An Intuitive Framework for Real-Time Freeform Modeling

An Intuitive Framework for Real-Time Freeform Modeling Mario Botsc Leif Kobbelt Computer Grapics Group RWTH Aacen University Abstract We present a freeform modeling framework for unstructured triangle meses wic is based on constraint sape optimization. Te goal is to simplify te user interaction even for quite complex freeform or multiresolution modifications. Te user first sets various boundary constraints to define a custom tailored (abstract) basis function wic is adjusted to a given design task. Te actual modification is ten controlled by moving one single 9-dof manipulator object. Te tecnique can andle arbitrary support regions and piecewise boundary conditions wit smootness ranging continuously from C 0 to C 2. To more naturally adapt te modification to te sape of te support region, te deformed surface can be tuned to bend wit anisotropic stiffness. We are able to acieve real-time response in an interactive design session even for complex meses by precomputing a set of scalar-valued basis functions tat correspond to te degrees of freedom of te manipulator by wic te user controls te modification. CR Categories: I.3.5 [Computer Grapics]: Computational Geometry and Object Modeling Pysically based modeling; I.3.6 [Computer Grapics]: Metodology and Tecniques Interaction Tecniques Keywords: surface editing, freeform design, user interaction 1 Introduction Computer aided geometric design tecniques ave become an important key tecnology in te industrial design and development process. Te availability of digital 3D models during te various stages of te design process triggers a large number of applications ranging from numerical simulation and assembly planning to design studies and product presentation. Tis variety of applications implies tat tere are many different (and sometimes contradicting) requirements for ow tese 3D models ave to be accessed in an interactive design session. Especially in te earlier (conceptual) design stages, te major bottleneck is still te freeform design metapor wic sould allow te user to convey an imaginary sape to te computer system in an intuitive fasion. Te fundamental problem ere is tat te space of possible geometric sapes is extremely ig dimensional and as a very complex structure wit estetically pleasing sapes sometimes lying surprisingly close to unacceptable sapes. Te designer as to explore tis ric space of sapes by just pressing buttons or clicking and dragging 2D positions on te screen. More sopisticated interaction tecnology like aptic input devices, immersive displays [Sckolne et al. 2001], or two-anded input metapors [Llamas et al. 2003] are available today but tey did not replace te well-establised PCbased working place as it can be found in every industrial design company. In an interactive design session, te user usually drags control vertices wic ave tree degrees of freedom (translation) or e moves more general manipulator objects aving six (translation and rotation) or nine (plus scaling) degrees of freedom. For complex sape modifications several control andles ave to be dragged sequentially or simultaneously. Tis rudimentary interface is te reason wy it takes igly skilled experts to operate a CAD system. Neverteless it is te widely accepted standard and implemented in many NURBS or subdivision based modeling frameworks. Let S be te given sape tat te designer wants to modify into anoter sape S and let us assume tat S and S do not differ too muc. Obviously, every extreme modification can be decomposed into a sequence of smaller modifications and tis is wat designers usually do even traditional designers working wit real clay, not just CAD designers. If we accept tat interactive design is about dragging manipulators and verifying te result by visual feedback ten any sape modification can be caracterized by S = S + B(δC) (1) were B represents an abstract basis function and δc someow represents te cange of position and orientation of te control andles. For a modeling tool based on NURBS surfaces, e.g., B could be te set of tensor-product basis functions and δc te displacement vectors by wic we sift te corresponding control vertices. A sape modification is complex if te update B(δC) is complex. Tis can be acieved by eiter providing a complicated andle object wit many degrees of freedom or by using special basis functions tat are adapted to te desired modification. Since our goal is to keep te user interaction simple, te possible types of modifications tat we can apply to te object S are ence limited by te abstract basis functions tat our system associates wit te control andles. In te case of NURBS or subdivision surfaces, tis means tat wit eac elementary modification we can add a smoot bump wit rectangular or polygonal support to te surface. Every more sopisticated modeling operation as to be built from tese elementary modifications. An important limitation wit most of te existing modeling frameworks is tat te underlying matematical surface representation is tigtly linked to te type and number of control andles. Tis problem is more tan obvious if our geometry representation is based on unstructured triangle meses since ere, sifting a (control) vertex just adds a tiny at function to te surface. In order to perform some non-trivial modification one as to manually move a larger number of control vertices simultaneously. In te setting of (1) tis would correspond to a igly complex δc, i.e. a igly complex user interaction.

Figure 1: In our modeling metapor, we define a custom tailored basis function by selecting a support region (blue) and a andle region (green). Te smootness conditions at te inner and outer boundary can be controlled independently and continuously blended between C 0 and C 2. From left to rigt we sow te initial configuration, C 2 at inner and outer boundary, C 0 at inner and C 2 at outer, as well as C 2 at inner and C 0 at outer boundary. Te blue, green and gray regions correspond to te sets of vertices p,, and f in equation (5), respectively. Tis is wy for polygon meses, control andle based modification metapors ave been developed wic are mostly independent from te underlying tessellation of te surface. Tis means we simplify te structure of C in (1) by making B sligtly more complicated. Freeform deformation [Sederberg and Parry 1986; Coquillart 1990; MacCracken and Joy 1996] is probably one of te most prominent examples were sifting a control vertex in a spatial grid causes a deformation of te embedding space around a 3D model and tereby induces a global modification on te model itself. Wile tis is a very intuitive modeling metapor, is does not provide significantly more degrees of freedom (bumps over simple support regions). Moreover, te support of te modification is sometimes difficult to predict as it is determined by intersecting a volumetric basis function s support wit te modified surface. Te purpose of tis sort paper is to describe anoter modeling metapor wic provides te maximum flexibility wit respect to te set of potential basis functions. Te idea is to elp te designer to define is own custom tailored basis function tat is optimally adapted to te intended modification. Ten tis basis function is associated wit a manipulator object tat te user can move interactively to do te actual sape editing operation in real-time. Our goal is to provide as many degrees of freedom as possible for te definition of te basis function B to offer enoug flexibility for non-trivial modifications. Yet, we always keep in mind tat te resulting sape modeling system sould be simple and intuitive enoug for an average (not specialized) user. Hence, after te definition of te boundary constraints, te user interaction is restricted to moving a single manipulator object C. Wile we are integrating various known geometry processing concepts into our freeform modeling framework, we also extend tese tecniques to meet te central requirements of our modeling system. Te two major innovations are tat we obtain flexible sape control by using an anisotropic discretization of te energy functional wic determines ow te surface bends under deformation and tat by precomputing a set of linear basis functions, we easily acieve real-time feedback even wen modifying large surface areas. 2 Te modeling metapor In Eq. (1) we represented an arbitrary freeform modification by an abstract basis function B wic is added to an existing sape S. In order to specify tis basis function for a particular modification we ave to define its support, i.e., te region of te surface S tat sould be affected by tis modification and its caracteristic sape. Flexibility wit respect to te support of B means tat we can select an arbitrary region, convex or non-convex wit smoot or nonsmoot boundary, and wic can be aligned to any feature on te surface. A simple way to let te user define tis region is by letting im draw directly on te surface eiter te outline of te region or te complete region (wit some painting tool). Te caracteristic sape of te basis function B is most intuitively defined by terms like smootness, stiffness, or fullness. Here, smootness is a property tat describes ow te deformed part of te surface connects wit te unmodified part, stiffness or tension describes ow te curvature is distributed (equally distributed vs. clustered near te constraints), and fullness rates te relation between eigt and volume of te basis function (pointed vs. blobby). To control stiffness and smootness, we let te user select eiter te interior of te support region or (a segment) of its boundary and ten provide a simple slider to increase or decrease te respective parameter. Even if te matematical meaning of tese terms migt not be obvious for te non-expert user, te visual feedback wen moving te slider still allows im to quickly set te corresponding scalar parameters according to is design intend. In order to map te control of te modification to a 9-dof manipulator object, te user selects anoter region, te andle region, in te interior of te support region (cf. Fig. 1, left). Te manipulator is ten rigidly attaced to tis surface patc and ence moving te manipulator moves te surface patc accordingly. Te remaining part of te surface, i.e., support region minus andle region is supposed to smootly bend according to te translation, rotation and scaling of te andle region. Te fullness of te basis function B can ence be controlled by te size and te sape of te andle region. Te modeling metapor as we described it so far, i.e. te definition of an abstract basis function B wic is ten controlled by a simple manipulator object, provides a flexible tool for freeform sape editing. We can easily integrate tis metapor into a multiresolution modeling framework in order to preserve te local detail information of an object wen applying a global modification. For tis we need a decomposition operator wic separates te ig-frequency detail from te low-frequency global sape. Te freeform modeling tecnique is ten applied to te low-frequency component and finally te detail information is added back to te modified surface by a reconstruction operator. Te multiresolution decomposition and reconstruction can be idden from te user suc tat e seems to interact wit te detailed surface wile te frequency of te modification is controlled by te size of te support region.

Figure 2: Te order k of te energy functional defines te stiffness of te surface in te support region and te maximum smootness C k 1 of te boundary conditions. From left to rigt: membrane surface (k = 1), tin-plate surface (k = 2), minimal curvature variation (k = 3). 3 Te matematical realization Te matematical tecniques tat we need in order to implement te design metapor described in te last section ave to be flexible enoug to allow for arbitrary support and caracteristics but tey also ave to be efficient enoug to give real-time responses wen te user moves te manipulator. Smoot deformation of a surface wit respect to boundary conditions is most elegantly modelled by an energy minimization principle [Moreton and Sequin 1992; Welc and Witkin 1992; Kobbelt et al. 1998; Du and Qin 2000]. Te surface is assumed to beave like a pysical skin wic stretces and bends as forces are acting on it. Matematically tis beavior can be captured by an energy functional wic penalizes stretc or bending. Ten te optimal surface is te one tat minimizes tis energy wile satisfying all te prescribed boundary conditions. Te advantage of tis formulation is tat it allows us to take arbitrary boundary conditions into account and te optimal solution is known to ave certain smootness properties. Wen canging te boundary conditions, te optimal surface canges accordingly and tis is wy we call tis approac boundary constraint modeling (BCM). An alternative approac wic mimics boundary constraint modeling is to compute a blending function tat propagates te constraints into te interior of te support region [Sing and Fiume 1998; Bendels and Klein 2003; Pauly et al. 2003]. Wile tese metods are extremely efficient, tey usually do not provide te full generality since tey often assume a certain topology or sape of te support region. Linear system derivation For efficiency reasons te energy functionals tat are used most often are quadratic functionals wit te generic form [Kobbelt 1997] E k (S) = F k (S u...u,s u...uv,...,s v...v ) (2) were te expressions S stand for te partial derivatives of order k wit respect to a surface parametrization S : Ω IR 3 wic is locally as close as possible to isometric. In order to actually compute te solution to te above optimization problem one usually applies variational calculus to derive te corresponding Euler-Lagrange equation wic caracterizes te minimizers of (2). For te most common quadratic energy functionals, te resulting linear differential equation as te form k S(x) = 0, j S(x) = b j (x), x Ω \ δω x δω, j < k (3) were is te Laplace operator and te boundary constraints b j of order j < k on δω imply a non-trivial solution. For k = 1 tis equation caracterizes membrane surfaces wic minimize surface area, for k = 2 it caracterizes tin plate surfaces wic minimize surface bending and for k = 3 we obtain surfaces tat minimize te variation of linearized curvature (cf. Fig. 2). Higer order equations are usually not recommended because of numerical instabilities. Te types of boundary conditions tat we can impose on (3) can be up to C k 1. Hence for k 2 we can coose between inged boundaries (C 0, j = 0) and clamped boundaries (C 1, j = 1). Moreover we can continuously blend boundary conditions between C 0 and C k 1 wit a smootness parameter c(p) [0,k 1] for constrained boundary vertices p by modifying te recursive definition of te iger order Laplacian from [Kobbelt et al. 1998] to be ( ) k (p) := λ k (p) := λ k 1 (p) k 1 (p) 1, c(p) > k c(p) k, k 1 c k 0, c(p) < k 1 Since we want to use a triangle mes as te underlying surface representation, we ave to discretize te Laplace operator [Desbrun et al. 1999; Meyer et al. 2003] by (p i ) := 2 ( )( ) A(p i ) cotαi j + cotβ i j p j p i, (4) p j N(p i ) were α i j = (p i, p j 1, p j ) and β i j = (p i, p j+1, p j ) for a vertex p i and its one-ring neigbors p j and A(p i ) denotes te Voronoi area around te vertex p i. By tis (3) becomes a sparse linear system k 0 I F+H p f = 0 f., (5) were p = (p 1,..., p P ) is te vector of free vertices in te interior of te support region, f = ( f 1,..., f F ) are te fixed vertices outside te support region and = ( 1,..., H ) are te vertices inside te andle region. Tese sets of vertices correspond to te blue, gray and green surface regions sown in Fig. 1, respectively. Since f and are fixed, tey impose te boundary conditions on te system. Notice tat only k + 1 rings of fixed vertices are used to prescribe C k boundary constraints. For te sake of simplicity we combined te optimality conditions and te boundary conditions into one equation. In te following we refer to te matrix in (5) as L. Wen te user moves te andle region, te vertices in cange teir position and provide a new rigt and side for te linear system. By solving (5) again we ence compute te vertex positions in p as a linear function of.

Anisotropic bending If we use te standard discretization (4) of te Laplace operator ten te resulting optimal surface bends isotropically even if te support region is anisotropic. Since te sape of te andle and support regions are considered as design parameters wen defining te basis function for a particular modification, we would rater like te resulting surface to better adapt its bending beavior to te boundary conditions. As sown in Fig. 3 te basis function looks more natural if te impact of te andle region is propagated troug te support region in suc a way tat its iso-contours it te outer boundary everywere wit approximately te same slope. Tis can be acieved by discretizing te Laplace operator (4) not wit respect to te mes itself (Laplace-Beltrami) but rater wit respect to a special parametrization tat factors out te anisotropy. First we compute a conformal parametrization for te support region [Lévy et al. 2002; Ray and Levy 2003]. Tis yields a planar triangulation wit te same connectivity as te support region. Ten we apply a principal axis transform and scale tis planar triangulation along its principal axes suc tat its diameter is approximately te same in eac direction. On tis scaled triangulation we finally compute te weigts for te Laplace operator tat we ten use in (5). Fig. 3 sows te effect of minimizing tis anisotropic energy functional. During interactive sape editing we are using a 9-dof manipulator wic provides an intuitive interface to control an affine map T tat is applied to te andle region. If we pick four affinely independent vertices a, b, c, and d from te andle region ten tese define an affine frame and tere exists a matrix Q IR H 4 of affine combinations suc tat = Q [a,b,c,d] T. Due to affine invariance, applying an affine map T (controlled by te manipulator) to te andle vertices, is ten equivalent to applying T to te affine frame, i.e. T (Q[a,b,c,d] T ) = Q T ([a,b,c,d] T ). As a consequence we can rewrite te non-constant term in (6) as p f + L 1 0 0 [a,b,c,d] T. 0 Q Tis owever means tat we only ave to solve te system (5) in a pre-processing step for 7 different rigt and sides: te tree columns of [0,f,0] T and te four columns of [0,0,Q] T. Te latter allows us to precompute te basis function matrix B = L 1 [0,0,Q] T. Ten for every frame we simply apply te manipulator transformation to te four andle points a, b, c, and d and add te current displacement vectors B[a,b,c,d] T to te constant part in (6). 4 Results Figure 3: Isotropic (left) and anisotropic basis functions (rigt) wit te corresponding parameter domains over wic te Laplace operator is discretized. Precomputed basis functions Te tecnique as we described it so far requires to solve a linear system for te free vertex positions wenever te manipulator (and ence te andle region) moves. Altoug we can use a igly efficient multi-grid solver for tis task we still do not acieve a sufficiently ig frame rate, especially wen minimum curvature variation surfaces (k = 3) are computed and te number of vertices in te support region is on te order of 10 4 or iger. By precomputing a special set of basis functions tat directly correspond to te degrees of freedom of te manipulator, we can significantly reduce te per-frame computing costs. Since te solution of (5) can be expressed explicitly in terms of te inverse matrix L 1, tis set of basis functions is represented by (a combination of) column vectors of L 1. A similar tecnique was used in te pysically-based modeling system ArtDefo [James and Pai 1999], were also a set of column vectors of an inverse matrix is precomputed in order to speed up te surface updating. First we observe tat te explicit solution of (5) is p f + L 1 0 0 0 (6) wit te first term on te rigt and side being constant and te second term depending on te vertices in te andle region. We integrated our freeform modeling metapor into a multiresolution mes editing framework like te one described in [Kobbelt et al. 1998]. Te decomposition and editing operators are bot based on BCM and te representation of ig-frequency detail is implemented in terms of normal displacement vectors [Kobbelt et al. 1999]. A real world example is sown in Fig. 5, were te sillboard of a car is to be lowered. Exploiting te flexibility provided by continuous boundary smootness avoids te generation of an unwanted point of inflection along te feature line and te anisotropic Laplace discretization propagates te displacement naturally over te support region. Features are preserved due to multiresolution decomposition and reconstruction (idden from te user). Notice tat multiple independent andle regions eac controlled by teir own manipulator object are no problem for te setup described Figure 4: Example of a non-disc saped modification using multiple andles. Te left cap (green) is defined as a andle component tat is not to be moved. Te actual modification is done by defining additional andle components as rings (green).

Figure 5: Multiresolution modification of te sillboard using flexible boundary conditions and anisotropic Laplacian: An unwanted point of inflection (center left) is avoided by reducing te smootness constraint near te andle region to C 0 continuity. Switcing from te isotropic discretization of te Laplace operator (center rigt) to te anisotropic one finally leads to te intended natural displacement propagation. in tis paper. All we ave to do is to split into several components and precompute te corresponding four basis functions for eac of tem. By tis it is even possible to generate modifications wit a support region tat is not topologically equivalent to a disk. We simply exclude some regions from te deformable area by labelling tem as special andle regions tat cannot be moved (cf. Fig. 4). More complex support regions can be used if we restrict to te isotropic Laplacian because tis avoids te parametrization step (cf. Sect. 3). Te last example sows a complex modification of a car s ood (cf. Fig. 6, 250k triangles). Here we use multiple andle regions placed at te weel ouses and te grill, enabling us to stretc te ood wile keeping te weel ouses circular. Te support region of tis modification contains 35k vertices, te complete precomputation (multigrid ierarcy, basis functions, multiresolution ierarcy) took less tan 15s. Te actual surface editing can be done wit 12 fps, were only 25ms are required to compute te BCM surface and te remaining time is used for detail reconstruction and rendering. In an industrial evaluation te presented modeling system proved to be bot sufficiently flexible as well as very intuitive, enabling also te non-experts to perform teir desired sape modifications. Due to te precomputed basis functions, deformations even on complex models could be performed in real-time. Figure 6: Stretcing te ood using multiple independent andle components. 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