Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Mathematical methods in medical image processing


Authors: Sigurd Angenent, Eric Pichon and Allen Tannenbaum
Journal: Bull. Amer. Math. Soc. 43 (2006), 365-396
MSC (2000): Primary 92C55, 94A08, 68T45; Secondary 35K55, 35K65
DOI: https://doi.org/10.1090/S0273-0979-06-01104-9
Published electronically: April 28, 2006
MathSciNet review: 2223011
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we describe some central mathematical problems in medical imaging. The subject has been undergoing rapid changes driven by better hardware and software. Much of the software is based on novel methods utilizing geometric partial differential equations in conjunction with standard signal/image processing techniques as well as computer graphics facilitating man/machine interactions. As part of this enterprise, researchers have been trying to base biomedical engineering principles on rigorous mathematical foundations for the development of software methods to be integrated into complete therapy delivery systems. These systems support the more effective delivery of many image-guided procedures such as radiation therapy, biopsy, and minimally invasive surgery. We will show how mathematics may impact some of the main problems in this area, including image enhancement, registration, and segmentation.


References [Enhancements On Off] (What's this?)

  • 1. L. Alvarez, F. Guichard, P. L. Lions, and J-M. Morel, Axiomes et équations fondamentales du traitement d`images, C. R. Acad. Sci. Paris 315 (1992), 135-138. MR 1197224 (94d:47066)
  • 2. -, Axioms and fundamental equations of image processing, Arch. Rat. Mech. Anal. 123 (1993), no. 3, 199-257. MR 1225209 (94j:68306)
  • 3. L. Alvarez, P. L. Lions, and J-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal. 29 (1992), 845-866. MR 1163360 (93a:35072)
  • 4. L. Alvarez and J-M. Morel, Formalization and computational aspects of image analysis, Acta Numerica 3 (1994), 1-59. MR 1288095 (95c:68229)
  • 5. L. Ambrosio, A compactness theorem for a special class of functions of bounded variation, Boll. Un. Math. It. 3-B (1989), 857-881. MR 1032614 (90k:49005)
  • 6. -, Lecture notes on optimal transport theory, Euro Summer School, Mathematical Aspects of Evolving Interfaces, CIME Series of Springer Lecture Notes, Springer, July 2000.
  • 7. S. Ando, Consistent gradient operators, IEEE Transactions on Pattern Analysis and Machine Intelligence 22 (2000), no. 3, 252-265.
  • 8. S. Angenent, S. Haker, and A. Tannenbaum, Minimizing flows for the Monge-Kantorovich problem, SIAM J. Math. Anal. 35 (2003), no. 1, 61-97 (electronic). MR 2001465 (2004g:49070)
  • 9. S. Angenent, G. Sapiro, and A. Tannenbaum, On the affine heat flow for non-convex curves, J. Amer. Math. Soc. 11 (1998), no. 3, 601-634. MR 1491538 (99d:58039)
  • 10. J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numerische Mathematik 84 (2000), 375-393. MR 1738163 (2000m:65111)
  • 11. -, Mixed $ L^2$-Wasserstein optimal mapping between prescribed density functions, J. Optimization Theory Applications 111 (2001), 255-271. MR 1865668 (2002h:49069)
  • 12. Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), 375-417. MR 1100809 (92d:46088)
  • 13. D. Brooks, Emerging medical imaging modalities, IEEE Signal Processing Magazine 18 (2001), no. 6, 12-13.
  • 14. J. Canny, Computational approach to edge detection, IEEE Transactions on Pattern Analysis and Machine Intelligence 8 (1986), no. 6, 679-698.
  • 15. V. Caselles, F. Catte, T. Coll, and F. Dibos, A geometric model for active contours in image processing, Numerische Mathematik 66 (1993), 1-31. MR 1240700 (94k:65195)
  • 16. V. Caselles, R. Kimmel, and G. Sapiro, Geodesic active contours, International Journal of Computer Vision 22 (1997), no. 11, 61-79.
  • 17. V. Caselles, J. Morel, G. Sapiro, and A. Tannenbaum, Introduction to the special issue on partial differential equations and geometry-driven diffusion in image processing and analysis, IEEE Trans. on Image Processing 7 (1998), no. 3, 269-273.
  • 18. F. Chabat, D.M. Hansell, and Guang-Zhong Yang, Computerized decision support in medical imaging, IEEE Engineering in Medicine and Biology Magazine 19 (2000), no. 5, 89-96.
  • 19. T. Chan and L. Vese, Active contours without edges, IEEE Trans. Image Processing 10 (2001), 266-277.
  • 20. T.F. Chan, J. Shen, and L. Vese, Variational PDE models in image processing, Notices of AMS 50 (2003), no. 1, 14-26. MR 1948832 (2003m:94008)
  • 21. Y. G. Chen, Y. Giga, and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geometry 33 (1991), 749-786. MR 1100211 (93a:35093)
  • 22. K.-S. Chou and X.-P. Zhu, The curve shortening problem, Chapman & Hall/CRC, Boca Raton, 2001. MR 1888641 (2003e:53088)
  • 23. L. D. Cohen, On active contour models and balloons, CVGIP: Image Understanding 53 (1991), no. 2, 211-218.
  • 24. C. L. Epstein and M. Gage, The curve shortening flow, Wave Motion: Theory, Modeling and Computation (A. Chorin and A. Majda, eds.), Springer-Verlag, New York, 1987. MR 0920831 (89f:58128)
  • 25. L. C. Evans and J. Spruck, Motion of level sets by mean curvature, I, J. Differential Geometry 33 (1991), no. 3, 635-681. MR 1100206 (92h:35097)
  • 26. J.R. Fram and E.S. Deutsch, On the quantitative evaluation of edge detection schemes and their comparisions with human performance, IEEE Transaction on Computers 24 (1975), no. 6, 616-627.
  • 27. D. Fry, Shape recognition using metrics on the space of shapes, Ph.D. thesis, Harvard University, 1993.
  • 28. M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geometry 23 (1986), 69-96. MR 0840401 (87m:53003)
  • 29. W. Gangbo and R. McCann, The geometry of optimal transportation, Acta Math. 177 (1996), 113-161. MR 1440931 (98e:49102)
  • 30. E.S. Gerson, Scenes from the past: X-Ray mania, the X-Ray in advertising, circa 1895, Radiographics 24 (2004), 544-551.
  • 31. E. Giusti, Minimal surfaces and functions of bounded variation, Birkhäuser-Verlag, 1984. MR 0775682 (87a:58041)
  • 32. R. Gonzalez and R. Woods, Digital image processing, Prentice Hall, 2001.
  • 33. M. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geometry 26 (1987), 285-314. MR 0906392 (89b:53005)
  • 34. -, Shortening embedded curves, Annals of Mathematics 129 (1989), 71-111. MR 0979601 (90a:53050)
  • 35. F. Guichard, L. Moisan, and J.M. Morel, A review of PDE models in image processing and image analysis, Journal de Physique IV (2002), no. 12, 137-154.
  • 36. S.R. Gunn, On the discrete representation of the Laplacian of Gaussian, Pattern Recognition 32 (1999), no. 8, 1463-1472.
  • 37. J. Hajnal, D.J. Hawkes, D. Hill, and J.V. Hajnal (eds.), Medical image registration, CRC Press, 2001.
  • 38. S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, Optimal mass transport for registration and warping, Int. Journal Computer Vision 60 (2004), no. 3, 225-240.
  • 39. R. Haralick and L. Shapiro, Computer and robot vision, Addison-Wesley, 1992.
  • 40. S. Helgason, The Radon transform, Birkhäuser, Boston, MA, 1980. MR 0573446 (83f:43012)
  • 41. W. Hendee and R. Ritenour, Medical imaging physics, 4th ed., Wiley-Liss, 2002.
  • 42. A.O. Hero and H. Krim, Mathematical methods in imaging, IEEE Signal Processing Magazine 19 (2002), no. 5, 13-14.
  • 43. R. Hobbie, Intermediate physics for medicine and biology (third edition), Springer, New York, 1997.
  • 44. B.K.P. Horn, Robot vision, MIT Press, 1986.
  • 45. G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geometry 20 (1984), 237-266. MR 0772132 (86j:53097)
  • 46. R. Hummel, Representations based on zero-crossings in scale-space, IEEE Computer Vision and Pattern Recognition, 1986, pp. 204-209.
  • 47. Insight Segmentation and Registration Toolkit, http:itk.org.
  • 48. B. Julesz, Textons, the elements of texture perception, and their interactions, Nature 12 (1981), no. 290, 91-97.
  • 49. L. V. Kantorovich, On a problem of Monge, Uspekhi Mat. Nauk. 3 (1948), 225-226.
  • 50. S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi, Conformal curvature flows: from phase transitions to active vision, Arch. Rational Mech. Anal. 134 (1996), no. 3, 275-301. MR 1412430 (97j:58023)
  • 51. M. Knott and C. Smith, On the optimal mapping of distributions, J. Optim. Theory 43 (1984), 39-49. MR 0745785 (86a:60026)
  • 52. J. J. Koenderink, The structure of images, Biological Cybernetics 50 (1984), 363-370. MR 0758126
  • 53. W. Köhler, Gestalt psychology today, American Psychologist 14 (1959), 727-734.
  • 54. S. Osher L. I. Rudin and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), 259-268.
  • 55. H. Ishii M. G. Crandall and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society 27 (1992), 1-67. MR 1118699 (92j:35050)
  • 56. A. Witkin M. Kass and D. Terzopoulos, Snakes: active contour models, Int. Journal of Computer Vision 1 (1987), 321-331.
  • 57. F. Maes, A. Collignon, D. Vandermeulen, G. Marchal, and P. Suetens, Multimodality image registration by maximization of mutual information, IEEE Transactions on Medical Imaging 16 (1997), no. 2, 187-198.
  • 58. J. Maintz and M. Viergever, A survey of medical image registration, Medical Image Analysis 2 (1998), no. 1, 1-36.
  • 59. S. Mallat, A wavelet tour of signal processing, Academic Press, San Diego, CA, 1998. MR 1614527 (99m:94012)
  • 60. D. Marr, Vision, Freeman, San Francisco, 1982.
  • 61. D. Marr and E. Hildreth, Theory of edge detection, Proc. Royal Soc. Lond. B (1980), no. 207, 187-217.
  • 62. R. McCann, A convexity theory for interacting gases and equilibrium crystals, Ph.D. Thesis, Princeton University, 1994.
  • 63. T. McInerney and D. Terzopoulos, Topologically adaptable snakes, Int. Conf. on Computer Vision (Cambridge, MA), June 1995, pp. 840-845.
  • 64. -, Deformable models in medical image analysis: a survey, Medical Image Analysis 1 (1996), no. 2, 91-108.
  • 65. J. Milnor, Morse theory, Princeton University Press, 1963. MR 0163331 (29:634)
  • 66. J-M. Morel and S. Solimini, Variational methods in image segmentation, Birkhäuser, Boston, 1995. MR 1321598 (96b:68184)
  • 67. D. Mumford, Geometry-driven diffusion in computer vision, ch. The Bayesian Rationale for Energy Functionals, pp. 141-153, Kluwer Academic Publishers, 1994. MR 1339987 (97e:68147)
  • 68. D. Mumford and J. Shah, Boundary detection by minimizing functionals, IEEE Conference on Computer Vision and Pattern Recognition, 1985, pp. 22-26.
  • 69. -, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 42 (1989), no. 5, 577-685. MR 0997568 (90g:49033)
  • 70. S. Osher and R. P. Fedkiw, Level set methods: An overview and some recent results, Journal of Computational Physics 169 (2001), 463-502. MR 1836523 (2002c:65255)
  • 71. S. J. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Computational Physics 79 (1988), 12-49. MR 0965860 (89h:80012)
  • 72. Jacob Palis, Jr. and Welington de Melo, Geometric theory of dynamical systems. An introduction. Translated from the Portuguese by A. K. Manning, Springer-Verlag, New York, 1982. MR 0669541 (84a:58004)
  • 73. G.P. Penney, J. Weese, J.A. Little, P. Desmedt, D.L.O Hill, and D.J. Hawkes, A comparison of similarity measures for use in 2-D-3-D medical image registration, IEEE Transactions on Medical Imaging 17 (1998), no. 4, 586-595.
  • 74. P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Machine Intell. 12 (1990), 629-639.
  • 75. E. Pichon, A. Tannenbaum, and R. Kikinis, Statistically based flow for image segmentation, Medical Imaging Analysis 8 (2004), 267-274.
  • 76. J.P.W Pluim and J.M. Fitzpatrick (Editors), Special issue on image registration, IEEE Transactions on Medical Imaging 22 (2003), no. 11.
  • 77. J.P.W Pluim, J.B.A. Maintz, and M.A. Viergever, Mutual-information-based registration of medical images: a survey, IEEE Transactions on Medical Imaging 22 (2003), no. 8, 986-1004.
  • 78. J. Sethian R. Malladi and B. Vemuri, Shape modeling with front propagation: a level set approach, IEEE Trans. Pattern Anal. Machine Intell. 17 (1995), 158-175.
  • 79. S. Rachev and L. Rüschendorf, Mass transportation problems, Springer, 1998.
  • 80. Radiology Centennial Inc., A century of radiology, http:www.xray.hmc.psu.edu/rci/centennial.html.
  • 81. L. Roberts, Optical and electro-optical information processing, ch. Machine perception of 3-D solids, MIT Press, 1965.
  • 82. W.C. Roentgen, Ueber eine neue Art von Strahlen, Annalen der Physik 64 (1898), 1-37.
  • 83. G. Sapiro, Geometric partial differential equations and image analysis, Cambridge University Press, Cambridge, 2001. MR 1813971 (2002a:68142)
  • 84. G. Sapiro and A. Tannenbaum, Affine invariant scale-space, International Journal of Computer Vision 11 (1993), no. 1, 25-44.
  • 85. -, On invariant curve evolution and image analysis, Indiana Univ. Math. J. 42 (1993), no. 3, 985-1009. MR 1254129 (94m:58048)
  • 86. -, On affine plane curve evolution, Journal of Functional Analysis 119 (1994), no. 1, 79-120. MR 1255274 (94m:58049)
  • 87. J.A. Sethian, Levelset methods and fast marching methods, Cambridge University Press, 1999. MR 1684542 (2000m:65125)
  • 88. K. Siddiqi, Y. Lauziere, A. Tannenbaum, and S. Zucker, Area and length minimizing flows for shape segmentation, IEEE TMI 7 (1998), 433-443.
  • 89. L. Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, Canberra, 1983. MR 0756417 (87a:49001)
  • 90. I.E. Sobel, Camera models and machine perception, Ph.D. thesis, Stanford Univ., 1970.
  • 91. M. Sonka, V. Hlavac, and R. Boyle, Image processing: Analysis and machine vision, 2nd ed., Brooks Cole, 1998.
  • 92. 3D Slicer, http://slicer.org.
  • 93. A. Toga, Brain warping, Academic Press, San Diego, 1999.
  • 94. A. Tsai, A. Yezzi, and A. Willsky, A curve evolution approach to smoothing and segmentation using the Mumford-Shah functional, CVPR (2000), 1119-1124.
  • 95. R. von de Heydt and E. Peterhans, Illusory contours and cortical neuron responses, Science 224 (1984), no. 4654, 1260-1262.
  • 96. B. White, Some recent developments in differential geometry, Mathematical Intelligencer 11 (1989), 41-47. MR 1016106 (90k:53003)
  • 97. A. P. Witkin, Scale-space filtering, Int. Joint. Conf. Artificial Intelligence (1983), 1019-1021.
  • 98. L. Zhu, On visualizing branched surfaces: An angle/area preserving approach, Ph.D. thesis, Department of Biomedical Engineering, Georgia Institute of Technology, 2004.

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 92C55, 94A08, 68T45, 35K55, 35K65

Retrieve articles in all journals with MSC (2000): 92C55, 94A08, 68T45, 35K55, 35K65


Additional Information

Sigurd Angenent
Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
Email: angenent@math.wisc.edu

Eric Pichon
Affiliation: Department of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0250
Email: eric@ece.gatech.edu.

Allen Tannenbaum
Affiliation: Departments of Electrical and Computer and Biomedical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0250
Email: tannenba@ece.gatech.edu.

DOI: https://doi.org/10.1090/S0273-0979-06-01104-9
Keywords: Medical imaging, artificial vision, smoothing, registration, segmentation
Received by editor(s): June 15, 2005
Received by editor(s) in revised form: September 22, 2005
Published electronically: April 28, 2006
Additional Notes: The authors would like to thank Steven Haker, Ron Kikinis, Guillermo Sapiro, Anthony Yezzi, and Lei Zhu for many helpful conversations on medical imaging and to Bob McElroy for proofreading the final document.
This research was supported by grants from the NSF, NIH (NAC P41 RR-13218 through Brigham and Women’s Hospital), and the Technion, Israel Institute of Technology. This work was done under the auspices of the National Alliance for Medical Image Computing (NAMIC), funded by the National Institutes of Health through the NIH Roadmap for Medical Research, Grant U54 EB005149.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society