Geometric modeling in shape space

“introduced a Riemannian shape space in [13], in which shapes are represented by meshes with fixed connectivity.”

“High quality correspondences are also essential in morphing, and shape interpolation [KMP07].”

“Inspired by [44], we leverage the as-isometric-as-possible constraint [15] to restrict neighboring 3D Gaussian centers to preserve a similar distance after deformation.”

“…meshes [Sheffer and Kraevoy 2004; Sorkine et al. 2004; Lipman et al. 2005; Botsch et al. 2006], interpolating between meshes [Alexa et al. 2000; Kilian et al. 2007; Winkler et al. 2010; Chao et al. 2010] and for transferring deformations from one mesh to another [Sumner and Popović 2004;…”

“• Combining digital reconstruction, optimization and recently introduced algorithms for computing in shape space [Kilian et al. 2007] we have a rich toolbox for geometry processing with curved folds.”

“In our setting, the objects have only one part, with a number of corresponding parameters essentially representing a shape space (e.g., [KMP07]).”

“Third, linear methods such as PCA are most often used — a notable exception being [Kilian et al. 2007] — even when it is not clear that a flat approximation to the shape space, either locally or globally is indicated.”

“2005] or other creatures [Praun et al. 2001; Kilian et al. 2007].”

“Shape interpolation is a fundamental task and several classic meth-ods exist, such as [Solomon et al. 2015] and [Kilian et al. 2007].”

“Another paper by Kilian et al. [18] represents parameterized surfaces by discrete triangulated meshes, assumes a Riemannian metric on the space of such meshes, and computes geodesic paths between given meshes.”

“Note that since we provide closed-form formulas for geodesic paths, interpolation (or extrapolation) on our manifold is considerably simpler than the algorithms presented in [21].”

“Kilian et al. [21] presented a technique for computing geodesics between triangulated meshes (discretized surfaces) but at their given parameterizations.”

“This multi-scale optimization strategy, which is motivated by classical nonlearning interpolation algorithms [27, 23], leads to an overall faster and more robust convergence.”

“Shape matching is a task that occurs in countless applications in computer graphics, including shape interpolation [Kilian et al. 2007], statistical shape analysis [Bogo et al.”

“Even seemingly simple handle-driven deformations restricted to such shape spaces turn out to be challenging, and re­main an active topic of research (see [Botsch et al. 2006; Kilian et al. 2007; Botsch and Sorkine 2008; Gal et al. 2009]).”

“It must be noted that, one sure way to avoid infinite-dimensional Riemannian geometry (and the pathologies associated with some metrics) is to model the shape space as a finite dimensional manifold: such is the case for example in the work of Kendall [24] (some recent works include, e.g. [26] [3]).”

“For example, [Kilian et al. 2007], [Allen et al. 2003], and others describe methods for interpolation between surfaces with consistent parameterizations.”

“Other shell energies, such as a thin-shell energy [GHDS03], or an energy favoring isometric deformations [KMP07] can be used as well, with similar results.”

“One exception is Kilian et al. [8] which represents parameterized 1625978-1-4244-6985-7/10/$26.00 ©2010 IEEE surfaces by discrete triangulated meshes, assumes a Riemannian metric on the space of such meshes, and computes geodesic paths between given meshes.”

“For example, approaches like [ 23, 29, 14 ] can be used on top of our method, and it can also be used for shape editing based on user-defined constraints, similar to how it is described in [ 15 ] .”

“Miller and Younes (2001) generalized this concept to the space of images and also examined the more general case of computing geodesics when a group acts on a Riemannian manifold so that this Riemannian manifold can be endowed with a new metric which incorporates also the group action (so-called metamorphosis, Trouvé…”

“Geometric modeling in " shape space " is another purely geometric approach to shape interpolation, where a notion of geodesics in shape space is used to provide an as-isometric-as-possible path between two poses [3].”

“On the other hand, assuming the availability of a dense correspondence, as in [KMP06,HRWW12], is not practical for large and complex shapes.”

“Such metrics measure the metric distortion or viscous dissipation required to physically deform an elastic object [Kilian et al. 2007; Chao et al. 2010; Wirth et al. 2011; Heeren et al. 2012].”

“Finding high quality correspondences is a key component in many tasks including statistical shape analysis [12], deformation transfer [30] and interpolation (morphing) [16] among others.”

“In computer graphics, this approach appears in work on shape interpolation including [KMP07], which defines Riemannian metrics for triangle stretching in shape interpolation, and [HRWW12], which interpolates between surfaces using a metric inspired by thin-shell deformation.”

“In other efforts, isometric deformations have been investigated as geodesics in appropriate shape spaces [Kilian et al. 2007; Wirth et al. 2011], or in the context of bending energies in the PriMo setup [Botsch et al. 2006].”

“Although regularizations such as the enforcement of local rigidity and isometry losses [30,56] help stabilize the learning process to a certain extent, the overall pipeline still requires a large number of input views for each step and struggles to reconstruct scenes faithfully from sparse…”

“Shape correspondence is one of the key problems in digital geometry processing with applications in fields ranging from manufacturing to shape morphing [KMP07], statistical shape analysis [HSS ∗ 09,BRLB14], texture mapping, animation and deformation transfer [SP04] among a wide variety of others.”

“motivates our work. There is a rich diversity of Riemannian shape spaces in the literature. While some of them are finite-dimensional and represent shapes as polygonal curves or triangulated surfaces [13,17], most deal with infinite-dimensional shapes such as planar curves with curvature-based or Sobolev-type metrics [1…”

“These concepts have a continuously increasing impact on the development of novel computational tools in computer vision and imaging, ranging from shape morphing and modeling [33], and shape statistics, e.”

“More abstractly, it is often useful in image processing to study whole sets of images themselves as Riemannian manifolds [42], or similarly in shape processing, to define a Riemannian structure on the set of all shapes [19, 9].”

“Finally, ARAPReg is relevant to early works on modeling tangent spaces of shape manifolds [23, 20, 58].”

“[24] represented a triangle mesh as a point in a high-dimensional space that treats each vertex coordinate as a free variable.”

“, in medical imaging [16], [31], robotics [32], [33], [34], geometric modeling [35] and fluid dynamics [36].”

“The same drawback holds for the shape space method in [Kilian et al. 2007] where an additional concern is computational efficiency, due to the involved non-linear and nonconvex optimization method.”

“Of the numerous solutions that have been proposed (see, e.g., [ACOL00, LSLCO05, KMP07]), our work is closest related to methods that interpolate in deformation or strain space: the scheme described by Martin et al. [MTGG11] interpolates between volumetric solids using the Green strain; Winkler et…”

“[8] represent surfaces by discrete triangulated meshes and compute geodesic paths (deformations) between them, while assuming that the meshes are registered.”

“Efficient multiresolution solvers for computing geodesics in shape spaces were proposed by Kilian et al. [2007] and by Wirth et al.”

“…non-rigid deformations, e.g. of a skirt, where traditional linear blend skinning-based models fail by incorporating regularization techniques such as as-isometric-as-possible [KMP07, HAWG08] and additionally supervising for highly dynamic and complex deformations with keypoint correspondences.”

“There exists a plethora of different numerical frameworks for shape analysis of parametrized surfaces, see e.g. Jermyn et al. (2017), Kilian et al. (2007), Rumpf and Wirth (2015a), Iglesias et al. (2018) and Pierson et al. (2022) and the references therein.”

“Dimension reduction: Instead of arranging points in a circle or another prescribed shape, it is also possible to place them in a data adaptive way using dimension reduction techniques. Kilian et al. [21] use a distance preserving embedding of points to represent shape descriptors to control different designs of shapes…”

“The authors also applied as-isometric-as-possible regularization [32] to the Gaussians to preserve geometric consistency.”

“Shape-space based methods, such as [Kilian et al. 2007], navigate the space of possible deformations between two shapes, by finding geodesics of a deformation metric.”

“nd correspondence-based shape retrieval [15,14,74]. From the perspective of graphics design, 3D object creation involves both coarse-level structural design as well as ne-grained geometric generation [39,54,30]. Specically, structural design choices re ect the discrete (e.g. number of chair legs to support the seat) a…”

“Once correspondence between two objects is achieved, the same measures can then be used to deform, morph, or warp one shape into another [31].”

“However, this technique can be complementary to our basic pipeline in case of the reconstruction from unoriented, noisy point clouds, as well as when a water-tight mesh reconstruction is required.”

“They include a metric that preservers local rigidity [11], a generalization of the elastic metric for curves to higher dimensions [18, 13], a metric inspired by a continuum mechanics which is defined in the interior of a two dimensional in ria -0 06 24 21 0, v er si on 1 16 S ep 2 01 1 Author manuscript, published in…”

“We have already mentioned the concept of Killing fields [37] and approximate Killing fields [6, 28].”

“[KMP07], and shape statistics, e.g .”

“We will see in Section 4 the need to identify points across multiple meshes of potentially varying connectivity, which is commonly referred to as shape correspondence or cross-parameterization [Kilian et al. 2007; Kraevoy and Sheffer 2004; Schreiner et al. 2004].”

“…little is known about how shapes are represented in the brain, work in mathematics (Kendall, 1984) and computer graphics (Blanz and Vetter, 1999; Kilian et al., 2007) show that it is possible to design shape spaces where different shapes can be synthesized from combinations of basic shapes in a…”

“Killian et al. [KMP07] treat shapes as points on the Riemannian surfaces derived from typical rigid or isometric transformations, and convert the plausible morphing problem to finding a shortest path in the shape space which can be solved using a variational approach.”

“Riemannian metrics on the shape space of triangular surface meshes (with a fixed connectivity) have been introduced by Kilian et al. [KMP07].”

“More generally, we can explore meaningful poses by interactively navigating this pose space and even create new animations from scratch (Section 4.4), in a similar way to the systems Kilian et al. [KMP07] and Yang et al. [YYPM11] propose for shape-space exploration.”

“, morphing and animation problems), in a way similar to [10].”

“[24] model the space of triangulated shapes.”

“[69] presented a technique for computing geodesics between triangulated meshes (discretized surfaces) but at their given parameterizations.”

“Killian et al. [KMP07] consider shapes as points on Riemannian surfaces derived from typical transformations (e.g. isometric), and formulate plausible morphing as finding a shortest path in the shape space.”

“In the work of Kilian et al. (2007), however, both the source and target meshes are fixed during optimization.”

“Shape analysis of surfaces in R 3 has been motivated by many applications in bioinformatics, computer graphics and medical imaging, see e.g., [16,18,2,11,13,28].”

“Since Kilian et al. [21]’s approach works on triangulated meshes then where s and t are two adjacent vertices on the tri-angulated mesh that represents the surface f .”

“in Kendall (1984), Cootes et al. (1995), Hafner et al. (2000), Kilian et al. (2007) .”

“The spaces of meshes can be equipped with Riemannian metrics [43, 44].”

“[17] presented an isometric deformation method based on Riemannian geometry, considering shape interpolation as a geodesic curve in shape space.”

“Thus far, the main applications of the model have been to analysis of magnetic resonance images of the human brain.”

“[20], where an inner product of deformations fields as the underlying metric measures the local distance from a rigid body motion.”

“Using generalized flows to compute continuous morphs that follow geodesics in shape spaces [25, 26] is another exciting avenue of future work.”

“The boundary value problem on functions (seen as parametrized surface) is closely related to mesh interpolation (Kilian et al. 2007; Fröhlich and Botsch 2011) and can be solved by the direct method of variational calculus or, for certain elastic metrics, by exploiting isometries to simpler spaces.”

“Kilian et al. [KMP07] showed how to compute geodesic paths between tri-angle meshes using a metric that measures changes in triangle edge lengths.”

“[2] represents parameterized surfaces by discrete triangulated meshes, assumes a Riemannian metric on the space of such meshes, and computes geodesic paths between given meshes.”

“Shape correspondence is also necessary in common applications such as modelling [KMP07], reconstruction [LSP08] and tracking [NFS15].”