Shape Deformation Models

Synonyms

Shape deformation computer imaging; Shape deformation curves; Shape deformation polygons; Shape deformation surfaces

Definition

Shape deformation models have occupied a prominent place in computer graphics and image processing. They possess superior geometric properties of flexible nature which make them an ideal candidate for several applications in graphics and imaging industry. In this article, some basic properties of shape deformation models are presented. These properties are highlighted though examples. An industry application with respect to topology driven structures through deformation with 3D shape correspondence of varying nature further highlights the strength of these models.

Introduction

The deformation of shapes corresponds as a basic problem in geometric modeling (González Hidalgo et al. 2013). Several methods have been developed to address rigid alignments, isometric shape articulations, part stretching nonrigid shape deformations, etc. Considering the inherent matching shapes dissimilarity, the search for correspondence becomes complex with respect to formulation of the problem computation cost. In order to deal with shapes varying in structure as well as geometry, the current techniques are basically data or knowledge based. They make use of various supervised learning or unsupervised co-analysis for collections of shape. In terms of design, these techniques give coarse as well as discrete correspondences which leaves structural discrepancies considering matched parts or their ensembles unsettled. Some examples worth mentioning are available in (Chaudhuri 2018).

The deformation process can be represented through hierarchy modeling which effectively represents object geometrically. Though this simple graphical primitives along with operators are combined together towards the formation of complicated objects. In the hierarchy of leaf nodes, there are hardware commands from where vectors are drawn, individual pixels’ colors change, and operations are performed on list of segmented polygons. With respect to proper algorithms, the users have strong intuitive feeling for manipulation of results. They can also think for every operation and can structure the objects they are looking for. These deformations enforce the user to consider an object from special topology that can be bent, twisted, tapered, compressed, expanded, or transformed towards a final construct. They have highly intuitive nature. They have visualized operations which simulate some kind of manufacturing processes towards objects’ fabrication. These deformations have also been considered towards traditional CAD/CAM modeling as well as surface patches. This decreases the data storage requirements, which simulate flexible objects geometrically. This makes it possible towards accurate modeling of physical properties for various elastic materials considering several partial differential equations. However, simple mathematical deformation techniques also exist. These simple techniques have reduced requirements in terms of computation. They model objects effectively.

A fair amount of work has been done on matching discrete structures, deformation driven correspondence, structure aware deformation, topology matching and blending, as well as co-analysis. This is in lines with deformations of shapes where the 3D geometrical structures are actively used. With this motivation, in this article we present some basic concepts related to shape deformation which are supported with illustrative examples. An industry application revolving around a deformation driven technique considering varying topology with shape correspondence in 3D is highlighted. This article is structured as follows. In proceeding section, the shape deformations overview is given. This is followed by an industry application in shape deformations. Finally, the conclusions are given.

Shape Deformations: Overview

The shape deformation can be defined as any object’s shape change which happens because of a force applied externally or temperature change. For shape change of an object, the deformation energy is transferred through work done. For temperature change, the deformation energy is transferred through heat. The first instance is attributed towards tensile and compressive forces as well as shear, bending, or torsion. The second instance is determined through temperature and is the structural defects mobility where the movement is thermally activated. The deformation also results from strain. The deformation is also associated with stability of an elastic material through incremental deformations. It is based on superposed small deformation considering an equilibrium solution. When deformation takes place, the internal forces of inter-molecular nature take place that opposes the external forces. If external force is not adequate, the forces are enough towards fully resisting said force. This also allows objects towards new equilibrium and return to actual state as the load withdraws. The greater external forces will go towards object’s full deformation as well as its structural distortion.

Figure 1 shows that compressive loading which is shown by arrow causes cylinder deformation such that original shape shown by broken lines is deformed considering sides which are bulging. The sides bulge as material because it is not strong enough to break or support any load. This results in forcing the material out laterally. The internal forces at right angles to deformation resists applied load. For negligible deformations, the object may be considered as rigid body.

Fig. 1

The object deforms because of stress (it shortens, and expansion takes place in outward direction)

We now present the basic mathematical formulations for shape deformations which are specified through global and local deformations. The global 3D solid deformation is mathematical function $\overline{H}$ that changes global coordinates towards points specified in space. The points for undeformed and deformed solids have been specified through $\overline{a}$ and $\overline{A}$, respectively. This is represented as:

$$ \overline{A}=\overline{H}\left(\overline{a}\right) $$

(1)

Let us consider a₁, a₂, and a₃ as the components of 3D vector $\overline{a}$ in three directions.

The local deformation changes tangent space for solid. The differential vectors for solid’s substance are circulated and skewed accordingly. There is an integration of these vectors considering the position globally.

There are separate chain links considering differential vectors that may stretch as well as rotate. The deformation’s local specification is given by matrix function which may be circulating as well as skewing. The end link’s position considering chain is vector sum for previous links and is discussed here.

Considering the modeling aspect, the two important vectors are tangent as well as normal. The first vector is used towards delineation and construction of local geometry. The latter is used towards orientation of surface as well as information for lighting. There is a transformation for the tangent as well as normal vectors from the undeformed to the deformed surface. The manipulations towards rules of transformation consider single multiplication with respect to jacobian matrix $\overline{\overline{JM}}$ for transformation function $\overline{H}$. The jacobian matrix $\overline{\overline{JM}}$ considering transformation function for Eq. (1) is function of $\overline{a}$ and is computed through partial derivatives for $\overline{H}$ with a_i; i = 1, 2, 3 such that:

$$ {\overline{\overline{JM}}}_i\left(\overline{a}\right)=\frac{\partial \overline{H}\left(\overline{a}\right)}{\partial {a}_i} $$

(2)

As such, ith column for $\overline{\overline{JM}}$ is achieved through partial derivative for $\overline{H}\left(\overline{a}\right)$ considering a_i. The object’s surface is represented through parametric function for variables p and q such that:

$$ \overline{a}=\overline{a}\left(p,q\right) $$

(3)

Any tangent vector considering the surface is reached through linear combinations of partial derivatives for $\overline{a}$ considering p and q. The normal vector direction can be obtained through cross product for linearly independent tangent vectors considering surface.

The transformation rule for tangent vector is restated in terms of a chain rule. As a result of this, new vector derivative is jacobian matrix times the previous derivative such that:

$$ \frac{\partial \overline{A}}{\partial p}=\overline{\overline{JM}}\frac{\partial \overline{a}}{\partial p} $$

(4)

In component form, the above equation can be written as:

$$ {A}_{i,p}=\sum \limits_{j=1}^3{JM}_{ij}{a}_{j,p} $$

(5)

The transformation rule for normal vector requires jacobian matrix inverse transpose such that:

$$ {\overline{\boldsymbol{n}}}^{(A)}=\det JM{\overline{\overline{JM}}}^{-1T}{\overline{\boldsymbol{n}}}^{(x)} $$

(6)

Since only normal vector’s direction is significant, it is not required to calculate determinant’s value. The jacobian’s determinant looks towards local volume ratio for each transformation point considering deformed as well as undeformed regions. Scaling is one of the simple examples of shape deformation (https://abaqus-docs.mit.edu/2017/English/SIMACAECAERefMap/simacae-t-customdeffactorframe.htm). Other notable examples include global tapering along any axis, global axial twists, global linear bends along any axis (https://abaqus-docs.mit.edu/2017/English/SIMACAEGSARefMap/simagsa-t-displayingandcustomizingadeformedshapeplot.htm), etc. We present few of these examples here.

Global Tapering Along Any Axis

The tapering operation is identical to scaling. It has been applied towards many flexible objects. It is achieved by differentially changing two global components’ length without changing third component’s length. The process is represented through piecewise linear function which decreases as the other axis increases. Figure 2 represents tapering of any ribbon.

Fig. 2

A ribbon’s tapering progressively

Global Axial Twists

The global axial twists basically simulate an object’s global twist. The twist is approximated through rotation achieved differentially. It draws analogy from the global basis vectors’ differential scaling. Here rotation is performed in terms of global basis vectors’ pair which is height’s function where third global basis vector is not altered. This deformation process is enforced through twisting a cards’ deck with every card being rotated somewhat more than the card below it. Figures 3, 4, 5, and 6 represent global axial twists process for several objects of interest such as ribbon’s progressive twist, two primitives’ progressive twist, tapered primitive twist, and twisted offset primitive tapering.

Fig. 3

A ribbon’s progressive twist

Fig. 4

The two primitives’ progressive twist

Fig. 5

The tapered primitive twist

Fig. 6

The twisted offset primitive tapering

Global Linear Bends Along Any Axis

The global linear bends along any axis mimics the bending simulation process. It is basically represented through bend of isotropic nature considering centerline parallel with respect to any specified axis. During bending process, centerline stretch does not change. The angle at bending is always constant considering extremities. However, all the changes take place linearly considering central part. Considering region at bent, rate of bending is kept fixed, and differential basis vectors are rotated and translated with respect to the third local basis vector. The deformation comprises of rigid body rotation as well as translation outside bent region. Here range of bending deformation is controlled through specified axis. Figures 7, 8, 9, 10, 11, and 12 represent global linear bends along any axis for several objects of interest such as region’s progressive bending, region’s progressive change in bending range, moebius band produced through twist and bend, tapered primitive with bents and twists, bent and twisted primitive and chair’s model considering six primitives with seven bends.

Fig. 7

A region’s progressive bending

Fig. 8

The bending range’s progressive change for a region

Fig. 9

The mobius band that is developed through bends and twists

Fig. 10

The tapered primitive with bents and twists

Fig. 11

A bent and twisted primitive

Fig. 12

The model of chair considering six primitives with seven bends

Now we discuss the process for generating more generalized shapes. In this direction, it converts local representations towards global representations with respect to shape deformations. The Jacobian matrix is generally known beforehand as a function of its parameters. However, the expression with closed form considering deformation function is not specified. The basic method revolves around the following steps:

• (a) The undeformed input shape’s conversion into its tangent vectors considering differentiation

• (b) The tangent vectors’ transformation through tangent transformation considering tangent vectors with respect to object deformed

• (c) The integration of new tangent vectors towards new position vectors for deformed curve, surface, or solid spaces

The local towards global operations changes local tangent vectors and Jacobian matrix in terms of position vectors globally. The space’s absolute position for deformed object is specified within arbitrary constant vector integration. This gives complete deformation description which can be directly coupled towards output considering elasticity equations or finite element analysis for entities of deformable nature describing general shapes’ composition. To further augment this discussion, two significant applications include space curves as well as 3D surfaces and solids transformations available in (Chaudhuri 2018).

Another application of shape deformation worth mentioning is available in rendering (Chaudhuri 2018). To achieve the control points and normal vectors set for which the surface patches with polygons or splines are developed, the surface deformed is sampled parametrically. Considering proper sampling, patches can deform the target object in more detailed manner when surface curvature is high and lesser detail where there is flatter surface. With the parametric values for raw grid, the object is at first sampled. The raw parametric surface’s sampling is refined through vector criteria normally which is compute considering rule of transformation. There is recursive subdivision for adjacent normal vectors where divergence is large. The dot products that are wide enough from unity indicate that large recursive detail is required for that part. Therefore, patch-based methods such as depth buffer as well as scan line encoding schemes are good. There is linearity in these algorithms with respect to total surface area as well as number of patches. The direct subdivision approach is not suitable towards ray tracing as the total number of operations becomes quadratic considering number of ray comparisons and objects. In order to reduce the number of objects, the incident ray is intersected with deformed primitives analytically. The inverse deformations can also be used to undeform primitives as well as tracing along deformed rays. Figures 13 and 14 represent the primitive of deformed nature in undeformed space and undeformed primitive with respect to undeformed coordinate system considering ray’s path. This reduces dimensionality for parameter search considering three towards one which indicates considerable benefits in terms of computational complexity.

Fig. 13

The primitive of deformed nature in undeformed space

Fig. 14

The undeformed primitive with respect to undeformed coordinate system considering the ray’s path

Industry Application

In this section, a novel industry application revolving around a topology driven through deformation with 3D shape correspondence of varying nature. The 3D geometry enforces shape deformation algorithm to consider 3D shapes as inputs (Alhashim et al. 2015). This allows more generalization for the deformation models. The input shapes are upright oriented and segmented as significant parts adhering to all the symmetries. The shapes are represented through shape’s topology encoding considering parts’ medial abstractions and structural relations with respect to various connectives and symmetries. A combinatorial tree may be traversed to reach part correspondence with pruning of priority nature. The priority is enforced through self-distortion energy. The evaluation is performed considering deformation from one shape to another under previously matched parts’ constraints. This produces piecewise continuous part-to-part mapping. Figure 15 highlights a typical correspondence approach (Alhashim et al. 2015). For shapes having large number of parts, course towards fine approach is adopted when search is performed for optimum correspondence. The correspondence may be from all the source parts to some target parts, but the reverse is not valid. Therefore, several source parts can be matched towards identical target part. However, there might be unmatching for certain target parts. The deformation energy determines which source to target designation yields the best solution.

Fig. 15

A typical correspondence approach (with target shapes and curve-sheet abstractions on extreme left, the search tree in middle part, and final correspondence result on extreme right)

This deformation model considers geometric as well as topological operations like part split, duplication, merging, etc. which leads towards fine-grained and piecewise results. The key aspect here is the deformation energy which affects geometric distortion. It also encourages preservation of structures as well as allows changes in topology. It accomplishes this by connecting shape parts through structural rods that behave in a similar manner as virtual springs. It allows encoding for energies coming from geometric, structural, and topological shapes. The optimal shape correspondence is reached considering the pruned beam search.

This process can be sketched as consisting of four steps, viz., initial segmentation, graphical structured representation, self-distortion energy, and correspondence search. The initial segmentation looks towards computation of fine-grained correspondence parts. The graphical structured representation defines the shape of segments as significant concrete structures achieved through various methods. The self-distortion energy is measured considering shape of the source which allows several shape distortions. The correspondence search allows one-to-one as well as one-to-many part correspondences. Figure 16 shows an array of fine-grained part correspondences calculated using this method.

Fig. 16

The array of fine-grained part correspondences calculated using this method (Considering each pair of input 3D models, the best-rated correspondence is shown; The matching parts have identical color and others non-matching parts are in gray)

Conclusion

In this article, we have presented some of the important conceptual aspects related to shape deformation models. The shape deformations find immense applications in computer graphics and image processing. Some of the basic properties of shape deformations are initially highlighted followed by illustrative examples. An industry application considering topology driven structures through deformation with 3D shape correspondence of varying nature justifies the significance of the shape deformation models.