## Elastic Matching: Continuum Mechanical and Probabilistic Analysis

Venue: | IN BRAIN WARPING |

Citations: | 29 - 10 self |

### BibTeX

@INPROCEEDINGS{Gee_elasticmatching:,

author = {James C. Gee and Ruzena K. Bajcsy},

title = {Elastic Matching: Continuum Mechanical and Probabilistic Analysis},

booktitle = {IN BRAIN WARPING},

year = {},

pages = {18--3},

publisher = {Academic Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

In 1981, Broit in collaboration with Bajcsy introduced a method for the "optimal registration of deformed images" [2], innovating the physics-based approach to shape modeling in image analysis [28, 34] and simultaneously enabling a breakthrough in computational anatomy. Originally developed to ameliorate the difficult task of anatomic localization in tomographic scans of the human brain, the solution approach has since been adopted in a wide variety of related problems involving the extraction, measurement, or visualization of image features important in facilitating clinical diagnosis, therapy, and research. To realize his insight that the various problems of localizing, segmenting, or visualizing cerebral structures in an image study are reducible to one of matching the labeled anatomy depicted in a reference study, or atlas, to the subject anatomy in the study, Broit invented a general registration procedure in which one image volume is modeled

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Citation Context ... can be specified in a principled way as probability measures on the space of admissible spatial transformations. Our formulation follows the general framework for Bayesian image analysis laid out in =-=[23, 32, 5, 37]-=-, and shares some elements of Grenander's pattern theory, which pioneered the probabilistic interpretation of flexible templates such as Broit's deformable atlas [24, 25, 26]. To illustrate the basic ... |

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Citation Context ...uction In 1981, Broit in collaboration with Bajcsy introduced a method for the "optimal registration of deformed images" [2], innovating the physics-based approach to shape modeling in image=-= analysis [28, 34]-=- and simultaneously enabling a breakthrough in computational anatomy. Originally developed to ameliorate the difficult task of anatomic localization in tomographic scans of the human brain, the soluti... |

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Citation Context ...ior and thus utilizes all the information that is available about the unknown mapping. Decision theory, for example, guides us in choosing the optimal mapping in light of this statistical information =-=[4]-=-. 4.3.1 Point Estimation To perform matching, point estimates of u are inferred by minimizing an expected loss with respect to the posterior distribution [4]. The solutions obtained with Broit's elast... |

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Citation Context ...ngs to be developed, which in turn can substantially simplify the matching calculation. Note that the strain energy compares with the first-order quadratic stabilizers used in standard regularization =-=[40, 36, 38]-=-. Moreover, the elastic constants are related to the regularization parameter: by varying their values, we can modulate the stiffness of the body and thus the degree of smoothness in the deformations.... |

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Citation Context ...spatial transformation x specifies for each point X in I s its corresponding point x(X) in I t , and f models the compound effect of processes which modify the values of the original intensity signal =-=[10]-=-. In our application, mappings are sought between images that arise from distinct but topologically similar sources, so (8) does not strictly apply---the success of atlas matching therefore depends bo... |

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Citation Context ...udy of biological shape, the imposition of C 1 continuity has been advocated in performing morphometrics, based on the observation that spatial variation in the proportions of shape tend to be graded =-=[6]-=-. For atlas matching, in addition to adopting such general assumptions about the spatial mappings, the analysis can be performed using actual statistics of the anatomic variation observed in a populat... |

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Citation Context ...likelihood of false matches that arise from this situation, a standard approach is to solve the problem at different spatial scales, as in the multiresolution version of elastic matching described in =-=[3]-=-: large scale displacements are first determined by matching the lower spatial frequencies in the images and these are then used to remove their confounding effect in the alignment of the higher frequ... |

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Citation Context ...the degree of smoothness in the deformations. It is evident that a range of regularizing behaviors may be effected by varying the particular idealized continuum on which the image volumes are modeled =-=[11, 8]-=-. The variational formulation of matching strongly hints at a Bayesian rationale, which we develop fully in the next section. We recall that computational anatomy aims at establishing probability meas... |

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Citation Context ...ngs to be developed, which in turn can substantially simplify the matching calculation. Note that the strain energy compares with the first-order quadratic stabilizers used in standard regularization =-=[40, 36, 38]-=-. Moreover, the elastic constants are related to the regularization parameter: by varying their values, we can modulate the stiffness of the body and thus the degree of smoothness in the deformations.... |

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Citation Context ... example, the exploration of a solution's reliability or the incorporation of empirical information that may be available about the spatial transformations into an analysis. For computational anatomy =-=[7, 35, 33, 15, 12, 39, 13]-=-, these aspects of the probabilistic approach will figure importantly in the development of a comprehensive methodology. 2 Anatomy Atlas At the time of Broit's development of the elastic matching in t... |

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Citation Context ...e analysis laid out in [23, 32, 5, 37], and shares some elements of Grenander's pattern theory, which pioneered the probabilistic interpretation of flexible templates such as Broit's deformable atlas =-=[24, 25, 26]-=-. To illustrate the basic ideas, we begin with a general definition of the image matching problem: find x and f such that I s (X) = f(I t (x(X)); (8) where the spatial transformation x specifies for e... |

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Citation Context ...e analysis laid out in [23, 32, 5, 37], and shares some elements of Grenander's pattern theory, which pioneered the probabilistic interpretation of flexible templates such as Broit's deformable atlas =-=[24, 25, 26]-=-. To illustrate the basic ideas, we begin with a general definition of the image matching problem: find x and f such that I s (X) = f(I t (x(X)); (8) where the spatial transformation x specifies for e... |

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Citation Context ... can be specified in a principled way as probability measures on the space of admissible spatial transformations. Our formulation follows the general framework for Bayesian image analysis laid out in =-=[23, 32, 5, 37]-=-, and shares some elements of Grenander's pattern theory, which pioneered the probabilistic interpretation of flexible templates such as Broit's deformable atlas [24, 25, 26]. To illustrate the basic ... |

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Citation Context ... can be specified in a principled way as probability measures on the space of admissible spatial transformations. Our formulation follows the general framework for Bayesian image analysis laid out in =-=[23, 32, 5, 37]-=-, and shares some elements of Grenander's pattern theory, which pioneered the probabilistic interpretation of flexible templates such as Broit's deformable atlas [24, 25, 26]. To illustrate the basic ... |

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Citation Context ...uction In 1981, Broit in collaboration with Bajcsy introduced a method for the "optimal registration of deformed images" [2], innovating the physics-based approach to shape modeling in image=-= analysis [28, 34]-=- and simultaneously enabling a breakthrough in computational anatomy. Originally developed to ameliorate the difficult task of anatomic localization in tomographic scans of the human brain, the soluti... |

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Citation Context ...nts in the target object. The first object is deformed in this way until an equilibrium configuration is reached or, equivalently, until the total potential energy of the system is at a local minimum =-=[31]-=-. Note that if the elastic strain energy is used as the measure of deformation in (1), the cost would be identical to the forementioned total potential energy. Therefore, the elastostatic configuratio... |

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Citation Context ...the degree of smoothness in the deformations. It is evident that a range of regularizing behaviors may be effected by varying the particular idealized continuum on which the image volumes are modeled =-=[11, 8]-=-. The variational formulation of matching strongly hints at a Bayesian rationale, which we develop fully in the next section. We recall that computational anatomy aims at establishing probability meas... |

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Citation Context ...e analysis laid out in [23, 32, 5, 37], and shares some elements of Grenander's pattern theory, which pioneered the probabilistic interpretation of flexible templates such as Broit's deformable atlas =-=[24, 25, 26]-=-. To illustrate the basic ideas, we begin with a general definition of the image matching problem: find x and f such that I s (X) = f(I t (x(X)); (8) where the spatial transformation x specifies for e... |

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Citation Context ... example, the exploration of a solution's reliability or the incorporation of empirical information that may be available about the spatial transformations into an analysis. For computational anatomy =-=[7, 35, 33, 15, 12, 39, 13]-=-, these aspects of the probabilistic approach will figure importantly in the development of a comprehensive methodology. 2 Anatomy Atlas At the time of Broit's development of the elastic matching in t... |

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Citation Context ... example, the exploration of a solution's reliability or the incorporation of empirical information that may be available about the spatial transformations into an analysis. For computational anatomy =-=[7, 35, 33, 15, 12, 39, 13]-=-, these aspects of the probabilistic approach will figure importantly in the development of a comprehensive methodology. 2 Anatomy Atlas At the time of Broit's development of the elastic matching in t... |

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Citation Context ...t group. the areas of higher intensity value indicate dilatation and the darker regions correspond to compression of the atlas as it was warped into alignment with the subject callosa. As detailed in =-=[30, 14]-=-, the mean Jacobian at each point provides a useful measure with which to quantify the shape differences between two groups. Toward a model that characterizes the callosal shapes of our subject popula... |

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Citation Context ... subject anatomy in the study, Broit invented a general registration procedure in which one image volume is modeled as an elastic continuum that is deformed to match the appearance of a second volume =-=[9]-=-. In this chapter 1 , we review the seminal ideas underlying Broit's elastic matching and then discuss their generalization within a probabilistic framework. We introduce the probabilistic approach by... |

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Citation Context .... Although the energy-based approach yields the same equations obtained from the classical derivation, the variational formulation opens the problem to numerical solution by the finite element method =-=[41]-=-, in which the search space comprises linear combinations of basis functions defined piecewise to fit the, possibly irregular, geometry of the problem domain. For images with sparse features, like tho... |

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Citation Context ...RASP Laboratory University of Pennsylvania, Philadelphia, Pennsylvanias1 Introduction In 1981, Broit in collaboration with Bajcsy introduced a method for the "optimal registration of deformed ima=-=ges" [2]-=-, innovating the physics-based approach to shape modeling in image analysis [28, 34] and simultaneously enabling a breakthrough in computational anatomy. Originally developed to ameliorate the difficu... |

28 | On matching brain volumes
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Citation Context ...omy in structural scans of the brain, Broit asserted that external knowledge had to be introduced. Noting that approaches based on shape primitives 1 This chapter is adapted from material reported in =-=[20, 22, 17, 18, 19, 16]-=-. were unable to simultaneously describe the complex shapes of anatomical structures and the complicated topological relationships between them, he proposed that a voxel array containing a labeled ima... |

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Citation Context ...e their confounding effect in the alignment of the higher frequency content. When even the multi-resolution search fails, additional external knowledge must be introduced into the problem formulation =-=[27]-=-. Acknowledgements Many people have contributed to the development and success of the work reported in this chapter, several of whom deserve special mention: David Haynor, Lionel Le Briquer, Pedro Per... |

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Citation Context ...omy in structural scans of the brain, Broit asserted that external knowledge had to be introduced. Noting that approaches based on shape primitives 1 This chapter is adapted from material reported in =-=[20, 22, 17, 18, 19, 16]-=-. were unable to simultaneously describe the complex shapes of anatomical structures and the complicated topological relationships between them, he proposed that a voxel array containing a labeled ima... |

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Citation Context ...omy in structural scans of the brain, Broit asserted that external knowledge had to be introduced. Noting that approaches based on shape primitives 1 This chapter is adapted from material reported in =-=[20, 22, 17, 18, 19, 16]-=-. were unable to simultaneously describe the complex shapes of anatomical structures and the complicated topological relationships between them, he proposed that a voxel array containing a labeled ima... |

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Citation Context ...t group. the areas of higher intensity value indicate dilatation and the darker regions correspond to compression of the atlas as it was warped into alignment with the subject callosa. As detailed in =-=[30, 14]-=-, the mean Jacobian at each point provides a useful measure with which to quantify the shape differences between two groups. Toward a model that characterizes the callosal shapes of our subject popula... |

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Citation Context ...always preserved after deformation. Suppose the displacements also take 2 Although the notation is from [31, chapter 2], our overview of the continuum mechanics closely follows the treatment found in =-=[29]-=-. a neighboring point Q at X + dX to x + dx = X + dX + u(X + dX; t). The relative position of P and Q becomes dx = dX + u(X +dX; t) \Gamma u(X; t); or dx = dX + (u ( r) \Delta dX ; where u ( r is the ... |

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Citation Context .... Another commonly cited point summary is the posterior mean, which optimizes the quadratic loss. To estimate its value, we sample from a series of normal approximations to the posterior distribution =-=[21]-=-. In contrast, the posterior mode is computed deterministically, but in iterative fashion as well. The main difficulty in both implementations stems from the highly nonlinear nature of the likelihood ... |

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