@article{vonTycowiczAmbellanMukhopadhyayetal., author = {von Tycowicz, Christoph and Ambellan, Felix and Mukhopadhyay, Anirban and Zachow, Stefan}, title = {An Efficient Riemannian Statistical Shape Model using Differential Coordinates}, series = {Medical Image Analysis}, volume = {43}, journal = {Medical Image Analysis}, number = {1}, doi = {10.1016/j.media.2017.09.004}, pages = {1 -- 9}, abstract = {We propose a novel Riemannian framework for statistical analysis of shapes that is able to account for the nonlinearity in shape variation. By adopting a physical perspective, we introduce a differential representation that puts the local geometric variability into focus. We model these differential coordinates as elements of a Lie group thereby endowing our shape space with a non-Euclidean structure. A key advantage of our framework is that statistics in a manifold shape space becomes numerically tractable improving performance by several orders of magnitude over state-of-the-art. We show that our Riemannian model is well suited for the identification of intra-population variability as well as inter-population differences. In particular, we demonstrate the superiority of the proposed model in experiments on specificity and generalization ability. We further derive a statistical shape descriptor that outperforms the standard Euclidean approach in terms of shape-based classification of morphological disorders.}, language = {en} } @inproceedings{AmbellanZachowvonTycowicz, author = {Ambellan, Felix and Zachow, Stefan and von Tycowicz, Christoph}, title = {An as-invariant-as-possible GL+(3)-based Statistical Shape Model}, series = {Proc. 7th MICCAI workshop on Mathematical Foundations of Computational Anatomy (MFCA)}, volume = {11846}, booktitle = {Proc. 7th MICCAI workshop on Mathematical Foundations of Computational Anatomy (MFCA)}, publisher = {Springer}, doi = {10.1007/978-3-030-33226-6_23}, pages = {219 -- 228}, abstract = {We describe a novel nonlinear statistical shape model basedon differential coordinates viewed as elements of GL+(3). We adopt an as-invariant-as possible framework comprising a bi-invariant Lie group mean and a tangent principal component analysis based on a unique GL+(3)-left-invariant, O(3)-right-invariant metric. Contrary to earlier work that equips the coordinates with a specifically constructed group structure, our method employs the inherent geometric structure of the group-valued data and therefore features an improved statistical power in identifying shape differences. We demonstrate this in experiments on two anatomical datasets including comparison to the standard Euclidean as well as recent state-of-the-art nonlinear approaches to statistical shape modeling.}, language = {en} } @misc{AmbellanZachowvonTycowicz, author = {Ambellan, Felix and Zachow, Stefan and von Tycowicz, Christoph}, title = {A Surface-Theoretic Approach for Statistical Shape Modeling}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-74497}, abstract = {We present a novel approach for nonlinear statistical shape modeling that is invariant under Euclidean motion and thus alignment-free. By analyzing metric distortion and curvature of shapes as elements of Lie groups in a consistent Riemannian setting, we construct a framework that reliably handles large deformations. Due to the explicit character of Lie group operations, our non-Euclidean method is very efficient allowing for fast and numerically robust processing. This facilitates Riemannian analysis of large shape populations accessible through longitudinal and multi-site imaging studies providing increased statistical power. We evaluate the performance of our model w.r.t. shape-based classification of pathological malformations of the human knee and show that it outperforms the standard Euclidean as well as a recent nonlinear approach especially in presence of sparse training data. To provide insight into the model's ability of capturing natural biological shape variability, we carry out an analysis of specificity and generalization ability.}, language = {en} } @misc{AmbellanZachowvonTycowicz, author = {Ambellan, Felix and Zachow, Stefan and von Tycowicz, Christoph}, title = {An as-invariant-as-possible GL+(3)-based Statistical Shape Model}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-74566}, abstract = {We describe a novel nonlinear statistical shape model basedon differential coordinates viewed as elements of GL+(3). We adopt an as-invariant-as possible framework comprising a bi-invariant Lie group mean and a tangent principal component analysis based on a unique GL+(3)-left-invariant, O(3)-right-invariant metric. Contrary to earlier work that equips the coordinates with a specifically constructed group structure, our method employs the inherent geometric structure of the group-valued data and therefore features an improved statistical power in identifying shape differences. We demonstrate this in experiments on two anatomical datasets including comparison to the standard Euclidean as well as recent state-of-the-art nonlinear approaches to statistical shape modeling.}, language = {en} } @article{AmbellanZachowvonTycowicz, author = {Ambellan, Felix and Zachow, Stefan and von Tycowicz, Christoph}, title = {Rigid Motion Invariant Statistical Shape Modeling based on Discrete Fundamental Forms}, series = {Medical Image Analysis}, volume = {73}, journal = {Medical Image Analysis}, doi = {10.1016/j.media.2021.102178}, abstract = {We present a novel approach for nonlinear statistical shape modeling that is invariant under Euclidean motion and thus alignment-free. By analyzing metric distortion and curvature of shapes as elements of Lie groups in a consistent Riemannian setting, we construct a framework that reliably handles large deformations. Due to the explicit character of Lie group operations, our non-Euclidean method is very efficient allowing for fast and numerically robust processing. This facilitates Riemannian analysis of large shape populations accessible through longitudinal and multi-site imaging studies providing increased statistical power. Additionally, as planar configurations form a submanifold in shape space, our representation allows for effective estimation of quasi-isometric surfaces flattenings. We evaluate the performance of our model w.r.t. shape-based classification of hippocampus and femur malformations due to Alzheimer's disease and osteoarthritis, respectively. In particular, we achieve state-of-the-art accuracies outperforming the standard Euclidean as well as a recent nonlinear approach especially in presence of sparse training data. To provide insight into the model's ability of capturing biological shape variability, we carry out an analysis of specificity and generalization ability.}, language = {en} } @misc{AmbellanZachowvonTycowicz, author = {Ambellan, Felix and Zachow, Stefan and von Tycowicz, Christoph}, title = {Geodesic B-Score for Improved Assessment of Knee Osteoarthritis}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-81930}, abstract = {Three-dimensional medical imaging enables detailed understanding of osteoarthritis structural status. However, there remains a vast need for automatic, thus, reader-independent measures that provide reliable assessment of subject-specific clinical outcomes. To this end, we derive a consistent generalization of the recently proposed B-score to Riemannian shape spaces. We further present an algorithmic treatment yielding simple, yet efficient computations allowing for analysis of large shape populations with several thousand samples. Our intrinsic formulation exhibits improved discrimination ability over its Euclidean counterpart, which we demonstrate for predictive validity on assessing risks of total knee replacement. This result highlights the potential of the geodesic B-score to enable improved personalized assessment and stratification for interventions.}, language = {en} } @inproceedings{AmbellanZachowvonTycowicz, author = {Ambellan, Felix and Zachow, Stefan and von Tycowicz, Christoph}, title = {Geodesic B-Score for Improved Assessment of Knee Osteoarthritis}, series = {Proc. Information Processing in Medical Imaging (IPMI)}, booktitle = {Proc. Information Processing in Medical Imaging (IPMI)}, doi = {10.1007/978-3-030-78191-0_14}, pages = {177 -- 188}, abstract = {Three-dimensional medical imaging enables detailed understanding of osteoarthritis structural status. However, there remains a vast need for automatic, thus, reader-independent measures that provide reliable assessment of subject-specific clinical outcomes. To this end, we derive a consistent generalization of the recently proposed B-score to Riemannian shape spaces. We further present an algorithmic treatment yielding simple, yet efficient computations allowing for analysis of large shape populations with several thousand samples. Our intrinsic formulation exhibits improved discrimination ability over its Euclidean counterpart, which we demonstrate for predictive validity on assessing risks of total knee replacement. This result highlights the potential of the geodesic B-score to enable improved personalized assessment and stratification for interventions.}, language = {en} } @misc{AmbellanLameckervonTycowiczetal., author = {Ambellan, Felix and Lamecker, Hans and von Tycowicz, Christoph and Zachow, Stefan}, title = {Statistical Shape Models - Understanding and Mastering Variation in Anatomy}, issn = {1438-0064}, doi = {10.1007/978-3-030-19385-0_5}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-72699}, abstract = {In our chapter we are describing how to reconstruct three-dimensional anatomy from medical image data and how to build Statistical 3D Shape Models out of many such reconstructions yielding a new kind of anatomy that not only allows quantitative analysis of anatomical variation but also a visual exploration and educational visualization. Future digital anatomy atlases will not only show a static (average) anatomy but also its normal or pathological variation in three or even four dimensions, hence, illustrating growth and/or disease progression. Statistical Shape Models (SSMs) are geometric models that describe a collection of semantically similar objects in a very compact way. SSMs represent an average shape of many three-dimensional objects as well as their variation in shape. The creation of SSMs requires a correspondence mapping, which can be achieved e.g. by parameterization with a respective sampling. If a corresponding parameterization over all shapes can be established, variation between individual shape characteristics can be mathematically investigated. We will explain what Statistical Shape Models are and how they are constructed. Extensions of Statistical Shape Models will be motivated for articulated coupled structures. In addition to shape also the appearance of objects will be integrated into the concept. Appearance is a visual feature independent of shape that depends on observers or imaging techniques. Typical appearances are for instance the color and intensity of a visual surface of an object under particular lighting conditions, or measurements of material properties with computed tomography (CT) or magnetic resonance imaging (MRI). A combination of (articulated) statistical shape models with statistical models of appearance lead to articulated Statistical Shape and Appearance Models (a-SSAMs).After giving various examples of SSMs for human organs, skeletal structures, faces, and bodies, we will shortly describe clinical applications where such models have been successfully employed. Statistical Shape Models are the foundation for the analysis of anatomical cohort data, where characteristic shapes are correlated to demographic or epidemiologic data. SSMs consisting of several thousands of objects offer, in combination with statistical methods ormachine learning techniques, the possibility to identify characteristic clusters, thus being the foundation for advanced diagnostic disease scoring.}, language = {en} } @inproceedings{AmbellanZachowvonTycowicz, author = {Ambellan, Felix and Zachow, Stefan and von Tycowicz, Christoph}, title = {A Surface-Theoretic Approach for Statistical Shape Modeling}, series = {Proc. Medical Image Computing and Computer Assisted Intervention (MICCAI), Part IV}, volume = {11767}, booktitle = {Proc. Medical Image Computing and Computer Assisted Intervention (MICCAI), Part IV}, publisher = {Springer}, doi = {10.1007/978-3-030-32251-9_3}, pages = {21 -- 29}, abstract = {We present a novel approach for nonlinear statistical shape modeling that is invariant under Euclidean motion and thus alignment-free. By analyzing metric distortion and curvature of shapes as elements of Lie groups in a consistent Riemannian setting, we construct a framework that reliably handles large deformations. Due to the explicit character of Lie group operations, our non-Euclidean method is very efficient allowing for fast and numerically robust processing. This facilitates Riemannian analysis of large shape populations accessible through longitudinal and multi-site imaging studies providing increased statistical power. We evaluate the performance of our model w.r.t. shape-based classification of pathological malformations of the human knee and show that it outperforms the standard Euclidean as well as a recent nonlinear approach especially in presence of sparse training data. To provide insight into the model's ability of capturing natural biological shape variability, we carry out an analysis of specificity and generalization ability.}, language = {en} } @incollection{AmbellanLameckervonTycowiczetal., author = {Ambellan, Felix and Lamecker, Hans and von Tycowicz, Christoph and Zachow, Stefan}, title = {Statistical Shape Models - Understanding and Mastering Variation in Anatomy}, series = {Biomedical Visualisation}, volume = {3}, booktitle = {Biomedical Visualisation}, number = {1156}, editor = {Rea, Paul M.}, edition = {1}, publisher = {Springer Nature Switzerland AG}, isbn = {978-3-030-19384-3}, doi = {10.1007/978-3-030-19385-0_5}, pages = {67 -- 84}, abstract = {In our chapter we are describing how to reconstruct three-dimensional anatomy from medical image data and how to build Statistical 3D Shape Models out of many such reconstructions yielding a new kind of anatomy that not only allows quantitative analysis of anatomical variation but also a visual exploration and educational visualization. Future digital anatomy atlases will not only show a static (average) anatomy but also its normal or pathological variation in three or even four dimensions, hence, illustrating growth and/or disease progression. Statistical Shape Models (SSMs) are geometric models that describe a collection of semantically similar objects in a very compact way. SSMs represent an average shape of many three-dimensional objects as well as their variation in shape. The creation of SSMs requires a correspondence mapping, which can be achieved e.g. by parameterization with a respective sampling. If a corresponding parameterization over all shapes can be established, variation between individual shape characteristics can be mathematically investigated. We will explain what Statistical Shape Models are and how they are constructed. Extensions of Statistical Shape Models will be motivated for articulated coupled structures. In addition to shape also the appearance of objects will be integrated into the concept. Appearance is a visual feature independent of shape that depends on observers or imaging techniques. Typical appearances are for instance the color and intensity of a visual surface of an object under particular lighting conditions, or measurements of material properties with computed tomography (CT) or magnetic resonance imaging (MRI). A combination of (articulated) statistical shape models with statistical models of appearance lead to articulated Statistical Shape and Appearance Models (a-SSAMs).After giving various examples of SSMs for human organs, skeletal structures, faces, and bodies, we will shortly describe clinical applications where such models have been successfully employed. Statistical Shape Models are the foundation for the analysis of anatomical cohort data, where characteristic shapes are correlated to demographic or epidemiologic data. SSMs consisting of several thousands of objects offer, in combination with statistical methods ormachine learning techniques, the possibility to identify characteristic clusters, thus being the foundation for advanced diagnostic disease scoring.}, language = {en} } @misc{AmbellanHanikvonTycowicz, author = {Ambellan, Felix and Hanik, Martin and von Tycowicz, Christoph}, title = {Morphomatics: Geometric morphometrics in non-Euclidean shape spaces}, doi = {10.12752/8544}, abstract = {Morphomatics is an open-source Python library for (statistical) shape analysis developed within the geometric data analysis and processing research group at Zuse Institute Berlin. It contains prototype implementations of intrinsic manifold-based methods that are highly consistent and avoid the influence of unwanted effects such as bias due to arbitrary choices of coordinates.}, language = {en} } @inproceedings{MayerBaumAmbellanetal., author = {Mayer, Julius and Baum, Daniel and Ambellan, Felix and von Tycowicz, Christoph}, title = {A Soft-Correspondence Approach to Shape-based Disease Grading with Graph Convolutional Networks}, series = {Proceedings of Machine Learning Research}, volume = {194}, booktitle = {Proceedings of Machine Learning Research}, pages = {85 -- 95}, abstract = {Shape analysis provides principled means for understanding anatomical structures from medical images. The underlying notions of shape spaces, however, come with strict assumptions prohibiting the analysis of incomplete and/or topologically varying shapes. This work aims to alleviate these limitations by adapting the concept of soft correspondences. In particular, we present a graph-based learning approach for morphometric classification of disease states that is based on a generalized notion of shape correspondences in terms of functional maps. We demonstrate the performance of the derived classifier on the open-access ADNI database for differentiating normal controls and subjects with Alzheimer's disease. Notably, our experiment shows that our approach can improve over state-of-the-art from geometric deep learning.}, language = {en} } @misc{NavaYazdaniHanikAmbellanetal., author = {Nava-Yazdani, Esfandiar and Hanik, Martin and Ambellan, Felix and von Tycowicz, Christoph}, title = {On Gradient Formulas in an Algorithm for the Logarithm of the Sasaki Metric}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-87174}, abstract = {The Sasaki metric is the canonical metric on the tangent bundle TM of a Riemannian manifold M. It is highly useful for data analysis in TM (e.g., when one is interested in the statistics of a set of geodesics in M). To this end, computing the Riemannian logarithm is often necessary, and an iterative algorithm was proposed by Muralidharan and Fletcher. In this note, we derive approximation formulas of the energy gradients in their algorithm that we use with success.}, language = {en} } @article{NavayazdaniAmbellanHaniketal., author = {Navayazdani, Esfandiar and Ambellan, Felix and Hanik, Martin and von Tycowicz, Christoph}, title = {Sasaki Metric for Spline Models of Manifold-Valued Trajectories}, series = {Computer Aided Geometric Design}, volume = {104}, journal = {Computer Aided Geometric Design}, doi = {10.1016/j.cagd.2023.102220}, pages = {102220}, abstract = {We propose a generic spatiotemporal framework to analyze manifold-valued measurements, which allows for employing an intrinsic and computationally efficient Riemannian hierarchical model. Particularly, utilizing regression, we represent discrete trajectories in a Riemannian manifold by composite B{\´e}zier splines, propose a natural metric induced by the Sasaki metric to compare the trajectories, and estimate average trajectories as group-wise trends. We evaluate our framework in comparison to state-of-the-art methods within qualitative and quantitative experiments on hurricane tracks. Notably, our results demonstrate the superiority of spline-based approaches for an intensity classification of the tracks.}, language = {en} } @article{KoflerWaldKolbitschetal., author = {Kofler, Andreas and Wald, Christian and Kolbitsch, Christoph and von Tycowicz, Christoph and Ambellan, Felix}, title = {Joint Reconstruction and Segmentation in Undersampled 3D Knee MRI combining Shape Knowledge and Deep Learning}, series = {Physics in Medicine and Biology}, journal = {Physics in Medicine and Biology}, doi = {10.1088/1361-6560/ad3797}, abstract = {Task-adapted image reconstruction methods using end-to-end trainable neural networks (NNs) have been proposed to optimize reconstruction for subsequent processing tasks, such as segmentation. However, their training typically requires considerable hardware resources and thus, only relatively simple building blocks, e.g. U-Nets, are typically used, which, albeit powerful, do not integrate model-specific knowledge. In this work, we extend an end-to-end trainable task-adapted image reconstruction method for a clinically realistic reconstruction and segmentation problem of bone and cartilage in 3D knee MRI by incorporating statistical shape models (SSMs). The SSMs model the prior information and help to regularize the segmentation maps as a final post-processing step. We compare the proposed method to a state-of-the-art (SOTA) simultaneous multitask learning approach for image reconstruction and segmentation (MTL) and to a complex SSMs-informed segmentation pipeline (SIS). Our experiments show that the combination of joint end-to-end training and SSMs to further regularize the segmentation maps obtained by MTL highly improves the results, especially in terms of mean and maximal surface errors. In particular, we achieve the segmentation quality of SIS and, at the same time, a substantial model reduction that yields a five-fold decimation in model parameters and a computational speedup of an order of magnitude. Remarkably, even for undersampling factors of up to R=8, the obtained segmentation maps are of comparable quality to those obtained by SIS from ground-truth images.}, language = {en} }