Artistic Rendering of Functionally Defined Volumetric Objects

Transcription

1 Artistic Rendering of Functionally Defined Volumetric Objects by Masashi Mizota Graduate School of Computer and Information Sciences Hosei University Supervisor: Alexander Pasko March 2006

2 Abstract We propose approaches to artistic (non-photorealistic) rendering of functionally defined volumetric objects, namely gaseous object without clear boundaries. The function-based three-dimensional models are described in the HyperFun language. An outside model of a volumetric object (steam or smoke) is defined with a group of skeletal implicit surface primitives. Internal density is defined with the Noise or Turbulence functions. The volumetric object is converted into volume data. An artistic image is generated by volume ray-tracing using the density value obtained from the volume data. The proposed method makes it possible to create cartoon images and animations of gaseous objects.

3 Contents Chapter 1 Introduction 1 Chapter 2 Related Work 3 Chapter 3 HyperFun Modeling Language 5 Chapter 4 Proposed Approach Gaseous object model Conversion to a volume data set Artistic rendering by using density threshold colors Artistic rendering using density values and strokes Chapter 5 Implementation and Experiments Artistic image of steam Artistic image of thin smoke Animation Chapter 6 Conclusion 22 Chapter 7 Future Work 23 References 24

4 Acknowledgement I would like to thank Professor Alexander Pasko for his support and help. I would like to thank research students of Prof. Pasko.

5 Chapter 1 Introduction In this paper, we propose a method for creating an artistic (non-photorealistic) rendering and animation of functionally defined volumetric objects. In a typical example of volumetric objects, there are clinical diagnosis image (CT, MRI, and PET, etc.), earthquake data, fluid analyses (CFD), and a general non-structural mesh data. More general examples are fire, fog and cloud. In this paper, we select steam and smoke as typical gaseous objects for the experiments. Recently, Non-photorealistic Rendering (NPR) research area is progressing very well. However, there is little research on NPR for volume objects without clear boundaries like cloud or smoke. Furthermore, in the research on some clouds and smokes, particles are allocated in volume, and they are supplemented with some elementary shape (e.g., sphere or another primitive). The related rendering techniques operate only on the surface and do not put consideration in a peculiar density of a gaseous object. In the method we propose, there are three steps for creating artistic images. First, a gaseous object is defined by some functions for overall shape and density. Second, the function model is converted to volume data. Finally, an artistic image is rendered. We propose two methods for artistic rendering. One method uses the density value requested from the volume data and generates Cartoon rendering for several isosurfaces inside the volume. Another method randomly arranges points on several isosurfaces, and the image of a hand-drawn style is generated with the supplementation these points by strokes. For the implementation, we use HyperFun that is a programming language designed for modeling geometric objects defined by single real continuous functions of point coordinates as f (x)>=0 [10] and its extension for modeling objects with attributes [14]. For rendering, we use and modify POV-Ray, which is free ray-tracing software. 1

6 In the following chapters, the related works on NPR rendering and volume rendering are described. In chapter 3, an explanation of HyperFun language is given. In chapter 4, approach to artistic rendering is described in three steps: object model, conversion to volume data, and artistic rendering. In chapter 5, implementation and experiments are described. The images of steam and thin smoke generated by using the proposed algorithms are shown, and animation is generated and illustrated. Conclusions and future work discussion are given in chapter 6. 2

7 Chapter 2 Related Work There are some papers about non-photorealistic rendering of clouds or smoke. Selle and Mohr [1] describe a technique for generating cartoon style animations of smoke. In their method, dynamics of the smoke is generated with a standard liquid simulator and output is in the form of particles. These particles are then rendered as texture mapped 2D stencils, and silhouette edges are added using a depth difference technique to emphasize shape and depth. However, the technique does not work well with varying viewpoints because the feature lines move as the depth direction changes, introducing artifacts due to the viewer motion, rather than the flow. In addition, there is no suitable shading. Hoshino et al. [2] proposed a method for creating cartoon-images and animation of smoke. At first, they set particles from the volume data given by the simulation. Clouds or smokes are represented by allocating a sphere to each particle. The size of the sphere is optimized according to the density distribution of the particles. The shadow is calculated by using the graphics hardware arranging the source of light and the image is generated. The image is acquired from the frame buffer, and converted to a phased shading image that is feature of cell rendering and extracted the outline. The final image is obtained by putting the phased shading image and the outline image together. The authors accelerate rendering by optimizing particles, but the number of particles may differ and formal adjustment cannot be completely taken as a picture changes while optimizing particle in animation. Selle [1] and Hoshino [2] treat smoke simply as a surface without considering the internal distribution of density, that is the main feature of a volumetric object. In our research, we propose the technique for expressing NPR paying attention to the internal distribution of density not treated them. Perry [4] proposed a novel framework for NPR based on adaptively sampled distance fields. A distance field is a scalar field that specifies the minimum distance to the surface of a shape. When the distance field is signed, the sign can be used to distinguish between the inside and outside of the shape. 3

8 Lu et al. [5] presented a framework for an interactive direct volume illustration system that simulates traditional stipple drawing. They explore several feature enhancement techniques to create effective visualizations of scientific and medical datasets. Rheingans [7] introduced the volume illustration approach, combining the familiarity of a physics-based illumination model with the ability to enhance important features using non-photorealistic rendering techniques. The properties that can be incorporated into the volume illustration procedures include the following: volume sample location and value, local volumetric properties, such as gradient and minimal change direction, view direction, and light information. Nagy [8] proposed a non-photorealistic rendering technique for volumetric data sets. For a number of seed points that are placed appropriately to represent selected volume structures, curvature lines are traced and encoded by a sparse set of control points. These curves are finally drawn as hatching strokes modulated by anisotropic lighting and transparency. Ebert [9] proposed a volumetric modeling and animation technique that combines implicit surfaces with turbulence-based procedural techniques. This technique can produce a realistic cloud image. A cumulus is created by positioning some implicit spheres with varying radii and blending amounts to define the overall shape of the cloud. The difference between these researches and our research is that we use functionally defined volume for a gaseous object and various levels of density inside the volume in the rendering techniques. 4

9 Chapter 3 HyperFun Modeling Language For implementation, we use HyperFun that is a programming language designed for modeling geometric objects defined by single real continuous functions of point coordinates as F(x) 0 [1]. This representation is called function representation (FRep) [8, 9]. In FRep, complex geometric objects are constructed using simple primitives from the FRep library and operations on them. Since FRep models are more general than traditional skeletal implicit surfaces, convolution surfaces, distance-based models, CSG (Constructive Solid Geometry), sweeps, and voxel models, HyperFun can deal with all these geometric models. In HyperFun, FRep objects are described using assignments, conditional selections, and iterations as in traditional programming languages. In addition to arithmetic and relational operators, HyperFun has built-in set-theoretic operators such as union, intersection, subtraction, and others. Some predefined library primitives and operators are available in HyperFun. Using these functionalities, quite complex objects can be modeled. HyperFun can define a multidimensional object by a function of several variables. For example, an object changing in time can be defined by F(x,y,z,t) with t representing time. In HyperFun, a geometric object in 3D space is described by: the inside, the surface, and the outside. F(x) > 0: the inside of an object F(x) = 0: the surface of an object F(x) < 0: the outside of an object In addition, HyperFun can define object properties. Object properties are modeled as attributes: photometric (opacity, color and reflectance), material, density, temperature. The attribute is defined by array of S(x) [14]. For example, when an array of s(x) is (1,1,1), an object is colored to white. This is a rule which determines where a given point belongs. The user can create own library of object by basing on this rule. 5

10 Chapter 4 Proposed Approach We propose a technique for generating cartoon images of steam and smoke. As we deal with the volumetric object, it is not enough only to render some surface. We select several volume density values and thus several isosurfaces inside the volume. These isosurfaces are rendered with parameters depending on the density value to simulate the visual effect of the semi-transparent media. For the given object, we select several threshold intervals T n (n=1, 2, 3...) of the density value to be used for changing rendering parameters. Our rendering system consists of three main components: define a gaseous object model by using the function representation; convert the model into a volume data; and rendering system for making an artistic image. We propose two rendering methods, namely a density threshold colors method and a strokes method Gaseous object model Let us describe a functional definition of a gaseous object. In the functionbased model, the outward form of the model is defined by the function f(x), and the inner density distribution model is defined by the function s(x). We define the outward form of the gaseous object by using blobby and convolution surface models, because ordinarily, a handwritten cartoon image of smoke or cloud is seen like a set of blended spheres. However, it is possible to use any primitive for defining the outward shape. The inside of the object is defined by using the function of turbulence for representing density. The turbulence-based [13] volumetric procedures allow for control of the density and edge sharpness of the object. 6

11 Attributes in the function representation. In FRep, physical and abstract objects and phenomena are modeled as point sets. Object properties are modeled as attributes: photometric (opacity, color and reflectance), material, density, temperature as follows: Volume model = {3D point set, point attributes}. Each point: {(x, y, z), (s1, s2,, sn)}. Figure 1 shows samples of attributes usage. Figure 1(a) illustrates simple change of color by coordinate x. Figure 1(b) shows the use of color attribute library function hfa_noiseg(x), which defines the Gardner s noise function. Figure 1(c) is an image of a cube with the turbulence procedure used to define gray color. (a) (b) (c) Figure 1. Samples of attributes Turbulence. Turbulence is a random noise function. The inner part of a gaseous object can be filled by turbulence making randomly scattered volume density. In our method, we use the Perlin s turbulence method [13]. The Perlin's turbulence function is defined as the sum of noise functions with different amplitudes over a range of different frequencies: 7

12 Turb(point, n) = n t Noise(point.2 ) t t = 0 2 where point is a point in space of any dimension, n is the number of frequencies, and Noise is a noise function. A noise function produces values that are apparently stochastic. A lattice noise is used as a noise function. A lattice noise function outputs a pseudo random number for the given space point and it is a continuous function. Since the turbulence function is a sum of continuous noise functions, and then the turbulence function is equally a continuous function. The turbulence library function in HyperFun is hfa_turbulence(x, freq), where x is a point coordinates array, and freq is frequency of the turbulence. Output is a value in [0, 1]. Figure 2 is a sample of turbulence. Frequency at the left picture is 1.25 and at the right picture is Figure 2. Sample of hfa_turbulence Conversion to a volume data set The defined functional model is converted into the volume data set (voxel array). The size of the volume is assumed to be nx*ny*nz, and the density is recorded in each voxel. In our technique, first, the function model is converted into the slice image grayscale images similar to CT data. Next, the density data is acquired by reading 8

13 the intensity values of the 2D image slices and using them as 3D volume data in the form acceptable for the renderer. The point coordinates and the density value are converted to a voxel stored in the volume data set. Figure 3 shows an illustration of slice data generation and one slice of a cylinder with internal turbulence attribute. Table 1 is a relation list of color and density value in the slice image. Moreover, the shape and density changes of the object according to the passage of time are followed by continuously updating the volume data according to the time series. Figure 3. Slices generation (left) and a slice image of cylinder with turbulence (right). Table 1. Relation of color and density value. RGB color Density value Color is Black 0, 0, Color is Gray 127, 127, Color is White 255, 255,

14 4.3. Artistic rendering by using density threshold colors. We propose a method for generating cartoon images of gaseous objects by using volume data. There are two steps: evaluation of the density and calculation of the intersection density threshold. In the process of ray tracing, the density value at each point is calculated from the volume data set. For example, the density data is obtained for a simple cylinder as shown in Figure 4. Figure 4. Example of a cylinder and a density graph. The ray-isosurface intersection for each threshold interval T n (n=1,2,3...) is calculated for the density data as shown in Figure 5. The left is the pure cylinder density function and the right is the cylinder density function with added turbulence. Figure 5. Finding density levels by threshold intervals. 10

15 Color is assigned to the respective threshold levels T n, and the color of the pixel for the ray-isosurface intersection is taken as the color of the corresponding T n. Colors close to the white level are assigned to the threshold with high density, and colors close to black are assigned to the low one. Thus, ray tracing produces the image with the cartoon style discrete grey levels. Figure 6 shows ray-traced images of a cylinder with the smooth and turbulent density attributes. Figure 6. Example of an artistic cylinder image Artistic rendering using density values and strokes. We propose another technique for rendering gaseous objects in a hand drawn style by generating strokes using density values. To make strokes, points are selected on the isosurface contour for the given density threshold value. We select points with maximum and minimum values of X-coordinates with constant Y coordinates on the contour (Figure 4). The minimum value is stored in the array of Left_Line, and the maximum value is stored in the array of Right_Line. 11

16 Figure 7. Finding points for strokes. Strokes are generated between detected points of the contour using the arrays Left_Line and Right_Line. Figure 5 (b) shows straight line strokes for one isosurface contour. Because general hand drawn steam is usually by curves, we supplement detected points with the curved strokes. Left_Line strokes are defined to bend to the left, and Right_Line is defined oppositely. Moreover, the first points of each array are connected by the curve too. Figure 5(c) shows a contour rendered with curved strokes. To simulate steam appearance, the following conditions are put for each density isosurface contour: - The color and the thickness of strokes are changed according to the density value. - The number of the strokes is decreased with random variation while the isosurface becoming deeper internal, because outside outline is clarified and the inside part will be rendered as indistinct fuzzy media. (a) (b) (c) Figure 8. Points and strokes for steam rendering. 12

17 Chapter 5 Implementation and Experiments 5.1 Artistic image of steam. The implementation includes steps of the model definition, generation of volume data, and the modification of POV-Ray for artistic rendering Definition of steam model We implement the steam model as follows: f(x) = 1; s(x) = hfconvcurve(x,vect,s,t) & hfa_turbulence(x,freq ); hfconvcurve is a convolution primitive with the curve skeleton (set of connected line segments), x is a point coordinates array, vect is coordinate array for vertices of the curve, S is an array of kernel width control parameters for each line segment, and T is a threshold; freq is frequency of the turbulence. Normally, the attributes are defined by the array s(x) when f(x) > threshold (ex.-0.01) in HyperFun. However with f(x)=1 we can consider distribution of the density attribute in the entire space. For the test model, we add the following expression to simulate diffusion of steam going up with thinning of the density. s[x] = s[x] * (1 - (y + α) * 0.1); Figure 9 is a source code of the Steam.hf model. 13

18 my_model(x[3], a[1], s[3]) { array xt[3], vertex[30], Sval[9]; xt[1]=x[1]; xt[2]=x[2]; xt[3]=x[3]; vertex = [ 0.0, -6.0, 0.0, 0.0, -5.0, 0.0, 0.0, -4.0, 0.0, 0.0, -2.0, 0.0, 0.0, -0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 4.0, 0.0, 0.0, 6.0, 0.0, 0.0, 8.0, 0.0, 0.0, 10.0, 0.0 ]; Sval = [1.5, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5]; smoke = hfconvcurve(xt, vertex, Sval, 0.1); noise =hfa_turbulence(x,1.5); s[1] = smoke & noise; s[2] = smoke & noise; s[3] = smoke & noise; if(x[2] > -6) then s[1] = s[1] * (1 - (x[2]+6)*0.05); s[2] = s[2] * (1 - (x[2]+6)*0.05); s[3] = s[3] * (1 - (x[2]+6)*0.05); endif; my_model = 1; } Figure 9. Steam model in HyperFun Generation of volume data set For making volume data, the steam model is converted into 200 bitmap images of 200x200 pixels by using the HFSlicer software. HFslicer is a slice viewer for HyperFun. Slices are performed along the Z axis. Figure 10 is one of slice images. When the model is sliced, the following command is used: hfslicer steam.hf -nslice 200 -bmp steam -b -8,-8,-8,8,8,8 -w

19 The -nslice defines the number of slices. The -bmp parameter is radix for the bmp file output. The -b parameter defines the bounding box. The -w parameter specifies the image size (always a square). The density value in each voxel is calculated from the color obtained by reading the bitmap image, and the density is stored as df3 file, which is a file format for volume data. As a result, we can get the volume data of 200x200x200 size. We used the density file enhancing patch of Mr. Suzuki in the National Institute of Advanced Industrial Science and Technology to mount in POV-Ray. In POV- Ray, external data of three dimensions can be read with density file (DF3), and media, isosurface, and texture can be changed according to the value. However, it was inconvenient to use it to display experimental data and the computer simulation because the density file was able to use only the integers of 8bit. Then, the format of the Density_File parameter was enhanced by applying Mr. Suzuki's patch and the integer in two bytes, the real numbers in four bytes, and the data such as text files can be treated. In addition, the option of the interpolation between data was added and smoother rendering was enabled (see Figure 10). The following headers are necessary for the DF3 file besides the density data: [Type] Float //data type: Float, Int(2byte), Char(1byte) [Format] Text //data format: Text, Binary [X_size] 200 //Dimension of data of direction of X. Integer value 4 or more. [Y_size] 200 //Dimension of data of direction of Y. Integer value 4 or more. [Z_size] 200 //Dimension of data of direction of Z. Integer value 1 or more [Endian] Little // The order of Binary data when Format is Binary [Cyclic] On // On : when data is repeated periodically beyond the limits of 0<(x y z)<1 [Data] //density data. 0.0 < data <1.0 15

20 Figure 10. Slice image of steam Artistic image generation. The generated volume data is read to POV-Ray. Figure 11 is a source code in which steam volume data is specified for POV-Ray. The volume is called by using the Media object supported by POV-Ray as an interior of a box. The emission is a color of the light issued from the corpuscle of media. Scattering is a parameter that gives the character to scatter light in media. The color of the scattered type and the scattered light is specified there. The interval and sample of necessary parameters are set to a little larger than 20. These parameters define intervals advanced by a ray during tracing. The method is a sampling method. The interpolate in density does the third supplementation by 2, and makes the volume smoother. The frequency does the volume data clipped between from 0 to 1 when not specifying it for 0. The hollow is a parameter to make contents of the box that is the container empty. 16

21 object{ } Smoke = object{ box {0,1 texture { pigment {rgbt 1} } interior { media { emission 1 scattering {2, 1} intervals 20 samples 20,20 method 3 density { density_file df3 "df_smoke.df" interpolate 2 frequency 0 } } } hollow } } Figure 11. Volume data specification in POV-Ray The modification part of the existing POV-Ray source code is the following. For the density threshold color method, we changed the media.cpp module to calculate the density value, and to detect the intersection with the threshold isosurface. The render.cpp is changed for cartoon rendering of the entire image. The color was gradually made to darken based on a threshold corresponding to the most inside isosurface where the density is high. For the stroke method, the module render.cpp is changed for the points generation and rendering of strokes. The interval was reduced at intervals of the line to take the point from the threshold as the density rose. The stroke is classified into three kinds: left line, right line and top line. The difference of the left line and the right line isthe direction of the swelling of the stroke, and the top line is a stroke connecting the highest points of the right and the left strokes by the curve. For the isosurfaces other than the most outside one, we specify the number of strokes at random. As for the thickness of the stroke, the stroke is gradually made to thicken from the outside to inside. 17

22 Figure 12 (a) is a realistic image of steam. Figure 12(b) shows artistic rendering with density threshold colors (see 4.3). In this image, there are five given threshold intervals. When a certain threshold is taken, the line that expresses the isosurface of steam might become notched. However, it is expectable that the line with peculiar width seems to be steam. Figure 12(c) is an artistic image obtained using the density values and strokes (see 4.4). There are four given threshold intervals and the thickness of the stroke is used from 1 dot to 3 dots. (a) (b) (c) Figure 12. Realistic (a) and artistic (b, c) images of steam Artistic image of thin smoke. Figure 13 is an artistic image of thin smoke rendered by using the density threshold colors method. This model is a convolution primitive with a curve 18

23 skeleton with added turbulence. With NPR applied only to the surface, this thin smoke object is not expressible and would be seen only as a bent cylinder, because there is no ruggedness on the surface. Figure 13. Thin smoke. Figure 14 is a source code of ThinSmoke.hf, where Sval is an array of kernel width control parameters for each line segment, and we change Sval to smaller values than for steam. The frequency of turbulence is small. 19

24 my_model(x[3], a[1], s[3]) { array xt[3], vertex[30], Sval[9]; xt[1]=x[1]; xt[2]=x[2]; xt[3]=x[3]; vertex = [ 0.0, -8.0, 0.0, -2.0, -6.0, 0.0, -1.7, -4.0, 0.0, -1.0, -2.0, 0.0, -0.3, -0.0, 0.0, 0.0, 2.0, 0.0, -0.3, 4.0, 0.0, -1.0, 6.0, 0.0, -1.4, 8.0, 0.0, -1.0, 10.0, 0.0 ]; Sval = [1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0]; smoke = hfconvcurve(xt, vertex, Sval, 1.0); noise =hfa_turbulence(x,1.05); } s[1] = smoke & noise; s[2] = smoke & noise; s[3] = smoke & noise; if(x[2] > -6) then s[1] = s[1] * (1 - (x[2]+6)*0.01); s[2] = s[2] * (1 - (x[2]+6)*0.01); s[3] = s[3] * (1 - (x[2]+6)*0.01); endif; my_model = 1; Figure 14. ThinSmoke.hf 5.3 Animation We simulate animation by rotating the camera. The animations with general NPR techniques using strokes typically have flicker. The coherence between frames cannot be maintained because the strokes are calculated again with changing the view. Our method of paragraph 3.3 can make flicker-free animation because it uses density values only without strokes. However, there is a part where the notched line that is the problem of density threshold color method similar to the jaggy. The model used for making animation is a steam model of a density threshold color method. The camera is rotated about y axis by 360 degrees. Bitmap images 20

25 of 100 frames were generated with applying the following options to the command line option of POVRay. +W320 +H240 +KFF100 The +W320 is width of image, and the +H240 is height of image. The +KFF is key final frame. Other parameters were all set by default, key initial frame is 1, initial clock is 0 and final clock is 1. The sample animation plays image with five frames per second. Only the camera is moved in this animation with steam fixed. The volume data changing at every frame is necessary for animation where steam moves. Figure 15. Animation frames. 21

26 Chapter 6 Conclusion In this paper, we proposed a new artistic rendering method for functionally defined volumetric objects. Gaseous object is defined by function with turbulence, and the defined functional model is converted into the volume data set. The technique using the density threshold colors is able to produce cartoon-like images that express gaseous objects with long lines of peculiar strength. In addition to steam rendering, it was possible to express thin smoke. The method of stroke style rendering is able to express hand-drawn steam. The density threshold color method can be used for animation. 22

27 Chapter 7 Future Work Our research and development has five future work directions from simple to more complex: First, in the density threshold color method, a notched line that goes out according to a certain threshold has to be changed to a gradual line. Second, the method of stroke style concerns only simple curve stroke. Our future work is to render various stroke types such as brush strokes, and we consider it possible to calculate a color of smoke from the density and the color of the background object. Third, we need to propose an approach to efficiently and probably locally convert a functional model into a voxel array (or ADF). This approach can improve direct volume rendering of functional models. Fourth, it is necessary to automatically define thresholds of the density values to generate better images. Moreover, it is improved that a notched line is generated depending on a certain threshold. Finally, for making animation, the generation of volume changing in time is needed because the object changes at every frame. As a result, the number of volumes becomes huge. It is possible to solve that problem by improving program to supplement the volume from the beginning and the last volume. 23

28 References [1] Andrew Selle, Alex Mohr, Stephen Chenney, Cartoon Rendering of Smoke Animations, International Symposium on Non-Photorealistic Animation and Rendering NPAR, 2004, pp [2] Takahashi Hoshino, Yoshinori Dobashi, Tsuyoshi Yamamoto, Cartoon Rendering of Smoke, IPSJ SIG Technical Report, [3] Oliver Deussen, Thomas Strothotte, Computer-Generated Pen-and-Ink Illustration of Trees, Proc. SIGGRAPH 2000, 2000, pp [4] Ronald N. Perry, Sarah F. Frisken, A New Framework For Non-Photorealistic Rendering, Technical Report 2001-TR , [5] Aidong Lu, C. Morris, D. Ebert, P. Rheingans, C. Hansen, Non-photorealistic Volume Rendering Using Stippling Techniques, Proceedings of IEEE Visualizaton 2002, pp [6] L. Neumann, B. Csebfalvi, A. Konig, and E. Groller, Gradient estimation in volume data using 4D linear regression. Computer Graphics Forum, 19(3), August 2000, pp [7] Penny Rheingans, David Ebert, Volume Illustration: Non-photorealistic Rendering of Volume Models. IEEE Transactions on Visualization and Computer Graphics, vol 7, no 3, pp [8] Zoltan Nagy, Jens Schneider, Rudiger Westermann, Interactive Volume Illustration, Vision Modeling and Visualization, 2002, pp [9] David S. Ebert, Volumetric Modeling with Implicit Function (A Cloud is Born), ACM SIGGRAPH 97 Technical Sketches Program, August [10] V. Adzhiev, R. Cartwright, E. Fausett, A. Ossipov, A. Pasko, V. Savchenko, "HyperFun project: a framework for collaborative multidimensional F-rep modeling", Implicit Surfaces '99 Workshop, 1999, pp

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