## Surface matching via currents (2005)

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Venue: | Proceedings of Information Processing in Medical Imaging (IPMI 2005), number 3565 in Lecture Notes in Computer Science |

Citations: | 63 - 1 self |

### BibTeX

@INPROCEEDINGS{Vaillant05surfacematching,

author = {Marc Vaillant and Joan Glaunès},

title = {Surface matching via currents},

booktitle = {Proceedings of Information Processing in Medical Imaging (IPMI 2005), number 3565 in Lecture Notes in Computer Science},

year = {2005},

pages = {381--392}

}

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### Abstract

Abstract. We present a new method for computing an optimal deformation between two arbitrary surfaces embedded in Euclidean 3-dimensional space. Our main contribution is in building a norm on the space of surfaces via representation by currents of geometric measure theory. Currents are an appropriate choice for representations because they inherit natural transformation properties from differential forms. We impose a Hilbert space structure on currents, whose norm gives a convenient and practical way to define a matching functional. Using this Hilbert space norm, we also derive and implement a surface matching algorithm under the large deformation framework, guaranteeing that the optimal solution is a one-to-one regular map of the entire ambient space. We detail an implementation of this algorithm for triangular meshes and present results on 3D face and medical image data. 1

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Citation Context ... in image analysis applications is to perform a non-rigid matching (deformation) between two occurrences of the same structure. For example, it has been recognized as early as 1917 by D’Arcy Thompson =-=[1]-=-, that given representations of a particular anatomic structure in two subjects, an appropriate methodology for comparing their gross morphological differences is to study a transformation–uniquely ch... |

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Citation Context ...hing as a “point correspondence” task: for each point on the discretized template surface find its corresponding point on the target. Fully automated approaches to this problem have been developed in =-=[7]-=-. However, a fundamental issue with this point of view is that, due to discretization, a point on one surface need not have a homologous point on the other. This problem is handled in [7] by simultane... |

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Citation Context ...: φ ♯ ω(x)(η, ν) = ω(φ(x)) ((dxφ)η, (dxφ)ν) . The push forward φ♯S of a current S is φ♯S(ω) = S(φ ♯ ω). The change of coordinates for integration of differential forms [13] states S(φ ♯ ω) = φ(S)(ω). =-=(2)-=- That is, φ♯S is indeed the current associated with φ(S), which is exactly the natural property we would like our representations to have. 2.3 Vectorial representation It will be convenient to use a v... |

134 |
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Citation Context ...deformation setting, it can be shown that the optimal vector fields vt are of the form vt(x) = � j f,q (6) kV (xj, x)α j t, (7) where kV denotes the reproducing kernel of the deformation space V (see =-=[4, 5]-=-). The vectors α j t are referred to as momentum vectors do to the connection of the large deformation setting to Hamiltonian mechanics (see [17, 18]). It follows from the flow equation that the match... |

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Citation Context ...kW (x, y)ξ = KW δ ξ x(y). Thus it is in fact the reproducing kernel of W , the space of vector fields corresponding to W . From this definition follows the formula: 〈δ ξ x, δ η y〉W ∗ = kW (x, y)ξ · η =-=(3)-=- We impose a slightly stronger constraint than continuity of the evaluation functionals: W is constructed so that it is continuously embedded in the space of continuous bounded 2-forms. That is, there... |

113 |
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Citation Context ...ector fields is of the form ∇Et(x) = � j kV (x j t, x)d j x E. t φ ∗ t1∇ x j tsIndeed, for a variation vt,ɛ = vt + ɛ�vt of the vector field vt, the corresponding variation of x j 1 = φ1(x j ) is (see =-=[19]-=-) and thus the variation of E is �x j j 1 = ∂ɛx1 |ɛ=0 = ∂ɛE|ɛ=0 = � = j � 1 0 ∂ j x E t � x j 1 = � 1 0 � 1 0 〈kV (x j t, ·)d j x t d j x φt1�vt(x t j t)dt, ∂ j x Ed j t x φt1�vt(x t j t)dt φ ∗ j t1∇x... |

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Citation Context ...n recognizing currents as an appropriate mathematical modeling object for surfaces. Given the recent active developments in exterior calculus [22] and current based approaches to curvature estimation =-=[23]-=-, we expect the representations to become more sophisticated (perhaps incorporating second order geometric information), and that discretization will continue to get better. A promising and exciting i... |

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Citation Context ...pendent families of elements of V , written vt ∈ V for t ∈ [0, 1], such that � 1 0 |vt|V dt < ∞, the solution φt at time t = 1, of ∂φ ∂t = vt ◦ φt, (4) with φ0(x) = x, is a unique diffeomorphism (see =-=[15, 16]-=-). The collection of all such solutions defines our subgroup of diffeomorphisms GV , and the inner product 〈·, ·〉V equips it with a Riemannian structure. We will sometimes denote φv for an element of ... |

87 | Discrete exterior calculus
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Citation Context ...ce on different types of data. The main contribution was in recognizing currents as an appropriate mathematical modeling object for surfaces. Given the recent active developments in exterior calculus =-=[22]-=- and current based approaches to curvature estimation [23], we expect the representations to become more sophisticated (perhaps incorporating second order geometric information), and that discretizati... |

54 | 3D Statistical Shape Models Using Direct Optimisation of Description Length
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Citation Context ... surface curvature in their matching criterion. Its advantage is that both triangulation and correspondence is established simultaneously. Another elegant approach includes the work of Davies et. al. =-=[9]-=- in which the correspondence problem is tackled by building the “best” model via a set of optimality criteria. We develop a surface matching approach in which the two issues mentioned above are overco... |

52 | Statistics on diffeomorphisms via tangent space representations
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(Show Context)
Citation Context ...e reproducing kernel of the deformation space V (see [4, 5]). The vectors α j t are referred to as momentum vectors do to the connection of the large deformation setting to Hamiltonian mechanics (see =-=[17, 18]-=-). It follows from the flow equation that the matching functional (5) is a function only of the trajectories x j t. Gradient of the data attachment term The gradient of the data attachment term, E, in... |

46 | Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching
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Citation Context ...criteria. We develop a surface matching approach in which the two issues mentioned above are overcome naturally in the fundamental theoretical framework. Our approach follows most closely the work of =-=[10]-=-, but differs in that we represent surfaces as the generalized distributions of deRham called currents [11], instead of the classical distributions of Schwartz. As in [10], distribution representation... |

29 |
Landmark matching via large deformation diffeomorphisms on the sphere
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(Show Context)
Citation Context ...[0, 1], V ) Let x j index the vertices of S. Like all point-based matching problems in the large deformation setting, it can be shown that the optimal vector fields vt are of the form vt(x) = � j f,q =-=(6)-=- kV (xj, x)α j t, (7) where kV denotes the reproducing kernel of the deformation space V (see [4, 5]). The vectors α j t are referred to as momentum vectors do to the connection of the large deformati... |

23 |
Geodesic shooting for computational anatomy
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(Show Context)
Citation Context ...e reproducing kernel of the deformation space V (see [4, 5]). The vectors α j t are referred to as momentum vectors do to the connection of the large deformation setting to Hamiltonian mechanics (see =-=[17, 18]-=-). It follows from the flow equation that the matching functional (5) is a function only of the trajectories x j t. Gradient of the data attachment term The gradient of the data attachment term, E, in... |

22 |
An infinite dimensional group approach for physics based models in patterns recognition
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(Show Context)
Citation Context ...pendent families of elements of V , written vt ∈ V for t ∈ [0, 1], such that � 1 0 |vt|V dt < ∞, the solution φt at time t = 1, of ∂φ ∂t = vt ◦ φt, (4) with φ0(x) = x, is a unique diffeomorphism (see =-=[15, 16]-=-). The collection of all such solutions defines our subgroup of diffeomorphisms GV , and the inner product 〈·, ·〉V equips it with a Riemannian structure. We will sometimes denote φv for an element of ... |

14 |
Variétés différentiables, formes, courants, formes harmoniques
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(Show Context)
Citation Context ...ally in the fundamental theoretical framework. Our approach follows most closely the work of [10], but differs in that we represent surfaces as the generalized distributions of deRham called currents =-=[11]-=-, instead of the classical distributions of Schwartz. As in [10], distribution representations allow us to get away from a strict pointwise representation of surfaces and therefore enable us to treat ... |

14 |
Spline models for observational data. CBMS-NSF regional conference serie in applied mathematics
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Citation Context ... via (1), can be equipped with a Hilbert space structure having an easily computable norm. The machinery of Reproducing kernel Hilbert space (r.k.h.s.) theory is fundamental in this construction (see =-=[14]-=-). Background in the somewhat uncommon setting of differential forms is given next, together with the derivation of the norm.sLet (W, 〈·, ·〉W ) be a Hibert space of differential 2-forms. The dual spac... |

12 | 3D brain surface matching based on geodesics and local geometry, Comput. Vision Image Understand. 89
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(Show Context)
Citation Context ...”. A second issue is that geometric information is necessarily discarded when reducing surfaces–inherently 2D objects–to 0-dimensional point sets. Another related approach is the work of Wang et. al. =-=[8]-=-. This approach does use local geometric constraints by including surface curvature in their matching criterion. Its advantage is that both triangulation and correspondence is established simultaneous... |

11 |
Manfredo P., Differential forms and applications
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(Show Context)
Citation Context ...e the pull back of a 2-form ω by: φ ♯ ω(x)(η, ν) = ω(φ(x)) ((dxφ)η, (dxφ)ν) . The push forward φ♯S of a current S is φ♯S(ω) = S(φ ♯ ω). The change of coordinates for integration of differential forms =-=[13]-=- states S(φ ♯ ω) = φ(S)(ω). (2) That is, φ♯S is indeed the current associated with φ(S), which is exactly the natural property we would like our representations to have. 2.3 Vectorial representation I... |

1 |
Geodesic interpolating splines. EMMCVPR
- Camion, Younes
(Show Context)
Citation Context ...deformation setting, it can be shown that the optimal vector fields vt are of the form vt(x) = � j f,q (6) kV (xj, x)α j t, (7) where kV denotes the reproducing kernel of the deformation space V (see =-=[4, 5]-=-). The vectors α j t are referred to as momentum vectors do to the connection of the large deformation setting to Hamiltonian mechanics (see [17, 18]). It follows from the flow equation that the match... |

1 |
Geometric measure theory, 2nd ed
- Morgan
- 1995
(Show Context)
Citation Context ...such as triangular meshes, by replacing the surface measure with 2-dimensional Hausdorff measure. In fact a wide class of geometric subsets of R 3 , called rectifiable sets, can be viewed as currents =-=[12]-=-. In the sequel, we continue to abuse notation by using the same letter to denote both a surface as well as its associated representation as a current. 2.2 Push forward of a current The fundamental pr... |

1 |
3D faces database, courtesy of Professor Sudeep
- HumanID
(Show Context)
Citation Context ...orithm.s4 Experiments Fig. 1. Left: template, right: target, center: mapped template. 4.1 Experiments with faces dataset In this experiment we used 10 segmented surfaces from the USF HumanID database =-=[20]-=- together with manually selected landmarks. The landmarks are used only for validation purposes. The first face was chosen as the template S to be matched to the other 9 surfaces. For each experiment ... |

1 |
Competitive segmentation of the hippocampus and the volumetry in alzheimer’s disease
- Chupin, Hasboun, et al.
(Show Context)
Citation Context ...m: deformations of template through the action of the optimal diffeomorphisms. 4.2 Experiments on hippocampus data Next we applied the matching algorithm to 15 left hippocampi segmented surfaces (see =-=[21]-=- for the method used); the first 7 belong to patients with Alzheimer disease and the others belong to normal subjects. In this experiment, the surfaces were downsampled to 500 triangles. Figure 3 disp... |