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{{Short description|Processing quantum-encoded images}}
'''Quantum image processing (QIMP)'''
Due to some of the ==Background==
A. Y. Vlasov's work<ref name="Vlasov Quantum 2003">{{cite
In 2003, Salvador Venegas-Andraca and S. Bose presented Qubit Lattice, the first published general model for storing, processing and retrieving images using quantum systems.<ref name="Venegas-AndracaIJCAI2003">{{cite journal |title=Quantum Computation and Image Processing: New Trends in Artificial Intelligence |journal=Proceedings of the 2003 IJCAI International Conference on Artificial Intelligence |pages=1563–1564 |year=2003 |last1=Venegas-Andraca |first1=S.E.|last2=Bose|first2=S.|url=https://www.ijcai.org/Proceedings/03/Papers/276.pdf}}</ref><ref name="Venegas Storing 2003">{{cite book |pages=134–147 |year=2003 |last1=Venegas-Andraca |first1=S.E.|last2=Bose |first2=S.|title=Proceedings Volume 5105, Quantum Information and Computation |chapter=Storing, processing, and retrieving an image using quantum mechanics |editor3-first=Howard E |editor3-last=Brandt |editor2-first=Andrew R |editor2-last=Pirich |editor1-first=Eric |editor1-last=Donkor |bibcode=2003SPIE.5105..137V |doi=10.1117/12.485960 |s2cid=120495441 |publisher=[[SPIE]]}}</ref> Later on, in 2005, Latorre proposed another kind of representation, called the Real Ket,<ref name="Latorre Image 2005">{{cite arXiv |title=Image compression and entanglement |eprint=quant-ph/0510031 |year=2005 |last1=Latorre |first1=J. I. }}</ref> whose purpose was to encode quantum images as a basis for further applications in QIMP. Furthermore, in 2010 Venegas-Andraca and Ball presented a method for storing and retrieving [[Well-known text representation of geometry|binary geometrical shapes]] in quantum mechanical systems in which it is shown that maximally entangled [[qubit]]s can be used to reconstruct images without using any additional information.<ref name="Venegas-Andraca2010">{{cite journal |title=Processing Images in Entangled Quantum Systems |journal=Quantum Informatiom Processing |volume=9 |issue=1 |pages=1–11 |year=2010 |last1=Venegas-Andraca |first1=S.E.|last2=Ball |first2=J.|doi=10.1007/s11128-009-0123-z |bibcode=2010QuIP....9....1V |s2cid=34988263 }}</ref>
Technically, these pioneering efforts with the subsequent studies related to them can be classified into three main groups:<ref name="Yan Quantum 2017"/>
A survey of quantum image representation has been published in.<ref name="Yan2016">{{cite journal |title=A survey of quantum image representations |journal=Quantum Informatiom Processing |volume=15 |issue=1 |pages=1–35 |year=2016 |last1=Yan |first1=F.|last2=Iliyasu |first2=A.M.|last3=Venegas-Andraca |first3=S.E.| bibcode=2016QuIP...15....1Y|doi=10.1007/s11128-015-1195-6 |s2cid=31229136 }}</ref> Furthermore, the recently published book ''Quantum Image Processing'' <ref name="Yan2020">{{cite book |last1=Yan |first1= Fei| last2=Venegas-Andraca |first2= Salvador E.|date= 2020|title= Quantum Image Processing|url= https://www.springer.com/gp/book/9789813293304 |publisher= Springer|isbn= 978-9813293304}}</ref> provides a comprehensive introduction to quantum image processing, which focuses on extending conventional image processing tasks to the quantum computing frameworks. It summarizes the available quantum image representations and their operations, reviews the possible quantum image applications and their implementation, and discusses the open questions and future development trends.
== Quantum image representations ==
There are various approaches for quantum image representation, that are usually based on the encoding of color information. A common representation is FRQI (''Flexible Representation for Quantum Images''), that captures the color and position at every pixel of the image, and defined as:<ref name=":0">{{Citation |last1=Yan |first1=Fei |title=Quantum Image Representations |date=2020 |work=Quantum Image Processing |pages=19–48 |url=http://link.springer.com/10.1007/978-981-32-9331-1_2 |access-date=2024-10-31 |place=Singapore |publisher=Springer Singapore |language=en |doi=10.1007/978-981-32-9331-1_2 |isbn=978-981-329-330-4 |last2=Venegas-Andraca |first2=Salvador E.|url-access=subscription }}</ref><math display="block">\vert I \rangle = \frac{1}{2^{n}} \sum^{2^{2n-1}}_{i=0} \vert c_{i} \rangle \otimes \vert i \rangle</math>where <math display="inline">| i \rangle </math> is the position and <math display="inline">\vert c_{i} \rangle = cos \theta_{i} \vert 0 \rangle + sin \theta_{i} \vert 1 \rangle</math> the color with a vector of angles <math display="inline">\theta_{i} \in \left[0, \pi/2 \right]</math>. As it can be seen, <math display="inline">\vert c_{i} \rangle </math> is a regular [[Qubit#Qubit states|qubit state]] of the form <math>\vert \psi\rangle = \alpha \vert 0 \rangle + \beta \vert 1 \rangle</math>, with basis states <math display="inline">\vert 0 \rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}</math> and <math display="inline">\vert 1 \rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} </math>, as well as amplitudes <math display="inline">\alpha </math> and <math display="inline">\beta </math> that satisfy <math display="inline">\left|\alpha\right|^{2} + \left|\beta\right|^{2} = 1</math>.<ref>{{Citation |last1=Yan |first1=Fei |title=Introduction and Overview |date=2020 |work=Quantum Image Processing |pages=1–17 |url=http://link.springer.com/10.1007/978-981-32-9331-1_1 |access-date=2024-10-31 |place=Singapore |publisher=Springer Singapore |language=en |doi=10.1007/978-981-32-9331-1_1 |isbn=978-981-329-330-4 |last2=Venegas-Andraca |first2=Salvador E.|url-access=subscription }}</ref>
Another common representation is MCQI (''Multi-Channel Representation for Quantum Images''), that uses the [[RGB color model|RGB]] channels with quantum states and following FRQI definition:<ref name=":0" /><math display="block">\vert I\rangle = \frac{1}{2^{n+1}} \sum^{2^{2n-1}}_{i=0} \vert C^{i}_{RGB}\rangle \otimes \vert i\rangle</math><math display="block">\begin{aligned}
\begin{aligned}
\vert C_{RGB}^i \rangle &=
{\cos \theta_R^i \vert000 \rangle} + {\cos \theta_G^i \vert001 \rangle} + {\cos \theta_B^i \vert010 \rangle} \\ &
\quad + {\sin \theta_R^i \vert100 \rangle} + {\sin \theta_G^i \vert101 \rangle} + {\sin \theta_B^i \vert110 \rangle} \\ &
\quad + {\cos{\theta_\alpha}\vert 011 \rangle} + {\sin\theta_\alpha\vert 111\rangle}
\end{aligned}
\end{aligned}</math>
Departing from the angle-based approach of FRQI and MCQI, and using a qubit sequence, NEQR (''Novel Enhanced Representation for Quantum Images'') is another representation approach, that uses a function <math display="inline">f \left( y,x \right) = C^{q-1}_{yx} C^{q-2}_{yx} \ldots C^{1}_{yx} C^{0}_{yx}</math> to encode color values for a <math>2^n \times 2^n</math> image:<ref name=":0" /><math display="block">\vert I\rangle = \frac{1}{2^{n}} \sum^{2^{n} - 1 }_{y=0} \sum^{2^{n} - 1 }_{x=0} \vert f \left( y,x \right) \rangle \vert yx \rangle</math>
==Quantum image manipulations==
A lot of the effort in
To illustrate the feasibility and capability of
In general, the work pursued by the researchers in this area are focused on expanding the applicability of
==Quantum image transform==
By [[encoding]] and processing the image information in [[Quantum mechanics|quantum-mechanical]] systems, a framework of quantum image processing is presented, where a pure [[quantum state]] encodes the image information: to encode the [[pixel]] values in the probability amplitudes and the pixel positions in the computational basis states.
Given an image <math>F=(F_{i,j})_{M \times L}</math>, where <math>F_{i,j}</math> represents the pixel value at position <math>(i,j)</math> with <math>i = 1,\dots,M</math> and <math>j = 1,\dots,L</math>, a vector <math>\vec{f}</math> with <math>ML</math> elements can be formed by letting the first <math>M</math> elements of <math>\vec{f}</math> be the first column of <math>F</math>, the next <math>M</math> elements the second column, etc.
A large class of image operations is [[linear]], e.g., unitary transformations, convolutions, and linear filtering. In the quantum computing, the linear transformation can be represented as <math>|g\rangle =\hat{U} |f\rangle </math> with the input image state <math>|f\rangle </math> and the output image state <math>|g\rangle </math>. A unitary transformation can be implemented as a unitary evolution.
==See also==
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{{Reflist|30em|refs=
<ref name="2017_Yao">{{cite journal | last1 = Yao | first1 = Xi-Wei | last2 = Wang | first2 = Hengyan | last3 = Liao | first3 = Zeyang | last4 = Chen | first4 = Ming-Cheng | last5 = Pan | first5 = Jian | last6 = Li | first6 = Jun | last7 = Zhang | first7 = Kechao | last8 = Lin | first8 = Xingcheng | last9 = Wang | first9 = Zhehui | last10 = Luo | first10 = Zhihuang | last11 = Zheng | first11 = Wenqiang | last12 = Li | first12 = Jianzhong | last13 = Zhao | first13 = Meisheng | last14 = Peng | first14 = Xinhua | last15 = Suter | first15 = Dieter | display-authors = 5 | date = 2017-09-11 | title = Quantum Image Processing and Its Application to Edge Detection: Theory and Experiment | journal = [[Physical Review X]] | language = en
}}
[[Category:
[[Category:Image processing]]
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