§1. Vision Modeling, Analysis, and Computing (ViMAC): Biological, Cognitive, and Pattern-Theoretic

1. [Multi-Agent Flocking] Jianhong (Jackie) Shen, Cucker-Smale Flocking under Hierarchical Leadership, SIAM J. Appl. Math., 68:3, pp. 694-719, 2008 (electronic 2007).   The PDF file.   1-Page Intro.
   [Keywords: (Bird) Flocking, (Fish) Schooling, (Robotic) Networking, Leaders, Dynamic Connectivity & Graphs, Graph Laplacian, Convergence Rates, Free Will]
   See the remarkable mathematical works on flocking/emergence by Cucker and Smale (2005/2006) and on swarming by   Andrea Bertozzi's group (UCLA).

2. [Cognition and Perception] Yoon-Mo Jung and Jianhong (Jackie) Shen,   First-order modeling and stability analysis of illusory contours,   J. Visual Commun. Image Representation, 19:42-55, 2008.     The PDF file.   A sample figure.     A sample movie for Kanizsa Triangle.   (via the level-set method of Osher and Sethian).
   [Keywords: illusion, illusory contours, energy model, local stability, contour decomposition, real and imaginary, kinks, spans, turns, level-set method, supervision]
   Most closely related works on modeling illusory contours:   Sarti, Malladi, and Sethian (Berkeley, 2002)   and   Zhu and Chan (UCLA, 2003/04).
   Also please experience the enormous literature on qualitative, neurological, or cognitive illusory contours (via Google).
   Dedicated to David Mumford on the Occasion of His 70th Birthday --- a Wellspring of Inspiration to the Young.

3. [Material Science vs. Visual Perception] Yoon-Mo Jung, Sung-Ha Kang, and Jianhong (Jackie) Shen,   Multiphase image segmentation by Modica-Mortola phase transition,   SIAM J. Appl. Math., 67(5):1213-1232, 2007.     The PDF file.   A sample figure.
   [Keywords: Multiphase, segmentation, Mumford-Shah, phase transition, Gamma-convergence, Modica-Mortola, integer progamming, convex splitting/convex-concave procedure (CCCP). ]

4. [Perception vs. Printing Technology] Jianhong (Jackie) Shen,   Least-square halftoning via human vision system and Markov gradient descent (LS-MGD): Algorithm and analysis,   to appear in SIAM Review, 2007.     The PDF file.   A sample figure.
   [Keywords: Halftoning, Human Vision System (HVS), mixing, entropy, least square, stochastic gradient descent, Markov random walk, random fields, Bernoulli flipping, convergence analysis, blue noise]
   See the closely related works on Analog/Digital signal conversion and quantization by   Ingrid Daubechies (Princeton)   and   Sinan Güntürk (Courant), et al.
   Substantially consolidates the earlier PDE-stochastic model/algorithm (see paper § 3.5 below) based on Perona-Malik diffusion and stochastic flipping.

5. [Pattern Mixture and Separation] Jianhong (Jackie) Shen,   A stochastic-variational model for  Soft  Mumford-Shah segmentation,  Int'l J. Biomedical Imaging, special issue on "Recent Advances in Mathematical Methods for the Processing of Biomedical Images," vol. 2006, Article ID 92329, 2006.
   The PDF file from the journal.        A sample figure.         Another sample of a brain image.
   [Keywords: soft vs. hard, Mumford-Shah, pattern, ownership, probability simplex, Modica-Mortola, phase-field, Egorov's theorem, Poincaré inequality, existence theorems, AM algorithm]
   For stochastic researchers less familiar with variational-PDE's: could treat the work as geometrically regularized   K-Mean Clustering and   Mixture Gaussians.

6. [Pattern Synthesis and Analysis] Jianhong (Jackie) Shen,   Piecewise H^-1+H⁰+H¹ images and the Mumford-Shah-Sobolev model for segmented image decomposition, Applied Math. Research Exp., 4: 143-167, 2005.   A sample figure.
   [Keywords: vision, patterns, synthesis vs. analysis, segmentation, decomposition, variational, free boundary, textures, wavelength, mathematical modeling of painting and artists]
   
   
7. [Unified Modeling & Algorithms]   Tony F. Chan and Jianhong Shen, Theory and computation of variational image deblurring, IMS (Inst. Math. Sci., Singapore) Lecture Notes Series, "Mathematics and Computation in Imaging Science and Information Processing," World Scientific Publishing Co., 2007.     The PDF file.  
   [Note: 2003/04 preprint revised; new: stochastic signals, Bayesian approach to Wiener filtering, iterated wavelet-shrinkage algorithm of Daubechies et al.]
   
   See the remarkable comprehensive review closely related to ours by three astrophysicists: Puetter, Gosnell, and Amos Yahil (Stony Brook).

8. [Generative Pattern Modeling] Jianhong Shen, On the foundations of vision modeling III.   Noncommutative monoids of occlusive preimages.   J. Math. Imaging Vision, 24:5-17, 2006.     Two sample preimages.
   [Keywords: Depth, occlusion, segmentation, preimages, monoids (semi-groups), noncommutativity, topology, invariants, knot theory]
   
   
Segmentation has been universally treated as an INVERSE problem.     But has anyone ever come up with a sound FORWARD problem?   Is this really a trivial matter?
9. [Geometric Pattern Detection] Jianhong Shen, On the foundations of vision modeling II. Mining of mirror symmetry of 2-D shapes , J. Visual Commun. Image Rep., 16(3), pp. 250-270, 2005.     Click here for a JPEG sample.
   [Keywords: Mirror symmetry, principle component analysis (PCA), Lebesgue, Hausdorff, features, existence, convex shape, support function, normal shooting algorithm]

10. [Psychophysics into Modeling] Jianhong Shen, On the foundations of vision modeling I. Weber's law and Weberized TV (total variation) restoration , Physica D: Nonlinear Phenomena , 175(3/4), pp. 241-251, 2003.
    [Thank my dear teachers and friends in vision psychology, psychophysics, and computation here at UMN: Dan Kersten and Paul Schrater.]
11. [Psychophysics and Quantum Physics] Jianhong Shen and Yoon-Mo Jung,   Weberized Mumford-Shah model with Bose-Einstein photon noise,   Appl. Math. Optim., 53(3):331-358, 2006.   ( here to the journal. )     The PDF file.
    [Keywords: Weber's Law, light adaptivity, retina, Mumford-Shah, segmentation, Bose-Einstein statistics, Gamma-convergence]
    
    
12. [Biochemical Turing Patterns] Jianhong Shen and Yoon-Mo Jung, Geometric and stochastic analysis of reaction-diffusion patterns , Int'l J. Pure Applied Math., 19(2):195-248, 2005.     The PDF file.
    [Keywords: data mining, pattern, Turing instability, reaction, diffusion, entropy, skewness, kurtosis, isoperimetric ratio, curvature measure.]

§2. Image/Signal Processing & Analysis (iSPA): Variational-PDE, Bayesian/Stochastic, and Wavelet Methods

1. [Oscillatory & Singular Patterns] Jianhong (Jackie) Shen,   Beamlets are densely embedded in H ^-1,   to appear in Adv. Comput. Math., 2008.   The PDF file.   A sample figure.
   [Keywords: texture, distribution, oscillation, singular Radon measure, beamlets, isoperimetric inequality, Haar molecules, Haar potentials]
   Two works having directly inspired ours:   Donoho and Huo's beamlets (2002, 2004) and   Osher, Sole, and Vese's H ^-1 texture (2003).

2. [Unified Models/Algorithms for Quantization/Segmentation] Jianhong (Jackie) Shen and Sung Ha Kang, Quantum TV and Applications in Image Processing, Inverse Problems and Imaging, 1(3):557-575, 2007.     The PDF file.
   [Keywords: quantization, quantum, quanta set, TV (total variation), discrete programming, Markov gradient descent, maximum likelihood, geometric, photometric, segmentation.]
3. [Invited Book Review] Jackie (Jianhong) Shen, Invited book review for    Deblurring Images--Matrices, Spectra, and Filtering, by Christian Hansen, James G. Nagy, and Dianne P. O'Leary, by SIAM Publisher (2006).     Math. Comput.,   76: 2256-2258, 2007.       The PDF file for the Review.
4. Sung-Ha Kang and Jianhong (Jackie) Shen, Image Dejittering Based on Slicing Moments, in Image Processing Based on Partial Differential Equations, pp. 35-55, Springer Series on Mathematics and Visualization ," Springer, 2007.   Click here for a JPEG sample.     The PDF file.
   [Keywords: BV functions, slicing moments, codimension, inverse problem, Bayesian, regularization, dejitter, existence, uniqueness.]
   
   
5. Sung-Ha Kang and Jianhong Shen, Video Dejittering by Bake and Shake, (UCLA Math CAM Tech. Report 04-60) Image and Vision Computing , 24(2): 143-152, 2006.  Click here for a JPEG sample.
   [Keywords: Line jitteers, Perona-Malik, bake, edge adaptive, total variation, PDE method, Newton-Raphson, shake]
   
   
6. Tony F. Chan and Jianhong Shen, Variational Image Inpainting, Comm. Pure Applied Math., vol. LVIII, pp. 579-619, 2005.     Click here to go to the journal article

7. Jianhong (Jackie) Shen, Progressive Halftoning by Perona-Malik Error Diffusion and Stochastic Flipping, Proc. SPIE on Image Processing: Algorithms and Systems, Neural Networks, and Machine Learning, vol. 6064, pp. 0301-0313, 2006.     The PDF file.
   [Keywords: Halftone, sigma-delta, error diffusion, Perona-Malik, stochastic flipping, blue noise, random fields, parallel computing]
   [  Thanks go to Ingrid Daubechies (Princeton), Sinan Güntürk (Courant), and Chai Wah Wu (IBM)  ]
   
   Also click here for a one-page invited letter exclusively for SPIE's     Electronic Imaging Newsletter, vol. 16, No. 2, page 6, 2005.
   [  Special thanks to Dr. Gabriel Marcu at the Apple Computer, Inc.]

8. Tony F. Chan, Jianhong Shen, and Hao-Min Zhou, Total Variation Wavelet Inpainting, UCLA Comp. Appl. Math. (CAM) Tech. Report   04-47 (pdf), 2004.   J. Math. Imag. Vision, in press.     Click here for a JPEG sample.
   [Keywords: error concealment, JPEG2000, loss, wavelets, total variation, BV space, Besov, geometry, existence, uniqueness, Bayesian]
   
   
9. Jianhong (Jackie) Shen, Γ-Convergence Approximation to Piecewise Constant Mumford-Shah Segmentation, J. Blanc-Talon et al. (Eds.): ACIVS 2005 (Int'l Conf. Advanced Concepts Intell. Vision Systems), Lec. Notes Comp. Sci. 3708 , pp. 499-506, 2005.  
   Click here for a mpeg movie (the Pathfinder),     a jpeg image (the Milky Way),     final PDF file (8-page limit).
   [Keywords: Mumford-Shah segmentation, Gamma convergence, Ginzburg-Landau, phase field approximation, Poisson equation.]
   
   
10. J. Shen, Bayesian video dejittering by the BV image model,   SIAM J. Appl. Math., 64(5), pp. 1691-1708, 2004.   Click here to visit SIAM.     The dedicatory line of the paper:
    ``Dedicated to all pioneering mathematicians in image and vision analysis,
    on whose shoulders we the younger generations are standing.''


11. Jianhong Shen, Inpainting and the fundamental problem of image processing, SIAM News, 36(5), June 2003. (Invited unrefereed survey article for the entire community of applied mathematicians.)
    Related recent work by Smale and Zhou, on sampling and learning theory, Bulletin of the AMS , 2004.
    Interesting tie to digital watermarking by Flesia and Donoho on the theory of digital watermarking, 2003.
    (Also see many other papers citing our inpainting works (as of April, 2004):   click here .)
    
    
12. Tony F. Chan, Jianhong Shen, and Luminita Vese, Variational PDE models in image processing, Notices Amer. Math. Soc. , 50,   pp. 14-26, January 2003.
13. T. F. Chan and J. Shen, On the role of the BV image model in image restoration, Amer. Math. Soc. Contemp. Math., vol. 330, pp. 25-41, 2003. (Research paper refereed by two referees.)
    
14. Tony F. Chan, Sang-Ha Kang and Jianhong Shen, Euler's elastica and curvature-based image inpainting, SIAM J. Appl. Math., 63(2), pp. 564-592, 2002.
15. T. F. Chan and J. Shen, Inpainting based on nonlinear transport and diffusion,  in Inverse Problems, Image Analysis, and Medical Imaging, Ed. Z. Nashed and O. Scherzer, Amer. Math. Soc. Contemp. Math., 313, pp. 53-65, 2002. (Research paper refereed by three referees.)
16. J. Shen, Geometric image inpainting and applications, Proc. SPIE, vol. 4792, pp. 102-113, 2002. (Refereed conference research paper for the associated invited talk at SPIE's 47th annual meeting, Seattle, July 2002. Note: SPIE=Int'l Society of Optical Engineering.)
    
17. Selim Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model, European J. Appl. Math., 13, pp. 353-370, 2002.
18. T. F. Chan and J. Shen, Bayesian inpainting based on geometric image models, Recent Progress in Computational & Applied PDEs, Kluwer Academic, pp. 73-98, 2002. (Refereed conference research paper.)
19. J. Shen, The Mumford-Shah digital filter pair (MS-DFP) and applications, 2002 IEEE Int'l Conf. Image Proc., 2, pp. 849-852, 2002. (Refereed conference research paper.)
    
20. T. F. Chan and J. Shen, Mathematical models of local non-texture inpaintings, SIAM J. Appl. Math., 62(3), pp. 1019-1043, 2001.
21. T. F. Chan and J. Shen, Non-texture inpainting by curvature-driven diffusions (CDD), J. Visual Comm. Image Rep., 12(4), 436-449, 2001.
22. T. F. Chan, S.-H. Kang, and J. Shen, Total variation denoising and enhancement of color images based on the CB and HSV color models, J. Visual  Comm. Image Rep., 12(4), pp. 422-435, 2001.
23. T. F. Chan and J. Shen, PDE models for image inpaintings and applications, Proc. Int'l Conf. Imaging Sci. Sys. Tech.'2001, pp. 30-36, Ed. H. R. Arabnia, 2001. (Refereed conference research paper.)
24. T. F. Chan, S. Osher, and J. Shen, The digital TV filter and nonlinear denoising, IEEE Trans. Image Process., 10(2), pp. 231-241, 2001.
    
25. T. F. Chan and J. Shen, Restoration of non-flat image features: models and algorithms, SIAM J. Appl. Math., 61(4),  pp. 1338-1361, 2000.
26. Stanley Osher and Jianhong Shen, Digitized PDE method for data restoration, in Analytical-Computational Methods in Applied Mathematics, Ed. G. A. Anastassiou, pp. 751-771, Chapman & Hall/CRC, FL, 2000.
    
27. Jianhong Shen, Gilbert Strang, and Andy Wathen, The potential theory of several intervals and its applications, Appl. Math. Optim., 44, 67-85, 2001.
    [Keywords: equiripple optimal filters, asymptotic convergence of matrix iterations, indefinite Stokes equation in fluid dynamics, Schwarz-Christoffel mapping. Thanks go to Nick Trefethen and Mark Embree.]
28. J. Shen and G. Strang, Asymptotics of optimal (equiripple) filters, IEEE Trans. Signal Process., 47(4), pp. 1087-1098, 1999.
    [Keywords: Kaiser's (Bell Lab) formula, Chebyshev polynomials on multi-intervals, Remez-Parks-McClellan algorithm. Thanks go to Alan Oppenheim's group.]
29. J. Shen, A note on wavelets and diffusions, J. Comp. Anal. Appl., 5, pp. 147-159, 2003.
30. J. Shen, On some quantum and analytical properties of fractional Fourier transforms, in Wavelet Analysis: Twenty Year's Developments, Ed. D.-X. Zhou, World Scientific, pp. 252-265, 2002.
31. J. Shen, Compactification of a set of matrices with convergent infinite products, Linear Alg. Appl., 311, pp. 177-186, 2000.
    [Keywords: wavelets matrices, cascading, Rota-Strang spectral radius, König chain.]
32. Jianhong Shen and Gilbert Strang, On wavelet fundamental solutions to the heat equation - heatlets, J. Diff. Eqn., 161(2), 403-421, 2000.
    [Keywords: fundamental solution, heat equation, scaling and translation invariance.]
33. J. Shen, Combinatorics for wavelets: umbral refinement equation, Studies Appl. Math., 103(2), 121-147, 1999.
34. T. Cai and J. Shen, Boundedness is redundant in a theorem of Daubechies, Appl. Comp. Harmon. Anal., 6(3), 400-404, 1999.
    [Keywords: orthogonality and vanishing moments.]
35. J. Shen, Refinement differential equations and wavelets, Methods Appl. Anal., 5(3), pp. 283-316, 1998.
36. J. Shen and G. Strang, Asymptotics of Daubechies filters, scaling functions, and wavelets, Appl. Comp. Harmon. Anal., 5(3), 312-331, 1998.
    [Keywords: pure phase, asymptotics, stationary phase, Airy function, multiscale transition. Thanks go to H. Cheng.]
37. J. Shen and G. Strang, Asymptotic analysis of Daubechies polynomials, Proc. Amer. Math. Soc., 124(12), 3819-3833, 1996.
    [Keywords: maxflat, lowpass, spectral factorization, zeros and vanishing moments, Szegö's asymptotics, error function. ]

§3 Applied Probability & Stochastic Methods: Random Algebraic Variables (Umbrae) and Matrices

1. J. Shen, On the singular values of Gaussian random matrices, Linear Alg. Appl., 326(1-3), 1-14, 2001.
   [Keywords: random matrices, singular values, Gaussian ensemble, Wishart ensemble, thermodynamic limit, pseudo-Coulomb gas, circle law, and quadrant law.]
2. Gian-Carlo Rota and Jianhong Shen, On the combinatorics of cumulants, J. Comb. Theory (A) , 91(1), 283-304, 2000.
   [Keywords:  cumulants, umbrae, exponential (moment) generating function, Schur symmetric functions, orthogonal polynomials, binomial sequence, Moebius inversion. In memory of my beloved mentor and friend - Gian-Carlo Rota.]
3. J. Shen, A geometric approach to ergodic non-homogeneous Markov chains, in Wavelet Analysis and Multiresolution Methods, Ed. T.-X. He, Lecture Notes in Pure and Applied Mathematics, 212, Marcel Dekker, pp. 341-366, 2000.
4. J. Shen, On the generalized "Lanczo's generalized derivative," Amer. Math. Monthly, 106(8), pp. 766-768, 1999.
5. G.-C. Rota, J. Shen, and B. D. Taylor, All polynomials of binomial type are represented by Abel polynomials, Ann. Scuola Norm. Sup. Pisa. Cl. Sci. (IV), 25(3-4), pp. 731-738, 1998.
   [Keywords: moment generating functions, symbolic random variables (i.e. umbrae), binomial polynomials, Lagrange inversion. In memory of De Giorgi.]