This paper presents recent advances in the use of diffeomorphic active shapes which incorporate the conservation laws of large deformation diffeomorphic metric mapping. The equations of evolution satisfying the conservation law are geodesics under the diffeomorphism metric and therefore termed geodesically controlled diffeomorphic active shapes (GDAS). Our principal application in this paper is on robust diffeomorphic mapping methods based on parameterized surface representations of subcortical template structures. Our parametrization of the GDAS evolution is via the initial momentum representation in the tangent space of the template surface. The dimension of this representation is constrained using principal component analysis generated from training samples. In this work, we seek to use template surfaces to generate segmentations of the hippocampus with three data attachment terms: surface matching, landmark matching, and inside-outside modeling from grayscale T1 MR imaging data. This is formulated as an energy minimization problem, where energy describes shape variability and data attachment accuracy, and we derive a variational solution. A gradient descent strategy is employed in the numerical optimization. For the landmark matching case, we demonstrate the robustness of this algorithm as applied to the workflow of a large neuroanatomical study by comparing to an existing diffeomorphic landmark matching algorithm. 1. Introduction There have been many approaches to segmentation in medical imaging, including both the active shape methods pioneered by Kass et al. [1] and template based approaches pioneered by Dann et al. [2]. For studying images made up of simple homogeneous structures such as anatomical structures, local active evolution methods [1, 3–5] which are encoded through their boundary representations are natural. In such methods, the complexity of the representation is reduced from an encoding based on the dimension of the extrinsic background space containing the object, to the dimension of the boundary. Given the line of work in template based computational anatomy which has emphasized the important role of diffeomorphisms for defining bijective correspondence between coordinate systems, it is natural to constrain the iterative methods of active shapes so that shape evolution preserves the original topology of the template. This is the intention of the diffeomorphic active contour (DAC) approaches taken by Younes et al. [6–8], including in the local evolution equations the diffeomorphism constraint. DAC methods, in a form similar to the
References
[1]
M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: active contour models,” International Journal of Computer Vision, vol. 1, no. 4, pp. 321–331, 1988.
[2]
R. Dann, J. Hoford, S. Kovacic, M. Reivich, and R. Bajcsy, “Evaluation of elastic matching system for anatomic (CT, MR) and functinal (PET) cerebral images,” Journal of Computer Assisted Tomography, vol. 13, no. 4, pp. 603–611, 1989.
[3]
L. H. Staib and J. S. Duncan, “Boundary finding with parametrically deformable models,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, no. 11, pp. 1061–1075, 1992.
[4]
T. F. Cootes, C. J. Taylor, D. H. Cooper, and J. Graham, “Active shape models-their training and application,” Computer Vision and Image Understanding, vol. 61, no. 1, pp. 38–59, 1995.
[5]
U. Grenander and M. I. Miller, “Representations of knowledge in complex systems,” Journal of the Royal Statistical Society B, vol. 56, no. 4, pp. 549–603, 1994.
[6]
Z. Sirong, L. Younes, J. Zweck, and J. T. Ratnanather, “Diffeomorphic surface flows: a novel method of surface evolution,” SIAM Journal on Applied Mathematics, vol. 68, no. 3, pp. 806–824, 2007.
[7]
L. Younes, F. Arrate, and M. I. Miller, “Evolutions equations in computational anatomy,” NeuroImage, vol. 45, no. 1, pp. S40–S50, 2009.
[8]
F. Arrate, J. Tilak Ratnanather, and L. Younes, “Diffeomorphic active contours,” SIAM Journal on Imaging Sciences, vol. 3, no. 2, pp. 176–198, 2010.
[9]
G. E. Christensen, R. D. Rabbitt, and M. I. Miller, “Deformable templates using large deformation kinematics,” IEEE Transactions on Image Processing, vol. 5, no. 10, pp. 1435–1447, 1996.
[10]
A. Trouvé, “Diffeomorphisms groups and pattern matching in image analysis,” International Journal of Computer Vision, vol. 28, no. 3, pp. 213–221, 1998.
[11]
P. Dupuis, U. Grenander, and M. I. Miller, “Variational problems on flows of diffeomorphisms for image matching,” Quarterly of Applied Mathematics, vol. 56, no. 3, pp. 587–600, 1998.
[12]
M. I. Miller, A. Trouvé, and L. Younes, “On the metrics and Euler-Lagrange equations of computational anatomy,” Annual Review of Biomedical Engineering, vol. 4, pp. 375–405, 2002.
[13]
V. I. Arnold, “Sur la geomerie differentielle des groupes de lie de dimension infinie et ses applications a l'hydrodynamique des uides parfaits,” Annales De L'Institut Fourier, vol. 16, no. 1, pp. 319–361, 1966.
[14]
D. D. Holm, J. E. Marsden, and T. S. Ratiu, “The euler-poincaré equations and semidirect products with applications to continuum theories,” Advances in Mathematics, vol. 137, no. 1, pp. 1–81, 1998.
[15]
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer, 1999.
[16]
M. I. Miller, A. Trouvé, and L. Younes, “Geodesic shooting for computational anatomy,” Journal of Mathematical Imaging and Vision, vol. 24, no. 2, pp. 209–228, 2006.
[17]
M. Vaillant and J. Glaunès, “Surface matching via currents,” in Proceedings of Information Processing in Medical Imaging, vol. 3565, 2005.
[18]
A. Qiu, L. Younes, and M. Miller, “Principal component based diffeomorphic surface mapping,” IEEE Transactions on Medical Imaging, vol. 31, pp. 302–3311, 2012.
[19]
T. Cootes and C. Taylor, “Active shape models smart snakes,” in Proceedings of the British Machine Vision Conference, pp. 266–275, Citeseer, 1992.
[20]
A. M. Baumberg and D. C. Hogg, “An efficient method for contour tracking using active shape models,” in Proceedings of the IEEE Workshop on Motion of Non-Rigid and Articulated Objects, pp. 194–199, Citeseer, November 1994.
[21]
T. Cootes, A. Hill, C. Taylor, and J. Haslam, “Use of active shape models for locating structures in medical images,” Image and Vision Computing, vol. 12, no. 6, pp. 355–365, 1994.
[22]
N. Duta and M. Sonka, “Segmentation and interpretation of MR brain images: an improved active shape model,” IEEE Transactions on Medical Imaging, vol. 17, no. 6, pp. 1049–1062, 1998.
[23]
B. Van Ginneken, A. F. Frangi, J. J. Staal, B. M. Ter Haar Romeny, and M. A. Viergever, “Active shape model segmentation with optimal features,” IEEE Transactions on Medical Imaging, vol. 21, no. 8, pp. 924–933, 2002.
[24]
C. Davatzikos, X. Tao, and D. Shen, “Hierarchical active shape models, using the wavelet transform,” IEEE Transactions on Medical Imaging, vol. 22, no. 3, pp. 414–423, 2003.
[25]
D. Rueckert, A. Frangi, and J. Schnabel, “Automatic construction of 3d statistical deformation models using non-rigid registration,” in Proceedings of the Medical Image Computing and Computer-Assisted Intervention (MICCAI '01), pp. 77–84, Springer, 2010.
[26]
U. Grenander, Y. Chow, and D. M. Keenan, Hands: A Pattern Theoretic Study of Biological Shapes, Springer, 1991.
[27]
U. Grenander and M. I. Miller, “Representation of knowledge in complex systems (with discussion section),” Journal of the Royal Statistical Society, vol. 56, no. 4, pp. 569–603, 1994.
[28]
S. C. Zhu, “Region competition: unifying snakes, region growing, and bayes/mdl for multiband image segmentation,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 18, no. 9, pp. 884–900, 1996.
[29]
T. F. Chan and L. A. Vese, “Active contours without edges,” IEEE Transactions on Image Processing, vol. 10, no. 2, pp. 266–277, 2001.
[30]
G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, 2001.
[31]
R. Guillemaud and M. Brady, “Estimating the bias field of MR images,” IEEE Transactions on Medical Imaging, vol. 16, no. 3, pp. 238–251, 1997.
[32]
W. M. Wells III, W. E. L. Crimson, R. Kikinis, and F. A. Jolesz, “Adaptive segmentation of mri data,” IEEE Transactions on Medical Imaging, vol. 15, no. 4, pp. 429–442, 1996.
[33]
M. Vaillant, M. I. Miller, L. Younes, and A. Trouvé, “Statistics on diffeomorphisms via tangent space representations,” NeuroImage, vol. 23, no. 1, pp. S161–S169, 2004.
[34]
L. Younes, Shapes and Diffeomorphisms, Applied Mathematical Sciences, Springer, 2010.
[35]
J. Ma, M. I. Miller, and L. Younes, “A bayesian generative model for surface template estimation,” International Journal of Biomedical Imaging, vol. 2010, Article ID 974957, 14 pages, 2010.
[36]
H. Le, “Mean size-and-shapes and mean shapes: a geometric point of view,” Advances in Applied Probability, vol. 27, pp. 44–55, 1995.
[37]
E. Klassen, A. Srivastava, W. Mio, and S. H. Joshi, “Analysis of planar shapes using geodesic paths on shape spaces,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 3, pp. 372–383, 2004.
[38]
P. T. Fletcher, S. Venkatasubramanian, and S. Joshi, “Robust statistics on Riemannian manifolds via the geometric median,” in Proceedings of the 26th IEEE Conference on Computer Vision and Pattern Recognition (CVPR '08), pp. 1–8, June 2008.
[39]
C. Bishop, Pattern Recognition and Machine Learning, Springer, New York, NY, USA, 2006.
[40]
J. W. Haller, A. Banerjee, G. E. Christensen et al., “Three-dimensional hippocampal MR morphometry with high-dimensional transformation of a neuroanatomic atlas,” Radiology, vol. 202, no. 2, pp. 504–510, 1997.
[41]
J. G. Csernansky, L. Wang, D. Jones et al., “Hippocampal deformities in schizophrenia characterized by high dimensional brain mapping,” American Journal of Psychiatry, vol. 159, no. 12, pp. 2000–2006, 2002.
[42]
J. A. Posener, L. Wang, J. L. Price et al., “High-dimensional mapping of the hippocampus in depression,” American Journal of Psychiatry, vol. 160, no. 1, pp. 83–89, 2003.
[43]
J. G. Csernansky, M. K. Schindler, N. R. Splinter et al., “Abnormalities of thalamic volume and shape in schizophrenia,” American Journal of Psychiatry, vol. 161, no. 5, pp. 896–902, 2004.
[44]
J. G. Csernansky, L. Wang, J. Swank et al., “Preclinical detection of Alzheimer's disease: hippocampal shape and volume predict dementia onset in the elderly,” NeuroImage, vol. 25, no. 3, pp. 783–792, 2005.
[45]
M. A. Munn, J. Alexopoulos, T. Nishino et al., “Amygdala volume analysis in female twins with major depression,” Biological Psychiatry, vol. 62, no. 5, pp. 415–422, 2007.
[46]
M. M. Mielke, N. J. Haughey, V. V. Ratnam Bandaru et al., “Plasma ceramides are altered in mild cognitive impairment and predict cognitive decline and hippocampal volume loss,” Alzheimer's and Dementia, vol. 6, no. 5, pp. 378–385, 2010.
[47]
B. Fischl, “FreeSurfer,” Neuroimage, vol. 62, no. 2, pp. 774–781, 2012.
[48]
A. Qiu and M. I. Miller, “Multi-structure network shape analysis via normal surface momentum maps,” NeuroImage, vol. 42, no. 4, pp. 1430–1438, 2008.
[49]
S. C. Joshi and M. I. Miller, “Landmark matching via large deformation diffeomorphisms,” IEEE Transactions on Image Processing, vol. 9, no. 8, pp. 1357–1370, 2000.
[50]
J. R. Landis and G. G. Koch, “The measurement of observer agreement for categorical data,” Biometrics, vol. 33, no. 1, pp. 159–174, 1977.
[51]
D. L. Pham, C. Xu, and J. L. Prince, “Current methods in medical image segmentation,” Annual Review of Biomedical Engineering, vol. 2, no. 2000, pp. 315–337, 2000.
[52]
S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” Journal of Computational Physics, vol. 79, no. 1, pp. 12–49, 1988.
[53]
S. Z. Li, Markov Random Field Modeling in Computer Vision, Springer, 1995.
[54]
F. L. Bookstein, “Principal warps: Thin-plate splines and the decomposition of deformations,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 11, pp. 567–585, 1992.
[55]
D. Rueckert, “Nonrigid registration using free-form deformations: application to breast mr images,” IEEE Transactions on Medical Imaging, vol. 18, no. 8, pp. 712–721, 1999.
[56]
B. Fischl, D. H. Salat, E. Busa et al., “Whole brain segmentation: automated labeling of neuroanatomical structures in the human brain,” Neuron, vol. 33, no. 3, pp. 341–355, 2002.
[57]
B. Fischl, D. H. Salat, A. J. W. Van Der Kouwe et al., “Sequence-independent segmentation of magnetic resonance images,” NeuroImage, vol. 23, no. 1, pp. S69–S84, 2004.
[58]
X. Han and B. Fischl, “Atlas renormalization for improved brain MR image segmentation across scanner platforms,” IEEE Transactions on Medical Imaging, vol. 26, no. 4, pp. 479–486, 2007.
[59]
M. Unser, “Multigrid adaptive image processing,” in Proceedings of the IEEE International Conference on Image Processing, vol. 1, pp. 49–52, October 1995.
[60]
A. Kumar, A. Yezzi, S. Kichenassamy, P. Olver, and A. Tannenbaum, “Active contours for visual tracking: a geometric gradient based approach,” in Proceedings of the 34th IEEE Conference on Decision and Control, vol. 1, pp. 4041–4046, December 1995.
[61]
C. E. Priebe, M. I. Miller, and J. Tilak Ratnanather, “Segmenting magnetic resonance images via hierarchical mixture modelling,” Computational Statistics and Data Analysis, vol. 50, no. 2, pp. 551–567, 2006.
[62]
T. F. Cootes, G. J. Edwards, and C. Taylor, “Active appearance models,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 23, pp. 681–685, 2001.
[63]
A. Srivastava, A. Jain, S. Joshi, and D. Kaziska, “Statistical shape models using elastic-string representations,” in Proceedings of the 7th Asian conference on Computer Vision (ACCV '06), vol. 3851 of Lecture Notes in Computer Science, pp. 612–621.
[64]
D. Cremers, T. Kohlberger, and C. Schn？rr, “Shape statistics in kernel space for variational image segmentation,” Pattern Recognition, vol. 36, no. 9, pp. 1929–1943, 2003.
[65]
U. Grenander and M. I. Miller, “Computational anatomy: an emerging discipline,” Quarterly of Applied Mathematics, vol. 56, no. 4, pp. 617–694, 1998.
