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## Brain Mapping with the Ricci Flow Conformal Parameterization and Multivariate Statistics on Deformation Tensors

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Citations: | 3 - 1 self |

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340 | High-resolution intersubject averaging and a coordinate system for the cortical surface.
- Fischl, Sereno, et al.
- 1999
(Show Context)
Citation Context ...ecific anatomical points, curved landmarks, or subregions lying within the two surfaces. This is often achieved by first mapping each of the 3D surfaces to canonical parameter spaces such as a sphere =-=[1, 2]-=- or a planar domain [3]. A flow, computed in the parameter space of the two surfaces [4, 5], then induces a correspondence field in 3D. This flow can be constrained using anatomic landmark points or c... |

217 | Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn Reson Med.
- Arsigny, Fillard, et al.
- 2006
(Show Context)
Citation Context ...ine the deformation tensors as S = (J T J) 1/2 . Instead of analyzing shape change based on the eigenvalues of the deformation tensor, we consider a new family of metrics, the “Log-Euclidean metrics” =-=[18]-=-. These metrics make computations on tensors easier to perform, as they are chosen such that the transformed values form a vector space, and statistical parameters can then be computed easily using st... |

205 |
Differential Geometry,
- Guggenheimer
- 1963
(Show Context)
Citation Context ... and then generalize it to the discrete setting. 2.1 Ricci Flow on Continuous Surfaces Riemannian Metric and Gaussian Curvature All the concepts used here may be found, with detailed explanations, in =-=[21]-=-. Suppose S is a C 2 smooth surface embedded in R 3 with local parameters (u1,u2). Let r(u1,u2) be a point on S and dr = r1du1 +r2du2 be the tangent vector defined at that point, where r1,r2 are the p... |

202 | A minimum description length approach to statistical shape modeling.
- Davies, Twining, et al.
- 2002
(Show Context)
Citation Context ...types of feature matches [8]. Finally, correspondences may be determined by using a minimum description length (MDL) principle, based on the compactness of the covariance of the resulting shape model =-=[9]-=-. Anatomically homologous points can then be forced to match across a dataset. Thodberg [10] identified problems with early MDL approaches and extended them to an MDL appearance model, when performing... |

188 | Genus zero surface conformal mapping and its application to brain surface mapping. - Gu, Wang, et al. - 2004 |

156 |
Geometry and Topology of Three-Manifolds. Princeton lecture notes,
- Thurston
- 1979
(Show Context)
Citation Context ...variant: vi∈V Ki = 2πχ(M). Discrete Conformal Deformation In the discrete setting, conformal deformation is carried out using the concept of circle packing metric, which was introduced by Thurston in =-=[24]-=-. By approximating infinitesimal circles using circles with finite radii, a circle packing metric of Σ can be denoted as (Γ,Φ), where Γ is a vertex function, Γ : V → R +, which assigns a radius γi to ... |

128 |
The Ricci flow on the 2-sphere,
- Chow
- 1991
(Show Context)
Citation Context ...g(∞) will induce the user-defined curvature ¯ K. The Ricci flow has been proven to converge. For surfaces with non-positive and positive Euler numbers, the proofs were given by Hamilton [22] and Chow =-=[23]-=- respectively. For a closed surface, if the total area is preserved during the flow, the Ricci flow will converge to a metric such that the Gaussian curvature is constant everywhere. 2.2 Ricci Flow on... |

119 |
The ricci flow on surfaces. Mathematics and general relativity,
- Hamilton
- 1988
(Show Context)
Citation Context ...ulting metric g(∞) will induce the user-defined curvature ¯ K. The Ricci flow has been proven to converge. For surfaces with non-positive and positive Euler numbers, the proofs were given by Hamilton =-=[22]-=- and Chow [23] respectively. For a closed surface, if the total area is preserved during the flow, the Ricci flow will converge to a metric such that the Gaussian curvature is constant everywhere. 2.2... |

80 | Combinatorial ricci flows on surfaces.
- Chow, Luo
- 2003
(Show Context)
Citation Context ...c. The above integration (Eq. 5) is called the discrete Ricci energy, which is welldefined. The discrete Ricci energy has been proved to be strictly convex (i.e., its Hessian is positive definite) in =-=[26]-=-. The global minimum uniquely exists, which gives the desired discrete metric that induces ¯ k. The discrete Ricci flow is the negative gradient flow of this energy, and it converges to the global min... |

76 | Spatial normalization of 3D brain images using deformable models.
- Davatzikos
- 1996
(Show Context)
Citation Context ...is is often achieved by first mapping each of the 3D surfaces to canonical parameter spaces such as a sphere [1, 2] or a planar domain [3]. A flow, computed in the parameter space of the two surfaces =-=[4, 5]-=-, then induces a correspondence field in 3D. This flow can be constrained using anatomic landmark points or curves, by constraining the mapping of surface regions represented implicitly using levelBr... |

58 | Cortical cartography using the discrete conformal approach of circle packings. - Hurdal, Stephenson - 2004 |

47 | Minimum description length shape and appearance models
- Thodberg
- 2003
(Show Context)
Citation Context ... description length (MDL) principle, based on the compactness of the covariance of the resulting shape model [9]. Anatomically homologous points can then be forced to match across a dataset. Thodberg =-=[10]-=- identified problems with early MDL approaches and extended them to an MDL appearance model, when performing unsupervised image segmentation. All oriented surfaces have conformal structures. The confo... |

44 | Generalized tensor-based morphometry of HIV/AIDS using multivariate statistics on deformation tensors.
- Lepore, Brun, et al.
- 2008
(Show Context)
Citation Context ...omy as a means to understand shape variation between structural brain images. Techniques based on Riemannian manifolds to compare deformation tensors or strain matrices were introduced in [16–18]. In =-=[19]-=-, the full deformation tensors were used in the context of tensor-based morphometry. In a conformal parameterization, the original metric tensor is preserved up to a constant. The conformal parametriz... |

43 | Riemannian elasticity: A statistical regularization framework for non-linear registration, MICCAI, - Pennec - 2005 |

36 |
Abnormal cortical complexity and thickness profiles mapped in Williams syndrome,”
- Thompson, Lee, et al.
- 2005
(Show Context)
Citation Context ...ents with Williams syndrome (WS) and a group of healthy control subjects. WS is a genetic disorder in which the cortex develops abnormally, but the scope and type of systematic differences is unknown =-=[20]-=-. In our experimental results, we identified several significantly different areas on the left and right cortical surfaces between WS patients and control subjects.38 Y. Wang et al. 2 Ricci Flow Conf... |

27 |
Growth patterns in the developing human brain detected using continuum-mechanical tensor mapping. Nature 404
- Thompson, Giedd, et al.
- 2000
(Show Context)
Citation Context ...is is often achieved by first mapping each of the 3D surfaces to canonical parameter spaces such as a sphere [1, 2] or a planar domain [3]. A flow, computed in the parameter space of the two surfaces =-=[4, 5]-=-, then induces a correspondence field in 3D. This flow can be constrained using anatomic landmark points or curves, by constraining the mapping of surface regions represented implicitly using levelBr... |

27 |
Inferring brain variability from diffeomorphic deformations of currents: an integrative approach. Med Image Anal 12(5):626–637
- Durrleman, Pennec, et al.
- 2008
(Show Context)
Citation Context ...ints or curves, by constraining the mapping of surface regions represented implicitly using levelBrain Mapping with the Ricci Flow 37 sets [3], or by using currents to represent anatomical variation =-=[6]-=- Feature correspondence between two surfaces can be optimized by using the L 2 -norm to measure differences in curvature profiles or convexity [1] or by using mutual information to align scalar fields... |

26 | Automated Surface Matching using Mutual Information Applied to Riemann Surface Structures
- Wang, Thompson
- 2005
(Show Context)
Citation Context ...g the L 2 -norm to measure differences in curvature profiles or convexity [1] or by using mutual information to align scalar fields of various differential geometric parameters defined on the surface =-=[7]-=-. Artificial neural networks may also be used to rule out or favor certain types of feature matches [8]. Finally, correspondences may be determined by using a minimum description length (MDL) principl... |

25 | Brain surface conformal parameterization using riemann surface structure - Wang, Lui, et al. - 2007 |

20 | Brain surface conformal parameterization with the ricci flow.
- Wang, Shi, et al.
- 2012
(Show Context)
Citation Context ...he Riemannian metric but places more restrictions on the the surface morphology than the topological structure. The Ricci flow method can conformally map an open boundary surface to a multi-hole disk =-=[11]-=-. Compared with other conformal parameterization methods [12–15], the Ricci flow method can handle cortical surfaces with complicated topologies without singularities. The continuous Ricci flow confor... |

19 | Incorporating statistical measures of anatomical variability in atlas-to-subject registration for conformal brain radiotherapy - Commowick, Stefanescu, et al. - 2005 |

18 |
Landmark matching on brain surfaces via large deformation diffeomorphisms on the sphere.
- Bakircioglu, Joshi, et al.
- 1999
(Show Context)
Citation Context ...ecific anatomical points, curved landmarks, or subregions lying within the two surfaces. This is often achieved by first mapping each of the 3D surfaces to canonical parameter spaces such as a sphere =-=[1, 2]-=- or a planar domain [3]. A flow, computed in the parameter space of the two surfaces [4, 5], then induces a correspondence field in 3D. This flow can be constrained using anatomic landmark points or c... |

18 | Learning object correspondences with the observed transport shape measures
- Pitiot, Delingette, et al.
- 2003
(Show Context)
Citation Context ...ation to align scalar fields of various differential geometric parameters defined on the surface [7]. Artificial neural networks may also be used to rule out or favor certain types of feature matches =-=[8]-=-. Finally, correspondences may be determined by using a minimum description length (MDL) principle, based on the compactness of the covariance of the resulting shape model [9]. Anatomically homologous... |

17 |
Brain structural mapping using a novel hybrid implicit/explicit framework based on the level-set method.
- Leow
- 2005
(Show Context)
Citation Context ...curved landmarks, or subregions lying within the two surfaces. This is often achieved by first mapping each of the 3D surfaces to canonical parameter spaces such as a sphere [1, 2] or a planar domain =-=[3]-=-. A flow, computed in the parameter space of the two surfaces [4, 5], then induces a correspondence field in 3D. This flow can be constrained using anatomic landmark points or curves, by constraining ... |

4 |
Differential Forms: A Complement to Vector Calculus
- Weitraub
- 2007
(Show Context)
Citation Context ...herical (Euclidean or hyperbolic) background geometry. For arbitrary two vertices vi and vj, the following symmetric relation holds: ∂Ki/∂uj = ∂Kj/∂ui. Let ω = ∑n i=1 Kidui be a differential one-form =-=[25]-=-; the symmetric relation guarantees that this one-form is closed (curl free) in the metric space:Brain Mapping with the Ricci Flow 41 dω = 0. Then by Stokes theorem, the following integration is path... |