Approximation with Active B-spline Curves and Surfaces

Transcription

1 Approximation with Active B-spline Curves an Surfaces Helmut Pottmann, Stefan Leopolseer, Michael Hofer Institute of Geometry Vienna University of Technology Wiener Hauptstr. 8 10, Vienna, Austria Abstract An active contour moel for parametric curve an surface approximation is presente. The active curve or surface aapts to the moel shape to be approximate in an optimization algorithm. The quasi-newton optimization proceure in each iteration step minimizes a quaratic function which is built up with help of local quaratic approximants of the square istance function of the moel shape an an internal energy which has a smoothing an regularization effect. The approach completely avois the parametrization problem. We also show how to use a similar strategy for the solution of variational problems for curves on surfaces. Examples are the geoesic path connecting two points on a surface an interpolating or approximating spline curves on surfaces. Finally we inicate how the latter topic leas to the variational esign of smooth motions which interpolate or approximate given positions. 1. Introuction Curve an surface approximation is a central topic of Computer Aie Geometric Design an there is a large boy of literature ealing with it. It is beyon the scope of this paper to give an overview of the many available algorithms. Thus we will just point to the work which is in close connection with our algorithms an even there we will mainly focus at the literature which comes not irectly from the CAGD community, but from areas like Computer Vision an applications of partial ifferential equations. At first, let us consier approximation with parametric curves or surfaces, which are usually expresse by means of B-splines [23, 44]. The usual approach uses a least squares formulation with a regularization term that expresses the fairness of the final result (see e.g. [23, 24]). The principle, explaine for surfaces, is as follows. Let p be the given ata points or samples on a given moel surface. We are looking for an approximating B-spline surface or another parametric surface with a representation of the form x! (1) The basis functions are usually polynomial, piecewise polynomial, or piecewise rational. We assume that the functions are given or precompute from the input ata; thus weights or knots are alreay etermine. Then one estimates the surface parameters " # $ % of those points x on the approximant which shoul be close to the corresponing ata points p. The approximant is compute as minimizer of a functional & x *) p -,/. &*0 (2) ( The first part is a quaratic function in the unknown control points, ( x *) p 1 2 ) p 43 The secon part &50 in (2) is a smoothing term (see e.g. [6]). A frequently use example is the simplifie thin plate energy, a quaratic function in the secon partial erivatives, &* x :;:<,/= x :;>?, x >"> A@B@C (3) It is also quaratic in the unknowns an thus the minimization of & is the minimization of a quaratic function an amounts to the solution of a linear system of equations. It is a ifficult task to estimate the parameters. This parameter choice largely effects the result (see e.g. [33] an the references therein). Therefore, iterative parameter correction proceures have been suggeste [23]. The final approximant shoul exhibit error vectors x D) p which are orthogonal to the approximating surface x.

2 Fortunately, there is a way to overcome the parameterization problem. The iea comes from Computer Vision an Image Processing, where so-calle active contour moels are use in a variety of applications (see e.g. [3]). The origin of this technique is a paper by Kass et al. [27], where a variational formulation of parametric curves, coine snakes, is use for etecting contours in images. An elegant formulation of curve an surface reconstruction an segmentation problems is the concept of geoesic active contours [8, 9, 49]. There, the curve to be reconstructe, e.g. from a meical image, is foun as geoesic in a Riemannian space whose metric is erive from the input (image). Analogously, surface reconstruction is reformulate as minimal surface computation in a Riemannian space. The literature on this highly interesting topic is rapily increasing. A goo overview of the methos is foun in the book by G. Sapiro [49]. Instea of a parametric representation of a planar curve or a surface, one may use an implicit form as zero set (level set) of a bivariate/trivariate function [5]. A major avantage of this approach is the simplicity with which one can moel arbitrary topologies. Often, the function which the level sets are taken from is assume to be polynomial. This yiels algebraic surfaces. For algebraic surface fitting, especially the fitting of algebraic tensor-prouct spline surfaces to scattere ata, we refer the reaer to the recent paper of Jüttler an Felis [26]. Some approaches to surface reconstruction via implicit surfaces efine a signe istance function to the ata set an enote the zero contour of the signe istance function as the reconstructe implicit surface [1, 4]. The formulation of active contour moels via level sets goes back to Osher an Sethian [42]. The level set metho [41, 53] has been successfully applie to the solution of a variety of problems, e.g. for segmentation an analysis of meical images [36]. There are also several extensions to surfaces. An application to the surface fitting problem to scattere ata sets has been given by Zhao et al. [61, 62]. For further work on the level set metho which is relate to surface reconstruction, see [17, 18, 60]. In the present investigation, we evelop further a concept for parametric curve an surface fitting which has recently been evelope by the authors [46]. We assume as input a rather ense set of points or even a given curve or surface representation. We refer to it as moel shape. The first situation arises for example when we are processing ata of moern 3D scanners, or if we woul like to fit a surface to a ense mesh. A curve or surface as input may arise when the representation of is not in the esire parametric representation. For example, we may have as input an implicit representation or a B-spline curve/surface with a too high egree or too many knots. Thus, we are also contributing to the problems of spline conversion, egree reuction an knot removal [23, 44]. Moreover, surfaces erive from a given surface in various ways might not be in the esire form. A well-known example are offset surfaces of NURBS surfaces, which are usually not NURBS surfaces themselves. The present approach is very well suite for offset surface approximation [35]. The basic iea of our approach is the application of a Newton type algorithm for the solution of the nonlinear problem of curve an surface approximation. Thus we use an iterative metho, i.e., an active curve or surface which aapts to the final moel shape. Moreover, for a Newton algorithm we require local quaratic (Taylor) approximants of the function to be minimize. Since we want to minimize the sum of square istances to the curve or surface to be approximate, we guie the shape change of the active curve/surface with help of local quaratic approximants of the square istance of the moel shape. The local approximants help to move the active curve/surface x to lower levels without having to specify which point x shoul move to which point of the moel shape. In this way we are avoiing the parametrization problem. Our metho is applicable to approximation with any linear curve or surface scheme, even subivision curves an surfaces. In this paper we restrict ourselves to a B-spline representation. The organization of our paper is as follows. In section 2, we review our recent work on local quaratic approximants of the square istance of curves an surfaces [45], since it forms the basis for the new approximation technique. In section 3, the new concept is outline for curve approximation an applie to egree reuction an offset approximation. In section 4, we escribe surface approximation an present some examples. Section 5 eals with curves on surfaces. It is shown how to compute the shortest path (geoesic) between two points on a surface an how to compute spline curves on surfaces. Moreover, we escribe how the new technique naturally leas to a variational formulation for the computation of smooth motions, which interpolate or appoximate a given set of positions. Finally, in section 6, we escribe possible extensions an inicate irections for future research. 2. Local quaratic approximants of the square istance function of curves an surfaces The algorithmic concept we are proposing heavily relies on local quaratic approximants of the square istance function of the curve/surface or point clou to which we woul like to fit a B-spline curve/surface. Let us first consier the istance function of a curve or surface, which assigns to each point p the shortest istance of p to. A variety of contributions eals with the 2

3 computation of this function; in many cases this computation aims towars the singular set of the function, i.e., towars points where the function is not smooth since those points lie on the meial axis (or skeleton) of the input shape. Early work on the geometry of the istance function comes from the classical geometric literature of the 19th century. One looks at its graph surface, which consists of evelopable surfaces of constant slope an applies results of classical ifferential geometry, line an sphere geometry (for a moern presentation, see e.g. [47]). For more recent work on istance transforms an the closely relate meial axis transform, see [10, 11, 29, 40, 51, 52, 56]. The istance function is also the (viscosity) solution of the so-calle eikonal equation. Its numerical computation is not trivial because the eikonal equation is a hyperbolic equation an an initially smooth front may evelop singularities (shocks) as it propagates. Precisely the latter belong to the meial axis an are of particular interest. The computation of viscosity solutions with the level set metho of Osher an Sethian [42, 41] prove to be a very powerful metho (see e.g. [49, 54, 53]). For our approach to surface approximation, not the istance function itself but the square istance function is important. We are especially intereste in local quaratic approximants of that function. For a erivation an proofs of the following results we refer the reaer to [45]. For a better unerstaning, we first present local quaratic approximants of planar curves an then move to surfaces an space curves. PSfrag replacements 2.1. Local quaratic approximants of the square istance function of a planar curve In a Eucliean plane we consier a curve c with parameterization ". The Frenet frame at a curve point c consists of the unit tangent vector e c c an its normal vector e. Those two vectors form a righthane orthonormal basis in the plane. We are intereste in the square istance which assigns to each point p in the square of its shortest istance to the curve c. In the following we give the formula for a local quaratic (Taylor) approximant of the square istance function with respect to a local Frenet coorinate system. Note that the square istance function is not smooth in points of the meial axis. Thus, we will not compute local quaratic Taylor approximants for points of the meial axis. Consier a point p in whose coorinates with respect to the Frenet frame at the normal footpoint c The curvature center k at c ; has coorinates. Here, is the inverse curvature an thus has the same sign as the curvature, which epens on the orientation of the curve (see Fig. 1). 3 c Figure 1. k p e c Graph of the square istance to a planar curve c. With respect & to the Frenet frame, the secon orer Taylor approximant of the square istance function@ is given by ) (4) For a erivation of this result an a iscussion of the ifferent types of & we refer the reaer to [45]. We just point out that the Taylor approximants may be inefinite. As shown in [45], we can use as appropriate nonnegative quaratic approximant e e, (5) are taken as positive. Equ. (5) is not vali for points beyon the curvature centers, but they will not arise anyway when we consier global istances Local quaratic approximants of the square istance function of a surface Consier an oriente surface s with a unit normal vector fiel n e. At each point s, we have a local right-hane Cartesian system whose first two vectors e e etermine the principal curvature irections. The latter are not uniquely etermine at an umbilical point, but in that case we can take any two orthogonal tangent vectors e e. We will refer to the thereby efine frame as principal frame D. Let be the (signe) principal curvature to the principal curvature irection e, =, an let. Then the two principal curvature centers at the consiere surface point s are expresse in

4 @ &. The quaratic approximant is the following: as k Proposition 1 The secon orer Taylor approximant of the square istance function of a surface at the point is given by @ ), (6) when coorinates are given with respect to the principal frame at s. Let us look at two important special cases. For@8 we obtain & ; This means that the secon orer approximant of@ at a surface point p is the same for the surface s an for its tangent plane at p. Thus, if we are close to the surface, the square istance function from the tangent plane at the footpoint is a very goo approximant. At least at first sight it is surprising that the tangent plane, which is just a first orer approximant, yiels a secon orer approximant when we are consiering the square istance to surface an tangent plane, respectively. In the we obtain & 4-,, This is the square istance from the footpoint on the surface. We see that istances from footpoints yiel goo approximations if we are in the far fiel of the surface s. In the near fiel it is much better to use other local quaratic approximants. The simplest one is the square istance from the tangent plane at the footpoint. Analogous to the curve case, we may have an inefinite Taylor approximant. Then we can use as appropriate nonnegative quaratic approximant @,, (7) Here,@ are taken as positive. Equ. (7) is not vali for points beyon the principal curvature centers. Such points o not arise anyway when we consier global istances Local quaratic approximants of the square istance function to a space curve Given a point p in, the shortest istance to a space curve c occurs along a normal of the curve or at a bounary point of the curve. The latter case is trivial an thus we exclue it from our iscussion. At the normal footpoint c we efine a Cartesian coorinate system with e as tangent vector an e in irection of the vector p ) c. This canonical frame can be viewe as limit case of the principal frame for surfaces, when interpreting the curve as pipe surface with vanishing raius. By this limit process, we can also show the following result. Proposition 2 The secon orer Taylor approximant of the square istance function of a space curve c at the point is given by ),, (8) where coorinates are given with respect to the canonical frame. Here, are the coorinates of the intersection point of the curvature axis of c at the footpoint c ; with the perpenicular line pc ; from p to c. As expecte, with@ we obtain & ;, This means that the secon orer approximant at a curve point is the same for the curve c an for its tangent. 3. Approximation with an active curve in the square istance fiel Our approach to curve approximation has as input a moel shape. This can be a sufficiently ense point set along a curve. It can also be a curve in or which is given in any mathematical representation. For the sake of simplicity in our explanation, we confine ourselves at first to planar curves, but the concept works in higher imensions as well. From the moel shape, we compute for example by means of secon orer Taylor approximants local quaratic approximants of the square istance of. Thus, for any point p, we have & a way to compute such a local quaratic approximant p. In its simplest formulation, the metho outline below further assumes that & p is a nonnegative quaratic function. The active curve moel we are using is of the following nature. It is governe by control points, an there is a linear relation which computes from the control point set a larger set of curve points s 4

5 or points on a refine moel. For example, we may have a Bézier or B-spline curve of the form c - 2 (9) We now evaluate the curve at parameters an get curve points s c " We coul also work with a subivision curve: The points can be the vertices of a coarse subivision level an the points s can be vertices of a refine moel, after application of a few steps of the subivision rule [59]. The set of points s must be large enough to capture the shape of the active curve well. In the following we use the notation s (10) to express the linear computation of s from the control points. The key iea is to iteratively change the input control points so that the active curve eforms towars the moel shape. We o not use the graient flow in the square istance fiel but in each step solve a minimization problem which ensures that we quickly move to lower levels of the function@. The metho now procees in the following steps. 1. Initialize the active curve an etermine the bounary conitions. This requires the computation of an initial set of control points <, the proper treatment of bounaries (such as fixing en points of a curve segment) an the avoiance of moel PSfrag shrinking replacements uring the following steps. More etails on that are escribe in section Repeately apply the following steps a. c. until the approximation error or change in the approximation error falls below a preefine threshol:, the active curve point s an a local quaratic approximant & & s of the square istance function of the moel shape at the point s. This has to be a nonnegative quaratic function, & x x. - 5 a. With the current control points, compute, for b. Compute isplacement vectors c for the control points by minimizing the functional & &, c, c,. &*0 (11) Thus, our goal is that the new curve points, c, c " s which are linear combinations of the new control points, c, are closer to the moel shape than the ol active curve points s. The functional &*0 is a smoothing functional which shall be quaratic in &50 the control points of the active curve. Thus, is a, quaratic function in the new control points c, an also quaratic in the unknowns c. We see that step b. requires & the minimization of a quaratic function in the isplacement vectors c of the control points. This amounts to the solution of a linear system of equations. Note that the factor. in Equ. (11) etermines the influence of the smoothing term &50 in the optimization. In all our examples we starte with a high value of. in the first iteration an let the influence of the smoothing term fae out in the later iterations (. ). In this way unwante folovers or loops of the active curve can be avoie in the beginning of the eformation process. As the active curve comes closer to the moel shape the smoothing term gets less important. c. With c from the previous step, we replace the control points by, c. c s s c c Figure 2. One step in the curve approximation proceure. The curve is approximate by a B-spline curve. Fig. 2 illustrates the algorithm. The moel shape is a curve which is to be approximate by a B-spline curve. 5

6 Fig. 2 shows an initial position of the B-spline curve c, with control points, an the upate B-spline curve, with control points, after one iteration step. For one of the sample points s c the local quaratic approximant & of the square istance function is inicate by three of its level sets, which are concentric ellipses. Let us briefly iscuss the choice of the initial curve. In all our tests it turne out to be not critical. Apparently the choice of the initial curve c is safe if c lies close enough to the moel curve so as not to intersect the meial axis of. This implies that for each point of c there is a unique closest point on. The square istance function is smooth in a tubular neighborhoo of, which inclues c. In such a case, the initial shape rapily eforms towars the final solution. However, it is not guarantee that the solution is a global optimum. The lanscape of the function & to be minimize nees to be investigate in PSfrag more etail replacements in future research. We nee to get more information about the typical istribution of local minima. For a first result in this irection, see [12]. s c s s g replacements Figure 3. s Active curve flow of the B-spline curve c towars target curve. Even if the initial curve intersects the meial axis of, we obtain goo results in most cases. For an example see Fig. 3 an Fig. 4. In both figures the moel shape is the same curve (bol soli line), but ifferent initial positions c (bol ashe line) of the active curve c have been chosen. Both of the initial curves are very rough approximants s s s s c Figure 4. Active curve flow of the B-spline curve c towars target curve. of an have intersection points with the meial axis of (which is not plotte). Nevertheless, our iterative algorithm converges to a practical solution in both cases. The final position c of the approximating curve c is obtaine after eight iterations (ashe curve near ) an its eviation from is harly perceptible in the figures. The two solutions of Fig. 3 an 4 iffer from each other, which can be best observe by means of the control polygons (thin ashe lines) of the final approximating curve c. The fact, that we obtain ifferent approximating curves is to be expecte because the final result epens on the initial position c of the active B-spline curve c. In general, we have to expect a certain number of local minima which are ifficult to avoi. In Fig. 3 an 4 also the paths of several sample points s c are epicte, from the initial position s to the final position s. It can be observe clearly that the sample points s of the active curve are not simply move towars their closest point on. Many of the sample points move tangentially to, especially in later steps of the iterative algorithm. 6

7 6 eplacements 3.1. Degree reuction PSfrag replacements As a first example of an application of this metho we eal with the problem of approximate egree reuction. There are several contributions on this topic, see e.g. [12, 15, 16]. For Bézier curves, the best egree reuction by one egree with respect to -norm ( / = ) has been erive by Eck [16]. In that paper explicit solutions are given, also for the egree reuction with enpoint interpolation conitions. See Fig. 5 for the egree reuction of a Bézier curve c of egree six by a Bézier curve c of egree four. The control points are enote by an, respectively. In this example the -norm was use, i.e., c ) c c *) was minimize with bounary constraints (enpoint interpolation). Note that because of the reuction of the egree by two, Eck s best approximation scheme ha to be applie twice (egree ) an we obtain only a suboptimal solution. Figure 5. Best egree reuction accoring to Eck [16]. Bézier curve c of egree 6 (soli line) is approximate by Bézier curve c of egree 4 (ashe line). The general curve approximation scheme outline in the present paper oes not use the information that c is a Bézier curve, or any other aitional information. If we are intereste in the istance between curves as sets (not in the ifference as functions), our algorithm yiels much better results than the metho of [16]. Fig. 6 an Fig. 7 show the result of our active curve approximation scheme applie to the example of Fig. 5. In Fig. 6 the control points of the approximating Bézier curve of egree four have been chosen to lie on the target c c PSfrag replacements c c Figure 6. Bézier curve c of egree 6 (soli line) an initial position c of approximating Bézier curve of egree 4 (ashe line). curve c. In general this yiels a sufficiently goo initial position c of the active Bézier curve c. Fig. 7 shows the final result after just five iterations of the active curve flow. Let us efine the istance of the point set c to the point set c c c c *) c5 " (12) where c5 is the point of c closest to c. Compare to the result of Eck s approximation the c c is reuce from (Fig. 5) to (Fig. 7), i.e., by a factor of Similar results have been obtaine for several other examples. c c Figure 7. Bézier curve c of egree 6 (soli line) an final position of approximating Bézier curve c of egree 4 (ashe line). 7

8 3.2. Offset curve approximation The present concept of an active curve uner the influence of the square istance function of a moel curve is also well suite for offset curve approximation. This is a wiely investigate topic since offset curves have many applications (NC machining,... ), see e.g. [23, 35] for surveys on offset curves. For a point p, we have Eq. (4) to escribe a quaratic approximant of the square istance function of a curve c, expresse in the principal frame e e at the normal footpoint., Let us consier c s one-sie offset curve c c e at istance. In corresponing points c an c the principal irections e % =, an the curvature center k are the same. If points have coorinates " k with respect to the principal frame at c, then these points have coorinates p ), an k ) with respect to the principal coorinate frame at c. The secon orer Taylor approximant of the square istance function of the offset curve c at a point p is expresse (with respect to the principal frame at the normal footpoint) via & ) With the quaratic approximant (4) of the square istance from c it is therefore simple to erive the corresponing quaratic approximant for its offset curve c at istance. Fig. 8 shows an example where a close curve is given as input shape. Two of its inner an three of its outer offset curves have been approximate by active cubic B-spline curves. These offset curve approximations an their respective control polygons are epicte in Fig. 8. If an offset curve intersects the meial axis of the original shape, it evelops a corner point, i.e., a tangent iscontinuity, at this intersection point. This situation arises in our example for the outer two of the offset curves (see Fig. 9 for a closeup). In the implementation of the example of Fig. 8 an 9 the corner points have not been treate in any particular way. The compute offset curves just evelop points of very high curvature where the corner points shoul lie. It is straightforwar to etect such points of high curvature an enforce a tangent iscontinuity via multiple insertion of an appropriate knot value. In this way corner points can be moelle exactly. Note that our active curve approach as presente here uses parametric curves an oes not allow topological changes. Thus it is not possible to approximate those inner offsets of our moel curve which consist of two close curves. Detection of a topology change an the corresponing aaption of the active curve is possible [13]. Figure 8. Close planar curve an five of its offset curves approximate by active B-spline curves Approximation of a helix segment by a B-spline curve For the illustration of space curve approximation by active B-spline curves we have chosen a helix segment as moel shape, see Fig. 10. A helix has constant curvature an torsion an possesses many applications in computer aie geometric esign, kinematics, an computer graphics. Despite its geometric simplicity a helix segment cannot be exactly represente as a polynomial or rational curve. Thus there are many contributions on the approximation of a helix segment by (rational) Bézier or B-spline curves (see e.g. [38, 50]). Figure 9. Detail of Fig. 8. 8

9 Fig. 10 shows the approximation of a helix segment by an active B-spline curve compose of four cubic segments. The initial position of the active curve was chosen as the straight line connecting the enpoints of the helix segment. The eviations of the approximating B-spline curve from the helix are epicte in Fig. 10, exaggerate by a factor of Figure 10. Approximation of a helix with a cubic B-spline, an visualization of error vectors of the approximation, exaggerate by a factor of Approximation with an active surface Our approach to curve approximation has a straightforwar extension to surface approximation. The active surface moel we are using shall be of the form (1), so that surface points to given parameter values epen on the control points in a linear way. Thus, we coul also use a subivision surface. The quaratic function we are minimizing in each iteration step again consists of a istance part, set up via local quaratic approximants of the square istance function, an a regularization term. Instea of repeating the algorithm, let us have a look at some etails, improvements, an refinements, which are important for a successful implementation of the propose metho. 1. There are similar consierations about the choice of the initial shape as in the curve curve case. One ifference concerns topology. For curves, even in parametric representation, the etection of a topology change an the corresponing aaption of the active curve is possible [13]. For surface approximation, the initial shape must alreay exhibit the correct topology. A change in topology is harly possible with a parametric surface. It is simple, however, if one views the active surface as level set of a trivariate function; this is the approach taken in the level set metho [53] which has been applie to surface approximation by Zhao et al. [61, 62]. 2. One has to impose appropriate bounary conitions. For example, we may want to fix the vertices of a surface patch or want to approximate bounary curves of the moel shape. One way to o this is to apply the curve analogue of the present metho in a first step; then we keep corresponing control points fixe in the surface approximation proceure. However, it can be possible to reach an overall better surface quality by sacrificing some accuracy at the bounary. In this case it is better to a the functional for bounary approximation as a penalty term to & of equation (11). An example is shown in Fig With totally unrestricte flow, the active surface may shrink to a single point of the moel surface an then in a trivial way yiel zero approximation error. Strategies for shrinking avoiance epen on the special situation, an usually involve appropriate bounary conitions which avoi shrinking, or an appropriate quaratic penalty function & ae to & in equation (11). For a close surface, we can start with an initial shape which lies entirely outsie the moel shape. The active surface then eflates towars the moel shape, if we forbi that it enters the interior of. 4. If we have an active B-spline surface or another parametric surface an get moel points s by evaluation, it is not necessary to keep the parameter values, at which we evaluate, fixe. An aaptive evaluation which guarantees a nearly uniform istribution over the active surface, or one which emphasizes especially important regions with help of more moel points, will be useful. Moreover, we can introuce further knots an thus more control points uring the algorithm if the esire accuracy cannot be achieve with a coarser moel. One sees that the metho naturally supports a multiresultion moeling strategy. 5. The assumption of nonnegativity of the approximants & of the square istance function of can be avoie if the change of control points (an thus moel points) is restricte. & This can be achieve by aing a term to which expresses the istance of the new 9

10 control points to the ol ones, e.g. c. To make sure that the corresponing moel points s o not move outsie the region where their respective local quaratic approximants are positive, we have to solve a constraine minimization problem. Such algorithms are known in optimization as trust regions algorithms [28]. 6. For &*0 we can use any quaratic smoothing functional, which may change in each iteration step. Thus we can also buil Greiner s metho for the minimization of nonlinear fairness functionals [19] into our surface approximation technique Approximation with B-spline surfaces In the following we give an example for the approximation of a given surface by a B-spline surface. In this example the initial position of the active B-spline surface patch (ark gray) is chosen as the bilinear patch connecting the bounary vertices of the moel surface (light gray), see Fig. 11. The result of our algorithm after only 7 iterations is shown in Fig. 12. As a bounary conition, the vertices of the moel patch have been fixe. The objective function that guies the active surface flow has been chosen as a weighte sum of the quaratic functionals for bounary an surface approximation together with a quaratic smoothing term. Figure 11. Moel shape (light gray) an initial position of approximating B-spline surface (ark gray). Figure 12. Moel shape (light gray) an final position of approximating B-spline surface (ark gray) with bounary curve approximation Approximation of offset surfaces Offset surface approximation is a wiely investigate topic, see e.g. [23, 35, 43]. It has been note in section 3.2 that our present concept of active curves is applicable to offset curve approximation. The same hols true for the approximation of offset surfaces by active surfaces. Let us consier s s one-sie offset surface s 8 s, e at istance. In corresponing points s an s the principal irections e =, an the two principal curvature centers are the same. If a point p has coorinates with respect to the principal frame at s, then this point has coorinates ) with respect to the principal coorinate frame at s. The secon orer Taylor approximant of the square istance function of the offset surface s at a point p is expresse (with respect to the principal frame at the normal footpoint) via & ), 5. Variational problems for curves on surfaces Let us now iscuss the computation of surface curves which are solutions to certain variational problems. The general iea here is to use the embeing space for the evolution of the active contour. In an iterative proceure, we 10

11 solve the given variational problem & for a curve on a surface by minimizing a functional which is compose &0 of two essential components. The first component is the functional to be minimize within the variational problem. For example, in case of a geoesic curve, it is the total arc length. The secon component expresses the istance of the curve to the surface with help of local quaratic approximants &*0 of the square istance function to. Thus, as long as is quaratic in the unknowns, or can be replace by a quaratic function in the unknowns, we are within the framework iscusse before. We will present three examples, the thir of which concerning motion esign is a higher imensional one an is just briefly outline. It will be escribe in more etail in a subsequent paper. In our algorithms, the final curve will be given as a parametric curve c which lies very close to the surface. Thus, our algorithm computes, within the esire accuracy, the shape of the curve. This may be sufficient. If one wishes to have the preimage curve in the parameter omain of the surface, we can project a sufficient number of points of c onto, compute their preimages in the parameter omain an then fit a B-spline curve c to the resulting point set. The image of c uner the surface parameterization then is the final result Geoesics The computation of the shortest path (geoesic) between two points on a surface is a classical one, an also of importance in various problems of CAD an geometric moeling (see e.g. [43]). Typically, one uses the secon orer ifferential equations for geoesics known from ifferential geometry, an then solves the corresponing bounary value problem. Bounary value problems, however, are not so simple to solve an numerical strategies, such as shooting methos, have to be employe [43]. In Computer Vision an Image Processing, geoesics have been consiere not only on surfaces in 3-space, but also on general Riemannian manifols. The metric then epens on the application, e.g. on the texture in an image. Recent algorithms for the fast computation of the istance function from a point p in a manifol, a tringulation or even a point clou [30, 31, 32, 37], are use to compute the shortest path between p an any other point q in the manifol; one just follows the graient flow of the istance function. Clearly, this requires the computation of a bivariate function (at least in some neighborhoo of the expecte geoesic). Conceptually, this is similar to a shooting approach. Within our framework, we can procee as follows. Given two points p an q on a parametric surface, we take an initial shape c of an active contour, e.g., a B-spline curve representing the straight line segment between p an q, or a B-spline curve which approximates a surface curve from PSfrag replacements p to q. The curve c is evaluate at a sequence of parameters an we obtain points s c " & Now, nonnegative quaratic approximants of the square istance function of s. Displacement vectors c compute as minimizers of & are compute at the points for the control points are &, c, c,/. &*0 (13) As before, expresses the linear epenence of s on &*0 the control points. coul be taken as the arc length &*0 76 " (14) This woul not yiel a quaratic function in the unknown isplacement vectors c of the control points. The Cauchy- Schwartz inequality shows easily that minimization of &*0 76 (15) yiels precisely the geoesics, but parametrize with constant velocity (cf. [7], p. 307, or [39], p. 70). This functional now is quaratic in the unknowns c an thus our concept is fully applicable to the approximate computation of geoesics. q p q q Figure 13. Geoesic curves from p to several points q on a given surface. An example of our algorithm is given in Fig. 13. Several geoesic curves emanating from a surface point p to surface q q q q q 11

12 points q have been compute. For each of the geoesics the initial position of the active curve has been chosen as the straight line connecting &50 p an q. The function (13) has been minimize with from (15). It is important to note that our metho of minimizing (15) uner the constraint that c lies within a given tolerance to the given surface is a penalty function approach. We usually obtain a goo approximation of a geoesic, but we can improve it as follows: We replace c by a polygonal approximation. This polygon is now consiere active an moves such that its vertices remain on or close to an also a iscrete version of (14), (15), or any equivalent functional is minimize. One particular example is the following: We consier the polygon as a rope an exert equal forces at both ens. The forces in each segment will be of the same magnitue, an unless the vector sum of the two forces acting in a vertex is orthogonal to the surface, this vertex will be move tangentially to the surface. A simple iterative proceure will prouce a iscrete geoesic polygon, which fulfills a iscrete version of the ifferential equation of geoesics Spline curves on surfaces Spline curves in which interpolate or approximate a sequence of points are often compute as solutions of a variational problem. The most famous example is that of natural cubic splines which minimize the norm of the secon erivative, &*0 76 " (16) subject to the given interpolation conitions p PSfrag replacements c ". Other well-known examples are cubic smoothing splines, splines in tension an -splines [23]. We now iscuss interpolating or approximating spline curves c on a given surface. Thus, also the given points p lie on. Unlike in the classical case, the solution cannot be escribe anymore in an explicit way (see e.g. [57]). The solution concept is simple: we will work with an active curve from a space of curves all of which satisfy the given interpolation constraints. With curves of this space, we then minimize a function as in (13), however with &0 from (16) or with an &50 which appears in another variational spline formulation (smoothing spline, splines in tension,... ). We still have to escribe the choice of an appropriate space of splines. One may simply compute a cubic spline interpolating the points p at (estimate or given) parameter values. This spline c is completely etermine by this input an there woul be no flexibility anymore to move it towars the given surface. There are several simple ways to achieve more flexibility: 1. We may formally raise the egree of the segments. For example, we can raise the egree to five an express them in Hermite form. The unknowns are then the first an secon orer erivative vectors at the given points p. In the original position they are in such a special position that the spline curve is a cubic spline, but this will change uring the optimization algorithm. 2. We may split each segment of c between consecutive points p p into several segments (knot insertion). If we stay with cubics, it is convenient to use a B-spline representation. For such a B-spline curve between two input points, the en points are fixe (the given input points p p ). The control points next to the en points, epen linearly on the (unknown) first erivative vectors in the enpoints, an the remaining B-spline control points can move freely (are unknowns). For join at p, also the control points an of the B-spline epen linearly on the unknown first an secon erivative vectors at p p. 3. We can combine splitting an egree elevation. For an example of the construction of a spline curve on a surface see Fig. 14. The initial curve c is a natural cubic spline through the surface points p an oes not lie on the given surface. By knot insertion each cubic segment between p an p has been splitte into three cubic segments. The resulting free parameters have been use in the active curve flow of the cubic spline curve c towars the surface. p p p p c Figure 14. Initial position c an final position c of a cubic spline curve which lies very close to the surface an interpolates given surface points p. It is possible that the initial cubic spline is far away from the given surface. In that case it is avisable to start with c p p p 12

13 another curve. A simple choice is the restriction to a spline which is tangent to the given surface at the given points p. Using a basis q r of the tangent plane of p, the first erivative vector c is of the form c. q, r The epenence of c on the unknowns. is linear. We then minimize &50 within the space of quintic splines with these first erivative vectors an unknown secon erivative vectors at p. This is again the minimization of a quaratic function an yiels an appropriate starting shape for active contour propagation towars a spline curve on the given surface. A fine tuning of the shape analogous to the computation of geoesics is possible, but may be unnecessary. Since a functional such as (16) is just an approximation of the bening energy, we minimize a simplifie version anyway. However, the minimization of a geometric functional with a metho as in Greiner s work [19] can also be inclue in the present metho. Then, fine tuning of the shape might be more interesting. We will report on such algorithms in a subsequent paper Motion esign PSfrag replacements The following problem appears in computer animation an robot motion planning: Given a set of positions of a moving rigi boy, compute a smooth motion of the boy which assumes the given positions at given time instances. This problem has receive a lot of attention in various scientific communities (Computer Graphics, CAGD, Robotics, Computational Geometry). We o not review the literature, but just point to a survey paper [48] an to a recent paper [21], where a new concept is presente which is base on the following iea. It is simple to solve the problem with an affine motion, i.e., to at first amit affine istortions of the moving boy. This is so since we may choose any linear curve interpolation or approximation scheme. Moreover, we choose a set of four inepenent points (calle feature points f henceforth) on the boy an compute their locations f at time instances. This results in 4 sequences of homologous points to which we apply the curve scheme. Thereby, we obtain four curves f " f. For each, the four points f " f may be consiere as image points of f f uner an affine map. Applying to we obtain an affinely istorte copy D of the boy an thus an affine motion which interpolates or approximates the given positions. By the linearity of the chosen curve scheme, it oes not matter at all which four feature points we select. Another choice woul lea to the same affine motion. What we eventually want, however, is a rigi boy motion. In [21] it is suggeste to procee as follows: For each, approximate the affine map by a Eucliean motion (rigi boy transformation) -. As a quality measure for the fit, the chosen feature points (or more points on the boy) are consiere. The sum of square istances between the images of these points uner an is minimize. This is a special case of a well-known registration problem in Computer Vision [22], an has been recently investigate in epth by J. Wallner [58]. Thus, one knows exactly how to explicitly compute the best approximation of by a Eucliean motion - an uner which circumstances the solution is unique. Figure 15. a b The affine motion correspons to a curve a in an the esigne motion b is the orthogonal projection of a onto the 6-imensional manifol motions. of Eucliean Let us view the approach to motion esign which has been just outline in a more geometric way (Fig. 15). To each affine map in 3-space we may associate a point in 12-imensional affine space. For example, we may just collect the coorinates of the image points of the four chosen feature points uner. The image points of Eucliean motions form a 6-imensional submanifol. The input positions correspon to points on. The affine motion correspons to a curve a which interpolates the points, but oes not lie on. The measurement of istances between two affine maps with help of the sum of square istances between the images of selecte feature points is equivalent to the introuction of a 13

14 eplacements Eucliean metric in. Thus, we also have an orthogonality in. The way in which we compute a rigi boy motion - from the affine motion outline above correspons to the orthogonal projection b of the curve a onto. However, if the original affine motion has been obtaine with a variational formulation, the new solution b is in general not the minimizer of that functional uner the constraint of lying in. This view shows immeiately how to apply the concepts of the present paper to motion esign. We have an active curve in moving towars &50 the manifol in a way such that a given functional is minimize. For a solution, one nees local quaratic approximants of the square istance function to, e.g. square istances to tangent spaces at normal footpoints. The remaining algorithm is pretty much the same as in the previous subsection. As an example, Fig. 16 shows a motion compute via minimization of the cubic spline functional (16). Figure 16. Cubic spline motion interpolating given positions. 6. Conclusion an future research We have presente an active contour moel for curve an surface approximation which avois the parametrization problem. Accoring to the nonlinearity of the approximation problem, this moel yiels an iterative metho: In each iteration step we solve a linear system of equations, which arises from the minimization of a quaratic function. The new iea is that in this quaratic function we use local quaratic approximants of the square istance function to the moel shape which shall be approximate. In the present paper, we just outline the concept an applie it to a few typical problems such as offset approximation, egree reuction an conversion of arbitrary representations into B-spline form. Moreover, we showe how the computation of geoesics, of spline curves on surfaces an of interpolating or approximating smooth rigi boy motions can be hanle. There are many possibilities for extensions an future research: The convergence behaviour of the present metho, conitions on the choice of the initial shape an if possible conitions for reaching a global optimum shoul be investigate. We nee more research on quaratic approximants In particular, an appropriate space partitioning structure with cells carrying the approximants nees to be evelope. Also, a hierarchical representation of the square istance function with such a spatial ata structure woul be important. The concept is applicable to approximation with subivision surfaces. One can use an initial shape as in [34], but other choices an an appropriate hanling of etails require a lot of future research. An interesting extension concerns the incorporation of shape constraints such as convexity. For example, we can use the sufficient linear convexity conitions which have been erive by B. Jüttler for surface fitting with convex tensor prouct splines [24, 25]. In our framework, we woul then have to solve a quaratic programming problem in each iteration step. Special interpolation an approximation problems which appear in geometric moeling, for example the esign of skinning surfaces, may be aresse with the new concept. The new approach to motion esign nees to be stuie in more epth. Moreover, we want to investigate motions constraine by a gliing surface pair an we woul like to compute motions which avoi given obstacles. This can then be use for motion planning in connection with 5-axis NC machining. 14

15 The present active contour moel an its eformability form the lower levels of a so-calle artifical life moel [55]. The evaluation points s can be seen as sensors, an the control structure is the layer above that which allows us to coorinate the action of the sensors. Incluing phyiscal moeling an on top of that behavioral, perceptual an cognitive moeling yiels an artificial life moel. Those moels play an increasingly important role in Computer Graphics [55] an Meical Imaging [20]. We expect that this is also a promising research irection for the creation of intelligent moeling an reverse engineering systems, which inclue shape unerstaning abilities. Acknowlegements This research has been carrie out as part of the project P13938-MAT supporte by the Austrian Science Fun (FWF). We woul like to thank Johannes Wallner for valuable comments uring our work. References [1] Bajaj, C., Bernarini, F., Xu, G., Automatic reconstruction of surfaces an scalar fiels from 3D scans. SIGGRAPH 95 Proceeings (1995), [2] Besl, P. J., McKay, N. D., A metho for registration of 3D shapes, IEEE Trans. Pattern Anal. an Machine Intell. 14 (1992), [3] Blake, A., Isar, M., Active Contours, Springer, [4] Boissonnat, J.D., Cazals, F., Smooth shape reconstruction via natural neighbor interpolation of istance functions, In: Proc. 16th Ann. ACM Sympos. Comput. Geom. (2000), [5] Bloomenthal, J., e., Introuction to Implicit Surfaces, Morgan Kaufmann Publ., San Francisco, [6] Brunnet, G., Hagen, H., Santarelli, P., Variational esign of curves an surfaces, Surveys on Mathematics for Inustry 3, [7] o Carmo, M.P., Differential Geometry of Curves an Surfaces, Prentice-Hall, Englewoo Cliffs, New Jersey, [8] Caselles, V., Kimmel, R., Sapiro, G., Geoesic active contours, Intl. J. Computer Vision 22 (1997), [9] Caselles, V., Kimmel, R., Sapiro, G., Sbert, C.: Minimal surfaces: A geometric three-imensional segmentation approach, Numerische Mathematik 77 (1997), [10] Choi, H.I, Choi, S.W., Moon, H.P.: Mathematical theory of meial axis transform. Pacific J. Math. 181 (1997), [11] Choi, H.I., Choi, S.W., Moon, H.P.: New algorithm for meial axis transform of plane omain. Graphical Moels an Image Processing 59 (1997), [12] Degen, W.L.F., Best approximations of parametric curves by splines, In: T. Lyche an L. L. Schumaker, es., Mathematical Methos in Computer Aie Geometric Design II, Boston, Acaemic Pres, 1992, pp [13] Delingette, H., Montagnat, J., Shape an topology constraints on parametric active contours, Computer Vision an Image Unerstaning 83 (2001), [14] Desbrun, M., Cani-Gascuel, M.-P., Active implicit surface for animation, Graphics Interface (1998), [15] Eck, M., Degree reuction of Bézier curves, Computer Aie Geometric Design 10 (1993), [16] Eck, M., Approximative Degree Reuction in Bernstein-Bézier form, Preprint Nr. 1540, Fachbereich Mathematik, Technische Hochschule Darmstat, [17] Faugeras, O., Gomes, J., Dynamic shapes of arbitrary imension: the vector istance functions, In: R. Martin, e., The Mathematics of Surfaces IX, Springer, 2000, pp [18] Gomes, J., Faugeras, O., Reconciling istance functions an level sets, J. Visual Communication an Image Representation 11 (2000), [19] Greiner, G., Variational esign an fairing of spline surfaces, Computer Graphics Forum 13 (1994), [20] Hamarneh, G., McInerny, T., Terzopoulos, D., Deformable organisms for automatic meical image analysis, Proc. Meical Image Computing an Computer Assiste Intervention MICCAI 2001, Springer, pp [21] Hofer, M., Pottmann, H., Ravani, B., From curve esign algorithms to motion esign, Technical Report No. 95, Insitute of Geometry, Vienna Univ. of Technology, July [22] Horn, B.K.P., Close form solution of absolute orientation using unit quaternions, Journal of the Optical Society A 4 (1987),

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17 [52] Serra, J., Soille, J., es., Mathematical Morphology an its Applications to Image Processing, Kluwer, Dorrecht [53] Sethian, J. A., Level Set Methos an Fast Marching Methos, Cambrige University Press, [54] Siiqi, K., Tannenbaum, A., Zucker, S.W., A Hamiltonian approach to the eikonal equation, Workshop on Energy Minimization Methos in Computer Vision an Pattern Recognition, 1999, pp [55] Terzopoulos, D., Artifical life for Computer Graphics, Communications of the ACM 42 (1999), [56] Toriwaki, J., Yokoi, S., Distance transformations an skeletons of igitize pictures with applications, Progress in Pattern Recognition, L.N. Kanal an A. Rosenfel, es., North Hollang, 1981, pp [57] Wallner, J., Pottmann, H., Variational spline interpolation, Technical Report No. 84, Institute of Geometry, Vienna Univ. of Technology, [58] Wallner, J., approximation by Eucliean motions, Technical Report No. 93, Institute of Geometry, Vienna Univ. of Technology, [59] Warren, J., Weimer, H, Subivision Methos for Geometric Design: A Constructive Approach, Morgan Kaufmann Series in Computer Graphics, San Francisco, [60] Zhao, H.K., Chan, T., Merriman, B., Osher, S., A variational level set approach to multiphase motion, J. Comp. Physics 127 (1996), [61] Zhao, H.K., Osher, S., Merriman, B., Kang, M., Implicit, non-parametric shape reconstruction from unorganize ata using variational level set metho, Computer Vision an Image Unerstaning 80 (2000), [62] Zhao, H.K., Osher, S., Fekiw, R., Fast surface reconstruction an eformation using the level set metho, Proc. IEEE Workshop on Variational an Level Set Methos in Computer Vision, Vancouver,

18 Figure 11. Moel shape (yellow) an initial position of approximating B-spline surface (blue). Figure 13. Geoesics on a given surface. Figure 14. Initial (re) an final (blue) position of an interpolating cubic spline curve which lies very close to the given surface. Figure 12. Moel shape (yellow) an final position of approximating B-spline surface (blue) with bounary curve approximation. Figure 16. Cubic spline motion interpolating given positions (marke in re). 18