confines of today's technologies and the requirements of the R transform, as discussed in [21], only a 2 × 2-pixel image, i.e., the case where n = 1, can be realized. The sub-circuit to execute the R transform for such a small-sized image (as shown in Figure 10 in Section 3.2) requires two controlled-rotation and two controlled-NOT gates. As specified by the PPT theorem, preparing a 2 × 2 FRQI quantum image requires a total of 40 simple quantum operations [21,26]. ...

## Citations

... The additional control-conditions on the mode bit are necessary in order to control Hadamard operation [24]. The qubit q 8 -q 14 representing the three-dimensional position in the enlarged image, they are subjected to Hadamard gates to form all the position information in the 4 × 4 × 8 three-dimensional space, namely (00, 00, 000), (00, 00, 001 ),..., (11,11,11111). ...

The digital image representation model directly affects the performance of digital image processing based on the representation model. Most quantum image representations which are based on traditional digital images, require a large number of qubits, or the complicated and difficult operations of quantum image processing. In this paper, we record the value and position of each bit in the binary sequence to form a multimode quantum image representation (MQIR) based on three-dimensional coordinates, which effectively reduces the number of qubits required to store the image. However, the basic operations for traditional digital images (such as scale operations) cannot be applied to MQIR image. To solve this problem, this paper proposes the nearest-neighbor interpolation scaling schemes for MQIR image, then we design and implement the quantum image scaling circuits. Finally, the complexity analysis and experimental simulation results of the scaling circuits for MQIR image are given. The simulation results verify the correctness of the image scaling schemes and circuits for MQIR image.

... One of the main advantages of this method is the low number of qubits required to encode the image: a square image with 2 n × 2 n can be encoded with just 2n + 1 qubits. Moreover, as the intensity data is encoded in a single qubit, pixel transformations (known as CTQI) can be implemented by applying a single quantum gate [4]. ...

The application of quantum computing to the field of image processing has produced several promising applications: quantum image representation techniques have been developed showing how, by taking advantage of quantum properties like entanglement and superposition, many image processing algorithms could have an exponential speed-up in comparison to their "classical" counterparts. In this paper, after briefly discussing some of the main quantum image representation methods, we propose an improved version of a quantum edge detection algorithm.

... After establishing a QIR, many researches in different areas of image processing had been established based on these representations. This includes geometric transformation [12,14], image scaling [8,9,20,31,32], image translation [23,33], image segmentation [2,15], feature extraction [30] and image encryption [4,25]. ...

Mid-point filter is an order statistic filter which cannot be realized in the frequency domain. It is used for de-noising Gaussian noise effectively. In this paper, a new method for quantum realization mid-point filter in the spatial domain is proposed. An enhanced method for preparing multiple copies of the same image is also proposed. The modular design of the quantum circuit was utilized with an articulation on reducing the number of ancillary qubits. In this work, we present the quantum circuit for the three basic modules (cyclic shift, swap and division by two) and four composite modules (full adder, comparator, sort and maximum–minimum extraction). Also, the enhanced quantum preparation of multiple copies of an image is introduced. Moreover, the design of maximum–minimum extraction is modified to adapt our quantum circuit design. Finally, the complete quantum circuit which implements the mid-point filtering task is constructed and the results of several simulation experiments with different noise patterns are presented on some grayscale images. Apparently, the proposed approach has identical noise suppression of the classical version; however, there is a clear reduction in the complexity from exponential function of image size \(O(2^{2n})\) to the second-order polynomial \(O(n^{2} + q)\).

... In this section we give a brief review of the basic theories on state vector and the quantum representation of images [12][13][14]. ...

The dilation and erosion operations are the first fundamental step in classical image processing. They are important in many image processing algorithms to extract basic image features, such as geometric shapes; such shapes are then fed to higher level algorithms for object identification and recognition. In this paper, we present an improved quantum method to realize dilation and erosion in imaging processing. Unlike in the classical way, in the quantum version of imaging processing, all of the information is stored in quantum bits (qubits). We use qubits to code the location and other information of each pixel of the images and apply quantum operators (or quantum gates) to accomplish specific functions. Because of quantum entanglement and other nonintuitive features in quantum mechanics, qubits have many advantages over classical bits, but their nature presents challenges in designing quantum algorithms. We first built the quantum circuit theoretically, and then ran it on the IBM Quantum Experience platform to test and process real images. With this algorithm, we are looking forward to more potential applications in quantum computation.

... In the case of global optimisation methods for biomedical image registration, a set of fuzzy rules may be exploited to dynamically adapt the settings for each particle of the PSO, so resulting in proactive optimising agents [122], achieving encouraging performance on benchmark functions [123] as well as in the parameter estimation of biochemical systems [124]. Concerning other unconventional computation models, Quantum Computing-which studies the information processing tasks executed on quantum mechanical systems [125]-might be applied to basic and advanced medical image processing operations, by devising effective techniques of internal representation of the images involved in a quantum process [126]. ...

Natural phenomena and mechanisms have always intrigued humans, inspiring the design of effective solutions for real-world problems. Indeed, fascinating processes occur in nature, giving rise to an ever-increasing scientific interest. In everyday life, the amount of heterogeneous biomedical data is increasing more and more thanks to the advances in image acquisition modalities and high-throughput technologies. The automated analysis of these large-scale datasets creates new compelling challenges for data-driven and model-based computational methods. The application of intelligent algorithms, which mimic natural phenomena, is emerging as an effective paradigm for tackling complex problems, by considering the unique challenges and opportunities pertaining to biomedical images. Therefore, the principal contribution of computer science research in life sciences concerns the proper combination of diverse and heterogeneous datasets-i.e., medical imaging modalities (considering also radiomics approaches), Electronic Health Record engines, multi-omics studies, and real-time monitoring-to provide a comprehensive clinical knowledge. In this paper, the state-of-the-art of nature-inspired medical image analysis methods is surveyed, aiming at establishing a common platform for beneficial exchanges among computer scientists and clinicians. In particular, this review focuses on the main natureinspired computational techniques applied to medical image analysis tasks, namely: physical processes, bio-inspired mathematical models, Evolutionary Computation, Swarm Intelligence, and neural computation. These frameworks, tightly coupled with Clinical Decision Support Systems, can be suitably applied to every phase of the clinical workflow. We show that the proper combination of quantitative imaging and healthcare informatics enables an in-depth understanding of molecular processes that can guide towards personalised patient care.

... On the other side, in spite of the fact that quantum images can be easily prepared in physical laboratory, but how to specify the information of a quantum images in a quantum computer [6,7] remains a problem. Recently, some papers have discussed relevant topics, such as [8,9,16,17], where the authors propose or summarize different mathematic forms of the state representation of quantum images. Typically, for example, flexible representation of quantum images (FRQI) [8,9] which uses a single qubit to encode the grey level, novel enhanced quantum representation(NEQR) [10] which improves the expression with two qubits,binary key image generation [11], and flexible quantum representation for grey-level quantum images (FQRGI) [12] et al. ...

... Recently, some papers have discussed relevant topics, such as [8,9,16,17], where the authors propose or summarize different mathematic forms of the state representation of quantum images. Typically, for example, flexible representation of quantum images (FRQI) [8,9] which uses a single qubit to encode the grey level, novel enhanced quantum representation(NEQR) [10] which improves the expression with two qubits,binary key image generation [11], and flexible quantum representation for grey-level quantum images (FQRGI) [12] et al. Based on these types of state representation of quantum image, different operations to the image are explored, such as [13,14,15,18,19,20,21]. ...

... The state preparation of quantum images as Eq. (23) has already been discussed in [8], ...

A topic about synthesis of quantum images is proposed, and a specific phase rotation transform constructed is adopted to theoretically realise the synthesis of two quantum images. The synthesis strategy of quantum images comprises three steps, which include: (1) In the stage of phase extraction, we obtain the phases of the state of the quantum image by transforming the state of the quantum image to prepare the conditions for multiple phases extraction. (2) In the stage of rotation operator construction, the phases obtained in the first stage are used to construct the rotation operator where a mechanism is introduced into it to reduce the phase overflow. (3) In the stage of application of the rotation operator, we apply the operator constructed in the second stage on the state of quantum image to get a goal state. Additionally, numerical analysis gives the joint uncertainty relation of the pixel of the synthesized quantum image. The analysis result about the compression ratio indicates that the phase rotation transform and the overflow control mechanism are effective.

... Quantum computation has shown great potential for improving information processing speed and enhancing communication security [1][2][3]. The quantum image encryption technology exploits quantum mechanics principles, such as parallel and entanglement, to further protect the security of information transmission and decrease computational resource [4][5][6][7]. ...

... The first step of quantum image processing is to design a suitable representation model, which can be run on quantum computers for compiling digital image. Nowadays, several efficient representation models are proposed [3]. The FRQI representation model [42] is widely used, as it is similar with pixel representation in classical computer and accord with human perception of vision. ...

Quantum image encryption offers major advantages over its classical counterpart in terms of key space, computational complexity, and so on. A novel double quantum image encryption approach based on quantum Arnold transform (QAT) and qubit random rotation is proposed in this paper, in which QAT is used to scramble pixel positions and the gray information is changed by utilizing random qubit rotation. Actually, the independent random qubit rotation operates once, respectively, in spatial and frequency domains with the help of quantum Fourier transform (QFT). The encryption process accomplishes pixel confusion and diffusion, and finally the noise-like cipher image is obtained. Numerical simulation and theoretical analysis verify that the method is valid and it shows superior performance in security and computational complexity.

... Quantum image processing (QIMP), an emerging sub-discipline of QC & QI, is a field devoted to capturing, manipulating, and recovering visual information using quantum mechanical systems. QIMP was born with the publication of [32,33,34,40] and, since then, it has amassed a spurt of interest from researchers with diverse backgrounds who have advanced this field by proposing quantum algorithms for image storage and retrieval [49,46], image encryption/decryption [41], image segmentation [42,39], image watermarking [50] and image filtering [43], as well as quantum operations to manipulate quantum images like geometric transformations [36], image comparison [51] and image translation [48], among many other contributions. The advancement of QIMP brought the development of more intuitive and flexible representations of quantum images, among them the Flexible Representation of Quantum Images (FRQI) [35] as well as a FRQI-based, novel enhanced quantum representation (NEQR) [45] in which grayscale pixel values are stored on a sequence of qubits. ...

In this paper, a novel method of quantum image rotation (QIR) based on shear transformations on NEQR quantum images is proposed. To compute the horizontal and vertical shear mappings required for rotation, we have designed quantum self-adder, quantum control multiplier, and quantum interpolation circuits as the basic computing units in the QIR implementation. Furthermore, we provide several examples of our results by presenting computer simulation experiments of QIR under $30^\circ$, $45^\circ$, and $60^\circ$ rotation scenarios and have a discussion onto the anti-aliasing and computational complexity of the proposed QIR method.

... Overall, the objective of QIP is to utilise quantum computing technologies to capture, manipulate and recover quantum images in different formats and for different purposes. Detailed reviews on QIP can be found in [17][18][19]. ...

... Inspired by the utility of image search and retrieval on digital computers, the studies in [15,19,23] explored the possibility of undertaking similar tasks on quantum computers. ...

... On its left (i.e., in Figure 3B) is a non-destructive (or ancilla-driven) measurement as used in MBQC, and in this particular case the ancilla-driven quantum computation (ADQC) model. The ADQC [19] is a newer type of MBQC, which was utilised for the movie reader in [9], exploits the properties of single-qubit projective measurements and the entanglement-based interaction between the ancilla qubit and the register qubit (i.e., strip state encoding the 2 m -ending FRQI images). The measurements are performed subject to satisfying some predefined conditions [24]. ...

An enhanced quantum-based image fidelity metric, the QIFM metric, is proposed as a tool to assess the “congruity” between two or more quantum images. The often confounding contrariety that distinguishes between classical and quantum information processing makes the widely accepted peak-signal-to-noise-ratio (PSNR) ill-suited for use in the quantum computing framework, whereas the prohibitive cost of the probability-based similarity score makes it imprudent for use as an effective image quality metric. Unlike the aforementioned image quality measures, the proposed QIFM metric is calibrated as a pixel difference-based image quality measure that is sensitive to the intricacies inherent to quantum image processing (QIP). As proposed, the QIFM is configured with in-built non-destructive measurement units that preserve the coherence necessary for quantum computation. This design moderates the cost of executing the QIFM in order to estimate congruity between two or more quantum images. A statistical analysis also shows that our proposed QIFM metric has a better correlation with digital expectation of likeness between images than other available quantum image quality measures. Therefore, the QIFM offers a competent substitute for the PSNR as an image quality measure in the quantum computing framework thereby providing a tool to effectively assess fidelity between images in quantum watermarking, quantum movie aggregation and other applications in QIP.

... Recently, many algorithms have been proposed on the basis of quantum information processing such as image processing and transformations, quantum optimization and etc. Many quantum image steganography have been planned for the need of storing image information in quantum states [2], i.e., Qu-bit Lattice, and Flexible Representation of Quantum Images (FRQI) [3]. Information concealing embeds the additional secret information into media such as a condition in which the carrier undergoes little changed. ...

... 2) According to the definition of Quantum Fourier Transform [3]. We can rewrite Eq. (14) as the following quantum state: ...

Quantum steganography can solve some problems that are considered inefficient in image information concealing. It researches on Quantum image information concealing to have been widely exploited in recent years. Quantum image information concealing can be categorized into quantum image digital blocking, quantum image stereography, anonymity and other branches. Least significant bit (LSB) information concealing plays vital roles in the classical world because many image information concealing algorithms are designed based on it. Firstly, based on the novel enhanced quantum representation (NEQR), image uniform blocks clustering around the concrete the least significant Qu-block (LSQB) information concealing algorithm for quantum image steganography is presented. Secondly, a clustering algorithm is proposed to optimize the concealment of important data. Finally, we used Con-Steg algorithm to conceal the clustered image blocks. Information concealing located on the Fourier domain of an image can achieve the security of image information, thus we further discuss the Fourier domain LSQu-block information concealing algorithm for quantum image based on Quantum Fourier Transforms. In our algorithms, the corresponding unitary Transformations are designed to realize the aim of concealing the secret information to the least significant Qu-block representing color of the quantum cover image. Finally, the procedures of extracting the secret information are illustrated. Quantum image LSQu-block image information concealing algorithm can be applied in many fields according to different needs.