Quantum image processing

Quantum image processing (QIMP) is using quantum computing or quantum information processing to create and work with quantum images.^[1]^[2]

Due to some of the properties inherent to quantum computation, notably entanglement and parallelism, it is hoped that QIMP technologies will offer capabilities and performances that surpass their traditional equivalents, in terms of computing speed, security, and minimum storage requirements.^[2]^[3]

Background

A. Y. Vlasov's work^[4] in 1997 focused on using a quantum system to recognize orthogonal images. This was followed by efforts using quantum algorithms to search specific patterns in binary images^[5] and detect the posture of certain targets.^[6] Notably, more optics-based interpretations for quantum imaging were initially experimentally demonstrated in ^[7] and formalized in ^[8] after seven years.

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.^[9]^[10] Later on, in 2005, Latorre proposed another kind of representation, called the Real Ket,^[11] 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 binary geometrical shapes in quantum mechanical systems in which it is shown that maximally entangled qubits can be used to reconstruct images without using any additional information.^[12]

Technically, these pioneering efforts with the subsequent studies related to them can be classified into three main groups:^[3]

Quantum-assisted digital image processing (QDIP): These applications aim at improving digital or classical image processing tasks and applications.^[2]
Optics-based quantum imaging (OQI)^[13]
Classically inspired quantum image processing (QIMP)^[2]

A survey of quantum image representation has been published in.^[14] Furthermore, the recently published book Quantum Image Processing ^[15] 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:^[16] $\vert I\rangle ={\frac {1}{2^{n}}}\sum _{i=0}^{2^{2n-1}}\vert c_{i}\rangle \otimes \vert i\rangle$ where ${\textstyle |i\rangle }$ is the position and ${\textstyle \vert c_{i}\rangle =cos\theta _{i}\vert 0\rangle +sin\theta _{i}\vert 1\rangle }$ the color with a vector of angles ${\textstyle \theta _{i}\in \left[0,\pi /2\right]}$ . As it can be seen, ${\textstyle \vert c_{i}\rangle }$ is a regular qubit state of the form $\vert \psi \rangle =\alpha \vert 0\rangle +\beta \vert 1\rangle$ , with basis states ${\textstyle \vert 0\rangle ={\begin{pmatrix}1\\0\end{pmatrix}}}$ and ${\textstyle \vert 1\rangle ={\begin{pmatrix}0\\1\end{pmatrix}}}$ , as well as amplitudes ${\textstyle \alpha }$ and ${\textstyle \beta }$ that satisfy ${\textstyle \left|\alpha \right|^{2}+\left|\beta \right|^{2}=1}$ .^[17]

Another common representation is MCQI (Multi-Channel Representation for Quantum Images), that uses the RGB channels with quantum states and following FRQI definition:^[16] $\vert I\rangle ={\frac {1}{2^{n+1}}}\sum _{i=0}^{2^{2n-1}}\vert C_{RGB}^{i}\rangle \otimes \vert i\rangle$ ${\begin{aligned}{\begin{aligned}\vert C_{RGB}^{i}\rangle &={\cos \theta _{R}^{i}\vert 000\rangle }+{\cos \theta _{G}^{i}\vert 001\rangle }+{\cos \theta _{B}^{i}\vert 010\rangle }\\&\quad +{\sin \theta _{R}^{i}\vert 100\rangle }+{\sin \theta _{G}^{i}\vert 101\rangle }+{\sin \theta _{B}^{i}\vert 110\rangle }\\&\quad +{\cos {\theta _{\alpha }}\vert 011\rangle }+{\sin \theta _{\alpha }\vert 111\rangle }\end{aligned}}\end{aligned}}$

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 ${\textstyle f\left(y,x\right)=C_{yx}^{q-1}C_{yx}^{q-2}\ldots C_{yx}^{1}C_{yx}^{0}}$ to encode color values for a $2^{n}\times 2^{n}$ image:^[16] $\vert I\rangle ={\frac {1}{2^{n}}}\sum _{y=0}^{2^{n}-1}\sum _{x=0}^{2^{n}-1}\vert f\left(y,x\right)\rangle \vert yx\rangle$

Quantum image manipulations

A lot of the effort in QIMP has been focused on designing algorithms to manipulate the position and color information encoded using flexible representation of quantum images (FRQI) and its many variants. For instance, FRQI-based fast geometric transformations including (two-point) swapping, flip, (orthogonal) rotations^[18] and restricted geometric transformations to constrain these operations to a specified area of an image^[19] 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^[20] and quantum image scaling to resize a quantum image^[21] 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.^[22] Later, Multi-Channel Quantum Image-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 have been fully discussed.^[23]

To illustrate the feasibility and capability of QIMP 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,^[24] quantum image segmentation,^[25] quantum image morphology,^[26] quantum image comparison,^[27] quantum image filtering,^[28] quantum image classification,^[29] quantum image stabilization,^[30] 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,^[31]^[32]^[33] encryption,^[34] and steganography^[35] 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.

Quantum image transform

By encoding and processing the image information in 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 $F=(F_{i,j})_{M\times L}$ , where $F_{i,j}$ represents the pixel value at position $(i,j)$ with $i=1,\dots ,M$ and $j=1,\dots ,L$ , a vector ${\vec {f}}$ with $ML$ elements can be formed by letting the first $M$ elements of ${\vec {f}}$ be the first column of $F$ , the next $M$ 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 $|g\rangle ={\hat {U}}|f\rangle$ with the input image state $|f\rangle$ and the output image state $|g\rangle$ . A unitary transformation can be implemented as a unitary evolution. Some basic and commonly used image transforms (e.g., the Fourier, Hadamard, and Haar wavelet transforms) can be expressed in the form $G=PFQ$ , with the resulting image $G$ and a row (column) transform matrix $P(Q)$ .

The corresponding unitary operator ${\hat {U}}$ can then be written as ${\hat {U}}={Q}^{T}\otimes {P}$ . Several commonly used two-dimensional image transforms, such as the Haar wavelet, Fourier, and Hadamard transforms, are experimentally demonstrated on a quantum computer,^[36] with exponential speedup over their classical counterparts. In addition, a novel highly efficient quantum algorithm is proposed and experimentally implemented for detecting the boundary between different regions of a picture: It requires only one single-qubit gate in the processing stage, independent of the size of the picture.

References

^ Venegas-Andraca, Salvador E. (2005). Discrete Quantum Walks and Quantum Image Processing (DPhil thesis). The University of Oxford.
^ ^a ^b ^c ^d Iliyasu, A.M. (2013). "Towards realising secure and efficient image and video processing applications on quantum computers". Entropy. 15 (8): 2874–2974. Bibcode:2013Entrp..15.2874I. doi:10.3390/e15082874.
^ ^a ^b Yan, F.; Iliyasu, A.M.; Le, P.Q. (2017). "Quantum image processing: A review of advances in its security technologies". International Journal of Quantum Information. 15 (3): 1730001–44. Bibcode:2017IJQI...1530001Y. doi:10.1142/S0219749917300017.
^ Vlasov, A.Y. (1997). "Quantum computations and images recognition". arXiv:quant-ph/9703010. Bibcode:1997quant.ph..3010V. {{cite journal}}: Cite journal requires |journal= (help)
^ Schutzhold, R. (2003). "Pattern recognition on a quantum computer". Physical Review A. 67 (6): 062311. arXiv:quant-ph/0208063. Bibcode:2003PhRvA..67f2311S. doi:10.1103/PhysRevA.67.062311.
^ Beach, G.; Lomont, C.; Cohen, C. (2003). "Quantum image processing (QuIP)". 32nd Applied Imagery Pattern Recognition Workshop, 2003. Proceedings. pp. 39–40. doi:10.1109/AIPR.2003.1284246. ISBN 0-7695-2029-4. S2CID 32051928.
^ Pittman, T.B.; Shih, Y.H.; Strekalov, D.V. (1995). "Optical imaging by means of two-photon quantum entanglement". Physical Review A. 52 (5): R3429–R3432. Bibcode:1995PhRvA..52.3429P. doi:10.1103/PhysRevA.52.R3429. PMID 9912767.
^ Lugiato, L.A.; Gatti, A.; Brambilla, E. (2002). "Quantum imaging". Journal of Optics B. 4 (3): S176–S183. arXiv:quant-ph/0203046. Bibcode:2002JOptB...4S.176L. doi:10.1088/1464-4266/4/3/372. S2CID 9640455.
^ Venegas-Andraca, S.E.; Bose, S. (2003). "Quantum Computation and Image Processing: New Trends in Artificial Intelligence" (PDF). Proceedings of the 2003 IJCAI International Conference on Artificial Intelligence: 1563–1564.
^ Venegas-Andraca, S.E.; Bose, S. (2003). "Storing, processing, and retrieving an image using quantum mechanics". In Donkor, Eric; Pirich, Andrew R; Brandt, Howard E (eds.). Quantum Information and Computation. Vol. 5105. pp. 134–147. Bibcode:2003SPIE.5105..137V. doi:10.1117/12.485960. S2CID 120495441. {{cite book}}: |journal= ignored (help)
^ Latorre, J.I. (2005). "Image compression and entanglement". arXiv:quant-ph/0510031. Bibcode:2005quant.ph.10031L. {{cite journal}}: Cite journal requires |journal= (help)
^ Venegas-Andraca, S.E.; Ball, J. (2010). "Processing Images in Entangled Quantum Systems". Quantum Informatiom Processing. 9 (1): 1–11. Bibcode:2010QuIP....9....1V. doi:10.1007/s11128-009-0123-z. S2CID 34988263.
^ Gatti, A.; Brambilla, E. (2008). "Chapter 5 Quantum imaging". Quantum imaging. Progress in Optics. Vol. 51. pp. 251–348. doi:10.1016/S0079-6638(07)51005-X. ISBN 978-0-444-53211-4.
^ Yan, F.; Iliyasu, A.M.; Venegas-Andraca, S.E. (2016). "A survey of quantum image representations". Quantum Informatiom Processing. 15 (1): 1–35. Bibcode:2016QuIP...15....1Y. doi:10.1007/s11128-015-1195-6. S2CID 31229136.
^ Yan, Fei; Venegas-Andraca, Salvador E. (2020). Quantum Image Processing. Springer. ISBN 978-9813293304.
^ ^a ^b ^c Yan, Fei; Venegas-Andraca, Salvador E. (2020), "Quantum Image Representations", Quantum Image Processing, Singapore: Springer Singapore, pp. 19–48, doi:10.1007/978-981-32-9331-1_2, ISBN 978-981-329-330-4, retrieved 2024-10-31
^ Yan, Fei; Venegas-Andraca, Salvador E. (2020), "Introduction and Overview", Quantum Image Processing, Singapore: Springer Singapore, pp. 1–17, doi:10.1007/978-981-32-9331-1_1, ISBN 978-981-329-330-4, retrieved 2024-10-31
^ Le, P.; Iliyasu, A.; Dong, F.; Hirota, K. (2010). "Multi-dimensional color image storage and retrieval for a normal arbitrary quantum superposition state". IAENG International Journal of Applied Mathematics. 40 (3): 113–123.
^ Le, P.; Iliyasu, A.; Dong, F.; Hirota, K. (2011). "Strategies for designing geometric transformations on quantum images" (PDF). Theoretical Computer Science. 412 (15): 1406–1418. doi:10.1016/j.tcs.2010.11.029.
^ Wang, J.; Jiang, N.; Wang, L. (2015). "Quantum image translation". Quantum Information Processing. 14 (5): 1589–1604. Bibcode:2015QuIP...14.1589W. doi:10.1007/s11128-014-0843-6. S2CID 33839291.
^ Jiang, N.; Wang, J.; Mu, Y. (2015). "Quantum image scaling up based on nearest-neighbor interpolation with integer scaling ratio". Quantum Information Processing. 14 (11): 4001–4026. Bibcode:2015QuIP...14.4001J. doi:10.1007/s11128-015-1099-5. S2CID 30804812.
^ Le, P.; Iliyasu, A.; Dong, F.; Hirota, K. (2011). "Efficient colour transformations on quantum image". Journal of Advanced Computational Intelligence and Intelligent Informatics. 15 (6): 698–706. doi:10.20965/jaciii.2011.p0698.
^ Sun, B.; Iliyasu, A.; Yan, F.; Garcia, J.; Dong, F.; Al-Asmari, A. (2014). "Multi-channel information operations on quantum images". Journal of Advanced Computational Intelligence and Intelligent Informatics. 18 (2): 140–149. doi:10.20965/jaciii.2014.p0140.
^ Zhang, Y.; Lu, K.; Xu, K.; Gao, Y.; Wilson, R. (2015). "Local feature point extraction for quantum images". Quantum Information Processing. 14 (5): 1573–1588. Bibcode:2015QuIP...14.1573Z. doi:10.1007/s11128-014-0842-7. S2CID 20213446.
^ Caraiman, S.; Manta, V. (2014). "Histogram-based segmentation of quantum images". Theoretical Computer Science. 529: 46–60. doi:10.1016/j.tcs.2013.08.005.
^ Yuan, S.; Mao, X.; Li, T.; Xue, Y.; Chen, L.; Xiong, Q. (2015). "Quantum morphology operations based on quantum representation model". Quantum Information Processing. 14 (5): 1625–1645. Bibcode:2015QuIP...14.1625Y. doi:10.1007/s11128-014-0862-3. S2CID 44828546.
^ Yan, F.; Iliyasu, A.; Le, P.; Sun, B.; Dong, F.; Hirota, K. (2013). "A parallel comparison of multiple pairs of images on quantum computers". International Journal of Innovative Computing and Applications. 5 (4): 199–212. doi:10.1504/IJICA.2013.062955.
^ Caraiman, S.; Manta, V. (2013). "Quantum image filtering in the frequency domain". Advances in Electrical and Computer Engineering. 13 (3): 77–84. doi:10.4316/AECE.2013.03013.
^ Ruan, Y.; Chen, H.; Tan, J. (2016). "Quantum computation for large-scale image classification". Quantum Information Processing. 15 (10): 4049–4069. Bibcode:2016QuIP...15.4049R. doi:10.1007/s11128-016-1391-z. S2CID 27476075.
^ Yan, F.; Iliyasu, A.; Yang, H.; Hirota, K. (2016). "Strategy for quantum image stabilization". Science China Information Sciences. 59 (5): 052102. doi:10.1007/s11432-016-5541-9. S2CID 255200782.
^ Iliyasu, A.; Le, P.; Dong, F.; Hirota, K. (2012). "Watermarking and authentication of quantum images based on restricted geometric transformations". Information Sciences. 186 (1): 126–149. doi:10.1016/j.ins.2011.09.028.
^ Heidari, S.; Naseri, M. (2016). "A Novel Lsb based Quantum Watermarking". International Journal of Theoretical Physics. 55 (10): 4205–4218. Bibcode:2016IJTP...55.4205H. doi:10.1007/s10773-016-3046-3. S2CID 124870364.
^ Zhang, W.; Gao, F.; Liu, B.; Jia, H. (2013). "A quantum watermark protocol". International Journal of Theoretical Physics. 52 (2): 504–513. Bibcode:2013IJTP...52..504Z. doi:10.1007/s10773-012-1354-9. S2CID 122413780.
^ Zhou, R.; Wu, Q.; Zhang, M.; Shen, C. (2013). "Quantum image encryption and decryption algorithms based on quantum image geometric transformations. International". Journal of Theoretical Physics. 52 (6): 1802–1817. doi:10.1007/s10773-012-1274-8. S2CID 121269114.
^ Jiang, N.; Zhao, N.; Wang, L. (2015). "Lsb based quantum image steganography algorithm". International Journal of Theoretical Physics. 55 (1): 107–123. doi:10.1007/s10773-015-2640-0. S2CID 120009979.
^ Yao, Xi-Wei; Wang, Hengyan; Liao, Zeyang; Chen, Ming-Cheng; Pan, Jian; et al. (11 September 2017). "Quantum Image Processing and Its Application to Edge Detection: Theory and Experiment". Physical Review X. 7 (3): 31041. arXiv:1801.01465. Bibcode:2017PhRvX...7c1041Y. doi:10.1103/physrevx.7.031041. ISSN 2160-3308. LCCN 2011201149. OCLC 706478714. S2CID 119205332.

[Venegas-Andraca2005-1] Venegas-Andraca, Salvador E. (2005). Discrete Quantum Walks and Quantum Image Processing (DPhil thesis). The University of Oxford.

[Iliyasu_Towards_2013-2] Iliyasu, A.M. (2013). "Towards realising secure and efficient image and video processing applications on quantum computers". Entropy. 15 (8): 2874–2974. Bibcode:2013Entrp..15.2874I. doi:10.3390/e15082874.

[Yan_Quantum_2017-3] Yan, F.; Iliyasu, A.M.; Le, P.Q. (2017). "Quantum image processing: A review of advances in its security technologies". International Journal of Quantum Information. 15 (3): 1730001–44. Bibcode:2017IJQI...1530001Y. doi:10.1142/S0219749917300017.

[Vlasov_Quantum_2003-4] Vlasov, A.Y. (1997). "Quantum computations and images recognition". arXiv:quant-ph/9703010. Bibcode:1997quant.ph..3010V. {{cite journal}}: Cite journal requires |journal= (help)

[Schutzhold_Pattern_2003-5] Schutzhold, R. (2003). "Pattern recognition on a quantum computer". Physical Review A. 67 (6): 062311. arXiv:quant-ph/0208063. Bibcode:2003PhRvA..67f2311S. doi:10.1103/PhysRevA.67.062311.

[Beach_Quantum_2003-6] Beach, G.; Lomont, C.; Cohen, C. (2003). "Quantum image processing (QuIP)". 32nd Applied Imagery Pattern Recognition Workshop, 2003. Proceedings. pp. 39–40. doi:10.1109/AIPR.2003.1284246. ISBN 0-7695-2029-4. S2CID 32051928.

[7] Pittman, T.B.; Shih, Y.H.; Strekalov, D.V. (1995). "Optical imaging by means of two-photon quantum entanglement". Physical Review A. 52 (5): R3429–R3432. Bibcode:1995PhRvA..52.3429P. doi:10.1103/PhysRevA.52.R3429. PMID 9912767.

[Lugiato_quantum_2002-8] Lugiato, L.A.; Gatti, A.; Brambilla, E. (2002). "Quantum imaging". Journal of Optics B. 4 (3): S176–S183. arXiv:quant-ph/0203046. Bibcode:2002JOptB...4S.176L. doi:10.1088/1464-4266/4/3/372. S2CID 9640455.

[Venegas-AndracaIJCAI2003-9] Venegas-Andraca, S.E.; Bose, S. (2003). "Quantum Computation and Image Processing: New Trends in Artificial Intelligence" (PDF). Proceedings of the 2003 IJCAI International Conference on Artificial Intelligence: 1563–1564.

[Venegas_Storing_2003-10] Venegas-Andraca, S.E.; Bose, S. (2003). "Storing, processing, and retrieving an image using quantum mechanics". In Donkor, Eric; Pirich, Andrew R; Brandt, Howard E (eds.). Quantum Information and Computation. Vol. 5105. pp. 134–147. Bibcode:2003SPIE.5105..137V. doi:10.1117/12.485960. S2CID 120495441. {{cite book}}: |journal= ignored (help)

[Latorre_Image_2005-11] Latorre, J.I. (2005). "Image compression and entanglement". arXiv:quant-ph/0510031. Bibcode:2005quant.ph.10031L. {{cite journal}}: Cite journal requires |journal= (help)

[Venegas-Andraca2010-12] Venegas-Andraca, S.E.; Ball, J. (2010). "Processing Images in Entangled Quantum Systems". Quantum Informatiom Processing. 9 (1): 1–11. Bibcode:2010QuIP....9....1V. doi:10.1007/s11128-009-0123-z. S2CID 34988263.

[Pittman_Optical_1995-13] Gatti, A.; Brambilla, E. (2008). "Chapter 5 Quantum imaging". Quantum imaging. Progress in Optics. Vol. 51. pp. 251–348. doi:10.1016/S0079-6638(07)51005-X. ISBN 978-0-444-53211-4.

[Yan2016-14] Yan, F.; Iliyasu, A.M.; Venegas-Andraca, S.E. (2016). "A survey of quantum image representations". Quantum Informatiom Processing. 15 (1): 1–35. Bibcode:2016QuIP...15....1Y. doi:10.1007/s11128-015-1195-6. S2CID 31229136.

[Yan2020-15] Yan, Fei; Venegas-Andraca, Salvador E. (2020). Quantum Image Processing. Springer. ISBN 978-9813293304.

[:0-16] Yan, Fei; Venegas-Andraca, Salvador E. (2020), "Quantum Image Representations", Quantum Image Processing, Singapore: Springer Singapore, pp. 19–48, doi:10.1007/978-981-32-9331-1_2, ISBN 978-981-329-330-4, retrieved 2024-10-31

[17] Yan, Fei; Venegas-Andraca, Salvador E. (2020), "Introduction and Overview", Quantum Image Processing, Singapore: Springer Singapore, pp. 1–17, doi:10.1007/978-981-32-9331-1_1, ISBN 978-981-329-330-4, retrieved 2024-10-31

[Le_Fast_2010-18] Le, P.; Iliyasu, A.; Dong, F.; Hirota, K. (2010). "Multi-dimensional color image storage and retrieval for a normal arbitrary quantum superposition state". IAENG International Journal of Applied Mathematics. 40 (3): 113–123.

[Le_Strategies_2011-19] Le, P.; Iliyasu, A.; Dong, F.; Hirota, K. (2011). "Strategies for designing geometric transformations on quantum images" (PDF). Theoretical Computer Science. 412 (15): 1406–1418. doi:10.1016/j.tcs.2010.11.029.

[Wang_Quantum_2015-20] Wang, J.; Jiang, N.; Wang, L. (2015). "Quantum image translation". Quantum Information Processing. 14 (5): 1589–1604. Bibcode:2015QuIP...14.1589W. doi:10.1007/s11128-014-0843-6. S2CID 33839291.

[Jiang_Quantum_2015-21] Jiang, N.; Wang, J.; Mu, Y. (2015). "Quantum image scaling up based on nearest-neighbor interpolation with integer scaling ratio". Quantum Information Processing. 14 (11): 4001–4026. Bibcode:2015QuIP...14.4001J. doi:10.1007/s11128-015-1099-5. S2CID 30804812.

[22] Le, P.; Iliyasu, A.; Dong, F.; Hirota, K. (2011). "Efficient colour transformations on quantum image". Journal of Advanced Computational Intelligence and Intelligent Informatics. 15 (6): 698–706. doi:10.20965/jaciii.2011.p0698.

[Sun_Multi_2014-23] Sun, B.; Iliyasu, A.; Yan, F.; Garcia, J.; Dong, F.; Al-Asmari, A. (2014). "Multi-channel information operations on quantum images". Journal of Advanced Computational Intelligence and Intelligent Informatics. 18 (2): 140–149. doi:10.20965/jaciii.2014.p0140.

[Zhang_Local_2015-24] Zhang, Y.; Lu, K.; Xu, K.; Gao, Y.; Wilson, R. (2015). "Local feature point extraction for quantum images". Quantum Information Processing. 14 (5): 1573–1588. Bibcode:2015QuIP...14.1573Z. doi:10.1007/s11128-014-0842-7. S2CID 20213446.

[Caraiman_Histogram_2014-25] Caraiman, S.; Manta, V. (2014). "Histogram-based segmentation of quantum images". Theoretical Computer Science. 529: 46–60. doi:10.1016/j.tcs.2013.08.005.

[Yuan_Quantum_2015-26] Yuan, S.; Mao, X.; Li, T.; Xue, Y.; Chen, L.; Xiong, Q. (2015). "Quantum morphology operations based on quantum representation model". Quantum Information Processing. 14 (5): 1625–1645. Bibcode:2015QuIP...14.1625Y. doi:10.1007/s11128-014-0862-3. S2CID 44828546.

[Yan_A_2013-27] Yan, F.; Iliyasu, A.; Le, P.; Sun, B.; Dong, F.; Hirota, K. (2013). "A parallel comparison of multiple pairs of images on quantum computers". International Journal of Innovative Computing and Applications. 5 (4): 199–212. doi:10.1504/IJICA.2013.062955.

[Caraiman_Quantum_2013-28] Caraiman, S.; Manta, V. (2013). "Quantum image filtering in the frequency domain". Advances in Electrical and Computer Engineering. 13 (3): 77–84. doi:10.4316/AECE.2013.03013.

[Ruan_Quantum_2016-29] Ruan, Y.; Chen, H.; Tan, J. (2016). "Quantum computation for large-scale image classification". Quantum Information Processing. 15 (10): 4049–4069. Bibcode:2016QuIP...15.4049R. doi:10.1007/s11128-016-1391-z. S2CID 27476075.

[Yan_Strategy_2016-30] Yan, F.; Iliyasu, A.; Yang, H.; Hirota, K. (2016). "Strategy for quantum image stabilization". Science China Information Sciences. 59 (5): 052102. doi:10.1007/s11432-016-5541-9. S2CID 255200782.

[Iliyasu_Watermarking_2012-31] Iliyasu, A.; Le, P.; Dong, F.; Hirota, K. (2012). "Watermarking and authentication of quantum images based on restricted geometric transformations". Information Sciences. 186 (1): 126–149. doi:10.1016/j.ins.2011.09.028.

[Heidari_Watermarking_2016-32] Heidari, S.; Naseri, M. (2016). "A Novel Lsb based Quantum Watermarking". International Journal of Theoretical Physics. 55 (10): 4205–4218. Bibcode:2016IJTP...55.4205H. doi:10.1007/s10773-016-3046-3. S2CID 124870364.

[33] Zhang, W.; Gao, F.; Liu, B.; Jia, H. (2013). "A quantum watermark protocol". International Journal of Theoretical Physics. 52 (2): 504–513. Bibcode:2013IJTP...52..504Z. doi:10.1007/s10773-012-1354-9. S2CID 122413780.

[Zhou_Quantum_2013-34] Zhou, R.; Wu, Q.; Zhang, M.; Shen, C. (2013). "Quantum image encryption and decryption algorithms based on quantum image geometric transformations. International". Journal of Theoretical Physics. 52 (6): 1802–1817. doi:10.1007/s10773-012-1274-8. S2CID 121269114.

[Jiang_Lsb_2015-35] Jiang, N.; Zhao, N.; Wang, L. (2015). "Lsb based quantum image steganography algorithm". International Journal of Theoretical Physics. 55 (1): 107–123. doi:10.1007/s10773-015-2640-0. S2CID 120009979.

[2017_Yao-36] Yao, Xi-Wei; Wang, Hengyan; Liao, Zeyang; Chen, Ming-Cheng; Pan, Jian; et al. (11 September 2017). "Quantum Image Processing and Its Application to Edge Detection: Theory and Experiment". Physical Review X. 7 (3): 31041. arXiv:1801.01465. Bibcode:2017PhRvX...7c1041Y. doi:10.1103/physrevx.7.031041. ISSN 2160-3308. LCCN 2011201149. OCLC 706478714. S2CID 119205332.

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[32]

[33]

[34]

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Fahrer / Team für die Suche eingeben: