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- ArticleAugust 2023
On Canonical Parameterizations of 2D-Shapes
AbstractThis paper is devoted to the study of unparameterized simple curves in the plane. We propose diverse canonical parameterization of a 2D-curve. For instance, the arc-length parameterization is canonical, but we consider other natural ...
- research-articleJuly 2023
Motion from Shape Change
ACM Transactions on Graphics (TOG), Volume 42, Issue 4Article No.: 107, Pages 1–11https://doi.org/10.1145/3592417We consider motion effected by shape change. Such motions are ubiquitous in nature and the human made environment, ranging from single cells to platform divers and jellyfish. The shapes may be immersed in various media ranging from the very viscous to ...
- ArticleApril 2023
A Function Space Perspective on Stochastic Shape Evolution
AbstractModelling randomness in shape data, for example, the evolution of shapes of organisms in biology, requires stochastic models of shapes. This paper presents a new stochastic shape model based on a description of shapes as functions in a Sobolev ...
- ArticleMarch 2022
CTA Coronary Labeling through Efficient Geodesics between Trees Using Anatomy Priors
Medical Image Computing and Computer-Assisted Intervention – MICCAI 2014Pages 521–528https://doi.org/10.1007/978-3-319-10470-6_65AbstractWe present an efficient realization of recent work on unique geodesic paths between tree shapes for the application of matching coronary arteries to a standard model of coronary anatomy in order to label the coronary arteries. Automatically ...
- research-articleJanuary 2022
Universal approximation theorems for differentiable geometric deep learning
The Journal of Machine Learning Research (JMLR), Volume 23, Issue 1Article No.: 196, Pages 8896–8968This paper addresses the growing need to process non-Euclidean data, by introducing a geometric deep learning (GDL) framework for building universal feedforward-type models compatible with differentiable manifold geometries. We show that our GDL models ...
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- research-articleJanuary 2020
Convergence of the Time Discrete Metamorphosis Model on Hadamard Manifolds
SIAM Journal on Imaging Sciences (SJISBI), Volume 13, Issue 2Pages 557–588https://doi.org/10.1137/19M1247073Continuous image morphing is a classical task in image processing. The metamorphosis model proposed by Trouvé, Younes, and coworkers [M. I. Miller and L. Younes, Int. J. Comput. Vis., 41 (2001), pp. 61--84; A. Trouvé and L. Younes, Found. Comput. Math., 5 ...
- research-articleDecember 2018
The shape space of discrete orthogonal geodesic nets
ACM Transactions on Graphics (TOG), Volume 37, Issue 6Article No.: 228, Pages 1–17https://doi.org/10.1145/3272127.3275088Discrete orthogonal geodesic nets (DOGs) are a quad mesh analogue of developable surfaces. In this work we study continuous deformations on these discrete objects. Our main theoretical contribution is the characterization of the shape space of DOGs for ...
- research-articleJanuary 2018
Image Extrapolation for the Time Discrete Metamorphosis Model: Existence and Applications
SIAM Journal on Imaging Sciences (SJISBI), Volume 11, Issue 1Pages 834–862https://doi.org/10.1137/17M1129544The space of images can be equipped with a Riemannian metric measuring both the cost of transport of image intensities and the variation of image intensities along motion lines. The resulting metamorphosis model was introduced and analyzed in [M. I. Miller ...
- research-articleDecember 2016
Fast and reliable example-based mesh IK for stylized deformations
ACM Transactions on Graphics (TOG), Volume 35, Issue 6Article No.: 235, Pages 1–12https://doi.org/10.1145/2980179.2982433Example-based shape deformation allows a mesh to be easily manipulated or animated with simple inputs. As the user pulls parts of the shape, the rest of the mesh automatically changes in an intuitive way by drawing from a set of exemplars. This provides ...
- research-articleNovember 2015
Optimal Mass Transport for Shape Matching and Comparison
IEEE Transactions on Pattern Analysis and Machine Intelligence (ITPM), Volume 37, Issue 11Pages 2246–2259https://doi.org/10.1109/TPAMI.2015.2408346Surface based 3D shape analysis plays a fundamental role in computer vision and medical imaging. This work proposes to use optimal mass transport map for shape matching and comparison, focusing on two important applications including surface registration ...
- research-articleNovember 2015
Efficient Shape Priors for Spline-Based Snakes
IEEE Transactions on Image Processing (TIP), Volume 24, Issue 11Pages 3915–3926https://doi.org/10.1109/TIP.2015.2457335Parametric active contours are an attractive approach for image segmentation, thanks to their computational efficiency. They are driven by application-dependent energies that reflect the prior knowledge on the object to be segmented. We propose an energy ...
- research-articleJanuary 2015
Time Discrete Geodesic Paths in the Space of Images
SIAM Journal on Imaging Sciences (SJISBI), Volume 8, Issue 3Pages 1457–1488https://doi.org/10.1137/140970719In this paper the space of images is considered as a Riemannian manifold using the metamorphosis approach (see [M. I. Miller and L. Younes, Int. J. Comput. Vis., 41 (2001), pp. 61--84; A. Trouvé and L. Younes, SIAM J. Math. Anal., 37 (2005), pp. 17--59; and ...
- research-articleDecember 2013
Visualization of Shape Motions in Shape Space
IEEE Transactions on Visualization and Computer Graphics (ITVC), Volume 19, Issue 12Pages 2644–2652https://doi.org/10.1109/TVCG.2013.230Analysis of dynamic object deformations such as cardiac motion is of great importance, especially when there is a necessity to visualize and compare the deformation behavior across subjects. However, there is a lack of effective techniques for ...
- research-articleJuly 2013
Generating and exploring good building layouts
ACM Transactions on Graphics (TOG), Volume 32, Issue 4Article No.: 122, Pages 1–10https://doi.org/10.1145/2461912.2461977Good building layouts are required to conform to regulatory guidelines, while meeting certain quality measures. While different methods can sample the space of such good layouts, there exists little support for a user to understand and systematically ...
- research-articleJanuary 2013
Discrete Geodesic Calculus in Shape Space and Applications in the Space of Viscous Fluidic Objects
SIAM Journal on Imaging Sciences (SJISBI), Volume 6, Issue 4Pages 2581–2602https://doi.org/10.1137/120870864Based on a local approximation of the Riemannian distance on a manifold by a computationally cheap dissimilarity measure, a time discrete geodesic calculus is developed and applications to shape space are explored. The dissimilarity measure is derived from ...
- research-articleJuly 2012
Guided exploration of physically valid shapes for furniture design
ACM Transactions on Graphics (TOG), Volume 31, Issue 4Article No.: 86, Pages 1–11https://doi.org/10.1145/2185520.2185582Geometric modeling and the physical validity of shapes are traditionally considered independently. This makes creating aesthetically pleasing yet physically valid models challenging. We propose an interactive design framework for efficient and intuitive ...
- articleMarch 2012
Almost Local Metrics on Shape Space of Hypersurfaces in $n$-Space
SIAM Journal on Imaging Sciences (SJISBI), Volume 5, Issue 1Pages 244–310https://doi.org/10.1137/100807983This paper extends parts of the results from [P. W. Michor and D. Mumford, Appl. Comput. Harmon. Anal., 23 (2007), pp. 74-113] for plane curves to the case of hypersurfaces in $\mathbb R^n$. Let $M$ be a compact connected oriented $n-1$ dimensional ...
- research-articleJanuary 2012
Optimization Methods on Riemannian Manifolds and Their Application to Shape Space
SIAM Journal on Optimization (SIOPT), Volume 22, Issue 2Pages 596–627https://doi.org/10.1137/11082885XWe extend the scope of analysis for linesearch optimization algorithms on (possibly infinite-dimensional) Riemannian manifolds to the convergence analysis of the BFGS quasi-Newton scheme and the Fletcher--Reeves conjugate gradient iteration. Numerical ...
- research-articleDecember 2011
Shape space exploration of constrained meshes
SA '11: Proceedings of the 2011 SIGGRAPH Asia ConferenceArticle No.: 124, Pages 1–12https://doi.org/10.1145/2024156.2024158We present a general computational framework to locally characterize any shape space of meshes implicitly prescribed by a collection of non-linear constraints. We computationally access such manifolds, typically of high dimension and co-dimension, ...
- research-articleDecember 2011
Shape space exploration of constrained meshes
ACM Transactions on Graphics (TOG), Volume 30, Issue 6Pages 1–12https://doi.org/10.1145/2070781.2024158We present a general computational framework to locally characterize any shape space of meshes implicitly prescribed by a collection of non-linear constraints. We computationally access such manifolds, typically of high dimension and co-dimension, ...