2D Shape Transformation Using 3D Blending

Transcription

1 D Shape Transormation Using 3D Blending G. Pasko, A. Pasko,, M. Ikeda, and T. Kunii, IT Institute, Kanazawa Institute o Technology, Tokyo, Japan {gpasko, ikeda}@iti.kanazawa-it.ac.jp Hosei University, Tokyo, Japan {pasko, kunii}@k.hosei.ac.jp Abstract Computer animation is one o the key components o a multimedia document or presentation. Shape transormation between objects o dierent topology and positions is an open modeling problem in computer animation. We propose a new approach to solving this problem or two given D shapes. The key steps o the proposed algorithm are: dimension increase by converting input D shapes into halcylinders in 3D space, bounded blending with added material between the hal-cylinders, and making cross-sections or getting rames o the animation. We use the bounded blending set operations deined using R-unctions and displacement unctions with the localized area o inluence applied to the unctionally deined 3D hal-cylinders. The proposed approach is general enough to handle input shapes with arbitrary topology deined as polygons with holes and disjoint components, set-theoretic objects, or analytical implicit curves. The obtained unusual amoeba-like behavior o the D shape combines metamorphosis with the non-linear movement on the plane.. Introduction Shape transormation is one o the important operations in animation and ree-orm modeling. It includes simple linear transormations (translation, scaling, rotation), non-linear transormations such as ree-orm transormations and other non-linear space mappings, and metamorphosis or morphing (transormation o one given shape into another). Approaches to D metamorphosis are well developed or two given simple polygons. They use physically-based methods [S9, S93], starskeleton representation [SR95], warping and distance ield interpolation [CLS96], wavelet-based [ZH00] and surace reconstruction methods [SSBT0]. The existing approaches are based on one or several o the ollowing assumptions: equivalent topology (mainly topological disks are considered), polygonal shape representation, shape alignment (shapes have common coordinate origin and signiicantly overlap in most o the case studies), possibility o shape matching (establishing o shape vertex-vertex, control points or other eatures correspondence), the resulting transormation should be close to the motion o an articulated igure. Figure. Three steps o biological amoeba motion. The speciic aspects o the shape transormation problem considered in this paper are the ollowing: initial D shapes can have arbitrary topology not corresponding to each other; the shapes can be deined as D polygons, implicit curves, or constructive D solids; the shapes are not aligned or overlap, and can occupy dierent positions on the plane;

2 there is no correspondence established between the boundary points or other shape eatures; a combined transormation is considered including metamorphosis and non-linear motion similar to the behavior o a biological amoeba illustrated in Fig.. In this work, we propose a new method o shape transormation including non-linear motion and metamorphosis. We use implicit curves [B97] or simple shapes without holes and sel-intersections, and D FRep solids [PASS95] in more general cases. Both these models use real-valued unctions o two variables F(x,y) to represent D shapes. The boundary o such shape is an implicit curve F(x,y)=0. A D FRep solid is constructed using primitives and operations, and can be deined by the inequality F(x,y) 0, where the unction F is evaluated at the given point by a procedure traversing an FRep constructive tree. A given polygonal shape can be converted automatically in an FRep solid as it is discussed below. Function-based metamorphosis [FPA00] using the linear (or two initial solids) or bilinear (or our initial solids) unction interpolation can be applied to transorm unctionbased shapes, but it can produce poor results or not aligned objects with dierent topology. We propose an alternative way based on increasing the object dimension, unction-based blending, and consequent cross-sections as it is explained in the ollowing section. Short animations produced as illustrations o the proposed approach can be ound at URL The HyperFun modeling language [A99] and supporting sotware tools were used to produce these animations and most o the illustrations in the paper. a b c d Figure. Blending union o two disjoint solids (a) with added material results in a single solid (b-d) with the dierent shapes o the blend depending on parameters.. Shape transormation using higher-dimensional blending A blending operation in 3D shape modeling generates smooth transition between two suraces. Note that sometimes the term blending is used to designate metamorphosis o D shapes, but we use it here in the way traditional to geometric and solid modeling. Blending versions o set-theoretic operations (intersection, union, and dierence) on solids approximate exact results o these operations by rounding sharp edges and vertices. Such operations are usually used in computer-aided design or modeling illets and chamers. In the case o blending union o two disjoint solids with added material, a single resulting solid with a smooth surace can be obtained as it is shown in Fig.. This property o the blending union operation is the basis o our approach to the shape transormation.

3 a b c d e Figure 3. Steps o D shape transormation using 3D blending z=-0 z=0 z= z=0 Figure 4. Frames o animation o the disk to square transormation using unbounded blending (initial shapes cannot be obtained, see z=-0 and z=0 cross-sections) Let us ormulate step by step the approach we propose and illustrate it by Figs. 3, 4: ) two initial shapes are given on the xy-plane (a disk and a square in Fig. 3a); ) each shape is considered as a cross-sections o a hal-cylinder in 3D space (a cylinder bounded by a plane rom one side) as it is shown in Figs. 3b and 3c; 3) the axes o both cylinders are parallel to some common straight line in 3D space, or example, to the coordinate z-axis, and the bounding planes o two hal-cylinders are placed at some distance to give space or making the blend (Fig. 3d); 4) apply the added material blend operation (similar to one illustrated in Fig. ) to the halcylinders (Fig. 3e,); 5) adjust parameters o the blend such that a satisactory intermediate D shape is obtained in one or several D cross sections by planes orthogonal to z-axis (Fig. 4); 6) considering additional z-coordinate as time, make consequent orthogonal cross-sections along z-axis (Fig. 4) and combine them into D animation. As it is shown in Fig. 4, the initial shapes cannot be obtained even in cross-sections with big absolute values o z-coordinate, i global blending is applied. The description o the unction-based blending and its localized version solving the above problem are given in the ollowing sections.

4 . Function-based blending The major requirements to blending operations are tangency o the blend surace with the initial suraces; automatic clipping o unwanted parts o the blending surace; blending deinition o the basic set-theoretic operations; and support o added and subtracted material blends. An analytical deinition o blending satisying these requirements was proposed in [PS94]. This deinition covered added and subtracted material blends as well as symmetric and asymmetric blending. An analytical deinition o blending set-theoretic operations on solids with implicit suraces was proposed in [PS94] on the basis o properties o R-unctions [R87, PASS95], which provide exact analytical deinitions o set-theoretic operations with C and higher-level continuity. The ollowing deinition o the blending set-theoretic operation was proposed: Fb (, ) R(, ) dispb (, ), () where R(, ) is an R-unction corresponding to the type o the operation, the arguments o the operation ( X ) and ( X ) are deining unctions o two initial solids, and disp (, ) is a Gaussian-type displacement unction. The ollowing expression or the displacement unction was used: a0 disp b (, ) =, () a a where a 0, a, an a are parameters controlling the shape o the blend. The proposed deinition is suitable or blending union, intersection, and dierence, allows or generating added and subtracted material, symmetric and asymmetric blends. For example, the union operation can be exactly deined by the R-unction as R int, ) with the corresponding blending union described as F b int ( (3) a a This blending union operation was used to blend the hal-cylinders in Figs. 3,4. The exact initial shapes (a disk and a square) cannot be obtained at any cross-section, see, or example, z=-0 and z=0 cross-sections in Fig. 4. The proposed displacement unction does not get zero value anywhere in the space. This is the reason o the main disadvantage o this deinition - the blend has a global character and cannot be localized using its parameters. A dierent displacement unction is needed, which would allow or the localization o the blend and respectively or deining an exact time interval or the shape transormation. a 0

5 Figure 5. Displacement unction or bounded blending.. Bounded blending In this section, we apply a new ormulation or blending operations with the blend bounded by an additional bounding solid as proposed in [PPIK0]. The deinition () o a blending operation is used as the basis with the modiications introduced only or the displacement unction. First, a displacement unction with the local area o inluence is selected, then, a generalized distance unction or bounding blends using bounding solids is constructed. Displacement unction At the irst step o deining bounded blending, we select or the deinition () a displacement unction o one variable disp bb (r), which should satisy the ollowing conditions: ) disp bb ( r) 0 and disp bb (r) takes the maximal value or r=0 ; ) disp bb r 0, r disp 3) bb 0, r (the curve tangentially approaches axis x at r=). r A plot o the desired displacement unction is shown in Fig. 5. One can derive or ind several analytical expressions or such unction. In act, this work has been done by many researchers or localizing individual components o skeletal implicit suraces [B97]. One o the possible expressions, which we will use in the rest o the paper, is 3 ( r ), r disp bb ( r) r (4) 0, r Here, r is a generalized distance, which is constructed using deining unctions o the initial solids, as it will be shown in the next section. Bounding solid The introduction o an additional bounding solid can bring real local character to the blend. It is required that the blending surace exists only inside the bounding solid, and only original suraces exist outside the bounding solid. To apply the bounded blending deinition F bb (,, 3) R(, ) a0dispbb ( r) (5)

6 with the displacement unction (4), we have to provide the ollowing interconnected properties: ) The zero value o the displacement unction disp bb (r) outside the bounding solid; ) The maximal value o the displacement unction disp bb (r) near the intersection points o two initial suraces; 3) The zero value o the generalized distance unction r at the intersection points o two initial suraces; 4) The value o the unction r outside the bounding solid. The ollowing deinitions satisy the above requirements: 3 ( r ) r, r, 0 r disp bb ( r) r with r r, r (8) 0, r, r 0 where (, ) r a a, 3, 3 0 and ( 3) r a, 3 0, 3 0 with numerical parameters a and interactively control the inluence o the unction a 3 3 controlling the blend symmetry, and a allowing the user to on the overall shape o the blend. This deinition o the unction r and the deinition (4) o the displacement unction disp bb (r) are not unique and can be changed, i it is necessary in particular applications. Shape transormation using bounded blending Let us apply the described above bounded blending union to get smooth transitions between two hal-cylinders or the shape transormation (Figs. 6,7). The hal-cylinders are bounded by the planes z=0 and z= to make the gap [0,] along z-axis between them. The bounding solid or the blend in this case is an ininite slab orthogonal to z-axis and deined by the unction 3 as an intersection o two halspaces with the deinitions z 0 and z 0. As it can be seen rom Fig. 6, the blending displacement rom the exact union o two hal-cylinders takes zero value at the boundaries o the bounding solid (planes z=-0 and z=0). This results in the exact initial D shapes obtained at the cross-sections outside the bounding solid: the disk or z 0 and the square or z 0 (see Fig. 7 and compare with Fig. 4). The parameters a 0 - a 3 o the bounded blend inluence the blend shape and respectively the shape o the intermediate cross-sections. The selection o the boundary values or the blend mainly depends on the overall size o the initial shapes and their positions. The key point here is that whatever interval is selected or the bounded blend along z-axis in 3D space, it can always be scaled to match the required time interval or the shape transormation on a D plane.

7 Figure 6. Bounded blending o two hal-cylinders with a disk and a square as cross-sections. z=-0 z=0 z= z=0 Figure 7. Frames o animation o the disk to the square transormation using bounded blending (exact initial cross-sections at the boundaries o the blend: cross-sections with z=-0 and z=0). 3. Transormation o D polygonal shapes In this section, we illustrate application o the proposed approach to the transormation o two polygonal shapes o arbitrary topology. An arbitrary D polygon (convex or concave) can be represented by a real unction (x,y) taking zero value at polygon edges [R74, PSS96]. The polygonto-unction conversion problem is stated as ollows. A two-dimensional simple polygon is bounded by a inite set o segments. The segments are the edges and their extremes are the vertices o the polygon. A polygon is simple i there is no pair o nonadjacent edges sharing a point, and convex i its interior is a convex set. The polygon-to-unction conversion algorithm using a monotone settheoretic expression with R-unctions was proposed in [R74], implemented and tested in [PSS96]. The conversion satisies the ollowing requirements: It provides an exact polygon boundary description as the zero set o a real unction; No points with zero unction value exist inside or outside o the polygon; It allows or the processing o any arbitrary simple polygon without any additional inormation. In the case o a polygon with holes, each hole is processed as a separate polygon, and then the settheoretic subtraction is applied between the initial polygon and the hole model. In the case o several disjoint components, each component is processed as a separate polygon, and then the set-theoretic union is applied to all components. Let us present a short experimental animation showing a transormation between two arbitrary polygonal shapes with holes and disjoint components (Fig. 8). The idea o this test came rom the Dancing Buddhas project [G0] devoted to multimedia augmenting o amous Buddhist texts. The problem in this test is to get a new type o transormation o a Buddha shape into a Chinese character. First, two polygonal shapes were obtained rom the images: the Buddha shape (see the image in Fig. 8 let) consists o the main concave polygon (49 segments) and two simple polygonal holes, the Chinese character (see the image in Fig. 8 right) consists o two disjoint components, one o which is a simple concave polygon (let part o the character) and another (right part o the character) is a concave polygonal shape ( segments) with ive holes. As one can see, the initial shapes have quite dierent topology, they also were placed on the plane without overlapping. We applied the proposed

8 algorithm using bounded blending in 3D space and obtained an animation (see some o the rames in Fig. 9). The transormation is quite smooth except some acceleration within the interval [0,] along z-axis, where the most drastic transormations occur. The 3D shape o the blend and respectively the behavior o the D shape during the transormation can be controlled by the blend parameters, the length o the gap between the hal-cylinders along z-axis, and the positions o the planes bounding the blend. Figure 8. Initial Buddha shape (let) and a Chinese character (right). Figure 9. Frames o the animation: transormation o a Buddha shape into a Chinese character. 4. Conclusions We introduced a new approach and analytical ormulations or the D shape transormation on the basis o the dimension increase, bounded blending between 3D objects, and cross-sectioning the blend area or getting rames o the animation. It is shown that the blending operation with such simple bounds as two planes satisies the condition o the localization o the shape transormation at some predeined time interval. The proposed approach can handle non-overlapping D shapes with arbitrary topology deined by input polygons, set-theoretic or analytical expression. The obtained behavior o the D shape during the transormation process does not imitate the motion o an articulated igure, but rather has amorphous or amoeba-like character including non-linear motion and metamorphosis. The extension o this approach to the case o 3D input shapes and bounded blending o 4D space-time objects will be the topic o our urther research eorts in this direction. The D animations illustrating the proposed approach can be ound at URL

9 Acknowledgements The authors would like to acknowledge the help o the University o Aizu students Yukinobu Katou, Kensuke Masuda, and Tomoko Matsuoka, who participated in the early stages o this research. Reerences [A99] Adzhiev V., Cartwright R., Fausett E., Ossipov A., Pasko A., Savchenko V., HyperFun project: a ramework or collaborative multidimensional F-rep modeling, Implicit Suraces '99, Eurographics/ACM SIGGRAPH Workshop, Universite Bordeaux I, France, J. Hughes and C. Schlick (Eds.), 999, pp , [B97] Bloomenthal J. et al., Introduction to Implicit Suraces, Morgan Kauman, 997. [CLS96] Cohen-Or D., Levin D., Solomovici A., Contour blending using warp-guided distance ield interpolation, IEEE Visualization 96, IEEE Computer Society, 996, pp [FPA00] Fausett E., Pasko A., Adzhiev V., Space-time and higher dimensional modeling or animation, Computer Animation 000, University o Pennsylvania, PA USA, May , pp [G0] Goodwin J. R., Goodwin J. M., Pasko A., Pasko G., Vilbrandt C., Dancing Buddhas: new graphical tools or the presentation and preservation o cultural artiacts, Virtual Systems and Multimedia Conerence VSMM'0 (University o Caliornia Berkley, USA), IEEE Computer Society, 00, pp [PASS95] Pasko A., Adzhiev V., Sourin A., Savchenko V., Function representation in geometric modeling: concepts, implementation and applications, The Visual Computer, vol., No.8, 995, pp [PPIK0] Pasko G., Pasko A., Ikeda M., Kunii T., Bounded blending operations, Shape Modeling International 00, IEEE Computer Society, 00, pp [PS94] Pasko A., Savchenko V., Blending operations or the unctionally based constructive geometry, Settheoretic Solid Modeling: Techniques and Applications, CSG 94 Conerence Proceedings, Inormation Geometers, Winchester, UK, 994, pp.5-6. [PSS96] Pasko A., Savchenko A., Savchenko V. Polygon-to-unction conversion or sweeping, Implicit Suraces 96, Eurographics/SIGGRAPH Workshop, Eindhoven, The Netherlands, 996, pp [R74] Rvachev V. L., Methods o Logic Algebra in Mathematical Physics, Naukova Dumka, Kiev, 974 (in Russian). [S9] Sederberg T., Greenwood E., A physically-based approach to -D shape blending, SIGGRAPH 9, Computer Graphics, vol. 6, No., 99, pp [S93] Sederberg T., Gao P., Wang G., Mu H., -D shape blending: an intrinsic solution to the vertex path problem, SIGGRAPH 93, Computer Graphics Proceedings, 993, pp [SR95] Shapira M., Rappoport A., Shape blending using the star-skeleton representation, IEEE Computer Graphics and Applications, vol. 5, No., 995, pp [SSBT0] Surazhsky T., Surazhsky V., Barequet G., Tal A., Blending polygonal shapes with dierent topologies, Computers and Graphics, vol. 5, No., 00, pp [ZH00] Yueeng Zhang, Yu Huang, Wavelet shape blending, Visual Computer, vol. 6, No., 000, pp