Segmentation of Tensor Fields: Recent Advances and Perspectives

The segmentation of tensor-valued images or 3D volumes is a relatively recent issue in image processing, but a significant effort has been made in the last years. Most of this effort has been focused on the segmentation of anatomical structures from DT-MRI (Diffusion Tensor Magnetic Resonance Imaging), and some contributions have also been made for the segmentation of 2D textured images using the Local Structure Tensor (LST). In this chapter, we carefully review the state of the art in the segmentation of tensor fields. We will discuss the main approaches that have been proposed in the literature, with particular emphasis on the importance of the different tensor dissimilarity measures. Also, we will highlight the key limitations of the segmentation techniques proposed so far, and will provide some insight on the directions of current research.

[1]  Derek K Jones,et al.  Applications of diffusion‐weighted and diffusion tensor MRI to white matter diseases – a review , 2002, NMR in biomedicine.

[2]  Andreas Eidehall.  Tensor representation of 3D structures , 2002 .

[3]  V. Wedeen,et al.  Mapping fiber orientation spectra in cerebral white matter with Fourier-transform diffusion MRI , 2000 .

[4]  P. Grenier,et al.  MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders. , 1986, Radiology.

[5]  Guido Gerig,et al.  Nonlinear anisotropic filtering of MRI data , 1992, IEEE Trans. Medical Imaging.

[6]  Tony Lindeberg,et al.  Scale-Space Theory in Computer Vision , 1993, Lecture Notes in Computer Science.

[7]  P Perona,et al.  Preattentive texture discrimination with early vision mechanisms. , 1990, Journal of the Optical Society of America. A, Optics and image science.

[8]  Olivier D. Faugeras,et al.  Co-dimension 2 Geodesic Active Contours for MRA Segmentation , 1999, IPMI.

[9]  Suyash P. Awate,et al.  A Fuzzy, Nonparametric Segmentation Framework for DTI and MRI Analysis: With Applications to DTI-Tract Extraction , 2007, IEEE Transactions on Medical Imaging.

[10]  V. Wedeen,et al.  Diffusion MRI of Complex Neural Architecture , 2003, Neuron.

[11]  S. Arridge,et al.  Detection and modeling of non‐Gaussian apparent diffusion coefficient profiles in human brain data , 2002, Magnetic resonance in medicine.

[12]  Rachid Deriche,et al.  Geodesic Active Regions: A New Framework to Deal with Frame Partition Problems in Computer Vision , 2002, J. Vis. Commun. Image Represent..

[13]  Rachid Deriche,et al.  Variational frameworks for DT-MRI estimation, regularization and visualization , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.

[14]  Xavier Bresson,et al.  White Matter Mapping in DT-MRI Using Geometric Flows , 2003, EUROCAST.

[15]  Thomas Brox,et al.  Nonlinear structure tensors , 2006, Image Vis. Comput..

[16]  Rachid Deriche,et al.  Active unsupervised texture segmentation on a diffusion based feature space , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

[17]  Kalvis M. Jansons,et al.  Persistent angular structure: new insights from diffusion magnetic resonance imaging data , 2003 .

[18]  Rachid Deriche,et al.  Regularizing Flows for Constrained Matrix-Valued Images , 2004, Journal of Mathematical Imaging and Vision.

[19]  Carlos Alberto,et al.  Nuevos esquemas para el procesado de señales tensoriales = New schemes for tensor signal processing , 2007 .

[20]  Ruzena Bajcsy,et al.  Similarity Measures for Matching Diffusion Tensor Images , 1999, BMVC.

[21]  David E. Breen,et al.  Level set modeling and segmentation of diffusion tensor magnetic resonance imaging brain data , 2003, J. Electronic Imaging.

[22]  Zhizhou Wang,et al.  An affine invariant tensor dissimilarity measure and its applications to tensor-valued image segmentation , 2004, CVPR 2004.

[23]  Rachid Deriche,et al.  Unsupervised Segmentation Incorporating Colour, Texture, and Motion , 2003, CAIP.

[24]  C. Poupon.  Detection des faisceaux de fibres de la substance blanche pour l'etude de la connectivite anatomique cerebrale , 1999 .

[25]  Xavier Bresson,et al.  White matter fiber tract segmentation in DT-MRI using geometric flows , 2005, Medical Image Anal..

[26]  Westin C. F. Maier.  A Dual Tensor Basis Solution to the Stejskal-Tanner Equations for DT-MRI , 2002 .

[27]  Jitendra Malik,et al.  Normalized Cuts and Image Segmentation , 2000, IEEE Trans. Pattern Anal. Mach. Intell..

[28]  Anil K. Jain,et al.  Markov Random Field Texture Models , 1983, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[29]  Tony F. Chan,et al.  Active contours without edges , 2001, IEEE Trans. Image Process..

[30]  P. Basser,et al.  Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. 1996. , 1996, Journal of magnetic resonance.

[31]  J. A. Hartigan,et al.  A k-means clustering algorithm , 1979 .

[32]  J. E. Tanner,et al.  Spin diffusion measurements : spin echoes in the presence of a time-dependent field gradient , 1965 .

[33]  Ghassan Hamarneh,et al.  DT-MRI segmentation using graph cuts , 2007, SPIE Medical Imaging.

[34]  Thomas Brox,et al.  Nonlinear Matrix Diffusion for Optic Flow Estimation , 2002, DAGM-Symposium.

[35]  A. Ravishankar Rao,et al.  Computing oriented texture fields , 1991, CVGIP Graph. Model. Image Process..

[36]  B. Vemuri,et al.  Fiber tract mapping from diffusion tensor MRI , 2001, Proceedings IEEE Workshop on Variational and Level Set Methods in Computer Vision.

[37]  Maurice K. Wong,et al.  Algorithm AS136: A k-means clustering algorithm. , 1979 .

[38]  P. Sundgren,et al.  Diffusion tensor imaging of the brain: review of clinical applications , 2004, Neuroradiology.

[39]  P. Basser,et al.  Introduction to Diffusion MR , 2009 .

[40]  Kaleem Siddiqi,et al.  A geometric flow for white matter fibre tract reconstruction , 2002, Proceedings IEEE International Symposium on Biomedical Imaging.

[41]  Guillermo Sapiro,et al.  Geodesic Active Contours , 1995, International Journal of Computer Vision.

[42]  Carl-Fredrik Westin,et al.  Representing Local Structure Using Tensors II , 2011, SCIA.

[43]  Rachid Deriche,et al.  Statistics on the Manifold of Multivariate Normal Distributions: Theory and Application to Diffusion Tensor MRI Processing , 2006, Journal of Mathematical Imaging and Vision.

[44]  Carl-Fredrik Westin,et al.  Processing and visualization for diffusion tensor MRI , 2002, Medical Image Anal..

[45]  Daniel C. Alexander,et al.  Persistent Angular Structure: New Insights from Diffusion MRI Data. Dummy Version , 2003, IPMI.

[46]  O. Faugeras,et al.  Statistics on Multivariate Normal Distributions: A Geometric Approach and its Application to Diffusion Tensor MRI , 2004 .

[47]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[48]  Robert M. Gray,et al.  An Algorithm for Vector Quantizer Design , 1980, IEEE Trans. Commun..

[49]  N. Ayache,et al.  Log‐Euclidean metrics for fast and simple calculus on diffusion tensors , 2006, Magnetic resonance in medicine.

[50]  H. Karcher.  Riemannian center of mass and mollifier smoothing , 1977 .

[51]  Max A. Viergever,et al.  Efficient and reliable schemes for nonlinear diffusion filtering , 1998, IEEE Trans. Image Process..

[52]  Daniel C. Alexander,et al.  An Introduction to Computational Diffusion MRI: the Diffusion Tensor and Beyond , 2006, Visualization and Processing of Tensor Fields.

[53]  L. Frank.  Characterization of anisotropy in high angular resolution diffusion‐weighted MRI , 2002, Magnetic resonance in medicine.

[54]  P. Basser,et al.  MR diffusion tensor spectroscopy and imaging. , 1994, Biophysical journal.

[55]  David E. Breen,et al.  Level Set Segmentation and Modeling of DT-MRI human brain data , 2003 .

[56]  J. Bigun,et al.  Optimal Orientation Detection of Linear Symmetry , 1987, ICCV 1987.

[57]  Zhizhou Wang,et al.  Tensor Field Segmentation Using Region Based Active Contour Model , 2004, ECCV.

[58]  Silvano Di Zenzo,et al.  A note on the gradient of a multi-image , 1986, Comput. Vis. Graph. Image Process..

[59]  Maher Moakher,et al.  A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices , 2005, SIAM J. Matrix Anal. Appl..

[60]  David S Tuch,et al.  Automatic segmentation of thalamic nuclei from diffusion tensor magnetic resonance imaging , 2003, NeuroImage.

[61]  Anthony J. Yezzi,et al.  Gradient flows and geometric active contour models , 1995, Proceedings of IEEE International Conference on Computer Vision.

[62]  C. Atkinson.  Rao's distance measure , 1981 .

[63]  Carlos Alberola-López,et al.  Mixtures of Gaussians on Tensor Fields for DT-MRI Segmentation , 2007, MICCAI.

[64]  Simon R. Arridge,et al.  A Regularization Scheme for Diffusion Tensor Magnetic Resonance Images , 2001, IPMI.

[65]  Christopher M. Bishop,et al.  Neural networks for pattern recognition , 1995 .

[66]  R. Deriche,et al.  A variational framework for active and adaptative segmentation of vector valued images , 2002, Workshop on Motion and Video Computing, 2002. Proceedings..

[67]  D. Parker,et al.  Analysis of partial volume effects in diffusion‐tensor MRI , 2001, Magnetic resonance in medicine.

[68]  Hans Knutsson,et al.  Signal processing for computer vision , 1994 .

[69]  Johan Wiklund,et al.  Multidimensional Orientation Estimation with Applications to Texture Analysis and Optical Flow , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[70]  C. R. Rao,et al.  Information and the Accuracy Attainable in the Estimation of Statistical Parameters , 1992 .

[71]  Rachid Deriche,et al.  DTI segmentation by statistical surface evolution , 2006, IEEE Transactions on Medical Imaging.

[72]  Rachid Deriche,et al.  Segmentation of 3D Probability Density Fields by Surface Evolution: Application to Diffusion MRI , 2004, MICCAI.

[73]  T. Brox,et al.  Diffusion and regularization of vector- and matrix-valued images , 2002 .

[74]  P. Basser,et al.  Estimation of the effective self-diffusion tensor from the NMR spin echo. , 1994, Journal of magnetic resonance. Series B.

[75]  R. Kikinis,et al.  Magnetic resonance imaging shows orientation and asymmetry of white matter fiber tracts , 1998, Brain Research.

[76]  Carl-Fredrik Westin,et al.  Segmentation of Thalamic Nuclei from DTI Using Spectral Clustering , 2006, MICCAI.

[77]  Suyash P. Awate,et al.  A fuzzy, nonparametric segmentation framework for DTI and MRI analysis: with applications to DTI-tract extraction. , 2007, IEEE transactions on medical imaging.

[78]  Isabelle Bloch,et al.  Regularization of MR Diffusion Tensor Maps for Tracking Brain White Matter Bundles , 1998, MICCAI.

[79]  Nicholas Ayache,et al.  Fast and Simple Calculus on Tensors in the Log-Euclidean Framework , 2005, MICCAI.

[80]  H. Knutsson.  A Tensor Representation of 3-D Structures , 1987 .

[81]  M. R. Turner,et al.  Texture discrimination by Gabor functions , 1986, Biological Cybernetics.

[82]  M. Fréchet.  Les éléments aléatoires de nature quelconque dans un espace distancié , 1948 .

[83]  P. V. van Zijl,et al.  Orientation‐independent diffusion imaging without tensor diagonalization: Anisotropy definitions based on physical attributes of the diffusion ellipsoid , 1999, Journal of magnetic resonance imaging : JMRI.