[Robert Marks Home Page: http://RobertMarks.org/]

POCS: Alternating Projection onto Convex Sets

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Handbook of Fourier Analysis & Its Applications

Chapter 11 has the most complete treatment of POCS of which we are aware. Available in print or in electronic form.

For more information about this book, visit its web site:

http://www.HandbookOfFourierAnalysis.com/


Other resources

TUTORIAL: R.J. Marks II, "Alternating Projections onto Convex Sets", in Deconvolution of Images and Spectra, edited by Peter A. Jansson, (Academic Press, San Diego, 1997), pp.476-501 (pdf)

CIA POCS Related Publications.

To download pdf files of these papers, visit Robert Marks's publication page.

R.J. Marks II and J.F. Walkup, "Coherent optical processors for ambiguity function display and one-dimensional correlation/convolution operations," Proceedings of the SPIE Symposium/Workshop on the Effective Utilization of Optics in Radar Systems, Huntsville, Alabama, September 1977.

R.J. Marks II, "Coherent optical extrapolation of two-dimensional signals: processor theory," Applied Optics, vol. 19, pp.1670-1672 (1980).

R.J. Marks II and D.K. Smith, "An iterative coherent processor for bandlimited signal extrapolation," Proceedings of the 1980 International Computing Conference, SPIE Vol 231, Washington D.C., April 1980, pp.106-111

R.J. Marks II, "Gerchberg's extrapolation algorithm in two dimensions," Applied Optics, vol. 20, pp.1815-1820 (1981).

D.K. Smith and R.J. Marks II, "Closed form bandlimited image extrapolation," Applied Optics, vol. 20, pp.2476-2483 (1981).

R.J. Marks II and M.J. Smith, "Closed form object restoration from limited spatial and spectral information," Optics Letters, vol. 6, pp.522-524 (1981).

R.J. Marks II, "Posedness of a bandlimited image extension problem in tomography," Optics Letters, vol. 7, pp.376-377 (1982).

D. Kaplan and R.J. Marks II, "Noise sensitivity of interpolation and extrapolation matrices," Applied Optics, vol. 21, pp.4489-4492 (1982).

R.J. Marks II, "Restoring lost samples from an oversampled bandlimited signal," IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-31, pp.752-755 (1983).

R.J. Marks II, "Superresolution via analysis," Proceedings of the Limits of Passive Imaging Workshop, Mackinac Island, MI, pp.45-55, May 24-26, 1983

R.J. Marks II and D.K. Smith "Gerchberg - type linear deconvolution and extrapolation algorithms," in Transformations in Optical Signal Processing, edited by W.T. Rhodes, J.R. Fienup and B.E.A. Saleh, SPIE vol. 373, pp.161-178 (1984).

R.J. Marks II and S.M. Tseng, "Effect of sampling on closed form bandlimited signal interval interpolation," Applied Optics, vol. 24, pp.763-765 (1985); Erratum, vol. 24, p.2490 (1985).

M.H. Goldburg and R.J. Marks II, "Signal synthesis in the presence of an inconsistent set of constraints", IEEE Transactions on Circuits and Systems, vol. CAS-32 pp. 647-663 (1985).R.J. Marks II, "A class of continuous level associative memory neural nets", Applied Optics, vol.26, pp.2005-2010, (1987).

J.A. Ritcey, L.E. Atlas, A. Somani, D. Nguyen, F. Holt and R.J. Marks II, "A signal space interpretation of neural networks," Proceedings of the International Symposium on Circuits and Systems, pp.370-376, Philadelphia, May 1987.

K.F. Cheung, R.J. Marks II and L.E. Atlas, "Neural net associative memories based on convex set projections," Proceedings of the IEEE First International Conference on Neural Networks, San Diego, June 1987, pp.III-245-III252.

R.J. Marks II, L.E. Atlas and K.F. Cheung, "A class of continuous level neural nets," Proceedings of the Fourteenth Congress of the International Commission for Optics, pp.29-30, Quebec City, Quebec Canada, August 24-28, 1987.

R.J. Marks II, L.E. Atlas and K.F. Cheung, "Optical processor architectures for alternating projection neural networks", Optics Letters, vol. 13, pp.533-535 (1988).

K.F. Cheung, R.J. Marks II and L.E. Atlas, "Convergence of Howard's minimum negativity constraint extrapolation algorithm," Journal of the Optical Society of America A, vol.5, pp.2008-2009 (1988).

R.J. Marks II, L.E. Atlas, S. Oh and J.A. Ritcey, "The performance of convex set projection based neural networks", Neural Information Processing Systems, Dana Z. Anderson, editor, (American Institute of Physics, New York, 1988), pp. 534-543.

R.J. Marks II, L.E. Atlas, S. Oh and J.A. Ritcey, "The performance of convex set projection based neural networks," Neural Information Processing Systems, Dana Z. Anderson, editor, (American Institute of Physics, New York, 1988), pp. 534-543.

R.J. Marks II, L.E. Atlas and K.F. Cheung, "Architectures for a continuous level neural network based on alternating orthogonal projections," Proceedings of O-E/LASE '88 Conference on Neural Network Models for Optical Computing, Los Angeles, January 1988, SPIE volume 882, pp 90-92

R.J. Marks II, L.E. Atlas and S. Oh, "Generalization in layered classification neural networks". 1988 IEEE International Symposium on Circuits and Systems, pp. 503-506, Helsinki, 7-9 June, 1988.

R.J. Marks II, S. Oh and L.E. Atlas, "Alternating projection neural networks", IEEE Transactions on Circuits and Systems, vol.36, pp.846-857 (1989).

M. Aggoune, M.A. El-Sharkawi, D.C. Park, M.J. Damborg and R.J. Marks II, "Preliminary results on using artificial neural networks for security assessment," Proceedings of the 1989 Power Industry Computer Applications (PICA) Conference, pp.252-258, June 1989, Seattle, WA.

S. Oh and R.J. Marks II, "Noise sensitivity of projection neural networks," Proc. 1989 IEEE International Symposium on Circuits and Systems, pp.2201-2204, 9-11 May 1989, Portland

R.J. Marks II, S. Oh, L.E. Atlas and J.A. Ritcey, "Homogeneous and layered alternating projection neural networks," in Real-Time Signal Processing for Industrial Applications, edited by Bahram Javidi (SPIE Optical Engineering Press, Bellingham, WA. 1989), pp. 217-232.

R.J. Marks II, L.E. Atlas and S. Oh, "Optical neural net memory," U.S. Patent No. 4,849,940 (assigned to the Washington Technology Center, University of Washington, Seattle), July 18, 1989.

S.Oh, R.J. Marks II, L.E. Atlas and J.W. Pitton, "Kernel synthesis for generalized time-frequency distributions using the method of projection onto convex sets," SPIE Proceedings 1348, Advanced Signal Processing Algorithms, Architectures, and Implementation, F.T. Luk, Editor, pp.197-207, San Diego, July 10-12, 1990.

S. Oh, R.J. Marks II and D. Sarr, "Homogeneous alternating projection neural networks", Neurocomputing, volume 3, pp. 69-95 (1991).

M.E. Aggoune, M.A. El-Sharkawi, D.C. Park, M.J. Damborg and R.J. Marks II, "Preliminary results on using artificial neural networks for security assessment," IEEE Transactions on Power Engineering, vol.6, pp.890-896 (1991) and vol.6, pp.1324-1325 (1991)

C.Ramon, S.Oh, M.G. Meyer and R.J. Marks II, "Biomagnetic image reconstruction using the method of alternating projections," Proceedings of the SPIE, vol.1652, 1992.

S. Oh. C.Ramon, M.G. Meyer and R.J. Marks II, "Resolution enhancement of biomagnetic images using the method of alternating projections", IEEE Transactions on Biomedical Engineering, vol. 40, no. 4, pp.323-328 (1993).

S.Oh, R.J. Marks II and L.E. Atlas, "Kernel synthesis for generalized time-frequency distributions using the method of alternating projections onto convex sets", IEEE Transactions on Signal Processing, vol. 42, No.7, July 1994, pp.1653-1661.

C. Ramon, P. Czapski, R.J. Marks II, H.C. Lai and S. Lee, "Noninvasive Biomagnetic Sensing of Biological Currents," National Acadamies of Science and Engineering National Research Council of the United States, Radio Science Meeting, June 19-24, 1994, Seattle, p. 272.

P. Cho, S. Lee, R.J. Marks II and S. Oh, "Comparison of algorithms for intensity modulated beam optimization: projections onto convex sets and simulated annealing," Proceedings of the XII International Conference on the Use of Computers in Radiation Therapy, pp.310-312, May, 1997, Salt Lake City

P.S. Cho, S. Lee, R.J. Marks II, S.Oh, S.G. Sutlief, M.H. Phillips, "Optimization of Intensity Modulated Beams With Volume Constraints Using Two Methods: Cost Function Minimization and Projections Onto Convex Sets", Medical Physics, (Am. Assoc. Phys. Med.), Vol. 25, No.4, pp.435-443 (April 1998).

R.J. Marks II, L.E. Atlas and K.F. Cheung, "Architectures for a continuous level neural network based on alternating orthogonal projections", Proceedings of O-E/LASE '88 Conference on Neural Network Models for Optical Computing, Los Angeles, January 1988, SPIE volume 882, pp 90-92.

M.E. Aggoune, L.E. Atlas, D.A. Cohn, M.J. Damborg, M.A. El-Sharkawi and R.J. Marks II, "Artificial neural networks for static system security assessment", Proc. 1989 IEEE International Symposium on Circuits and Systems, pp.490-494, 9-11 May 1989, Portland - invited paper.

S. Oh and R.J. Marks II, "Noise sensitivity of projection neural networks", Proc. 1989 IEEE International Symposium on Circuits and Systems, pp.2201-2204, 9-11 May 1989, Portland.

S.Oh, R.J. Marks II, L.E. Atlas and J.W. Pitton, "Kernel synthesis for generalized time-frequency distributions using the method of projection onto convex sets", SPIE Proceedings 1348, Advanced Signal Processing Algorithms, Architectures, and Implementation, F.T. Luk, Editor, pp.197-207, San Diego, July 10-12, 1990.

C.Ramon, S.Oh, M.G. Meyer and R.J. Marks II, "Biomagnetic image reconstruction using the method of alternating projections", Proceedings of the SPIE, vol.1652, 1992.

S. Oh and R.J. Marks II, "Alternating projections onto fuzzy convex sets", Proceedings of the Second IEEE International Conference on Fuzzy Systems (FUZZ-IEEE `93), San Francisco, March 1993, vol.1, pp. 148-155.

R.J. Marks II, Loren Laybourn, Shinhak Lee and Seho Oh, "Fuzzy and extra crisp alternating projection onto convex sets (POCS)", Proceedings of the International Conference on Fuzzy Systems (FUZZ-IEEE), pp. 427-435, Yokohama, Japan, March 20-24, 1995.

R.J. Marks II, "Alternating Projections onto Convex Sets," in Deconvolution of Images and Spectra, edited by Peter A. Jansson, (Academic Press, San Diego, 1997).

P.S. Cho & R.J. Marks II, "Hardware-sensitive optimization for intensity modulated radiotherapy", Phys. Med. Biol, 2000 (pp. 429-440) .

Ceon Ramon, J. Schreiber, Jens Haueisen, Paul Schimpf, Robert J. Marks, Akira Ishimaru, "Reconstruction and Enhancement of Current Distribution on Curved Surfaces from Biomagnetic Fields Using POCS," Canadian Applied Mathematics Quarterly, vol. 10, No.2, Summer 2002.

Jiho Park, D.C. Park, R.J. Marks II, M.A. El-Sharkawi, "Block Loss Recovery in DCT Image Encoding Using POCS", IEEE International Symposium on Circuits and Systems, Scottsdale, Arizona, May 26-29, 2002, pp.V 245 - V 248.

Sreeram Narayanan, R.J. Marks II , John L. Vian, J.J. Choi, M.A. El-Sharkawi & Benjamin B. Thompson, "Set Constraint Discovery: Missing Sensor Data Restoration Using Auto-Associative Regression Machines," Proceedings of the 2002 International Joint Conference on Neural Networks, 2002 IEEE World Congress on Computational Intelligence, May12-17, 2002, Honolulu, pp. 2872-2877.

Sreeram Narayanan, John L. Vian, J.J. Choi, R.J. Marks II, M.A. El-Sharkawi, & Benjamin B. Thompson, "Missing Sensor Data Restoration for Vibration Sensors on a Jet Aircraft Engine," 2003 International Joint Conference on Neural Networks, July 20-24, 2003, Portland, Oregon (pp. 3007-3010).

M.A. El-Sharkawi and Robert J. Marks II, "Missing Sensor Restoration for Systems Control and Diagnosis," SDEMPED 2003 - Symposium for Diagnostics for Electric Machines, Power Electronics and Drives, Atlanta, GA, USA, 24-26 August 2003, pp. 338-341.

Jiho Park, R.J. Marks II, D.C. Park and M.A. El-Sharkawi, "Content Based Adaptive Spatio-Temporal Methods for MPEG Repair," IEEE Transactions on Image Processing, Vol. 13, # 8 , pp 1066-1077 (August 2004).

Jiho Park, D.C. Park, R.J. Marks II and M.A. El-Sharkawi, "Recovery of Image Blocks Using the Method of Alternating Projections," IEEE Transactions on Image Processing, Vol. 14, No. 4, pp. 461-471, (April 2005).

R.J. Marks II, "Alternating Projections onto Convex Sets," in Deconvolution of Images and Spectra, 2nd edition, edited by Peter A. Jansson, Dover Publications; (2012).

Liang Dong, Yanqing Liu, and Robert J. Marks II. ``Reduction of Out-of-Bound Power and Peak-to-Average Ration in OFDM-Based Cognitive Radio Using Alternating Projections,'' 2013 Texas Symposium on Wireless & Microwave Circuits & Systems, Waco, Texas, April 4-5, 2013

Dylan Eustice, Charles Baylis, Casey Latham, Robert J. Marks II, and Lawrence Cohen "Optimizing Radar Waveforms Using Generalized Alternating Projections," 2015 IEEE Texas Symposium on Wireless and Microwave Circuits and Systems (WMCS), April 23-24,2015. Waco, Texas (pp. 1-6)

Dylan Eustice, Charles Baylis, Lawrence Cohen, and Robert J. Marks II. "Waveform synthesis via alternating projections with ambiguity function, peak-to-average power ratio, and spectrum requirements." 2016 IEEE Radio and Wireless Symposium (RWS), pp. 190-192. IEEE, 2016.

 

POCS