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**1 - 9**of**9**### 174 Mathematical Foundations of Computational Anatomy (MFCA'06) Realizing Unbiased Deformation: A Theoretical Consideration

, 2011

"... Abstract — Maps of local tissue compression or expansion are often recovered by comparing MRI scans using nonlinear registration techniques. The resulting changes can be analyzed using tensor-based morphometry (TBM) to make inferences about anatomical differences. Numerous deformation techniques hav ..."

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Abstract — Maps of local tissue compression or expansion are often recovered by comparing MRI scans using nonlinear registration techniques. The resulting changes can be analyzed using tensor-based morphometry (TBM) to make inferences about anatomical differences. Numerous deformation techniques have been developed, although there has not been much theoretical development examining the mathematical/statistical validity of each technique. In this paper, we propose a basic principle that any registration technique should satisfy: realizing unbiased test statistics under null distribution of the displacement. In other words, any registration technique should recover zero change in the test statistic when comparing two images differing only in noise. Based on this principle, we propose a fundamental framework for the construction and analysis of image deformation. Moreover, we argue that logarithmic transform is instrumental in the analysis of deformation maps. Combined with the proposed framework, this leads to a theoretical connection between image registration and other branches of applied mathematics including information theory and grid generation. Index Terms-Mutual information, Image registration, Computational anatomy. 1.

### Mathematical Foundations of Computational Anatomy Geometrical and Statistical Methods for Modelling Biological Shape Variability

"... Non-linear registration and shape analysis are well developed research topic in the medical image analysis community. There is nowadays a growing number of methods that can faithfully deal with the underlying biomechanical behaviour of intra-subject shape deformations. However, it is more difficult ..."

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Non-linear registration and shape analysis are well developed research topic in the medical image analysis community. There is nowadays a growing number of methods that can faithfully deal with the underlying biomechanical behaviour of intra-subject shape deformations. However, it is more difficult to relate the anatomical shape of different subjects. The goal of computational anatomy is to analyse and to statistically model this specific type of geometrical information. In the absence of any justified physical model, a natural attitude is to explore very general mathematical methods, for instance diffeomorphisms. However, working with such infinite dimensional space raises some deep computational and mathematical problems. In particular, one of the key problem is to do statistics. Likewise, modelling the variability of surfaces leads to rely on shape spaces that are much more complex than for curves. To cope with these, different methodological and computational frameworks have been proposed. The goal of the workshop was to foster interactions between researchers investigating the combination of geometry and statistics for modelling biological shape variability from image and surfaces. A special emphasis was put on theoretical developments, applications and results being welcomed as illustrations. Contributions were solicited in the following areas:

### 1 Avoiding Symmetry-Breaking Spatial Non-Uniformity in Deformable Image Registration via a Quasi-Volume-Preserving Constraint

, 2014

"... The choice of a reference image typically influences the results of deformable image registration, thereby making it asymmetric. This is a consequence of a spatially non-uniform weighting in the cost function integral that leads to general registration inaccuracy. The inhomogeneous integral measure ..."

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The choice of a reference image typically influences the results of deformable image registration, thereby making it asymmetric. This is a consequence of a spatially non-uniform weighting in the cost function integral that leads to general registration inaccuracy. The inhomogeneous integral measure – which is the local volume change in the transformation, thus varying through the course of the registration – causes image regions to contribute differently to the objective function. More importantly, the optimization algorithm is allowed to minimize the cost function by manipulating the volume change, instead of aligning the images. The approaches that restore symmetry to deformable registration successfully achieve inverse-consistency, but do not eliminate the regional bias that is the source of the error. In this work, we address the root of the problem: the non-uniformity of the cost function integral. We introduce a new quasi-volume-preserving constraint that allows for volume change only in areas with well-matching image intensities, and show that such a constraint puts a bound on the error arising from spatial non-uniformity. We demonstrate the advantages of adding the proposed constraint to standard (asymmetric and symmetrized) demons and diffeomorphic demons algorithms through experiments on synthetic images, and real X-ray and 2D/3D

### Forward

, 2011

"... Non-linear registration and shape analysis are well developed research topic in the medical image analysis community. There is nowadays a growing number of methods that can faithfully deal with the underlying biomechanical behaviour of intra-subject shape deformations. However, it is more difficult ..."

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Non-linear registration and shape analysis are well developed research topic in the medical image analysis community. There is nowadays a growing number of methods that can faithfully deal with the underlying biomechanical behaviour of intra-subject shape deformations. However, it is more difficult to relate the anatomical shape of different subjects. The goal of computational anatomy is to analyse and to statistically model this specific type of geometrical information. In the absence of any justified physical model, a natural attitude is to explore very general mathematical methods, for instance diffeomorphisms. However, working with such infinite dimensional space raises some deep computational and mathematical problems. In particular, one of the key problem is to do statistics. Likewise, modelling the variability of surfaces leads to rely on shape spaces that are much more complex than for curves. To cope with these, different methodological and computational frameworks have been proposed. The goal of the workshop was to foster interactions between researchers investigating the combination of geometry and statistics for modelling biological shape variability from image and surfaces. A special emphasis was put on theoretical developments, applications and results being welcomed as illustrations. inria-00614989, version 1- 17 Aug 2011

### J Math Imaging Vis (2009) 34: 61–88 DOI 10.1007/s10851-008-0129-7 Symmetric Non-rigid Registration: A Geometric Theory and Some Numerical Techniques

, 2009

"... Abstract This paper proposes L2- and information-theory-based (IT) non-rigid registration algorithms that are exactly symmetric. Such algorithms pair the same points of two im-ages after the images are swapped. Many commonly-used L2 and IT non-rigid registration algorithms are only approx-imately sy ..."

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Abstract This paper proposes L2- and information-theory-based (IT) non-rigid registration algorithms that are exactly symmetric. Such algorithms pair the same points of two im-ages after the images are swapped. Many commonly-used L2 and IT non-rigid registration algorithms are only approx-imately symmetric. The asymmetry is due to the objective function as well as due to the numerical techniques used in discretizing and minimizing the objective function. This paper analyzes and provides techniques to eliminate both sources of asymmetry. This paper has five parts. The first part shows that objec-tive function asymmetry is due to the use of standard differ-ential volume forms on the domain of the images. The sec-ond part proposes alternate volume forms that completely eliminate objective function asymmetry. These forms, called graph-based volume forms, are naturally defined on the graph of the registration diffeomorphism f, rather than on the domain of f. When pulled back to the domain of f they involve the Jacobian Jf and therefore appear “non-standard”. In the third part of the paper, graph-based volume forms are analyzed in terms of four key objective-function properties: symmetry, positive-definiteness, invariance, and lack of bias. Graph-based volume forms whose associated L2 objective functions have the first three properties are

### doi:10.1155/2008/686875 Research Article Symmetric and Transitive Registration of Image Sequences

"... This paper presents a method for constructing symmetric and transitive algorithms for registration of image sequences from image registration algorithms that do not have these two properties. The method is applicable to both rigid and nonrigid registration and it can be used with linear or periodic ..."

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This paper presents a method for constructing symmetric and transitive algorithms for registration of image sequences from image registration algorithms that do not have these two properties. The method is applicable to both rigid and nonrigid registration and it can be used with linear or periodic image sequences. The symmetry and transitivity properties are satisfied exactly (up to the machine precision), that is, they always hold regardless of the image type, quality, and the registration algorithm as long as the computed transformations are invertable. These two properties are especially important in motion tracking applications since physically incorrect deformations might be obtained if the registration algorithm is not symmetric and transitive. The method was tested on two sequences of cardiac magnetic resonance images using two different nonrigid image registration algorithms. It was demonstrated that the transitivity and symmetry errors of the symmetric and transitive modification of the algorithms could be made arbitrary small when the computed transformations are invertable, whereas the corresponding errors for the nonmodified algorithms were on the order of the pixel size. Furthermore, the symmetric and transitive modification of the algorithms had higher registration accuracy than the nonmodified algorithms for both image sequences. Copyright © 2008 Oskar Škrinjar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.

### tensor-based

, 2006

"... 3D pattern of brain atrophy in HIV/AIDS visualized using ..."

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### ABSTRACT

, 2004

"... (Under the direction of Russell M. Taylor II) Current methods for surface reconstruction from AFM images do not enable one to incorporate constraints from other types of data. Current methods for surface reconstruction from SEM images are either unsuitable for nanometer scale shapes or limited to sh ..."

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(Under the direction of Russell M. Taylor II) Current methods for surface reconstruction from AFM images do not enable one to incorporate constraints from other types of data. Current methods for surface reconstruction from SEM images are either unsuitable for nanometer scale shapes or limited to shapes described by a small number of parameters. I have developed a new approach to surface reconstruction from combination AFM/SEM images that overcomes these limitations. A dilation model is used to model AFM image formation and a filter bank model is used to model SEM image formation. I construct noise models for both AFM and SEM images from real data. The image formation models including the noise descriptions are used to construct an objective function expressing the probability of observed images given a hypothetical surface reconstruction. The surface is modeled as a sum of Gaussian basis functions and I derive a formula to estimate the gradient of the objective function in the surface parameter space. The conjugate gradient method is used to optimize the surface parameters. My thesis is that this algorithm is more general and accurate than existing methods and

### USING INFORMATION THEORY Ming-Chang Chiang MD

"... Keywords: diffusion tensor imaging, Kullback-Leibler divergence, inverse consistency, fluid registration, TV regularization ..."

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Keywords: diffusion tensor imaging, Kullback-Leibler divergence, inverse consistency, fluid registration, TV regularization