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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 |last=Yan |first=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.}}</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 |last=Yan |first=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.}}</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==
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