Skip to main content
SpringerLink
Log in

Fast Regularization of Matrix-Valued Images

  • Conference paper
  • First Online:
Efficient Algorithms for Global Optimization Methods in Computer Vision

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 8293))

Abstract

Regularization of matrix-valued data is important in many fields, such as medical imaging, motion analysis and scene understanding, where accurate estimation of diffusion tensors or rigid motions is crucial for higher-level computer vision tasks. In this chapter we describe a novel method for efficient regularization of matrix- and group-valued images. Using the augmented Lagrangian framework we separate the total-variation regularization of matrix-valued images into a regularization and projection steps, both of which are fast and parallelizable. Furthermore we extend our method to a high-order regularization scheme for matrix-valued functions. We demonstrate the effectiveness of our method for denoising of several group-valued image types, with data in \(SO(n)\), \(SE(n)\), and \(SPD(n)\), and discuss its convergence properties.

This research was supported by European Community’s FP7-ERC program, grant agreement no. 267414.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 44.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Fingerprints Verification Competition database

    Google Scholar 

  2. Ali, S., Shah, M.: A Lagrangian particle dynamics approach for crowd flow segmentation and stability analysis. In: Computer Vision and Pattern Recognition, pp. 1–6 (2007)

    Google Scholar 

  3. Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka-Lojasiewicz inequality. Math. Oper. Res. 35, 438–457 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Basser, P.J., Mattiello, J., LeBihan, D.: MR diffusion tensor spectroscopy and imaging. Biophys. J. 66(1), 259–267 (1994)

    Article  Google Scholar 

  5. Basu, S., Fletcher, T., Whitaker, R.T.: Rician noise removal in diffusion tensor MRI. In: Larsen, R., Nielsen, M., Sporring, J. (eds.) MICCAI 2006. LNCS, vol. 4190, pp. 117–125. Springer, Heidelberg (2006)

    Google Scholar 

  6. Bayro-Corrochano, E., Ortegón-Aguilar, J.: Lie algebra approach for tracking and 3D motion estimation using monocular vision. Image Vision Comput. 25, 907–921 (2007)

    Article  Google Scholar 

  7. Bergmann, Ø., Christiansen, O., Lie, J., Lundervold, A.: Shape-adaptive DCT for denoising of 3D scalar and tensor valued images. J. Digit. Imaging 22(3), 297–308 (2009)

    Article  Google Scholar 

  8. Besl, P.J., McKay, N.D.: A method for registration of 3D shapes. IEEE Trans. Pattern Anal. Mach. Intell. 14(2), 239–256 (1992)

    Article  Google Scholar 

  9. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  10. Del Bue, A., Xavier, J., Agapito, L., Paladini, M.: Bilinear factorization via augmented lagrange multipliers. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) ECCV 2010, Part IV. LNCS, vol. 6314, pp. 283–296. Springer, Heidelberg (2010)

    Google Scholar 

  11. Burgeth, B., Didas, S., Florack, L.M.J., Weickert, J.: A generic approach to the filtering of matrix fields with singular PDEs. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 556–567. Springer, Heidelberg (2007)

    Google Scholar 

  12. Celledoni, E., Owren, B.: Lie group methods for rigid body dynamics and time integration on manifolds. Comput. Meth. Appl. Mech. Eng. 19, 421–438 (1999)

    MathSciNet  Google Scholar 

  13. Chen, B., Hsu, E.W.: Noise removal in magnetic resonance diffusion tensor imaging. Magn. Reson. Med.: Official J. Soc. Magn. Reson. Med./Soc. Magn. Reson. Med. 54(2), 393–401 (2005)

    Article  Google Scholar 

  14. Deriche, R., Tschumperle, D., Lenglet, C.: DT-MRI estimation, regularization and fiber tractography. In: ISBI, pp. 9–12 (2004)

    Google Scholar 

  15. Duits, R., Burgeth, B.: Scale spaces on lie groups. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 300–312. Springer, Heidelberg (2007)

    Google Scholar 

  16. Dumortier, Y., Herlin, I., Ducrot, A.: 4D tensor voting motion segmentation for obstacle detection in autonomous guided vehicle. In: IEEE Int. Vehicles Symp., pp. 379–384 (2008)

    Google Scholar 

  17. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. CMS Books in Mathematics. SIAM, Philadelphia (1999)

    Book  MATH  Google Scholar 

  18. Fletcher, R.: Semi-definite matrix constraints in optimization. SIAM J. Cont. Optim. 23(4), 493–513 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gibson, W.: On the least-squares orthogonalization of an oblique transformation. Psychometrika 27, 193–195 (1962)

    Article  MathSciNet  Google Scholar 

  20. Gur, Y., Sochen, N.A.: Regularizing flows over lie groups. J. Math. Imaging Vis. 33(2), 195–208 (2009)

    Article  MathSciNet  Google Scholar 

  21. Hall, B.C.: Lie Groups, Lie Algebras, and Representations. An Elementary Introduction. Springer, New York (2004)

    Google Scholar 

  22. Hesteness, M.R.: Multipliers and gradient methods. J. Optim. Theor. Appl. 4, 303–320 (1969)

    Article  Google Scholar 

  23. Higham, N.J.: Matrix nearness problems and applications. In: Applications of Matrix Theory, pp. 1–27. Oxford University Press, Oxford (1989)

    Google Scholar 

  24. Iserles, A., Munthe-kaas, H.Z., Nørsett, S.P., Zanna, A.: Lie group methods. Acta Numerica 9, 215–365 (2000)

    Article  Google Scholar 

  25. Kimmel, R., Sochen, N.: Orientation diffusion or how to comb a porcupine. J. Vis. Commun. Image Representation 13, 238–248 (2002). (special issue on PDEs in Image Processing, Computer Vision, and Computer Graphics)

    Article  Google Scholar 

  26. Kiryati, N., Riklin-Raviv, T., Ivanchenko, Y., Rochel, S.: Real-time abnormal motion detection in surveillance video. In: ICPR, pp. 1–4 (2008)

    Google Scholar 

  27. Koay, C., Carew, J., Alexander, A., Basser, P., Meyerand, M.: Investigation of anomalous estimates of tensor-derived quantities in diffusion tensor imaging. Magn. Reson. Med. 55, 930–936 (2006)

    Article  Google Scholar 

  28. Kobilarov, M., Crane, K., Desbrun, M.: Lie group integrators for animation and control of vehicles. ACM Trans. Graph. 28(2), 1–14 (2009)

    Article  Google Scholar 

  29. Larochelle, P.M., Murray, A.P., Angeles, J.: SVD and PD based projection metrics on SE(N). In: Lenarčič, J., Galletti, C. (eds.) On Advances in Robot Kinematics, pp. 13–22. Kluwer, Dordrecht (2004)

    Chapter  Google Scholar 

  30. Lenglet, C., Campbell, J.S.W., Descoteaux, M., Haro, G., Savadjiev, P., Wassermann, D., Anwander, A., Deriche, R., Pike, G.B., Sapiro, G.: Mathematical methods for diffusion MRI processing. Neuroimage 45(1), S111–S122 (2009)

    Article  Google Scholar 

  31. Lin, D., Grimson, W., Fisher, J.: Learning visual flows: a Lie algebraic approach. In: Computer Vision and Pattern Recognition, pp. 747–754 (2009)

    Google Scholar 

  32. Lundervold, A.: On consciousness, resting state fMRI, and neurodynamics. Nonlinear Biomed. Phys. 4(Suppl. 1), S9–S18 (2010)

    Article  Google Scholar 

  33. Manton, J.H.: Optimization algorithms exploiting unitary constraints. IEEE Trans. Signal Process. 50(3), 635–650 (2002)

    Article  MathSciNet  Google Scholar 

  34. Mehran, R., Moore, B.E., Shah, M.: A streakline representation of flow in crowded scenes. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) ECCV 2010, Part III. LNCS, vol. 6313, pp. 439–452. Springer, Heidelberg (2010)

    Google Scholar 

  35. Moisan, L.: Perspective invariant movie analysis for depth recovery. Proc. SPIE 2567, 84–94 (1995)

    Google Scholar 

  36. Nicolescu, M., Medioni, G.: A voting-based computational framework for visual motion analysis and interpretation. IEEE Trans. Pattern Anal. Mach. Intell. 27, 739–752 (2005)

    Article  Google Scholar 

  37. Park, F.C., Bobrow, J.E., Ploen, S.R.: A lie group formulation of robot dynamics. Int. J. Rob. Res. 14, 609–618 (1995)

    Article  Google Scholar 

  38. Park, W., Liu, Y., Zhou, Y., Moses, M., Chirikjian, G.S.: Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map. Robotica 26, 419–434 (2008)

    Article  MATH  Google Scholar 

  39. Pennec, X., Fillard, P., Ayache, N.: A riemannian framework for tensor computing. Int. J. Comput. Vis. 66(1), 41–66 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  40. Perona, P.: Orientation diffusions. IEEE Trans. Image Process. 7(3), 457–467 (1998)

    Article  Google Scholar 

  41. Powell, M.J.: A method for nonlinear constraints in minimization problems. In: Optimization, pp. 283–298. Academic Press (1969)

    Google Scholar 

  42. Rahman, I.U., Drori, I., Stodden, V.C., Donoho, D.L., Schroeder, P.: Multiscale representations of manifold-valued data. Technical report, Stanford (2005)

    Google Scholar 

  43. Raptis, M., Soatto, S.: Tracklet descriptors for action modeling and video analysis. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) ECCV 2010, Part I. LNCS, vol. 6311, pp. 577–590. Springer, Heidelberg (2010)

    Google Scholar 

  44. Rosman, G., Bronstein, M.M., Bronstein, A.M., Wolf, A., Kimmel, R.: Group-valued regularization framework for motion segmentation of dynamic non-rigid shapes. In: Bronstein, A.M., Bronstein, M.M., Bruckstein, A.M., Haar Romeny, B.M. (eds.) SSVM 2011. LNCS, vol. 6667, pp. 725–736. Springer, Heidelberg (2012)

    Google Scholar 

  45. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D Lett. 60, 259–268 (1992)

    MATH  Google Scholar 

  46. Sagiv, C., Sochen, N.A., Kimmel, R.: Stereographic combing a porcupine or studies on direction diffusion in image processing. SIAM J. Appl. Math. 64(5), 1477–1508 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  47. Salvador, R., Pena, A., Menon, D.K., Carpenter, T., Pickard, J., Bullmore, E.: Formal characterization and extension of the linearized diffusion tensor model. Hum. Brain Mapp. 24(2), 144–155 (2005)

    Article  Google Scholar 

  48. Steidl, G., Setzer, S., Popilka, B., Burgeth, B.: Restoration of matrix fields by second-order cone programming. Computing 81(2–3), 161–178 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  49. Stejskal, E.O., Tanner, J.E.: Spin diffusion measurements: spin echoes in the presence of a time-dependent field gradient. J. Chem. Phys. 42, 288–292 (1965)

    Article  Google Scholar 

  50. Stillwell, J.: Naive Lie Theory. Undergraduate texts in mathematics. Springer, New York (2008)

    Book  MATH  Google Scholar 

  51. Tai, X.-C., Wu, C.: Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 502–513. Springer, Heidelberg (2009)

    Google Scholar 

  52. Tang, B., Sapiro, G., Caselles, V.: Diffusion of general data on non-flat manifolds viaharmonic maps theory: the direction diffusion case. Int. J. Comput. Vis. 36, 149–161 (2000)

    Article  Google Scholar 

  53. Tschumperlé, D., Deriche, R.: Vector-valued image regularization with pdes: a common framework for different applications. IEEE Trans. Pattern Anal. Mach. Intell. 27, 506–517 (2005)

    Article  Google Scholar 

  54. Tseng, P.: Coordinate ascent for maximizing nondifferentiable concave functions. LIDS-P 1940, MIT (1988)

    Google Scholar 

  55. Turaga, P., Veeraraghavan, A., Srivastava, A., Chellappa, R.: Statistical computations on grassmann and stiefel manifolds for image and video-based recognition. IEEE Trans. Pattern Anal. Mach. Intell. 33, 2273–2286 (2011)

    Article  Google Scholar 

  56. Tuzel, O., Porikli, F., Meer, P.: Learning on lie groups for invariant detection and tracking, In: Computer Vision and Pattern Recognition (2008)

    Google Scholar 

  57. Vemuri, B.C., Chen, Y., Rao, M., McGraw, T., Wang, Z., Mareci, T.: Fiber tract mapping from diffusion tensor MRI. In: Proceedings of the International Conference on Variational, Geometry and Level Sets Methods in Computer Vision, pp. 81–88. IEEE Computer Society (2001)

    Google Scholar 

  58. Vese, L.A., Osher, S.J.: Numerical methods for p-Harmonic flows and applications to image processing. SIAM J. Numer. Anal. 40(6), 2085–2104 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  59. Žefran, M., Kumar, V., Croke, C.: On the generation of smooth three-dimensional rigid body motions. IEEE Trans. Robot. Autom. 14(4), 576–589 (1998)

    Article  Google Scholar 

  60. Žefran, M., Kumar, V., Croke, C.: Metrics and connections for rigid-body kinematics. I. J. Robotic Res. 18(2), 242 (1999)

    Google Scholar 

  61. Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imag. Sci. 1(3), 248–272 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  62. Weickert, J., Brox, T.: Diffusion and regularization of vector- and matrix-valued images. Inverse problems, image analysis, and medical imaging, vol. 313 (2002)

    Google Scholar 

  63. Wen, Z., Goldfarb, D., Yin, W.: Alternating direction augmented Lagrangian methods for semidefinite programming. CAAM TR09-42, Rice University (2009)

    Google Scholar 

  64. Westin, C.-F., Peled, S., Gudbjartsson, H., Kikinis, R., Jolesz, F.A.: Geometrical diffusion measures for MRI from tensor basis analysis. In: ISMRM (1997)

    Google Scholar 

  65. Wiest-Daesslé, N., Prima, S., Coupé, P., Morrissey, S.P., Barillot, Ch.: Non-local means variants for denoising of diffusion-weighted and diffusion tensor MRI. In: Ayache, N., Ourselin, S., Maeder, A. (eds.) MICCAI 2007, Part II. LNCS, vol. 4792, pp. 344–351. Springer, Heidelberg (2007)

    Google Scholar 

  66. Wu, C., Tai, X.-C.: Augmented lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models. SIAM J. Imaging Sci. 3(3), 300–339 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  67. Wu, C., Zhang, J., Tai, X.: Augmented lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Prob. Imaging 5(1), 237–261 (2011)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Technion, 32000, Haifa, Israel

    Guy Rosman, Yu Wang, Ron Kimmel & Alfred M. Bruckstein

  2. Department of Mathematics, University of Bergen, Johaness Brunsgate 12, 5007, Bergen, Norway

    Xue-Cheng Tai

Authors
  1. Guy Rosman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xue-Cheng Tai
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ron Kimmel
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Alfred M. Bruckstein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guy Rosman .

Editor information

Editors and Affiliations

  1. University of Stuttgart, Stuttgart, Germany

    Andrés Bruhn

  2. Graz University of Technology, Graz, Austria

    Thomas Pock

  3. University of Bergen, Bergen, Norway

    Xue-Cheng Tai

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Rosman, G., Wang, Y., Tai, XC., Kimmel, R., Bruckstein, A.M. (2014). Fast Regularization of Matrix-Valued Images. In: Bruhn, A., Pock, T., Tai, XC. (eds) Efficient Algorithms for Global Optimization Methods in Computer Vision. Lecture Notes in Computer Science(), vol 8293. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54774-4_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-54774-4_2

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-54773-7

  • Online ISBN: 978-3-642-54774-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation