Geometric modeling in shape space

“Kilian et al. (2007) ensures near-isometric shape deformation by preserving geodesics between surface points.”

“achieved three-dimensional shape deformation, shape morphing and deformation transfer with the shape space theories [15].”

“…quantities such as the pullback metric tensor (i.e., rotation-invariant right Cauchy-Green strain); Green strain or edge length; dihedral angles; or a geometric measures of shape distortion [Chen et al. 2013; Kilian et al. 2007; Levi and Gotsman 2015; Martin et al. 2011; Winkler et al. 2010].”

“And morphing between shapes can be tackled as an interpolation problem on shape space [12], [13].”

“Typical examples of applications include shape registration [29, 35, 33], tracking [29] and shape deformation [20].”

“[21] extend geodesics of an isometric energy in the deformation space, though it is restricted to exploring (and extrapolating) paths between shapes rather than the full deformation space.”

“[49] present a shape exploration based on barycentric 212 interpolation between example shapes.”

“[8] accounts for 3D shapes as points of a high-dimensional Riemannian shape space equipped with an isometric inner-product.”

“Among all existing approaches, a number of different concepts of a shape are employed, including landmark vectors [16, 39], planar curves [41, 52, 84], surfaces in R3 [24, 25, 40], boundary contours”

“There are multiple options, e.g. the usage of landmark vectors [18, 36, 44, 73, 88], plane curves [65, 66, 64, 67] or surfaces [9, 10, 45, 54, 63] in higher dimensions, boundary contours of objects [27, 59, 75], multiphase objects [100], characteristic functions of measurable sets [103] and…”

“The recent shape space metaphor [KMH07], which treats meshes of a given topology as points in a higher dimensional manifold, and, consequently, local deformations as tangent plane vectors, has been applied in a recent work [YYPM11], with a focus on planar mesh space exploration.”

“Our formulation combines the strengths of volume preserving fields [19] and projective dynamics [10] with those of classical interpolation methods [28,32].”

“For instance, dense maps between surfaces are often used in the definition of retrieval algorithms [10, 11, 9, 5, 24, 2], to perform shape interpolation [14] and to define statistical models of shapes [12, 3].”

“112 There is also a number of methods in the graphics liter113 ature, which provide tools for various shape modeling tasks 114 [17, 18, 19].”

“Applications include shape modeling in computer vision and graphics [1, 2] or in computational anatomy [3, 4], and shape clustering [5] as well as shape statistics [6].”

“We also compare to three optimization-based geometric methods, by building on the ideas from [30,41,12].”

“[41] studied geodesics in the space of triangular surfaces to interpolate between two given poses.”

“The “shape spaces” implied by collections of models can be used for visualization and analysis 3 and provide a principled approach for applications such as deformation transfer 4 or scan completion.”

“Our explicit neural representation is robust, flexible, and has the potential to be used for downstream geometry-processing tasks such as physically-based optimization [34], computational geometric design [6] or exploring topologically consistent shape and configuration spaces [12].”

“Recently, there has been a growing interest in analyzing variability in 3D shape collections using tools from differential and Riemannian geometry [4], [5], [18], [19], [20], [21], [22]; see [23] for a detailed survey.”

“[17] represent parameterized surfaces by discrete triangulated meshes, assume a Riemannian metric on the space of such meshes, and compute geodesic paths between given meshes.”

“[21] use a distance preserving embedding of points to represent shape descriptors to control different designs of shapes.”

“The constant need to implement automation algorithms to control mechanical systems while executing industrial R&D projects, and the continuous needs to implement applications that help understanding the mechanical devices in engineering education, motivated the creation of an interactive platform which combines these…”

“5, MCC+AIAP delivers similar results to MCC, while MCC+LLR consistently outperforms MCC+AIAP across different levels of occlusion.”

“Since there exists no canonical path, interpolation sequences are gauged by the extent to which intrinsic metrics are preserved, in particular isometric distortion [85].”

“They have been used explicitly in geometry [6,28,31,43], including vector field design on curved surfaces [2,3], and recently also in flow visualization [18,39].”

“The problem is critical for a wide range of applications such as shape interpolation (Baek et al., 2015; Kilian et al., 2007), statistical shape analysis and modeling (Allen et al.”

“The problem is critical for a wide range of applications such as shape interpolation [1, 2], statistical shape analysis and modeling [3, 4], geometry transfer [5], and such.”

“[KMP07] interprets the solution to interpolation as geodesics in the point-wise shape space.”

“[18] note that we can find the true point-to-point correspondences between two deformed triangular meshes over a set of possible candidate correspondence alignments by computing the distance in S for each alignment.”

“…where M gt is the foreground mask and M is the accumulated α value: Similar to [54, 56], we also utilize the as-isometric-as-possible constraint [28] to enforce neighboring distance similarity of 3D Gaussian centers and covariance matrices between canonical space and observation space: where N…”

“the usage of landmark vectors [16, 28, 37, 74, 85], plane curves [63, 65, 64, 66] or surfaces in higher dimensions [5, 6, 40, 46, 62], boundary contours of objects [22, 49, 80], multiphase objects [93], characteristic functions of measurable sets [94] and morphologies of images [17].”

“This data preprocessing step is then followed with an independent framework for the analysis of parametrized shapes, such as the As-Rigid-As-Possible (ARAP) energy [45] or the as isometric as possible energy [27].”

“In shape analysis, a shape is considered as an element on the so-called shape manifold (the set of all possible shapes) (Kendall 1984; Kilian et al. 2007).”

“Kilian et al. [39] introduced a Riemannian metric on spaces of triangle meshes and show that concepts from Riemannian geometry such as the exponential map and parallel transport can be used for geometry processing tasks like deformation transfer, shape interpolation, and extrapolation.”

“In future, we would also explore the application of the deformation energy in mesh skeleton, smoothing, shape extrapolation [Kilian et al. 2007], and the segmentation by modal analysis [Huang et al.”

“consider geodesics between consistently triangulated surfaces [55], where the underlying Riemannian metric measures the stretching of triangle edges (cf.”

“In contrast to finite deformation methods, continuous deformations require velocities as boundary constraints (cf. [29,21]).”

“A discretisation of an almost local metric was proposed in Kilian et al. (2007) in which the discrete L2 inner product is regularised by adding a second L2 inner product where each vertex contribution is weighted by the areas of triangles surrounding that vertex.”

“[12] use the local distortion of the mesh edges to define a global one.”

“Technically, the continuous morphing is obtained following the algorithm proposed in Kilian et al. (2007).”

“3) taken from the publicly available data(3) of [16] that is based on data of the Large Geometric Models Archive from the Georgia Institute of Technology.”

“The authors of [14] presented a novel framework to treat shapes in the setting of Riemannian geometry.”

“[12]) incorporate geodesic distances instead of Euclidean distances; while geodesic distances are beneficial in modeling motion and expression, since they are not easily transferable to color spaces we here focus on Euclidean distance.”

“Kircher and Garland [2008] propose to interpolate the map connection, that is the relative transformations between neighboring elements.”

“[7]) of the n oriented line segments PiBi.”

“Designing and analyzing shape deformations is a central problem in computer graphics and geometry processing, with applications in scenarios such as shape manipulation [47, 59], animation and deformation transfer [49], shape interpolation [24, 56], and even anisotropic meshing [39], among myriad others.”

“Unparametrized curves arise naturally in shape analysis and its many applications, including medical imaging [24, 32, 34], object tracking [30, 31], computer animation [5, 12], computer aided design [13], speech recognition [29], analysis of bird migration patterns and hurricane paths [28], biology [11, 18], and many…”

“L(2) shape metric to the proposed quasiconformal metric [13],”

“The former (Heeren et al., 2012; Kilian et al., 2007) formulates interpolation as geodesic paths on a recovered shape manifold, while the latter (Eisenberger et al., 2019; 2021) directly estimates transformations between shapes under isometric assumptions. which are limited by the need for strict…”

“3DGS [49] apply as-isometric-as-possible regularization [25].”

“The theory has been widely used in image processing due to its remarkable inter-pretability of data and insensitivity to limited datasets [19– 22].”

“In previous work, this problem was often solved by a 2-step process, in which the two surfaces first were registered using one criterion, while deformations and distances were then computed independently using a different criterion [2, 6, 16, 29, 12, 18, 22].”

“to [31, 37, 54, 62, 35].”

“However, the method based on the shape space [15] provides a novel and better solution both in theory and practice.”

“One needs to take into account that these Riemannian manifolds could be also infinite dimensional, e.g., the space of plane curves [37, 38, 39, 51], the space of piecewise-smooth curves [40], and the space of surfaces in higher dimensions [3, 4, 26, 30, 36].”

“Traditional methods try to recover the shape space manifold [24,25,31,54], and then the shape interpolation becomes a geodesic-path searching problem on the manifold.”

“Other shape representation strategies (e.g. (Kilian et al., 2007)) incorporate geodesic distances instead of Euclidean distances; while geodesic distances are benefi-cial in modeling motion and expression, since they are not easilytransferabletocolorspacesweherefocusonEuclidean distance.”

“Kilian et al. [31] represent surfaces by discrete triangulated meshes and compute geodesic deformation paths between them while assuming that the meshes are registered.”

“We can choose to combine shapes in a non-linear way, for example by using geodesics with respect to an appropriate Riemannian metric [KMP07] or by computing the deformation path with least energy dissipation [HRWW12].”

“[11] treated each pose of shapes as a point in a shape space.”

“Moreover, the subdivision results are as good as the those from [KMP07], which conducts geodesic interpolation based on the as-isometric-as-possible shape metric.”

“Traditionally, motion modeling was achieved through explicit constraints or hand-crafted priors [Eisenberger et al. 2018; Huang et al. 2008; Innmann et al. 2020; Kilian et al. 2007; Myronenko and Song 2010; Sorkine and Alexa 2007], ensuring spatial coherence and adherence to physical laws.”

“Inspired by the as-isometric-as-possible constraint (Kilian et al., 2007; Huang et al., 2008) in computer graphics, our refinement module enforces j p Figure 6: Patch matching refinement.”

“L iso is the intra-frame distance preserving loss, originating from [19], which enforces consistent distances between neighboring points across two consecutive frames, defined as: where j is the index of top-5 nearest neighboring points to point 𝒗 𝑖 .”

“Kilian et al. [14] recast shapes as points in a shape space endowed with Riemannian metrics that naturally enable tasks like shape morphing.”

“Another closely related concept is geodesic interpolation in shape space [Kilian et al. 2007].”

“Computing minimizing geodesics is useful to perform shape registration [39, 46, 44], tracking [39] and shape deformation [28].”