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Image processing via simulated quantum dynamics. (English) Zbl 1365.94053
Summary: We apply simulated quantum evolution to image processing, and examine its practicality in the context of image denoising. More specifically, in our approach image processing consists of three stages: First, a digitized gray-scale image is represented as a quantum variable – typically, a density matrix. Second, the quantum variable is evolved via the Markovian master equation in Lindblad form. Third, the quantum variable is back-converted into an image. Numerical experiments indicate remarkable denoising results are obtained in this way for a suitable choice of flow parameters. To our knowledge the proposed image processing technique is conceptually new.
##### MSC:
 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 60G35 Signal detection and filtering (aspects of stochastic processes) 68U10 Computing methodologies for image processing 81S22 Open systems, reduced dynamics, master equations, decoherence 93E11 Filtering in stochastic control theory
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##### References:
 [1] Alicki R. and Fannes M., Quantum Dynamical Systems, Oxford University Press, Oxford, 2009. · Zbl 0814.46055 [2] Baird P., Emergence of geometry in a combinatorial universe, J. Geom. Phys. 74 (2013), 185-195. · Zbl 1278.05079 [3] Boixo S., Rønnow T. F., Isakov S. V., Wang Z., Wecker D., Lidar D. A., Martinis J. M. and Troyer M., Evidence for quantum annealing with more than one hundred qubits, Nat. Phys. 10 (2014), 218-224. [4] Breuer H.-P. and Petruccione F., The Theory of Open Quantum Systems, Oxford University Press, Oxford, 2002. · Zbl 1053.81001 [5] Chan T. F. and Shen J., Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, Philadelphia, 2005. · Zbl 1095.68127 [6] Farhi E., Goldstone J., S. , Gutmann , Lapan J., Lundgren A. and Preda D., A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem, Science 292 (2001), 472-476. · Zbl 1226.81046 [7] Frigerio A., Simulated anenaling and quantum detailed balance, J. Stat. Phys. 58 (1990), 325-354. · Zbl 0716.47039 [8] Harrow A. W., Hassidim A. and Lloyd S., Quantum algorithm for linear systems of equations, Phys. Rev. Lett. 103 (2009), Article ID 150502. [9] Kato T., Perturbation Theory for Linear Operators, Springer, New York, 1966. · Zbl 0148.12601 [10] Krzakala F., Mézard M., Sausset F., Sun Y. F. and Zdebrová L., Statistical physics based reconstruction in compressed sensing, Phys. Rev. X 2 (2012), Article ID 021005. [11] Marsousi M., Abhari K., Babyn P. and Alirezaie J., An adaptive approach to learn overcomplete dictionaries with efficient numbers of elements, IEEE Trans. Signal Process. 62 (2014), 3272-3283. · Zbl 1394.94377 [12] Nielsen M. A. and Chuang I. L., Quantum Computation and Quantum Communication, Cambridge University Press, Cambridge, 2000. [13] Percival I., Quantum State Diffusion, Cambridge University Press, Cambridge, 1998. · Zbl 0946.60093 [14] Sowa A., Factorizing matrices by Dirichlet multiplication, Linear Algebra Appl. 438 (2013), 2385-2393. · Zbl 1263.15013 [15] Sowa A., The Dirichlet ring and unconditional bases in $${L_{2}[0,2π]}$$, Funct. Anal. Appl. 47 (2013), 227-232. · Zbl 1315.46015 [16] Sowa A., Encoding spatial data into quantum observables, preprint 2016, . [17] Sowa A. P., Everitte M. J., Samson J. H., Savilev S., Zagoskin A. M., Heidel S. and Zúñiga-Anaya J. C., Recursive simulation of quantum annealing, J. Phys. A 48 (2015), Article ID 415301. · Zbl 1326.81051
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