Designing and analyzing shape deformations is a central problem in computer graphics and geometry processing, with applications in scenarios such as shape manipulation [Yu et al. (2004), Sorkine and Alexa (2007)], animation and deformation transfer [Sumner and Popović (2004)]
, shape interpolation[Kilian et al. (2007), Von-Tycowicz et al. (2015)], and even anisotropic meshing [Panozzo et al. (2014)] among myriad others. Traditionally, shape deformation has been motivated by interactive applications in which the main goal is to design a deformation that satisfies some user-prescribed handle constraints while preserving the main structural properties of the shape. In other applications, such as shape interpolation and deformation transfer, that lack handle constraints, the goal is to design a global deformation field that would satisfy some structural properties as well as possible.
In both types of applications, most approaches are based on specifying a deformation energy and providing a method to optimize it. On the other hand, several works have demonstrated that by choosing an appropriate representation
for shape deformations, many tasks can become significantly easier, and in particular can help to enforce certain properties of the deformation field, which are otherwise very difficult to access and optimize for. In addition to the classical per-vertex displacement vectors, such representations have included gradient-based deformations[Yu et al. (2004), Zayer et al. (2005)], Laplacian-based approaches [Lipman et al. (2004), Sorkine et al. (2004)] and Möbius transformations in the context of conformal deformations [Crane et al. (2011), Vaxman et al. (2015)] among others.
At the same time, a number of recent works have shown that many basic operations in geometry processing can be viewed as linear operators acting on real-valued functions defined on the shapes. This includes the functional representation of mappings or correspondences acting through composition [Ovsjanikov et al. (2012), Pokrass et al. (2013)], representations of vector fields as derivations [Pavlov et al. (2011), Azencot et al. (2013)] and formulation of shape distortion via shape difference operators [Rustamov et al. (2013)]. One advantage of these representations is that linear operators can be naturally composed, which makes it easy to define, for example, the push-forward of a vector field with respect to a mapping, if both are represented as linear operators, or to solve for Killing vector fields, by composition between a derivation and the Laplacian operator. Moreover, by using a consistent functional representation these techniques often alleviate the need for point-wise correspondences, which can be difficult to obtain, as shown very recently for example in a work on joint cross-field design [Azencot et al. (2017)].
While tangent vector fields are classically understood as operators (derivations) in differential geometry, extrinsic vector fields do not enjoy a similar property. Our main goal is to provide a coordinate-free representation of extrinsic vector fields (that we also call deformation fields) as functional operators, which will prove useful for analysis and design of shape deformations. As we demonstrate below our representation greatly simplifies certain tasks such as intrinsic symmetrization, the computation of mappings by composition with other operators, and joint deformation design without requiring point-wise mappings. Moreover, it provides an explicit link between deformation fields and the changes in intrinsic metric quantities, which can be useful in a variety of analysis and deformation processing tasks.
For example, consider two shapes shown in Fig. 1 (left). By using our framework, it is possible to combine local deformation constraints with intrinsic objectives such as constructing a deformation field that is as-isometric-as-possible. Moreover, our representation allows to relate deformations on multiple shapes in a coordinate-free way, enabling deformation transfer and joint design using only soft, functional correspondences as shown in Fig. 1 (right).
2 Related Work
Shape deformation is one of the oldest and best-researched topics in computer graphics and geometry processing. We therefore only mention works most directly related to ours and refer the interested reader to surveys including [Nealen et al. (2006), Botsch and Sorkine (2008)] and [Botsch et al. (2010)] (Chapter 9).
A multitude of methods exists for surface deformation starting with the seminal work of [Terzopoulos et al. (1987)], its early follow-ups including [Celniker and Gossard (1991), Welch and Witkin (1992)] and the multi-scale variants, such as [Zorin et al. (1997), Kobbelt et al. (1998), Guskov et al. (1999)] among many others. Similarly to our approach, many of these techniques are based on optimizing the so-called elastic thin shell energy that measures stretching and bending, and which is often linearized for efficiency. In the majority of cases, deformations are represented explicitly as extrinsic vector fields defined on a surface, making deformation transfer difficult in the absence of precise pointwise correspondences.
A number of methods have proposed alternative representations for deformation fields, which greatly simplify certain tasks in design and analysis. This includes gradient-based techniques [Yu et al. (2004), Zayer et al. (2005)] which consider the deformation field by aligning its gradient with a set of local per-triangle transformations. By working in gradient space, constraints can be posed independently on the triangles and then optimized globally by solving the Poisson equation. Similarly, Laplacian-based techniques [Sorkine et al. (2004), Lipman et al. (2004), Nealen et al. (2005)] are based on defining shape deformations by manipulating per-vertex differential coordinates (Laplacians) in order to match some target Laplacian coordinates. Such differential coordinates enable direct editing of local shape properties, which can be especially beneficial for preserving and manipulating the high-frequency details of the surface. However, these coordinates are typically not rotationally invariant and additional steps are necessary to introduce invariance [Sorkine et al. (2004), Lipman et al. (2004), Paries et al. (2007)].
More recently, a number of methods have introduced representations for mesh deformations specifically geared towards particular shape manipulations, such as computing conformal transformations by designing special maps into the space of quaternions [Crane et al. (2011)] or by using face-based compatible Möbius transformations [Vaxman et al. (2015)]. These techniques are rotationally invariant and coordinate-free, while being restricted to special types of manipulations. Another technique, closely related to ours, designs shape deformations by constructing a continuous divergence-free vector field [von Funck et al. (2006)], and applying path line integration to obtain a deformed shape. We also consider the effect of the deformation on the metric, but both analyze the distortion of arbitrary extrinsic vector fields and show how they can be represented in coordinate-free way as linear functional operators.
Our use of spectral techniques and functional maps for representing deformation fields is also related to previous works in spectral shape processing, including the early approaches of Lévy and colleagues and their extensions [Lévy (2006), Vallet and Lévy (2008), Dey et al. (2012)] and more recent techniques such those based on coupled quasi-harmonic bases and functional maps [Kovnatsky et al. (2013), Yin et al. (2015)]. In these and related methods deformation fields are represented as triplets of functions, which encode displacement in each spatial coordinate. Although this representation is simple and naturally fits with the functional map framework, it suffers from several drawbacks. First, it is not rotationally invariant and induces artefacts if the shapes are not pre-aligned or are in different poses (see e.g., Figure 1). Perhaps more fundamentally, such a representation is not “shape-aware” since it does not reflect the change in the (e.g., metric) structure of the shapes induced by the deformation, which reduces its utility in deformation analysis and design. We demonstrate through extensive experiments, that by using our coordinate-free representation we can avoid these limitations and open the door to entirely novel design and analysis applications, such as intrinsic symmetrization (Section 8.2), which cannot be achieved using previous methods.
who manipulate shapes by explicitly editing their curvature properties. Moreover, our use of the strain tensor in characterizing metric distortion is closely related to the applications in various physically based deformation scenarios including[Thomaszewski et al. (2009), Müller et al. (2014)] among many others (see also the surveys on physically based elastic deformable models [Nealen et al. (2006), Rumpf and Wardetzky (2014)]). Our approach is also related to the works that aim to design as-isometric-as-possible shape deformations [Zhang et al. (2015), Solomon et al. (2011), Martinez Esturo et al. (2013)]. Similarly to the latter work, our framework is general and allows an arbitrary prescribed distortion, although our method works directly on surface representations and moreover enables applications such as joint deformation design.
Finally, our framework for joint design is related to the deformation transfer and interpolation techniques such as [Sumner and Popović (2004), Baran et al. (2009)] and [Kilian et al. (2007)] to name a few. Our approach is different in that we place special emphasis on relating deformations between shapes with only soft (or functional) correspondences, which are often much easier to obtain than detailed point matches. Moroever, rather than transporting Jacobian matrices associated with the deformation, which requires both a pre-alignment and an approximate triangle-to-triangle map (as done in [Sumner and Popović (2004)]) we study and transport the change in the intrinsic metric structure directly. As we show below, this results in better joint deformation design especially given approximate functional maps, and shapes in arbitrary poses.
Thus, in contrast to the majority of existing techniques our goal is to devise a coordinate-free representation of extrinsic deformations as linear functional operators, by making an explicit connection between the extrinsic deformations and the change in intrinsic metric quantities. As such, our representation fits within the recent line of work that represents many operations in geometry processing as functional operators, including mappings or correspondences [Ovsjanikov et al. (2012), Pokrass et al. (2013)], representations of vector fields as derivations [Pavlov et al. (2011), Azencot et al. (2013)] and the formulation of shape distortion via shape difference operators [Rustamov et al. (2013)]. Therefore, although we build on classical constructions such as the infinitesimal strain tensor, we show how they can be exploited to create a functional representation of shape deformation, which can be used in conjunction with other operators. As we demonstrate below, our representation is particularly useful for analysing and manipulating the effect of the deformation on the shape structure and for relating deformations across shapes, with only soft correspondences between them. In particular, it enables applications such as intrinsic symmetrization, joint deformation design and allows to introduce extrinsic information in the computation of functional maps. Remarkably, we prove that together with the classical Laplace-Beltrami operator, our approach leads to a complete (up to rigid motion) coordinate-free functional shape representation, which opens the door to new shape processing applications.
The rest of the paper is organized as follows: first, we define the functional deformation field representation using the classical notions of the Levi-Civita connection and the strain tensor, and list the main properties of this representation (Section 4). We then provide a link between this definition and the previously proposed shape difference operators, by considering their infinitesimal extensions, introducing a new unified operator, and proving the equivalence between the two definitions (Section 5). In Sections 6 and 7 we provide a discretization of all of these notions, and show that they preserve the main properties of the continous counterparts. Finally, we illustrate the utility of our representation by describing several novel application scenarios, which range from functional map inference, to intrinsic symmetrization and deformation field design that all exploit the properties of our representation and its relation to other previously proposed linear operators (Section 8). Note that Sections 5 and 7 can be skipped by readers that are not interested in the connection to shape difference operators.
To summarize, our main contributions include:
Introducing functional deformation fields as a way to represent extrinsic vector fields in a coordinate-free way as operators acting on functions, represented as matrices in the discrete setting.
Providing a link between functional deformation fields and the previously proposed shape difference oprators, which leads to both a new unified shape difference and alternative functional deformation fields, which can be made sensitive to specific (e.g., non-conformal) classes of distortions.
Showing how functional deformation can be used to naturally add extrinsic information (second fundamental form) into the computation and analysis of functional maps. We also prove that together with the Laplace-Beltrami operator, they provide a complete coordinate-free shape characterization up to rigid motions.
Describing how this representation enables a number of novel applications including intrinsic shape symmetrization, deformation design and functional deformation transfer without pointwise correspondences.
4 Extrinsic Vector Fields as Operators
In this section we provide a coordinate-free representation of extrinsic vector fields by considering their action on the underlying shape metric. Throughout this section we assume that we are dealing with a smooth surface without boundary embedded in The appropriate discretization of all the concepts introduced in this section will be given in Section 6.
The Levi-Civita Covariant Derivative
We first need to introduce some fundamental notions from differential geometry. In particular, we will use the classical Levi-Cevita connection to define derivatives on a surface. More precisely, given a tangent vector at some point , and an extrinsic vector field on , consider an arbitrary curve on such that and . Then, we let . Here is the standard covariant derivative of the ambient space. Note that at a fixed point , is a vector in . We can project the covariant derivative onto the tangent plane at to obtain a vector in the tangent plane, which is denoted simply by where is the Levi-Cevita connection on extended naturally to extrinsic vector fields, ([do Carmo (2013)] p. 126). We also remark that for any vector in the tangent space, , which we will use in our discretization.
The fundamental object that we consider below is the infinitesimal strain tensor, which can be understood as a bilinear form, acting on pairs of vectors in the tangent plane of a point . Namely, given an extrinsic vector field , the infinitesimal strain tensor is defined as:
This quantity has the advantage of being linear in the vector field , which makes it easy to handle for deformation and vector field design and therefore has been used in a wide variety of works in computer graphics [Nealen et al. (2006)].
With these definitions in hand we propose to consider a linear functional operator , which we will use to capture and manipulate a deformation field . Both the input and the output of our operator are smooth real-valued functions defined on the surface. This operator is defined implicitly, in the same spirit as the shape difference operators introduced by Rustamov et al. [Rustamov et al. (2013)] as follows: for every pair of real-valued functions we require:
The following proposition guarantees that is well-defined.
For any extrinsic vector field there is a unique linear functional operator that satisfies Eq. (2) above. Moreover, this operator is linear in both the vector field and function .
In the rest of the paper we call the linear functional operator , a functional deformation field representation of . Our main goal is to design, manipulate and analyze extrinsic vector fields through their associated linear functional operators . This approach has already proved useful in the context of manipulating maps or correspondences [Ovsjanikov et al. (2012)], tangent vector fields [Azencot et al. (2013)] and shape distortions [Rustamov et al. (2013)]. In particular, these works have helped to establish a general formalism of shape manipulation through the associated linear functional operators, which can “communicate” by composition. This allows, for example, to transfer tangent vector fields across shapes without assuming pointwise correspondences [Azencot et al. (2013)] or to design very efficient shape matching algorithms using the functional map representation [Ovsjanikov et al. (2016)]. Therefore, inspired by these works, we propose to extend this framework to also include extrinsic (or deformation) fields. As we show below, our representation naturally fits within the general functional operator formalism and enables a number of novel applications.
4.1 Key Properties of Functional Deformation Fields
Second-fundamental form representation
One interesting special case to consider is the interpretation of when the deformation field is the normal field . By using Eq. (1) it is possible to see ([do Carmo (2013)] p.128) that the covariant derivative of the normal yields the second fundamental form denoted by , more precisely . Therefore the operator captures the action of curvature on functions, since:
From a theoretical point of view the knowledge of the Laplace-Beltrami operator gives access to the first fundamental form and yields information about the second. Thus these two operators jointly provide a coordinate-free representation of the embedding.
The operator can be used to obtain a multi-scale representation of curvature information on the triangle mesh, as shown in Figure 2
. In particular, the eigenfunctions corresponding to the largest eigenvalues of, are those that align the best with the maximal principal curvature direction, and can be obtained even if is represented in a reduced functional basis, making the computation less sensitive to noise in the triangulation. Moreover, as we demonstrate in Section 8.1, the operator can be used to inject extrinsic information into the computation of functional maps.
|1st eigenfunction||2nd eigenfunction|
Composition with mappings
In many applications we are interested in the relation between deformations on multiple surfaces related by a mapping. In particular given a deformation field of shape and a diffeomorphism with the associated functional map (pullback) of functions from to , one can define a deformation field of shape that produces the same metric distortion. Instead of looking directly at the deformation of the metric, which might require a mapping between individual triangles [Sumner and Popović (2004)], we account for the action of the metric on functions:
In other words, can be obtained by considering an extrinsic vector field, whose operator representation has the same effect on functions when composed with the functional map as . This property allows us to relate deformation fields without requiring point-to-point correspondences between shapes, by simply considering the commutativity of the operators and . We illustrate this in Figure 5 and use it in Section 8.4 for deformation transfer and deformation symmetrization on meshes with different connectivities with only a functional map known between them. Furthermore, this approach is applicable to design deformations jointly on two shapes, such that they are consistent with the functional map and even as a regularizer in map computation.
Vector field representation
In general the operator does not uniquely define an extrinsic vector field. From Def. 2 it can be shown that the kernel of coincides with the vector fields satisfying . In case of a volumetric manifold (i.e. ) the kernel of our operator is restricted to infinitesimal rigid motions (see Theorem 1.7-3 in [Ciarlet (2000)]) and thus provides a complete representation of extrinsic vector fields. In the case of a surface embedded in the kernel of includes infinitesimal isometries such as Killing vector fields but also local normal fields in planar areas. No rigidity result seems to be known for smooth surfaces. However, as we demonstrate below, in the discrete case of shapes represented as triangle meshes, it can be shown that for almost all surfaces the kernel of consists only of rigid deformations (Prop. 6.1). Note that we place no restriction on the magnitude of the deformation fields. Thus, although our construction is based on the infinitesimal strain tensor, the extrinsic vector fields themselves are not limited to infinitesimal (or local) deformations. Finally, as we show below, our constructions can be easily extended to the case of tetrahedral meshes, resulting in a complete operator-based representation for deformation fields of volumes, not sensitive to the exceptional cases, present in the case of surfaces.
5 Relation to Shape Difference Operators
The functional deformation field representation introduced above is closely related to the previously proposed shape difference operators. In this section we describe this relation in detail, and highlight the following two key insights: 1) How our analysis leads to a novel unified shape difference operator, and 2) How alternative functional deformation field representations can be constructed, to be sensitive to only a particular class of metric distortions. Our analysis also sheds light on the discretization of functional deformation fields. Nevertheless, the discussion in this section is not required for the understanding of either the implementation or the results of our approach, apart from the intrinsic symmetrization application (Sec. 8.2), in which we use this relation. As such, this section can be skipped by readers not interested in these relations.
5.1 Shape Difference Operators
Introduced by [Rustamov et al. (2013)], the shape difference operators describe a shape deformation by considering the change of inner products between functions. Namely, given a pair of shapes and a diffeomorphism , with the associated linear functional map (pullback) defined by , the authors introduce the area-based and conformal shape difference operators and respectively, as linear operators acting on (and producing) real-valued functions on implicitly via the following equations:
where the inner products are defined as and
The existence and the linearity of the operators and is guaranteed by the Riesz representation theorem. As shown in [Rustamov et al. (2013)], for smooth surfaces, the map is area-preserving (resp. conformal) if and only if (resp. ) is the identity map between functions. From this it follows that is an isometry if and only if and are both identity.
Note that in the discrete setting the shape difference operators are obtained simply by considering transposes and inverses of the functional map and Laplacian matrices, as highlighted in [Rustamov et al. (2013)]. This makes properties such as existence and linearity trivial to see. Below we adopt the continuous (surface) formulation proposed in the original article as it helps to highlight both the generality of these concepts and also the relation to our representation of extrinsic vector fields.
Infinitesimal Shape Difference Operators
Our main goal in this section is to consider a one-parameter family of shapes , given by displacing the points of a base shape along some fixed deformation field. Specifically, given a surface embedded in we consider a family , parameterized by a scalar and given by where is a fixed point in , and is a vector in that represents the displacement of the point.
Now consider the family of maps , given trivially via , and the associated functional maps mapping functions from to . This gives rise to a one-parameter family of shape difference operators (which can be taken either to be the area or conformal-based operators). We then introduce the infinitesimal shape difference operator as follows: The infinitesimal area-based shape difference operator associated with an extrinsic vector field on a surface is defined as:
We define the infinitesimal conformal shape difference operator similarly by replacing by on the right side of Eq. (5).
Remark that since both and are defined as derivatives of a one-parameter family of linear operators acting on real-valued functions on a surface, both the range and the domain of these operators are also real-valued functions on . Moreover, as and reflect (or, equivalently, are sensitive to) changes in the area and conformal metric structure, this implies that and will only reflect extrinsic vector fields up to infinitesimally area-preserving or conformal deformations. This naturally raises the question of whether there exists another “unified” shape difference operator , which would be sensitive to general (non-isometric) metric changes. If so, would such lead to an infinitesimal shape difference that would agree with the definition of given in Eq. (2)? Below, we provide precisely such a definition which both extends the applicability of shape difference operators and helps to establish a deeper link with our functional deformation fields.
Unified shape difference
The main reason for which is only sensitive to conformal changes is that both the inner product and the integration are taken on the target shape. To define a unified shape difference taking into account all intrinsic changes one should compare the pullback metric to the metric on while keeping the integrating measure fixed. We thus propose a unified shape difference operator that fully characterizes isometric distortion.
Assuming that is a diffeomorphism, the unified shape difference is defined implicitly by:
The existence of is once again guaranteed by the Riesz representation theorem. Moreover, as we claimed above, the following proposition (proved in the supplemental material) shows that the unified shape difference fully characterizes isometric deformation.
for all if and only if is an isometry.
To illustrate the properties of the three shape differences we use a simple low-dimensional description of a shape collection in Figure 3. Here we choose a fixed base shape and compute the shape difference matrices with respect to the remaining shapes in a collection. Then, we represent each shape by its shape difference matrix and plot them as points in PCA space. Figure 3 represents the conformal deformation of a bunny into a sphere as viewed by the three shape differences. As expected is almost identity while the area and isometric shape differences both capture the distortion. In the second experiment, shown in Figure 3, we explore another collection obtained by the shearing of a plane patch. As this deformation is area preserving, the area-based shape difference provides no information, unlike the other two operators.
|Conformal deformation||Area preserving deformation|
With the definition of the unified shape difference in hand, we introduce its infinitesimal counterpart by following the same construction as done in (5) above. The following proposition (proved in the supplemental material) characterizes these new operators. Let be a smooth deformation field on , the derivatives of , and at time zero satisfy for all smooth functions :
As can be seen, the infinitesimal shape differences inherit the properties of the original operators. Namely, vanishes if and only if is equal to zero, i.e., whenever infinitesimally preserves the volume form. On the conformal side, finding an extrinsic vector field such that is equivalent to solving the conformal Killing equation: characteristic of a conformal vector field. Both properties combined lead to an isometric deformation induced by the vector field captured by .
Moreover Prop. 5.1 reveals a clear link between shape differences:
Thus, intuitively, the operator , representing isometric distortion, can be decomposed into an area and a conformal part. We note that linear dependence between shape operators shown in Eq. (6) can be understood as the decomposition of the matrix into a trace free part, linked to the conformal Killing equation, and a divergence part, related to the change in area.
Finally, this proposition shows that the functional deformation field representation introduced in Section 4 is exactly the same as the infinitesimal shape difference operator arising from the unified shape difference. Remarkably, this relation also holds exactly in the discrete setting as we show in Section 7.
To summarize, in this section we first showed that an alternative way for constructing a linear functional operator representation of extrinsic vector fields consists in considering a family of deformations of the shape, constructing the associated shape difference operators, and taking their derivative at zero, which leads to infinitesimal shape differences. This also suggests alternative functional deformation field operators, sensitive only to specific kinds of deformations (e.g., non area-preserving or non-conformal). Finally, we showed that by modifying the definition of shape differences, a new, unified difference operator can be constructed and that its derivative at time zero leads precisely to the functional deformation field formulation introduced in the previous section.
6 Discrete Setting
In this section we provide the discretization of functional deformation fields. For this, we first propose a particular discretization of the Levi-Civita connection and the Lie derivative of the metric on the triangle mesh, which leads to a simple formula for the operator . In the following section, Sec. 7, which can be skipped similarly to Sec. 5, we demonstrate that the deep connection between functional deformation fields and infinitesimal shape difference operators also holds in the discrete setting.
Throughout this section, we assume that we are given a manifold triangle mesh. We denote by respectively the set of vertices, edges and faces. We will consider the deformation field , which we also call an extrinsic vector field, to be given as a three-dimensional vector per vertex.
To build the discrete operator we need a consistent discretization of the Levi-Civita connection. While several discrete connections have been proposed (e.g. [Azencot et al. (2015), Liu et al. (2016)]), because of the special nature of our problem, we choose to build our own. This is because, applications such as parallel transport require that the vectors , and are expressed in the same space (at vertex or face or edge) so often an averaging step has to be introduced to transfer, for example, a face-based representation of a vector to an edge based representation. In our setting such a requirement is not needed and it is easier to distinguish tangent vector fields that will be expressed by one vector per face and extrinsic vector fields expressed at vertices. Thus, our goal is to obtain a connection of the ambient space where is a tangent vector and is an extrinsic vector field :
We build the connection using finite differences as follows. Since extrinsic vector fields are defined at vertices the differences are taken along the edges. In a given triangle the ambient covariant derivative along the edge is defined by
Thus the ambient connection in the directions can be stored in a matrix
Then, given any tangent vector , the covariant derivative in its direction can be computed as .
Given the expression above, the discrete Lie derivative of the metric at triangle follows immediately, using Eq. (1). Namely for any pair of tangent vectors in the triangle , we have:
If denotes the cotangent-weight Laplacian, which classically represents the inner products of (and is also called stiffness matrix), we obtain the discrete functional deformation field operator from its definition (2):
Then we obtain , where is a Laplacian matrix whose weights depend on the extrinsic vector field:
The computations can be found in the supplemental material.
Interestingly, many of the properties of the continuous operators are satisfied exactly by their discrete counterparts.
The discretization naturally preserves the linearity with respect to both and which is very convenient for practical purposes.
In practice, it is often convenient to use a functional basis, so that any function can be represented as a linear combination of some basis functions . Given such a basis, the operator can be seen as the (possibly infinite) matrix: . The choice of basis depends on the application. Since we are interested in smooth deformations of a surface, we take a subset of the smoothest functions given by the first eigenfunctions of the Laplace-Beltrami operator. In that case, will be represented simply as a matrix. As shown in Figure 13 the size of the basis affects the deformation field that we can represent and recover. Increasing allows a more faithful representation of high frequency deformation fields.
The linearity with respect to allows the same operation for vector fields. Therefore, if the deformation field is given in some basis then the operator reads . This means that when designing a deformation field we can consider an objective as a function of the coefficients .
Vector Fields representation
In the continuous setting the kernel of is the set of infinitesimal isometries. However, to the best of our knowledge, there is no characterization of how often this set is reduced to rigid motion. In the particular setting of our discretization some standard results can be applied, however.
For almost all triangle meshes without boundary, the operator uniquely defines the extrinsic vector field up to rigid motion.
Thanks to this proposition, we can guarantee that is almost always a complete coordinate-free representation of extrinsic vector fields . Triangle meshes containing perfectly flat neighborhoods fall in the category of shapes on which the map is not injective. Namely, since by definition of the strain tensor (Eq. 2), whenever and is normal to the surface for all , (as is the case when e.g. is a normal field on a flat part and zero elsewhere), the tensor will lead to the zero operator. Although we have found that for organic and natural shapes, such vector fields are rare or non-existent, they can nevertheless be important for coarse meshes or man-made objects with flat areas.
6.2 Construction for Tetrahedral Meshes
To remedy this problem, we extend our discretization to tetrahedral meshes thus avoiding ill-defined vector fields as the kernel of is of dimension (translations and infinitesimal rotations). For this we follow the construction provided in Section 6, by adapting it to tet meshes. Namely, we extend the ambient covariant derivative matrix in Eq. (7) to three dimensions, by considering the covariant derivative along three directions of a tet mesh, and thus storing a 3x3 matrix per simplex. We then use Eq. (8) without any modifications to obtain a discretization of the functional deformation fields on tet meshes. The final resulting formula for the matrix is provided in the supplementary material.
We compare the stability of our representation between tetrahedral and triangle meshes in Figure 4, by plotting the condition number of the linear system for recovering the vector field from its operator representation in the case of surface (triangle) and tet mesh reprentations of a cube. We note that although the condition number becomes unbounded for the triangle mesh representation as the shape approaches a flat cube, it nevertheless remains remarkably stable: even at where the sphere is almost a cube the condition number is about . In contrast the condition number for tet meshes remains bounded even for a perfectly flat shape.
An important consequence of Proposition 6.1 is that deformation fields are fully encoded by the operator up to infinitesimal rigid motions. Therefore any deformation can be recovered regardless of its scale and nature. For instance Figure 13 shows that non-infinitesimal global rotations are correctly encoded and recovered from our operator representations.
7 Discrete Infinitesimal Shape Differences
Similarly to the link between established in Section 5 between our initial definition for functional deformation fields and infinitesimal shape differences, we can consider an alternative discretization to the one above by considering a family of deformed meshes and taking the derivative of shape difference operators. In this section we show that this approach leads to exactly the same result, which means that remarkably Proposition 5.1 is satisfied exactly in the discrete setting. To demonstrate this result we first provide a discretization of the unified shape difference operator and then highlight the link between the infinitesimal shape difference operators and functional deformation fields. Similarly to Section 5, this section is primarily of conceptual interest and can be skipped by readers who wish to proceed to the practical results.
To compute the shape differences we start from the discretization of the inner product using standard first order finite elements. We will denote by the classical cotangent Laplacian matrix, the inner product of and the lumped mass matrix such that . As before is a measure and denotes the area of triangle .
7.1 Discrete unified shape difference
The discretization of the unified shape difference is straightforward when and are triangle meshes and share the same connectivity. In Definition 5.1 given above, the gradients and the point-wise scalar products are taken on while the measure comes from . Therefore the right hand side can be discretized by a modified cotangent weight formula:
Here are the two triangles adjacent to edge , which is opposite to angles and , while and are the triangle areas on shapes and respectively. Note that differs from the standard cotangent weight matrix only by the ratio of weights per triangle. Moreover,notice that if the transformation is area preserving for all triangles then reduces to the conformal shape difference defined in [Rustamov et al. (2013)] (Option 1 in Section 5).
From the expression above it follows that .
Expression in a basis
Similarly to the construction given in [Rustamov et al. (2013)] we can also express the unified shape difference when the basis on the source shape is given by the eigenfunctions of the Laplace-Beltrami operator. In that case, using a diagonal matrix of eigenvalues, becomes:
This expression has the advantage of avoiding the inverse of a large sparse matrix, and can be used to analyze deformation of a shape with fixed connectivity in multi-scale basis, which can make the computations resilient to local perturbations (see Option 3 in Section 5 of [Rustamov et al. (2013)] ).
Approximation with a functional map
Note that both expressions above assume that the source and target meshes share the same connectivity. When the meshes have different connectivity this discretization requires a map between triangles making it challenging to use in practice. To overcome this problem we approximate this discrete formulation by transferring the weights on triangles to lumped weights on vertices. The approximation then reduces to the usual discrete quantities:
We recognize here the cotangent Laplacian with lumped area weights, namely . In the case of meshes with different connectivity, this remark suggests the following approximation of the isometric shape difference, valid only in a discrete sense, for an arbitrary linear functional map between and :
In the reduced basis of the Laplacian eigenvectors, the approximation of the shape difference becomes , which preserves the principal property of the operator: is identity if and only if the deformation is an isometry since the Laplacian on has to be equal to the Laplacian on . We used this discretization in Figure 3 and observed that the two expressions given above typically produce similar results.
7.2 Shape difference derivative
Suppose that each vertex of the mesh is displaced by the vector by . This produces a family of triangle meshes with identical connectivity. It is now possible to take the derivative with respect to of Eq. (9) at time . This way we obtain a discretization of the infinitesimal shape differences. Remarkably the resulting discretization is strictly identical to the discrete functional operator proposed in Section 6 based on the discrete Levi-Civita connection. The discretization of based on the discrete Levi-Civita connection is equivalent to the one obtained by differentiating the unified shape difference operator.
Shape difference decomposition
Since the discretization using a discrete connection and through the time derivative agree, the decomposition described by Eq. (6) is also satisfied exactly. Namely, the matrix representing the discrete infinitesimal conformal shape difference splits into the discrete functional deformation field and an appropriately defined discrete divergence:
Thus, the decomposition of , representing isometric distortion, into area and conformal parts given in Eq. (6) in the continuous case holds exactly in the discrete case as well.
In this section we apply our constructions to various tasks in shape correspondence, deformation design and analysis. As our framework relies on manipulating and inverting moderately-sized matrices, all of the applications are very efficient, even when combining multiple objectives.
In some applications (Sec. 8.2 - 8.4), it is necessary to recover the deformation field from its function operator representation. For this, we use a reduced basis and recover the coefficients of the deformation field by solving a convex problem similar to basis pursuit. Namely, given a target functional deformation field operator represented as a matrix, we solve for via:
where is a vector of coefficients and are the functional representation of the deformation field in an overcomplete basis. Of course, the choice of basis is application dependent. The simplest and most general choice would be to consider a basis which consists of independent displacements at each vertex of the given mesh. For a mesh with vertices, this would result in 3 unknowns when solving for a deformation field, which is feasible when is small (and was used in the experiment in Figure 13), but can be inefficient for larger meshes. When needed (Sec. 8.2 - 8.4) we use the following deformation bases in the experiments below:
The simplest option is to take the eigenfunctions of the Laplace-Beltrami operator as the basis for each component of the deformation field. While simple, this basis might not preserve rotation invariance.
Alternatively we construct a basis via modal analysis of a deformation energy. In particular we consider an energy of the form . This corresponds to the energy on a particular discretization of the Bochner Laplacian of extrinsic vector fields. To obtain the basis we take the eigenvectors of the Hessian of the energy, which correspond to smooth deformation fields.
Lastly, we use the handle-based deformation model described in [Adams et al. (2008)]. Unlike the other families, the deformation fields arising from this model are compactly supported and therefore better suited to reproduce local deformations.
8.1 Functional map inference
|Same pose||Different Pose|
In our first application, we show how our functional deformation field representation can be used as a regularization in shape matching problems. In particular, we show how this representation can be used to add extrinsic information to the computation of functional maps [Ovsjanikov et al. (2012)]. The vast majority of the existing methods for shape correspondence with functional maps use the assumption of approximate intrinsic isometries (see [Ovsjanikov et al. (2016)]
for an overview) and are either purely intrinsic or inject extrinsic or embedding-dependent information by adding extrinsic descriptors. On the other hand, our functional deformation field representation provides a natural coordinate-free way to add embedding-dependent information into the map estimation pipeline. In particular, our approach below is based on the following key observation: Given a pair of surfacesembedded in 3D, and a diffeomorphism , let be the corresponding functional map Then and are related by a rigid motion in space if and only if:
where are the LB operators, while are functional deformation fields arising from the normal fields.
This proposition is simply a consequence of the fundamental theorem of surface theory and the relation between functional deformation fields and the second fundamental form described in Section 4.1. Note that enforcing the condition of this proposition in practice reduces simply to penalizing the lack of commutativity of the functional map with predefined operators, which can be done efficiently in practice. Therefore, we can see that functional deformation fields provide an effective way to capture embedding-dependent information in a coordinate-free way that fully characterizes the shape geometry up to rigid motions.
Inspired by this observation, we propose to solve the following problem: given two shapes and a sparse set of correspondences recover a dense map. The shapes come from the Faust dataset [Bogo et al. (2014)] and we are given five corresponding landmarks at the hands, feet and head. The baseline method following the logic of the original paper is to represent the landmark points as delta functions and and look for the most isometric functional map , by enforcing commutativity with the Laplace-Beltrami operator. Thus, the straightforward approach would be to solve the optimization problem:
The basic way to add extrinsic information to this problem is to constraint the map to preserve extrinsic descriptors noted respectively on :
We evaluate two commonly used descriptors:, 1) the normal vector field encoded as three independent functions and 2) the purely extrinsic descriptor SHOT [Tombari et al. (2010)] successfully used for solving partial matching problems [Litany et al. (2017)].
We compare these descriptor-based approaches to our coordinate-free extrinsic constraint. According to Prop. 8.1 if a diffeomorphism commutes with the Laplace-Beltrami operator and then the shapes admit the same embedding. Our new optimization problem thus reads:
Once the functional map are obtained they are converted to a point-to-point map using the knn-algorithm as described in[Ovsjanikov et al. (2012)]. The results are shown for two non-isometric shape matching problems: different characters taking the same pose and taking different pose. Figure 6 shows the percentage of correspondences within a given geodesic distance. Interestingly, SHOT provides valuables information on sharp features allowing accurate matching near salient points but tends to fail on featureless regions. In comparison our constraint provides information everywhere on the shape making the results less subject to obvious mismatching but it is less informative on the placement of salient features. This intuition is confirmed by Figure 7 which provides a visualization of the point-to-point correspondences by transferring the coordinates functions encoded as RGB channels. Finally, the combination of those two constraints overcomes the limitations of both methods taken independently.
8.2 Intrinsic Symmetrization
|Initial||Deformed||Intrinsic Sym.||Extrinsic Sym.|
In this section we show how our representation of deformation fields can be used to deform shapes to make them more intrinsically symmetric, while keeping their general pose. For example, Figure 8 shows a shape with important features which would be lost by an extrinsic symmetrization scheme. However an intrinsic symmetrization algorithm would preserve those features while recovering the symmetry. This way, our goal is similar to the one of [Zheng et al. (2015)] although our approach, unlike theirs, avoids the computation of a skeleton and is purely intrinsic. More precisely, given a base shape and a self-map we would like to compute the shape such that the self-map on is an isometry. If we denote by the map from to then the symmetry map on the deformed shape is given by . Using Prop. 5.1 the isometric constraint is satisfied if and only if the unified shape difference , computed with the map , equals identity. If is the functional map representation of a map , then after simplification this is equivalent to (see supplementary material).
Note, however that every intrinsically symmetric shape would be a solution of this equation. Therefore we regularize the problem by imposing that should be as-isometric-as possible. The equality conditions are enforced in the least squares sense leading to the optimization problem:
The optimization (10) is restricted to the set of diffeomorphisms so a direct approach is challenging to use in practice. A more tractable method is to use functional deformation fields as a first order approximation of shape differences and thus find the deformation field that solves (10) to first order. After linearization, Eq. (10) becomes:
|Initial||Iter. 1||Iter. 2||Iter. 3|
|Initial||Iter. 1||Iter. 2||Iter. 3|
|Initial||Iter. 1||Iter. 2||Iter. 3|
This linearization suggests an iterative algorithm (described in Algorithm 1) which alternates between solving the linearized problem (11) and computing the new vertex positions. In practice, we construct an over-complete dictionary of deformation fields, composed of the three bases described at the beginning of Section 8, compute the optimal deformation field by solving for the coefficients .
|Constraints||As-isometric-as possible||Symmetry||Anti-symmetry||Laplacian Reg.|
Figure 9 shows two examples where our method successfully recovers intrinsic symmetry from meshes with outstretched parts. In [Zheng et al. (2015)] the authors propose a method based on skeleton driven deformation to achieve intrinsic symmetry but limited to reflectional symmetries. Our method does not require such assumptions and works for any given self-mapping (e.g. bottom row in Figure 9). Note also that our deformation field representation is essential in this scenario, since for example, representing deformation fields through displacement functions would not provide information on the necessary (or induced) metric distortion.
8.3 Deformation design
Since our operator is linear with respect to the deformation field one can easily combine multiple constraints to the deformation vector field. In Figure 10 we show how multiple different constraints can be combined using our representation. First, we can easily require that at a point the deformation field matches a given vector , by setting , in addition to other global constraints. Second, we can find the most isometric (preserving the intrinsic metric) deformation by minimizing . At the same time, given a self-map represented as a functional map , we design a symmetric vector field by imposing a constraint of the form . Similarly, we can impose an anti-symmetry constraint by requiring . In comparison, Figure 11 shows an extrinsic deformation design method consisting in projecting each vector field component into the space of symmetric (respectively anti-symmetric) functions. The resulting shapes look quite distorted compared to our solution. Moreover the distance of the conformal shape difference (resp. area-based shape difference) to identity, measuring how far the map is from an isometry, is of (resp. ) for our design and (resp. ) for the extrinsic design. Thus, the extrinsic deformation design tends to distort the intrinsic structure of the shape. Finally, we test a regularization technique for the deformation field by imposing the commutativity with the Laplace-Beltrami operator, which tends to spread to the entire shape. Note that despite the diversity of these constraints, they can all be enforced easily using our operator-based representation. In contrast, the straightforward method shown in Figure 11, consisting in projecting the vector field onto the space of symmetric or anti-symmetric functions, fails in this tasks as it is fully extrinsic.
|Separate design||Joint design|
Figures 12 and 1 present an example of joint deformation design. Namely, we impose a set of directional constraints and on two different shapes and and we solve for two deformation fields, one on each shape, that are “informed” by the deformation of the other shape. On a single shape, our objective is designed to promote smoothness of the resulting deformation field and sparsity in the coefficients of the deformation field basis:
Therefore, on a single shape, the optimization becomes:
where the constraints enforce the given pointwise directions.
To design the deformation fields jointly, we propose to find a field on shape and on shape such that for a given functional map we have while respecting the local constraints on the respective shape. The resulting optimization problem reads:
As a result, the constraints as well as the structure of one shape is transferred onto the other. Moreover the area that could lead to contradictory deformation remains still.
8.4 Functional Deformation transfer
Given a deformation field on shape represented as an operator and a functional map from to shape , we can use our method to transfer the deformation to an arbitrary mesh. The transferred deformation on shape by solving:
In all of our experiments below, we represent the linear operators and in a reduced functional basis, consisting of the first eigenfunctions of the Laplace-Beltrami operator. We parameterize the space of deformations by computing the extrinsic vector fields in each of the three categories described in Section 8 to build an over-complete dictionnary. This implies that the number of unknowns is relatively small: and is independent of the resolution of the underlying mesh. We choose the parameter , controlling the sparsity of the representation, to be times the largest singular value of the linear map .
Using this setup we solve different instances of the deformation transfer problem, namely:
Style transfer: we transfer style across poses. Here, given two different shapes in a rest pose and a deformed version of one of them, we transfer the deformation to the other shape (Figure 14). This also shows that our vector field collection is not limited to a specific type of deformation.
We stress that although enabled by our representation, this is by no means the central application and therefore the results presented below simply serve as an illustration of the functionality that can be achieved using our functional deformation fields.
We use our approach to transfer style across the poses of different shapes in the Faust dataset [Bogo et al. (2014)], shown in Figure 14. Here first consider the deformation field given by the point displacements across two different shapes in approximately the same reference pose. We then use our framework to transfer to another shape in a different pose and with different mesh structure. In Figure 14 our method consistently preserves the global structure, although some high frequency details of the deformation are lost due to the projection onto a vector field basis.
One interesting feature of the functional representation of deformation fields is that it is “shape aware.” For example in Figure 15 we transfer the shrinking of the right leg to the left leg by looking for the operator which commutes with the operator representation of the symmetry map. Since both legs are in different positions this transfer is not easy to achieve by a simple point-to-point transfer of the vector field or even by transferring it using local coordinates. As shown in Figure 15 bottom row, our transferred deformation field adapts to the geometry.
|Initial shape||Deformation||Symmetric Def.|
8.5 Relation to existing techniques
An important property of our deformation transfer algorithm is that it relies fully on the deformation of the metric. This makes it fundamentally different from the spectral pose transfer described in [Lévy (2006)]. Those methods use the strong stability of the first eigenfunctions of the Laplace-Beltrami under deformation. Thus, a deformation field can be efficiently transferred by projecting its components into a reduced eigen-basis of the initial shape and reconstructed using the basis of the shape to be deformed. The new embedding is computed from the old embedding simply by
Recent improvements of this technique [Kovnatsky et al. (2013), Yin et al. (2015)] include pre-alignment of the spectral basis but the shortcoming are essentially the same. Figure 16 shows that this deformation transfer is by definition extrinsic, orientation dependent and furthermore completely agnostic to the intrinsic structure of the shape. Our method in contrast is rotation-invariant and directly linked to the induced changes in the geometry.
We also compare our method with the algorithm for deformation transfer described in [Sumner and Popović (2004)]. This method is based on reallocating Jacobian matrices defined on triangles of the source mesh to those of the target mesh. This method, however, is not without limitations. First, this transfer does not take into account changes in orientation from the source to the target thus ruling out any possibility of symmetric transfer and requiring a pre-alignment of the source and target meshes. This can be challenging to achieve in practice in case of non-rigid deformations (e.g. Figure 15). Secondly, it assumes as input a triangle-to-triangle map which can be cumbersome to obtain.
These limitations do not apply to our representation as our approach is based on transferring metric information, and is therefore immune to changes of orientation. Moreover, instead of a triangle-to-triangle map, an approximate functional map is enough. Furthermore, note that although in general reconstructing geometry from metric tensors is more difficult than a reconstruction from Jacobians as local rotations are no longer available (see e.g. [Boscaini et al. (2015)]) our method relies on solving a moderately-sized convex optimization problem.
Figure 17 shows that working within the functional map framework makes our algorithm more robust to noise usually encounter when using this representation. The computation of functional maps, as described in [Ovsjanikov et al. (2012)], is done by solving a least squares system incorporating intrinsic descriptors (HKS, WKS) therefore there often exists multiple solutions in presence of an intrinsic symmetry . We model a noisy functional map by a linear blending between the direct map (mapping the left to the left and the right to the right) and the anti-symmetric map (mapping the left to the right and the right to the left) represented as operators:
Our method outputs a non-linear interpolation between the deformation and its symmetric version while the method by Sumner et al. exhibits various artifacts.
|Source||Deformed||Target||Ours||Sumner et al.|
|[Sumner and Popović (2004)]|
9 Conclusion and Future Work
In this paper we presented a method for representing extrinsic vector fields as linear operators acting on functions on the shapes, by considering the metric distortion induced by the deformation. In particular, we base our representation on the infinitesimal strain tensor and show how it leads to a linear functional operator that can naturally be combined with other such operators including functional maps and the Laplace-Beltrami. We showed how this representation can be used to analyze and design deformations and to introduce extrinsic information into the computation of functional correspondences. In the future, we are planning to use the newly introduced functional representation for shape animation. In this context, it would be interesting to establish a connection between the metric on the space of functional deformation fields and different inner products as suggested in [Eckstein et al. (2007)]. It would also very interesting to use our representation within the framework of shape spaces, e.g. [Kilian et al. (2007)], for exploration and design.
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