Algorithms to Improve the Reparameterization of Spherical Mappings of Brain Surface Meshes
Rachel A. Yotter PhD
From the Department of Psychiatry, Friedrich-Schiller University, Jena, Germany (RAY, CG); Laboratory of Neuro Imaging, Department of Neurology, Division of Brain Mapping, UCLA School of Medicine, Los Angeles, California (PMT).
Search for more papers by this authorPaul M. Thompson PhD
From the Department of Psychiatry, Friedrich-Schiller University, Jena, Germany (RAY, CG); Laboratory of Neuro Imaging, Department of Neurology, Division of Brain Mapping, UCLA School of Medicine, Los Angeles, California (PMT).
Search for more papers by this authorChristian Gaser PhD
From the Department of Psychiatry, Friedrich-Schiller University, Jena, Germany (RAY, CG); Laboratory of Neuro Imaging, Department of Neurology, Division of Brain Mapping, UCLA School of Medicine, Los Angeles, California (PMT).
Search for more papers by this authorRachel A. Yotter PhD
From the Department of Psychiatry, Friedrich-Schiller University, Jena, Germany (RAY, CG); Laboratory of Neuro Imaging, Department of Neurology, Division of Brain Mapping, UCLA School of Medicine, Los Angeles, California (PMT).
Search for more papers by this authorPaul M. Thompson PhD
From the Department of Psychiatry, Friedrich-Schiller University, Jena, Germany (RAY, CG); Laboratory of Neuro Imaging, Department of Neurology, Division of Brain Mapping, UCLA School of Medicine, Los Angeles, California (PMT).
Search for more papers by this authorChristian Gaser PhD
From the Department of Psychiatry, Friedrich-Schiller University, Jena, Germany (RAY, CG); Laboratory of Neuro Imaging, Department of Neurology, Division of Brain Mapping, UCLA School of Medicine, Los Angeles, California (PMT).
Search for more papers by this authorConflict of Interest: The authors declare no conflicts of interests.
J Neuroimaging 2011;21:e134-e147.
ABSTRACT
A spherical map of a cortical surface is often used for improved brain registration, for advanced morphometric analysis (eg, of brain shape), and for surface-based analysis of functional signals recorded from the cortex. Furthermore, for intersubject analysis, it is usually necessary to reparameterize the surface mesh into a common coordinate system. An isometric map conserves all angle and area information in the original cortical mesh; however, in practice, spherical maps contain some distortion. Here, we propose fast new algorithms to reduce the distortion of initial spherical mappings generated using one of three common spherical mapping methods. The algorithms iteratively solve a nonlinear optimization problem to reduce distortion. Our results demonstrate that our correction process is computationally inexpensive and the resulting spherical maps have improved distortion metrics. We show that our corrected spherical maps improve reparameterization of the cortical surface mesh, such that the distance error measures between the original and reparameterized surface are significantly decreased.
Supporting Information
Table S1. Mean distance error and Hausdorff distances for 20 central surface meshes
Table S2. Mean distance error and Hausdorff distances for 20 white matter surface meshes
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References
- 1 Memoli F, Sapiro G, Thompson P. Implicit brain imaging. NeuroImage 2004; 23(Suppl 1): S179-S188.
- 2 Thompson PM, MacDonald D, Mega MS, et al. Detection and mapping of abnormal brain structure with a probabilistic atlas of cortical surfaces. J Comput Assist Tomogr 1997; 21(4): 567-581.
- 3 Thompson PM, Schwartz C, Toga AW. High-resolution random mesh algorithms for creating a probabilistic 3D surface atlas of the human brain. NeuroImage 1996; 3: 19-34.
- 4 Narr KL, Bilder RM, Kim S, et al. Abnormal gyral complexity in first-episode schizophrenia. Biol Psychiatry 2004; 55(8): 859-867.
- 5 Luders E, Narr KL, Thompson PM, et al. Gender differences in cortical complexity. Nat Neurosci 2004; 7(8): 799-800.
- 6 Thompson PM, Schwartz C, Lin RT, et al. Three-dimensional statistical analysis of sulcal variability in the human brain. J Neurosci 1996; 16(13): 4261-4274.
- 7 Desai R, Liebenthal E, Possing ET, et al. Volumetric vs. surface-based alignment for localization of auditory cortex activation. NeuroImage 2005; 26(4): 1019-1029.
- 8 Hinds OP, Rajendran N, Polimeni JR, et al. Accurate prediction of V1 location from cortical folds in a surface coordinate system. NeuroImage 2008; 39(4): 1585-1599.
- 9
Fischl B,
Sereno MI,
Tootell RBH, et al.
High-resolution intersubject averaging and a coordinate system for the cortical surface.
Hum Brain Mapp
1999; 8(4): 272-284.
10.1002/(SICI)1097-0193(1999)8:4<272::AID-HBM10>3.0.CO;2-4 CAS PubMed Web of Science® Google Scholar
- 10 Thompson PM, Toga AW. A surface-based technique for warping three-dimensional images of the brain. IEEE Trans Med Imaging 1996; 15(4): 402-417.
- 11 Luders E, Thompson PM, Narr KL, et al. A curvature-based approach to estimate local gyrification on the cortical surface. NeuroImage 2006; 29(4): 1224-1230.
- 12 Schaer M, Cuadra MB, Tamarit L, et al. A surface-based approach to quantify local cortical gyrification. IEEE Trans Med Imaging 2008; 27(2): 161-170.
- 13 Han X, Jovicich J, Salat D, et al. Reliability of MRI-derived measurements of human cerebral cortical thickness: the effects of field strength, scanner upgrade and manufacturer. NeuroImage 2006; 32(1): 180-194.
- 14 Fischl B, Dale AM. Measuring the thickness of the human cerebral cortex from magnetic resonance images. Proc Natl Acad Sci USA 2000; 97(20): 11050-11055.
- 15 Van Essen DC, Drury HA, Dickson J, et al. An integrated software suite for surface-based analyses of cerebral cortex. J Am Med Inform Assoc 2001; 8(5): 443-459.
- 16 Floater MS, Hormann K. Surface parameterization: a tutorial and survey. In: N Dodgson, MS Floater, M Sabin, eds. Advances in Multiresolution for Geometric Modelling. Vol IV. Berlin Heidelberg : Springer; 2005: 157-186.
- 17 Schwartz EL. Spatial mapping in the primate sensory projection: analytic structure and relevance to perception. Biol Cybern 1977; 25(4): 181-194.
- 18 Angenent S, Haker S, Tannenbaum A, et al. On the Laplace-Beltrami operator and brain surface flattening. IEEE Trans Med Imaging 1999; 18(8): 700-711.
- 19 Tosun D, Rettmann ME, Prince JL. Mapping techniques for aligning sulci across multiple brains. Med Image Anal 2004; 8(3): 295-309.
- 20 Lui LM, Wang Y, Chan TF, et al. Automatic landmark tracking applied to optimize brain conformal mapping. Proceedings of the IEEE International Symposium on Biomed Imaging. Arlington , VA , 2006.
- 21 Lui LM, Wang YL, Chan TF, et al. Brain anatomical feature detection by solving partial differential equations on a general manifold. J Discrete Cont Dyn Syst, Ser B 2007; 7(3): 605-618.
- 22 Lui LM, Wang Y, Chan TF, et al. Landmark constrained genus zero surface conformal mapping and its application to brain mapping research. Appl Numer Math 2007; 57(5-7): 847-858.
- 23 Gu X, Wang Y, Chan TF, et al. Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Trans Med Imaging 2004; 23(8): 949-958.
- 24 Wang Y, Chiang M-C, Thompson PM. 3D surface matching with mutual information and Riemann surface structures. Comput Graph Imaging. Honolulu , HI ; 2005: 94-99.
- 25 Joshi AA, Shattuck DW, Thompson PM, et al. Cortical surface parameterization by p-harmonic energy minimization. Proceedings of the IEEE International Symposium of Biomed Imaging. pp. 428-431. Arlington , VA , 2004.
- 26 Joshi AA, Shattuck DW, Thompson PM, et al. Surface-constrained volumetric brain registration using harmonic mappings. IEEE Trans Med Imaging 2007; 26(12): 1657-1669.
- 27 Hurdal MK, Stephenson K. Cortical cartography using the discrete conformal approach of circle packings. NeuroImage 2004; 23(Suppl 1): S119-S128.
- 28 Wang Y, Gu X, Chan TF, et al. Multivariate tensor-based brain anatomical surface morphometry via holomorphic one-forms. Med Image Comput Comput Assist Interv 2009; 5761: 337-344.
- 29 Wang Y, Gu X, Chan TF, et al. Brain surface conformal parameterization with the Ricci flow. Proc IEEE Int Symp Biomed Imaging. pp. 1312-1315. Arlington , VA , 2007.
- 30 Wang Y, Gu X, Chan TF, et al. Brain mapping with the Ricci flow conformal parameterization and multivariate statistics on deformation tensors. Proceedings of the 2nd MICCAI Workshop on Mathematical Foundations of Computational Anatomy. pp. 36-47. New York , NY , 2008.
- 31 Wang Y, Gu X, Chan TF, et al. Brain surface conformal parameterization with algebraic functions. Med Image Comput Comput Assist Interv 2006; 9(Pt 2): 946-954.
- 32 Wang Y, Gu X, Chan TF, et al. Brain surface conformal parameterization with the slit mapping. Proceedings of the IEEE International Symposium on Biomed Imaging. pp. 448-451. Paris , 2008.
- 33 Wang Y, Gu X, Chan TF, et al. Shape analysis with conformal invariants for multiply connected domains and its application to analyzing brain morphology. NeuroImage 2009; 47(Suppl 1): S39-S41.
- 34 Thompson PM, Hayashi KM, Sowell ER, et al. Mapping cortical change in Alzheimer's disease, brain development, and schizophrenia. NeuroImage 2004; 23(Suppl 1): S2-S18.
- 35 Leow A, Yu CL, Lee SJ, et al. Brain structural mapping using a novel hybrid implicit/explicit framework based on the level-set method. NeuroImage 2005; 24(3): 910-927.
- 36 Wang Y, Lui LM, Chan TF, et al. Optimization of brain conformal mapping with landmarks. Med Image Comput Comput Assist Interv 2005; 8(Pt 2): 675-683.
- 37 Shi Y, Thompson PM, Dinov I, et al. Direct cortical mapping via solving partial differential equations on implicit surfaces. Med Image Anal 2007; 11(3): 207-223.
- 38 Thompson PM, Hayashi KM, de Zubicaray GI, et al. Detecting dynamic and genetic effects on brain structure using high-dimensional cortical pattern matching. Proceedings of the IEEE International Symposium on Biomed Imaging. pp. 473-476. Washington , DC , 2002.
- 39 Fischl B, Sereno MI, Dale AM. Cortical surface-based analysis. II. Inflation, flattening, and a surface-based coordinate system. NeuroImage 1999; 9(2): 195-207.
- 40 Carman GJ, Drury HA, Van Essen DC. Computational methods for reconstructing and unfolding the cerebral cortex. Cereb Cortex 1995; 5(6): 506-517.
- 41 Kruggel F. Robust parametrization of brain surface meshes. Med Image Anal 2008; 12(3): 291-299.
- 42 Shen L, Makedon F. Spherical parameterization for 3D surface analysis in volumetric images. Proceedings of the International Conference on Information Technology: Coding and Computing. Vol 2, pp. 643-649. Las Vegas , NV , 2004.
- 43 Timsari B, Leahy RM. Optimization method for creating semi-isometric flat maps of the cerebral cortex. SPIE Symposium on Medical Imaging 2000: Image Processing. Vol. 3979, pp. 698-708. San Diego , CA , 2000.
- 44
Liseikin VD.
Grid Generation Methods.
Heidelberg
: Springer-Verlag; 1999
10.1007/978-3-662-03949-6 Google Scholar
- 45 Tinoco-Ruiz J-G, Barrera-Sanchez P, Cortes-Medina A. Some properties of area functionals in numerical grid generation. Proceedings of the 10th International Meshing Roundtable. Newport , pp. 43-54. Beach , CA , 2001.
- 46 Ju L, Stern J, Rehm K, et al. Cortical surface flattening using least square conformal mapping with minimal metric distortion. Proceedings of the IEEE International Symposium on Biomed Imaging. pp. 77-80. Arlington , VA , 2004.
- 47 MacDonald D. A method for identifying geometrically simple surfaces from three dimensional images. Montreal : School of Computer Science, McGill Unversity, 1998.
- 48 Drury HA, Van Essen DC, Anderson CH, et al. Computerized mappings of the cerebral cortex: a multiresolution flattening method and a surface-based coordinate system. J Cogn Neurosci 1996; 8(1): 1-28.
- 49 Praun E, Hoppe H. Spherical parametrization and remeshing. ACM Trans Graph 2003; 22(3): 340-349.
- 50 Schwartz EL, Shaw A, Wolfson E. A numerical solution to the generalized mapmaker's problem: flattening nonconvex polyhedral surfaces. IEEE Trans Pattern Anal Machine Intell 1989; 11(9): 1005-1008.
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