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Quantum image processing (QIMP) is an emerging sub-discipline that is primarily devoted to utilizing quantum computing technologies to capture, manipulate, and recover quantum images in different formats and for different purposes. Due to some of the astounding properties inherent to quantum computation, notably entanglement and parallelism, it is anticipated that QIMP technologies will offer capabilities and performances that are, as yet, unrivaled by their traditional equivalents. These improvements could be in terms of computing speed, guaranteed security, and minimal storage requirements, etc.

Background
Retrospectively, the first published material relating quantum mechanics to image processing can be traced to Vlasov’s work, in 1997, which focused on the use of a quantum system to recognize orthogonal images. This was followed by efforts using quantum algorithms to search specific patterns in binary images and detect the posture of certain targets. Notably, more optics-based interpretation for quantum imaging were initially experimentally demonstrated in and formalized in after seven years. However, the pioneering role of research that gave birth to what is today referred to as QIMP should be attributed to Venegas-Andraca and Bose’s Qubit Lattice description for quantum images in 2003. Following this, Lattorre proposed another kind of representation, called the Real Ket, whose purpose was to encode quantum images as a basis for further applications in QIMP.

Technically, these pioneering efforts with the subsequent studies related to them can be classified into three main groups, of which (1) and (2) are deemed outside the purview of this article:

(1) Quantum-assisted digital image processing (QDIP): These applications aim at exploiting some of the properties responsible for the potency of quantum computing algorithms in order to improve some well-known digital or classical image processing tasks and applications.

(2) Optics-based quantum imaging (OQI): These applications focus on devising novel techniques for optical imaging and parallel information processing at the quantum level by exploiting the quantum nature of light and the intrinsic parallelism of optical signals.

(3) Classically-inspired quantum image processing (QIMP): These applications derive their inspiration from the expectation that quantum computing hardware will soon be physically realized and, hence, such research focuses on extending classical image processing tasks and applications to quantum computing framework.

Quantum image representations
QIMP is devoted to utilizing the quantum computing technologies to capture, manipulate, and recover quantum images. Logically, percolating this requires that representations to encode images based on the quantum mechanical composition of any potential quantum computing hardware be conjured.

In particular, Venegas-Andraca posited that if we assume an apparatus that could detect electromagnetic frequencies and produce a quantum state as output, we could store color in a qubit by translating given frequencies to quantum states. And by updating the indices to specify the pixels in the image, a full image could be stored in a Qubit Lattice. In 2010, Le et al. proposed a flexible representation of quantum image (FRQI), which is a normalized state that captures the essential information (i.e., its color and position) about every point in an image. It utilizes the idea of using the angle to represent the color information of an image as in, and by using the two-dimensional position information (Y-axis and X-axis), the representation is more similar to the pixel representation for images on conventional computers. The FRQI representation maintains a normalized state and significantly decreases the number of qubits required to prepare images compared to the Qubit Lattice description. In addition, it is a very flexible representation because it facilitates both the geometric and the color transformations on an image.

Since the FRQI, many other quantum image representations (QIRs) have been proposed as well as an array of algorithmic frameworks that target the spatial or chromatic content of the image. They include multi-channel representation for quantum images (MCQI), the so-called Caraiman’s QIR approach (presented as CQIR for brevity); novel enhanced quantum representation of digital image model (NEQR); quantum image representation for log-polar images (QUALPI); simple quantum representation of infrared images (SQR); and multi-dimensional image representation using a normal arbitrary quantum superposition state (NAQSS). The NEQR representation uses two entangled qubit sequences to store the whole image, and in this manner, it reduces the time complexity and provides more accurate information retrieval and more complex operations compared to FRQI. Similar to NEQR, the basis states of a sequence of qubits are used to represent the color information, but different number of qubits with different gray levels to encode the color information of an image in CQIR so that different processing operations could be performed. The MCQI representation is an extension of the FRQI representation that facilitates more advanced color image processing by applying different operations on the R, G, and B channels of the image. Finally, three special quantum image representations are concluded here. QUALPI on its part is a representation that exhibits the image in log-polar coordinate. In this way, the affine transformations such as rotation and scaling could be easily performed. In addition, SQR is a representation for infrared quantum images where the color information (a quantum state) is produced from infrared radiation energy that could improve the visual capacity in almost any environment. Last, NAQSS, where the usual position notation/information is divided into k binary expansions for representing a coordinate in the k-dimensional space, was proposed to satisfy the image processing research in multi-dimensional space.

Quantum image manipulations
A lot of the effort in QIP has been focused on designing algorithms to manipulate the position and color information encoded using the FRQI and its many variants. For instance, FRQI-based fast geometric transformations including (two-point) swapping, flip, (orthogonal) rotations and restricted geometric transformations to constrain these operations to a specified area of an image were initially proposed. Recently, NEQR-based quantum image translation to map the position of each picture element in an input image into a new position in an output image and quantum image scaling to resize a quantum image were discussed. While FRQI-based general form of color transformations were first proposed by means of the single qubit gates such as X, Z, and H gates. Later, MCQI-based channel of interest (CoI) operator to entail shifting the grayscale value of the preselected color channel and the channel swapping (CS) operator to swap the grayscale values between two channels were fully discussed in.

To illustrate the feasibility and capability of QIP algorithms and application, researchers always prefer to simulate the digital image processing tasks on the basis of the QIRs that we already have. By using the basic quantum gates and the aforementioned operations, so far, researchers have contributed to quantum image feature extraction, quantum image segmentation, quantum image morphology, quantum image comparison, quantum image filtering, quantum image classification, quantum image stabilization, among others. In particular, QIMP-based security technologies have attracted extensive interest of researchers as presented in the ensuing discussions. Similarly, these advancements have led to many applications in the areas of watermarking, encryption, and steganography etc., which form the core security technologies highlighted in this area.

In general, the work pursued by the researchers in this area are focused on expanding the applicability of QIMP to realize more classical-like digital image processing algorithms; propose technologies to physically realize the QIMP hardware; or simply to note the likely challenges that could impede the realization of some QIMP protocols.