We study the extension of total variation (TV), total deformation (TD), and (second-order) total generalized variation ($\TGV^2$) to symmetric tensor fields. We show that for a suitable choice of finite-dimensional norm, these variational seminorms are rotation-invariant in a sense natural and well suited for application to diffusion tensor imaging (DTI). Combined with a positive definiteness constraint, we employ these novel seminorms as regularizers in Rudin--Osher--Fatemi (ROF) type denoising of medical in vivo brain images. For the numerical realization, we employ the Chambolle--Pock algorithm, for which we develop a novel duality-based stopping criterion which guarantees error bounds with respect to the functional values. Our findings indicate that TD and $\TGV^2$, both of which employ the symmetrized differential, provide improved results compared to other evaluated approaches.


  1. total variation
  2. total deformation
  3. total generalized variation
  4. regularization
  5. medical imaging
  6. diffusion tensor imaging

MSC codes

  1. 92C55
  2. 94A08
  3. 26B30
  4. 49M29

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S. Aja-Fernández, M. Niethammer, M. Kubicki, M. Shenton, and C. Westin, Restoration of DWI data using a Rician LMMSE estimator, IEEE Trans. Medical Imaging, 27 (2008), pp. 1389--1403.
M. Aksoy, C. Forman, M. Straka, S. Skare, S. Holdsworth, J. Hornegger, and R. Bammer, Real-time optical motion correction for diffusion tensor imaging, Magnetic Resonance in Medicine, 66 (2011), pp. 366--378.
M. Aksoy, C. Liu, M. E. Moseley, and R. Bammer, Single-step nonlinear diffusion tensor estimation in the presence of microscopic and macroscopic motion, Magnetic Resonance in Medicine, 59 (2008), pp. 1138--1150.
L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, UK, 2000.
V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, Fast and Simple Computations on Tensors with log-Euclidean Metrics, Tech. report 5584, INRIA, Villeneuve d'Ascq, France, 2005.
V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, Log-Euclidean metrics for fast and simple calculus on diffusion tensors, Magnetic Resonance in Medicine, 56 (2006), pp. 411--421.
R. Bammer, M. Aksoy, and C. Liu, Augmented generalized sense reconstruction to correct for rigid body motion, Magnetic Resonance in Medicine, 57 (2007), pp. 90--102.
P. J. Basser and D. K. Jones, Diffusion-tensor MRI: Theory, experimental design and data analysis---a technical review, NMR in Biomedicine, 15 (2002), pp. 456--467.
R. L. Bishop and S. I. Goldberg, Tensor Analysis on Manifolds, Dover, New York, 1980.
K. Bredies, Symmetric tensor fields of bounded deformation, Ann. Mat. Pura Appl., to appear. DOI 10.1007/s10231-011-0248-4.
K. Bredies, K. Kunisch, and T. Pock, Total generalized variation, SIAM J. Imaging Sci., 3 (2010), pp. 492--526.
K. Bredies, K. Kunisch, and T. Valkonen, Properties of $L^1$-$\mbox{TGV}^2$: The one-dimensional case, J. Math. Anal Appl., 398 (2013), pp. 438--454.
K. Bredies and T. Valkonen, Inverse problems with second-order total generalized variation constraints, in Proceedings of SampTA 2011---9th International Conference on Sampling Theory and Applications, Singapore, 2011. Available online from http://sampta2011.ntu.edu.sg/ łinebreak SampTA2011Proceedings/papers/Tu2S03.1-P0182.pdf.
A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004), pp. 89--97.
A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), pp. 120--145.
C. Chefd'hotel, D. Tschumperlé, R. Deriche, and O. Faugeras, Regularizing flows for constrained matrix-valued images, J. Math. Imaging Vision, 20 (2004), pp. 147--162.
O. Coulon, D. Alexander, and S. Arridge, Diffusion tensor magnetic resonance image regularization, Medical Image Anal., 8 (2004), pp. 47--67.
J. Diestel, J. H. Fourie, and J. Swart, The Metric Theory of Tensor Products: Grothendieck's Résumé Revisited, AMS, Providence, RI, 2008.
H. Federer, Geometric Measure Theory, Springer, New York, 1969.
P. Fillard, X. Pennec, V. Arsigny, and N. Ayache, Clinical DT-MRI estimation, smoothing, and fiber tracking with log-Euclidean metrics, IEEE Trans. Medical Imaging, 26 (2007), pp. 1472--1482.
L. Florack and L. Astola, A multi-resolution framework for diffusion tensor images, in IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, CVPRW '08, IEEE, Washington, DC, 2008, pp. 1--7.
L. Florack and H. van Assen, Multiplicative calculus in biomedical image analysis, J. Math. Imaging Vision, 42 (2012), pp. 64--75.
H. Gudbjartsson and S. Patz, The Rician distribution of noisy MRI data, Magnetic Resonance in Medicine, 34 (1995), pp. 910--914.
Y. Gur, O. Pasternak, and N. Sochen, Fast $\mbox{GL}(n)$-invariant framework for tensors regularization, Int. J. Comput. Vis., 85 (2009), pp. 211--222.
M. Herbst, J. Maclaren, M. Weigel, J. Korvink, J. Hennig, and M. Zaitsev, Prospective motion correction with continuous gradient updates in diffusion weighted imaging, Magnetic Resonance in Medicine, 67 (2012), pp. 326--338.
B. Hofmann, B. Kaltenbacher, C. Pöschl, and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), pp. 987--1010.
P. Kingsley, Introduction to diffusion tensor imaging mathematics: Parts I--III, Concepts in Magnetic Resonance Part A, 28 (2006), pp. 101--179.
F. Knoll, K. Bredies, T. Pock, and R. Stollberger, Second order total generalized variation (TGV) for MRI, Magnetic Resonance in Medicine, 65 (2011), pp. 480--491.
F. Knoll, Y. Dong, M. Hintermüller, and R. Stollberger, Total variation denoising with spatially dependent regularization, in Proceedings of the 18th Scientific Meeting and Exhibition of ISMRM, Stockholm, Sweden, 2010, p. 5088.
T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), pp. 455--500.
A. Lewis, The convex analysis of unitarily invariant matrix functions, J. Convex Anal., 2 (1995), pp. 173--183.
X. Pennec, P. Fillard, and N. Ayache, A Riemannian framework for tensor computing, Int. J. Comput. Vis., 66 (2006), pp. 41--66.
C. Poupon, J. Mangin, V. Frouin, J. Régis, F. Poupon, M. Pachot-Clouard, D. Le Bihan, and I. Bloch, Regularization of MR diffusion tensor maps for tracking brain white matter bundles, in Medical Image Computing and Computer-Assisted Interventation--MICCAI98, Springer, New York, 1998, pp. 489--498.
L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259--268.
S. Setzer, G. Steidl, B. Popilka, and B. Burgeth, Variational methods for denoising matrix fields, in Visualization and Processing of Tensor Fields, J. Weickert and H. Hagen, eds., Springer, New York, 2009, pp. 341--360.
S. M. Smith, M. Jenkinson, M. W. Woolrich, C. F. Beckmann, T. E. J. Behrens, H. Johansen-Berg, P. R. Bannister, M. D. Luca, I. Drobnjak, D. E. Flitney, R. K. Niazy, J. Saunders, J. Vickers, Y. Zhang, N. D. Stefano, J. M. Brady, and P. M. Matthews, Advances in functional and structural MR image analysis and implementation as FSL, Neuroimage, 23 (Suppl 1) (2004), pp. S208--S219.
R. Temam, Mathematical Problems in Plasticity, Gauthier-Villars, Paris, France, 1985.
J.-D. Tournier, S. Mori, and A. Leemans, Diffusion tensor imaging and beyond, Magnetic Resonance in Medicine, 65 (2011), pp. 1532--1556.
D. Tschumperlé and R. Deriche, Orthonormal vector sets regularization with PDEs and applications, J. Math. Imaging Vision, 50 (2002), pp. 237--252.
D. Tschumperle and R. Deriche, Vector-valued image regularization with PDEs: A common framework for different applications, IEEE Trans. Pattern Anal. Machine Intel., 27 (2005), pp. 506--517.
M. Uecker, A. Karaus, and J. Frahm, Inverse reconstruction method for segmented multishot diffusion-weighted MRI with multiple coils, Magnetic Resonance in Medicine, 62 (2009), pp. 1342--1348.
T. Valkonen, Diff-convex Combinations of Euclidean Distances: A Search for Optima, Jyväskylä Studies in Computing 99, University of Jyväskylä, Jyväskylä, Finland, 2008.
Z. Wang, B. Vemuri, Y. Chen, and T. Mareci, A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI, IEEE Trans. Medical Imaging, 23 (2004), pp. 930--939.
J. Weickert, Anisotropic Diffusion in Image Processing, ECMI Series, B.G. Teubner, Stuttgart, Germany, 1998.

Information & Authors


Published In

cover image SIAM Journal on Imaging Sciences
SIAM Journal on Imaging Sciences
Pages: 487 - 525
ISSN (online): 1936-4954


Submitted: 23 February 2012
Accepted: 26 December 2012
Published online: 26 February 2013


  1. total variation
  2. total deformation
  3. total generalized variation
  4. regularization
  5. medical imaging
  6. diffusion tensor imaging

MSC codes

  1. 92C55
  2. 94A08
  3. 26B30
  4. 49M29



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