Geometric modeling in shape space

“While the design of a good interpolation scheme is not the focus of this paper, one could use the approach of (Kilian et al., 2007) to minimize the number of self intersections in the morph.”

“Kilian et al. [25], in contrast, introduced an interesting approach that formulates the optimization as a path problem instead of deforming the reference shape.”

“…low-dimensional rig-based parameterizations, including linear blend skinning (LBS) [Lewis et al. 2000] and dual quaternion skinning [Kavan et al. 2008], often augmented with energy-based regularizers that preserve local rigidity or volume under articulation [Kilian et al. 2007; Lipman et al. 2007].”

“Additionally, to improve the smoothness of predicted sequences, we add an As-Isometric-As-Possible (AIAP) regularization loss L I based on [29] and used in [20] to avoid small non-isometric local deformations.”

“…using sagej.cls Then, for the 3D Gaussian part, following the previous work (Prokudin et al., 2023), we use the as-isometric-as-possible constrain (Kilian et al., 2007) to restrict the distance between neighboring 3D Gaussians to keep similar before and after deformation: where N is the total…”

“There exists a plurality of mathematical and numerical frameworks to study shape variability; examples include [2,16,24,57,84,96,102,104, 110, 142, 149, 156, 163].”

“In[Kilian et al. 2007], a geometric modeling based DT method is proposed.”

“In this regard, the work is more closely related to landmark based treatments of Kendall (1984), Bookstein (1989) and Kilian et al. (2007) than parametric ones of Charpiat et al. (2005), Michor and Mumford (2006), Mio et al. (2007) and Younes (1998).”

“Our approach is inspired by the success of studying organic shape collections using the lens of shape spaces as differential manifolds [Hart-man et al. 2023; Kilian et al. 2007; Klassen et al. 2004; Srivastava et al. 2011; Yang et al. 2023a].”

“…in vision and graphics have been applied to several optimization tasks, for instance, for total-variation denoising [3], texture parameterization and surface mappings with approximated Hessians [28], 3D shape manipulation [13], and image deconvolution tasks via (Hessian-free) Newton methods [14].”

“For instance, Kilian et al. [33] presented continuous deformation of 3D models using Geodesic interpolation in the Shape Space.”

“This representation has a long history in shape analysis Kendall (1984); Kilian, Mitra and Pottmann (2007).”

“[KMP07].”

“Applications include shape modeling in computer vision [17, 16], computational anatomy, where the morphing path establishes correspondences between a patient and a template [2, 26], shape clustering based on Riemannian distances [32], as well as shape statistics [9, 13], where geodesic paths in…”

“A different Riemannian metric defined on discrete meshes was presented in [45].”

“In this sense, we are aware of only two approaches to continuous deformations so far: [14] and its extension [15] which applies divergence-free vector fields for volume preserving shape deformation, and [18] which considers the deformations in a shape space.”

“In another approach [9] the authors use a semi-Riemann metric on a discrete surface (simplicial complex) to define a geodesic flow, modulo the space of rigid deformations.”

“[20], who utilized Riemannian metrics for both a rigid and an isometric deformation among shapes, would be an alternatively applicable approach for our problem.”

“[55], the results depend heavily on the initial rigid alignment of the shapes.”

“Many researches have traditionally been conducted for the shape morphing 26–30 .”

“Kilian et al. [11] focus on geodesic interpolation or extrapolation of parameterized surfaces and do not study regression.”

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

“…2016; Fef-ferman et al. 2018; Yao and Xia 2023; Yao et al. 2023), as in the examples of geometric shapes in the shape space (Turk and Levoy 1994; Kilian, Mitra, and Pottmann 2007; Bradley, Nowrouzezahrai, and Beardsley 2013; Dryden and Mardia 2016) and graphs in computer graphics (Phillips et…”

“Finally, our joint deformation design is related to the deformation transfer and interpolation techniques such as [117, 9] and [61] to name a few.”

“[22] design the metrics based on the shape space metrics, and propose a framework of model interpolation for multiresolution models.”

“(See [5] for an overview.)”

“It was previously shown [10] that we can associate a well defined Riemannian metric to the space of isometric deformations.”

“[9] used shape space representations to guide morphs and other more general deformations between shapes represented as triangular meshes.”

“[15] quantify shape differences using deviations from rigidity and isometry.”

“…or shape registration, is a central challenge in digital geometry processing, with applications spanning across various fields such as manufacturing, shape interpolation, shape morphing [1, 2], deformation transfer [3], statistical shape analysis [4] and segmentation of 3D shapes [5, 6 ].”

“Proposed solutions use geometric regularization [8, 37], model directly the shape space [2, 4, 16, 18, 21] or, more recently, learn shape interpolation by regularizing neural networks [10– 12].”

“The shape-based algorithm considers the 3D shape of the bones, representing them as points on Riemannian manifold [6]; while the anatomical structure-based algorithm incorporates local geometric measurements of the bones, as suggested by clinicians.”

“This representation has a long history in shape analysis [28, 29].”

“In the area of computer graphics and computer vision, shape correspondence have wide range of applications, such as shape morphing and interpolation [2, 3], shape segmentation [4], mesh parameterization [5], shape recognition and retrieval [6] and so on.”

“Other methods include the As-Similar-As-Possible method [129, 251] which preserves the tetrahedral meshes composing an object, or the Example-Driven method [84] which minimizes the deformations of edges, angles and dihedral angles.”

“, in [6], propose a geometric modeling method that projects a shape (mesh) in higher dimensions called the shape space and suggests its use for DT.”

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

“To solve these problems non-linear techniques are proposed [3, 4, 7, 11, 14, 18, 20, 21, 25, 28].”

“Dealing with the 3D shape correspondence problem is important because it lays the foundations of numerous practical applications such as shape morphing and interpolation [1][2], mesh parameterization [3][4], rigid or non-rigid shape registration [5][6], time-varying reconstruction [7][8], shape recognition and retriev…”

“Alternatively, literature [12] formulates the shape interpolation problem as computing geodesic curves in the shape space.”

“…component analysis, [9] which maintains the use of the Mahalanobis distance but consider the later as only valid locally on an open subset, [10] which defines the metric in terms of “ad hoc” constraints on the geometry transformations, [11] which calculates the Eigen vector in terms of the…”

“Some geometric structures have been established to provide the Isometric and Rigid metrics based on the Riemannian metric definition [7].”

“VI. 978-1-4244-7249-9/10/$26.00 ©2010 IEEE”

“For example, Kilian et al. [18] use a metric that penalizes deviations from rigidity and isometry.”

“Photometric supervision and regularization are applied after ras-terization: where R(·) enforces local rigidity or smoothness [Duan et al. 2024; Gao et al. 2024; Huang et al. 2023; Kilian et al. 2007; Oblak et al. 2024; Prokudin et al. 2023; Song et al. 2025; Sorkine and Alexa 2007].”

“Related work in computational shape analysis studies geometric metrics and geodesics for deformable shape spaces [23, 29, 34, 17, 41, 15, 14, 16, 22, 12]; GeoLatent [41], for example, derives a Riemannian metric from ARAP deformation geometry.”

“For example, Kilian et al. [21] compute geodesics by penalizing deviations from rigidity and isometry.”

“In Figure 5, following Kilian et al. (2007), we perform geodesic interpolation of anatomical poses.”

“For example, Kilian et al. [38] utilize Geodesic interpolation within the Shape Space to enable the continuous deformation of 3D models.”

“In most previous works the problem was either presumed or solved as an independent preprocessing step [6, 11, 19, 27].”

“We follow an approach similar to Kilian et al. (2007) to perform all computations on a coarsened mesh and prolongate solutions to a fine one only for visualizations.”

“Correspondence is a key component in tasks such as 3D alignment and spatiotemporal reconstruction, which is also an indispensable prerequisite for various applications such as attribute migration [5], shape interpolation [7], and statistical modeling [8].”

“This is closely related to shape interpolation [35, 27, 20, 19], which is typically based on the assumption that there exists a smooth deformation between two given shapes.”

“Here, the notion of shape includes planar curves [16, 15, 1], smooth surfaces [14, 2], images [17, 19], or triangle meshes [13].”

“6 and 7 depict the frame transition smoothness of sequences by Kilian et al. [16], Heeren et al. [11], Xia et al. [33] and our approach via stacking up the frames together.”

“This problem has many applications such as deformation transfer [91], shape interpolation [42] and even statistical shape modeling [35] among myriad others.”

“[Kilian 2007] proposed a numerical approach to calculate geodesic conductivities between triangulated surfaces using Euclidean metrics.”

“These areas include the measurement of the level of similarity between two objects, shape registration [1], indexing and retrieval [2], shape morphing, and detecting deformations of objects [3, 4], amongst others.”

“La résistance des films en PET au contact des acides est attribuée à la phase cristalline dans le PET [11], [12] .”

“Impressive results in shape processing have recently been obtained [Davis et al., 2007; Kilian et al., 2007] by imposing a Riemannian metric on the set of shapes, so that shape interpolation is reduced to the estimation of a smooth curve in the manifold of all shapes.”

“D shapes in the field of geometry processing, with specific applications including shape interpolation [5], texture and deformation transfer [7], and statistical shape modeling and analysis [8].”

“IsometricConstraint: Inspiredby[24],[57]andbasedon[95], we introduce an isometric constraint that preserves the relative distances between adjacent 3D Gaussian means during deformation.”

“We compare our results with Gamma expansion (GE) method [2] and Banterle’s inverse tone mapping operators (iTMO) method [11].”

“Also, [4] constructed a Riemannian space by imposing ad hoc constraints on the metric based on predetermined geometrical transformations.”