## Deformotion - Deforming Motion, Shape Average and the Joint Registration and Segmentation of Images (2002)

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Venue: | International Journal of Computer Vision |

Citations: | 108 - 16 self |

### BibTeX

@ARTICLE{Soatto02deformotion-,

author = {Stefano Soatto and Anthony J. Yezzi},

title = {Deformotion - Deforming Motion, Shape Average and the Joint Registration and Segmentation of Images},

journal = {International Journal of Computer Vision},

year = {2002},

volume = {53},

pages = {153--167}

}

### Years of Citing Articles

### OpenURL

### Abstract

What does it mean for a deforming object to be "moving" (see Fig. 1)? How can we separate the overall motion (a finite-dimensional group action) from the more general deformation (a di#eomorphism)? In this paper we propose a definition of motion for a deforming object and introduce a notion of "shape average" as the entity that separates the motion from the deformation. Our definition allows us to derive novel and e#cient algorithms to register non-equivalent shapes using region-based methods, and to simultaneously approximate and register structures in grey-scale images. We also extend the notion of shape average to that of a "moving average" in order to track moving and deforming objects through time.

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Citation Context ...ssive) moving average of y. Without loss of generality one may assume that k = 1 since the difference equation above can always be reduced to first-order by augmenting the dimension of the state (see =-=[26] for details-=-). 10 (26)sγi γi gi(µ) gi(µ) µ µ γi γi Figure 3: Euclidean (top) vs. affine (bottom) registration and average. For each pair of objects γ1, γ2, the registration g1(µ), g2(µ) relative to th... |

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Citation Context ... intuition. However, Mumford has critiqued current theories of shape on the grounds that they fail to capture the essential features of perception [32]. “Deformable Templates,” pioneered by Grenan=-=der [14], do not rel-=-y on “features” or “landmarks;” rather, images are directly deformed by a (possibly infinite-dimensional) group action and compared for the best match in an “image-based” approach [44]. Th... |

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Citation Context ...i.e. equivalent under the group action), but they also undergo deformations. Also related to this paper is the recent work of Paragios and Deriche, where active regions are tracked as they “move.”=-= In [34]-=- the notion of motion is not made distinct from the general deformation, and therefore what is being tracked is a general (infinite-dimensional) deformation. Our aim is to define tracking as a traject... |

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Citation Context ...the work of Matheron on “Stochastic Sets” [30] as well as that of Thom, Giblin and others [37, 13]. In statistics, the study of “Shape Spaces” was championed by Kendall, Mardia and Carne among=-= others [17, 24, 9, 29].-=- Shapes are defined as the equivalence classes of N points in R M under the similarity group, R MN /{SE(M) × R}. Shape spaces are thus organized in a fiber bundle where motion along the fibers corres... |

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Citation Context ... matching is, by construction, invariant with respect to either the affine or the Euclidean group, and the resulting match is based on “features” rather than on image intensity directly, similarly=-= to [10, 11]-=-. Koenderink [22] is credited with providing some of the key ideas involved in formalizing a notion of shape that matches our intuition. However, Mumford has critiqued current theories of shape on the... |

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Citation Context ...he earlier attempts to formalize a notion of shape include D’Arcy Thompson’s treatise “Growth and Form” [38], the work of Matheron on “Stochastic Sets” [30] as well as that of Thom, Giblin=-= and others [37, 13]. In-=- statistics, the study of “Shape Spaces” was championed by Kendall, Mardia and Carne among others [17, 24, 9, 29]. Shapes are defined as the equivalence classes of N points in R M under the simila... |

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Citation Context ...s and the solution of partial differential equations (PDEs) to model shape and to compute distances and similarity. In this framework, not only can the notion of alignment or distance be made precise =-=[5, 43, 31, 20, 35]-=-, but quite sophisticated theories that encompass perceptually relevant aspects, can be formalized in terms of the properties of the evolution of PDEs (e.g. [21]). The work of Kimia et al. [18] descri... |

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Citation Context ...der [14], do not rely on “features” or “landmarks;” rather, images are directly deformed by a (possibly infinite-dimensional) group action and compared for the best match in an “image-based�=-=�� approach [44]. There, the n-=-otion of “motion” (or “alignment” or “registration”) coincides with that of deformation, and there is no clear distinction between the two [7]. Grenander’s work sparked a current that ha... |

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Citation Context ...s a nice taxonomy of existing work on shape and deformation and a review of the state of the art as of 1994. The variational framework has also proven very effective in the analysis of medical images =-=[28, 39, 27]-=-. Although most of the ideas are develop in a deterministic setting, many can be transposed to a probabilistic context (e.g. Zhu et al. [45]). None of these approaches, however, distinguishes a notion... |

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Citation Context ...has also proven very effective in the analysis of medical images [23, 30, 22]. Although most of these ideas are developed in a deterministic setting, many can be transposed to a probabilistic context =-=[35, 7, 31]. Sc-=-alespace is a very active research area, and some of the key contributions as they relate to the material of this paper can be found in [14, 29, 17, 1, 2] and references therein. The “alignment,” ... |

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Citation Context ...s and the solution of partial differential equations (PDEs) to model shape and to compute distances and similarity. In this framework, not only can the notion of alignment or distance be made precise =-=[5, 43, 31, 20, 35]-=-, but quite sophisticated theories that encompass perceptually relevant aspects, can be formalized in terms of the properties of the evolution of PDEs (e.g. [21]). The work of Kimia et al. [18] descri... |

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Citation Context ...s and the solution of partial differential equations (PDEs) to model shape and to compute distances and similarity. In this framework, not only can the notion of alignment or distance be made precise =-=[5, 43, 31, 20, 35]-=-, but quite sophisticated theories that encompass perceptually relevant aspects, can be formalized in terms of the properties of the evolution of PDEs (e.g. [21]). The work of Kimia et al. [18] descri... |

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Citation Context ...art of the inference problem. Errors in the pre-processing stage can never be compensated, and the “registration” cannot be adapted as the 3soriginal contours evolve towards their average. Similar=-=ly, [41]-=- performs the joint segmentation of a number of images by assuming that their registration (stereo calibration) is given. We wish to extend these approaches to situations where the calibration/registr... |

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Citation Context ...ved in formalizing a notion of shape that matches our intuition. However, Mumford has critiqued current theories of shape on the grounds that they fail to capture the essential features of perception =-=[32]. “Deforma-=-ble Templates,” pioneered by Grenander [14], do not rely on “features” or “landmarks;” rather, images are directly deformed by a (possibly infinite-dimensional) group action and compared for... |

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Citation Context ...rature exists for the problem of “matching” or “aligning” objects based on their images, and space limitations do not allow us to do justice to the many valuable contributions. We refer the re=-=ader to [40] f-=-or a recent survey. A common approach consists of matching collections points organized in graphs or trees (e.g. [23, 12]). Belongie et al. [6] propose comparing planar contours based on their “shap... |

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Citation Context ...tion, their scale-space representation at some level may be. Scale-space is a very active research area, and some of the key contributions as they relate to the material of this paper can be found in =-=[16, 36, 19, 2, 3, 1] and referen-=-ces therein. The “alignment,” or “registration,” of curves has also been used to define a notion of “shape average” by several authors (see [25] and references therein). The shape average,... |

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Citation Context ...the work of Matheron on “Stochastic Sets” [30] as well as that of Thom, Giblin and others [37, 13]. In statistics, the study of “Shape Spaces” was championed by Kendall, Mardia and Carne among=-= others [17, 24, 9, 29].-=- Shapes are defined as the equivalence classes of N points in R M under the similarity group, R MN /{SE(M) × R}. Shape spaces are thus organized in a fiber bundle where motion along the fibers corres... |

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Citation Context ...tion, their scale-space representation at some level may be. Scale-space is a very active research area, and some of the key contributions as they relate to the material of this paper can be found in =-=[16, 36, 19, 2, 3, 1] and referen-=-ces therein. The “alignment,” or “registration,” of curves has also been used to define a notion of “shape average” by several authors (see [25] and references therein). The shape average,... |

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Citation Context ...any can be transposed to a probabilistic context [35, 7, 31]. Scalespace is a very active research area, and some of the key contributions as they relate to the material of this paper can be found in =-=[14, 29, 17, 1, 2] and referen-=-ces therein. The “alignment,” or “registration,” of curves has also been used to define a notion of “shape average” by several authors (see [21] and references therein). The shape average,... |

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Citation Context ...ematical morphology to statistics, geology, neuroanatomy, paleontology, astronomy etc. Some of the earlier attempts to formalize a notion of shape include D’Arcy Thompson’s treatise “Growth and =-=Form” [38], the wo-=-rk of Matheron on “Stochastic Sets” [30] as well as that of Thom, Giblin and others [37, 13]. In statistics, the study of “Shape Spaces” was championed by Kendall, Mardia and Carne among other... |

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Citation Context ...s a nice taxonomy of existing work on shape and deformation and a review of the state of the art as of 1994. The variational framework has also proven very effective in the analysis of medical images =-=[28, 39, 27]-=-. Although most of the ideas are develop in a deterministic setting, many can be transposed to a probabilistic context (e.g. Zhu et al. [45]). None of these approaches, however, distinguishes a notion... |

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Citation Context ...the work of Matheron on “Stochastic Sets” [30] as well as that of Thom, Giblin and others [37, 13]. In statistics, the study of “Shape Spaces” was championed by Kendall, Mardia and Carne among=-= others [17, 24, 9, 29].-=- Shapes are defined as the equivalence classes of N points in R M under the similarity group, R MN /{SE(M) × R}. Shape spaces are thus organized in a fiber bundle where motion along the fibers corres... |

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Citation Context ...ies from mathematical morphology to statistics, geology, neuroanatomy, paleontology, astronomy etc. In statistics, the study of “Shape Spaces” was championed by Kendall, Mardia and Carne among oth=-=ers [15, 20, 8, 24, 9]. Sha-=-pes are defined as the equivalence classes of N points in R M under the similarity group 2 , R MN /{SE(M) × R}. Although the framework clearly distinguishes the notion of “motion” (along the fibe... |

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Citation Context ...istance be made precise [5, 43, 31, 20, 35], but quite sophisticated theories that encompass perceptually relevant aspects, can be formalized in terms of the properties of the evolution of PDEs (e.g. =-=[21]). The w-=-ork of Kimia et al. [18] describes a scale-space that corresponds to various stages of evolution of a diffusing PDE, and a “reacting” PDE that splits “salient parts” of planar contours by gene... |

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Citation Context ...any can be transposed to a probabilistic context [35, 7, 31]. Scalespace is a very active research area, and some of the key contributions as they relate to the material of this paper can be found in =-=[14, 29, 17, 1, 2] and referen-=-ces therein. The “alignment,” or “registration,” of curves has also been used to define a notion of “shape average” by several authors (see [21] and references therein). The shape average,... |

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Citation Context ...s and the solution of partial differential equations (PDEs) to model shape and to compute distances and similarity. In this framework, not only can the notion of alignment or distance be made precise =-=[3, 33, 25, 28]-=-, but quite sophisticated theories that encompass perceptually relevant aspects can be formalized in terms of the properties of the evolution of PDEs (e.g. [18, 16]). Kimia et al. [16] describes a sca... |

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Citation Context ...the best match in an “imagebased” approach [34]. There, the notion of “motion” (or “alignment” or “registration”) coincides with that of deformation, and there is no clear distinction =-=between the two [5]-=-. Another line of work uses variational methods and the solution of partial differential equations (PDEs) to model shape and to compute distances and similarity. In this framework, not only can the no... |

5 |
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Citation Context ...ming that their registration (stereo calibration) is given. We wish to extend these approaches to situations where the calibration/registration is not known a-priori. A somewhat complementary work is =-=[42]-=-, where objects, assumed to be identical except for a group action, are registered by minimizing a regionbased cost functional. We wish to extend that approach to situations where the objects are not ... |

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Citation Context ...llow us to do justice to the many valuable contributions. We refer the reader to [40] for a recent survey. A common approach consists of matching collections points organized in graphs or trees (e.g. =-=[23, 12]). Belon-=-gie et al. [6] propose comparing planar contours based on their “shape context.” There, points are not bound to represent particular “landmarks” but are just a discrete representation of the c... |

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Citation Context ... matching is, by construction, invariant with respect to either the affine or the Euclidean group, and the resulting match is based on “features” rather than on image intensity directly, similarly=-= to [10, 11]-=-. Koenderink [22] is credited with providing some of the key ideas involved in formalizing a notion of shape that matches our intuition. However, Mumford has critiqued current theories of shape on the... |