Volume Modelling: Representations and Advanced Operations

Volume Modelling: Representations and Advanced Operations V.V. Savchenko1, A.A. Pasko1, A.I. Sourin2 , T.L. Kunii3 1) 2) The University of Aizu, Aizu-Wakamatsu City, Fukushima, 965-80, Japan, Nanyang Technological University, Singapore, 3) Hosei University, Tokyo, Japan E-mail: [email protected] Abstract This paper presents our approach to volume modelling which combines volume representations by voxel data and by continuous real functions. We discuss the main differences between direct volume visualization and modelling with voxel data, questions of conversion between two representations including volume reconstruction from contour data. We illustrate the approach by several advanced operations on a volumetric object: set-theoretic operations, sweeping, hypertexturing, feature-based sculpting, splitting, spatial and temporal transformations. 1. Introduction Increasingly developing volume graphics provides tools for visualization and manipulation of voxel data represented by scalar values in the nodes of a regular 3D grid. Its perspective is to replace "polygon pushers" by rendering hardware based on a volume buffer. In addition to rendering, volume graphics provides few operations on volumetric objects. They are Boolean operations and metamorphosis in discrete space. However, "very little has yet been done on the topic of volume modelling" [1]. There are such problems here as pointed by Kaufman et al. [2]: • Sculpturing in discrete space. • Feature mapping and warping. • Morphing and changing of the model. • Intermixing volumetric and analytically defined geometric objects. One can observe these topics being intensively investigated in geometric modelling. Above mentioned operations and many others like blending, sweeping, hypertexturing have found quite general solutions for geometric solids represented by continuous real functions as f ( x , y , z ) ≥ 0 . This representation is usually called an implicit model or a function representation. Set-theoretic solids have been successfully included in this representation with the application of R-functions and their modifications [3-11]. The paper addresses the following questions: • How to include a volumetric object as a primitive when using or designing a solid modeller? • How to visualize a functionally defined object using volume visualization? • Why does intermixing voxel objects and analytically defined geometric objects need voxelization of the latter ones resulting in the loss of accuracy? • How to apply well-known and recently proposed advanced geometric operations to volumetric objects? This paper mainly presents the state of our volume modelling project. However, we describe here our new results on function-to-voxel conversion, sweeping by volumetric objects, and improved hair modelling technique. In the following sections, we discuss the main differences between direct volume visualization and modelling with voxel data, describe methods of voxel-tofunction and function-to-voxel conversion and connected problems, illustrate our approach by some advanced operations on a volumetric object. 2. Volume visualization and modelling Volume visualization is one of the most interesting and fast-growing areas in computer graphics today. In terms of its impact on modern computer graphics, it is sometimes compared to "raster graphics revolution" that happened in the early 1980s. Today's volume visualization system creating high-quality images from scalar and vector multidimensional data are actually using the model that was classified as spatial occupancy enumeration more than 15 years ago [12]. Most volume visualization techniques are based on one of about five foundation algorithms [13]. Some graphics workstations recently arrived in the market can perform high-speed volume rendering that could be considered as a step toward replacing "polygon pushers" with "voxel pushers". By comparing raster graphics with its "next of kin" volume visualization, we would not avoid identifying some problem that is common for both of them. This problem is a limited set of operations that can be directly applied to pixels and voxels. To perform some complex geometric operation, one has to raise the level of abstraction and consider other geometric models that then can be visualized with the raster/voxel model. Volume visualization can be thought as mapping 3D objects onto 2D images. The scope of modelling is quite different. It starts with some 3D objects, manipulates them and results in new 3D objects. A generated object model can be used further in different ways: • Visualization. • Integral properties calculations (volume, center of inertia, and so on). • Rapid prototyping. • Manufacturing. • Application problems solving. There is a trend in volume graphics to distinguish between modelling and visualization. The research in modelling includes scattered data interpolation [19], transformation from one voxel data structure to another by manual 3D painting [14,15], Boolean operations and linear transformations [16,17], and metamorphosis [18]. We believe that the spatial occupancy enumeration requires unification with another model that would imply more sophisticated operations and ability to use voxel model together with another solid model. This higher level geometric abstraction would allow one to use all abilities of solid modelling with the advantages of volume visualization. Such fruitful alliance could be achieved if we combine together spatial occupancy enumeration and the function representation of solid objects. Here, some real function f ( x , y , z , vol ) ≥ 0 can be introduced, where vol is a voxel data parameter and x, y, z are Cartesian coordinates. This function has positive values for the internal points, equal to zero for the boundary points and less than zero for the points outside the solid. Elsewhere in [4], we show how this function representation (F-rep) can be used in solid modelling. In particular, we discussed that some solid models like Constructive Solid Geometry (CSG), sweeping, and skeleton-based implicits can be unified under the platform of F-rep. Modelling CSG objects, we define solid primitives with the inequalities f ( x , y , z ) ≥ 0 and use exact analytic representation for set-theoretic operations either in min/max form or in more complex one with R-functions [3,4]. The latter form provides any desirable continuity of the resulting function. Different solid object metamorpho- sis and sophisticated operations can be defined as function superposition aiming to get a result that is in turn defined with the real function f ( x , y , z ) ≥ 0 . This approach allowed us to attend and often to find quite interesting solutions to the problems of nonlinear metamorphosis, sculpting, sweeping by arbitrary CSG solid, NC machining, and hypertexturing. Using F-rep together with voxel data allows us to raise the level of abstraction from the direct rendering as used in volume visualization to modelling with voxel data. We are able to transform the existing voxel data and create new volumes applying geometric operations. In the following sections, we combine our F-rep model with spatialoccupancy enumeration technique and implement some advanced operations. 3. Conversions between object representations In this section, we discuss conversions between two representations: • Voxel-to-function conversion is useful when applying advanced operations to the object, especially while intermixing voxel data with functionally defined objects. • Function-to-voxel conversion can be applied to reduce time of function evaluations and to use volume visualization in applications. 3.1 Voxel-to-function conversion In this section, we observe methods of converting voxel data to a function, and discuss connected problems. Our concept is to treat volumetric objects as any other functionally represented primitives while applying advanced operations. The difference between traditional primitives (sphere, torus, blobby objects) and voxel objects is that the latter ones are defined by an unknown function of three variables sampled at the nodes of the regular grid or even at scattered points and its values are usually positive integers. To provide function continuity, one can apply the following well-known procedures to voxel data: Interpolation and smoothing. This can include the simple trilinear interpolation for a regular grid. In the case of scattered data, one can apply volume splines, multiquadrics and other methods [19]. Here, the problem of choosing an adequate function for local or global interpolation arises. To perform any further operations, a modeller has to keep rough data. Filtering technique. Rough data are processed by different kinds of filters. The modeller has to keep rough data for the function evaluation. If an approximate de- scription of data is known and it has high-frequently error distribution, filtering with the convolution can be guaranteed. These algorithms are simple and fast. Spectral algorithms. Voxel data can be replaced by analytical descriptions with the use of the Fourier transformation or wavelets [20]. This allows the modeller do not keep rough data but calculations become much slower. After getting a continuous positive function, a given level value is subtracted to make the defining function change its sign on the certain isosurface. After these procedures are done, the defining function can be used in settheoretic and other operations. However, several conceptual problems exist: • Discontinuous density function. In practice, discrete voxel data in some applications represent object's density. A density function of a 3D object is discontinuous on the object boundary. If we are given the exact density function (emergence of such models is predicted in [21]), some special measures will be necessary to convert it to the continuous defining function. • Distance property. Some operations, i.e., blending, require a defining function to have a distance property, which means monotonous descent according to the distance from the object. To provide this property outside the volume, one can intersect the object with the bounding volume using R-functions. However, to provide this property everywhere outside the object its bounding volume has to be as close to the surface as possible, at least in the area of interest. Note that in the conversion from a function to voxel data, it is preferable to generate not binary but integer voxel data to preserve the distance property to some extent. • Internal zeroes. The voxel-to-function conversion can result in nested disjoint objects. Moreover, isolated points, lines and surfaces with zero function value can appear inside the object. This is known as the "internal zeroes" problem, which can cause incorrect point membership classification. For example, if only function value is used for the classification, an "internal zero" point can be classified as a boundary point. Usually internal zeroes are avoided in constructing defining functions and special techniques (segmentation or space filling) have to be applied for this in the voxel-to-function conversion. 3.2 Function-to-voxel conversion: case study of reconstruction It can be necessary to convert a functionally represented object to voxel data. The simplest way to do this is to generate binary voxel data by assigning the value of 1 to the point where the defining function is not negative and the value of 0 otherwise. This leads to so-called ob- ject-space aliasing and requires a filtering technique to be applied to improve the surface quality [22]. On the other hand, having a real continuous function, one can try to use it to improve the resulting data. If the range [Fmin, Fmax] of the function values is known, the following scaling function gives one byte positive integers for the voxelization: I = 255 (F − Fmin ) (Fmax − Fmin ) The level value on the object surface corresponds to F = 0. In this section we describe our case study reconstruction of an object from cross-section contour points and its further voxelization for manipulation and rendering reconstructed objects stored in a volume raster. Here, the reconstruction includes the following steps: 1. Retrieving and editing contours represented by 2D point coordinates; 2. Calculation of parameters of a defining function for each contour; 3. Function evaluation on a sparse grid; 4. Sample-based approximation of function values on the given raster; 5. Generation of voxel data. Traditionally, CSG uses simple geometric objects for a base model which can be further changed by applying such operations like set-theoretic operations, blending or offsetting. We consider the problem of reconstruction of solids from cross-sectional contours as an example of creating a complex base model. In addition, it seems to us that the combination of real cross-sectional contours extracted from CT data and some additional, say handdrawn or user edited, contours looks promising and can bring benefit in practical applications. For the reconstruction/voxelization we use the combination of contour points from femur CT data and extra points simulating interior boundary of the femur to construct its real defining function and then to voxelize the object. Further, we demonstrate some operations on this voxel object. For the visualization of voxel objects, we use a simple ray-marching algorithm, however widely used volume-rendering tools can be exploited. Wang and Kaufman noticed in the paper [22] that due to the discrete nature of the model representation, volume graphics suffer some disadvantages. Particularly, if not properly synthesized, voxelized volume models are imprecise, they generate jagged surfaces. In our experiments, we confirmed by visual inspection that the 2D sample-based interpolation is faithful to avoid object-space aliasing. We examine here two our reconstruction algorithms: - reconstruction using monotone formula (see 3.2.1); - reconstruction using volume splines (see 3.2.2). Since the evaluation of defining functions is time consuming for both algorithms, we also discuss in 3.2.3 the accelerated voxelization using a sample-based interpolation scheme. A number of researchers have investigated the problem of the reconstruction of surfaces from cross-sectional contours. A brief review can be found in [5]. The main difficulty in these algorithms is that data are presented as polygonal contours. Connecting them into surface polygons is a very difficult combinatorial problem. The surface generated by the triangulation method depends heavily on the choice of the nodes for each contour. The main problem of this approach is the case where different numbers of contours appear in different cross-sections. A generalized model, called the homotopy model, was presented by Shinagawa and Kunii [23] to reconstruct surfaces from cross-sectional data of objects. Homotopy is used for the reconstruction of parametric surfaces with the help of the toroidal graph representation. Intuitively, it is clear that transition and scaling between the cross-sectional shapes can be defined using interpolation or homotopy map between functions. If we apply one-dimensional interpolation along the z-axis, the object can be described as F(x,y,z) = (1-g(z))F1(x,y) + g(z)F2(x,y), Figure 1b shows an object reconstructed from three polygonal cross-sections. Two triangles and one square contour are used, Figure 1a. Let a two-dimensional simple polygon is defined by a finite set of segments. The segments are the edges and their extremes are the vertices of the polygon. A polygon is simple if there is no pair of nonadjacent edges sharing a point and convex if its interior is a convex set. We consider the problem of representation of polygons by continuous real functions F(x,y) taking zero values at polygon edges. The algorithm has to satisfy the following requirements: ‚ It should provide exact polygon description as a zero set of a real function; ‚ No points with zero function value should be inside or outside a polygon; ‚ It should allow processing a simple arbitrary polygon without any additional information. Rvachev [26] and Peterson [27] independently proposed to represent a concave polygon by a set-theoretic formula where each of the supporting half-planes appears exactly once and no additional half-plane is used. This approach does not generate any internal or external zeroes when applying R-functions. where g(z) is a blending function with range [0,1]. Note that this method produces a connected surface if the areas F1(x,y) ≥ 0 and F2(x,y) ≥ 0 have nonempty intersection. The proposed algorithm can generate highly concave and branching objects. In the case of two or more branches, the cross-section is represented as a union of several regions separately converted to their defining functions. This method can be also applied for the reconstruction from nonparallel cross-sections [5]. (a) (b) 3.2.1 Reconstruction with a monotone formula The reconstruction from cross-sections requires a defining function for each individual cross-section. This function can be expressed by R-functions based on a monotone formula for a polygon in a cross-section. (a) (b) Figure 1. Reconstruction of a solid from three parallel polygonal cross-sections. Figure 2. A concave polygon (a) and a tree (b) representing its monotone boolean formula. The monotone formula for the defining function of the polygon in Figure 2a results in the tree structure shown in Figure 2b. To evaluate the defining function value at the given point, the algorithm traces the tree from the leaves to the root, evaluates defining functions of half-planes and applies corresponding R-function to them. The details of the algorithm are given elsewhere [6]. ∫∑ Ω m!/α! (Dαu)2dΩ→ min, |α |=m where m is a parameter of the variational functional. The zero set of the function f(x,y)=U(x,y) - fc(x,y) Figure 3. Femur reconstructed from 53 polygonal slices. As an illustration of the implementation of the monotone formula discussed above, Figure 3 presents the reconstruction of a femur from a set of 53 parallel crosssections. We apply the linear interpolation between real functions generated for cross-sections by our algorithm. Note the non smooth femur surface resulting from the polygonal nature of the cross-sectional data. The next approach lets us to provide a more smooth shape of an individual cross-section and as the result a more smooth overall shape. 3.2.2 Reconstruction with a volume spline Consider the formulation of the reconstruction problem. Let Ω be an n-dimensional domain of arbitrary shape that contains a set of points Pi = (xi 1, xi 2, ..., xi n): i = 1,2,...,N. We assume that the points Pi are distinct and lie on some surface. The goal of the reconstruction is to find a smooth function f(x1, x2, ..., xn) that approximately describes the surface. In the two-dimensional case, a contour is represented by the equation f(x,y) = 0. The general idea of our algorithm [5] is to introduce a carrier solid with a defining function fc and to construct a volume spline U interpolating values of the function fc in the points Pi . The algebraic difference between U and fc describes a reconstructed solid. The algorithm consists of two steps. At the first step, we introduce a carrier solid object which is an initial approximation of the object being searched. In the simplest case, it can be a ball. Then data set r associated with the points {ri = fc(Pi ): i = 1,2, ..., N} is calculated at all given points. At the second step, these values are approximated by the volume spline derived for random or unorganized points [28]. The problem is to construct the interpolation spline function U(Pi ) ∈ W2m(Ω), where W2m is the set of all functions whose squares of all derivatives of order ≤ m are integrable over Rn, so that U(Pi ) = ri , i = 1,2,...,N, and has minimum energy of all functions that interpolate the values ri . This conforms to the following minimum condition which defines operator T: for n = 2 approximates the unknown contour. 3.2.3 Sample-based voxelization In the function-to-voxel conversion, we face the problem of evaluating the function at grid points. The work required is proportional to the number of the grid nodes and the number of scattered points in the cross-section. The amount of computations becomes significant, even for moderate nodes number when the number of scattered points in the cross-section is increasing. The special methods to reduce processing time were developed for thin plate splines [25]. However, we need a more general acceleration scheme to be applied to scattered points with values of different functions (see 3.2.1 and 3.2.2). A possible way to reduce the computation cost is to use a finite element method (FEM) approximation. We do not consider this problem in detail, but only sketch a basic idea of the variational schema and describe our software algorithm. The problem of constructing interpolating functions by FEM for 2D space Ω is stated as follows: given data points Pi (x,y), i = 1,2,...,N, are scattered in the sense that there are no assumptions about the disposition of the independent data and data set r is associated with the points. In our case, we solve the general problem of smooth approximation of the scattered data (see [28]). We have to construct a smooth function σh(x,y) which takes on the value ri at points Pi (x,y) ∈ Ω, if it is possible, or satisfies the condition: ||Aσh - r||2 = min, where A is some linear bounded operator. Along with this condition the function σh(x,y) has minimum energy of all functions that interpolate the values ri . This conforms to the following minimum condition: ∫ Ω [(∂σh/∂x)2 + (∂σh/∂y)2]d Ω = min, where the integrating is taken over the two-dimensional space Ω. We use the piecewise linear approximation over the rectangular mesh. The bilinear basis function ωij(x,y) = ωi (x)ωj(y) corresponds to each node of the rectangular mesh. In fact, it is possible to use more smooth basis functions and more complicated minimum conditions. The solution of the data approximation can be found in the form: σh(x,y) = ∑ σ ω (x,y). 7. Quadratic interpolation over the voxel space. The object shape between cross-sections is defined with the quadratic interpolation of the grid values for three neighboring slices. Retrieval of contour data & construction of volume splines Sorting Approximation of FEM matrix Factorization Function evaluation in 53 slices Voxel approximation for 150 voxel layers 11.14 2.18 1.09 1.91 84.73 46.23 ij ij i, j The voxelization algorithm consists of the following steps: 1. Generation of original scattered data. The processed number of function evaluations in our experiments is 2601 for one cross-section or slice. The number of crosssections is equal 53. The size of voxel data is the 150x150x150. 2. Sorting the data. At this stage, we put calculated data on the grid of 50x50 size according to the chosen network pattern. For instance, a 3x3 pixel area applied in halftone dithering can be used. One point network pattern scheme was employed in the presented experiments to fill the 3x3 areas and to construct the logical structure of the FEM matrix. During the summation for calculating the matrix, all points with the coordinates which do not belong to the intersection of basis functions should be discarded. To avoid the multiple verification of the point membership in this collection, it is reasonable to provide sorting the initial data set that is to place points according to the cells numeration. Such sorting allows us to effectively calculate the components of the right part of the system of linear equations and to calculate the function values for each point in the slice. 3. Numerical assembly. This step provides calculating the values of all elements of the FEM matrix according to the logical structure. 4. Cholesky factorization. The problem of approximating the data comes to the solution of the system of linear algebraic equations: Ax = b, where A is a band matrix of coefficients, x is a vector of unknown node values and b is a vector of right parts. Cholesky decomposition constructs a lower triangular matrix L whose transpose Lt can itself serve as the upper triangular part. 5. Calculating the right part for the resulting linear algebraic system, solving the system by backsubstitution. 6. Interpolation over the mesh. The presented techniques thus far assumed that the array being calculated is smaller than the voxel array being displayed. To this end, the problem is to calculate the function values for each point in the slice. Table 1. Processing time. All described parts of our software tool have been implemented in C language with double precision. The measured processing time on the Silicon Graphics Indigo2 workstation for the above mentioned steps are presented in Table 1; time is given in seconds. We see from Table 1 that the most expensive part is evaluating the functions at the chosen network pattern points. The function evaluation includes constructing of the right parts and solving systems of linear algebraic equations, however we have to notice that this fraction of processing time is constant for each slice and this calculations take 0.21 seconds for each slice. In fact, as we mentioned above, the function evaluation depends on the number of nodes in a cross-section and the amount of computations can be significant. This algorithm allows us to decrease the number of expensive function evaluations approximately by a factor of nine with acceptable accuracy. (a) (b) (c) Figure 3. (a) 150x150x150 voxelization of a reconstructed femur with one point network pattern scheme; (b) splitting of the reconstructed femur; (c) blending between the split parts. Figure 3a shows a femur reconstructed using volume splines (3.2.2) and voxelized with the sample-based interpolation as described above. Figure 3b presents two separated parts of the reconstructed femur. Figure 3c demonstrates the result of blending [7] between two separated parts. It shows that the proposed voxelization procedure keeps the distance property of defining functions that is required for further operations. We have to underline that with the binary voxelized objects, blending and several other operations do not produce expected results. In the paper [5] we have investigated the question of the numerical error estimation. Our experiment lets us draw a conclusion that the error of the reconstruction with using volume splines and the linear interpolation between slices is about two percent. FEM interpolation gives the error approximately 0.7 percent for the worst case of interpolating extreme points of the voxel space. We suppose that it is sufficient for the voxelization of free forms in practical applications. Temporal transformation or metamorphosis can be useful in different applications, for instance, artistic animation or recognition tasks. Figure 5 illustrates shapes constructed by the linear interpolation of two shapes with using a linear blending function which specify the relative contribution of each shape on the resulting blended shape (examples of using nonlinear blending function are given , for example, in [29]). More complicated transformation will be considered later. 4. Advanced operations on volumetric objects We describe and illustrate the following operations on volumetric objects: set-theoretic operations, metamorphosis and sweeping by a volumetric object, hypertexturing, feature-based sculpting and splitting. A volumetric object was converted to a continuous real function by the trilinear interpolation and level value subtraction. All operations result in a new procedurally defined real function of three variables. 4.1 Set-theoretic operations and metamorphosis In Figure 4, we create a complex object made of a volumetric head and simple CSG primitives. The head is applied set-theoretic operations and affine transformations to make a drawer that is filled with CSG primitives. Figure 6. Volumetric head cut with a moving CSG cutter. In [8], we study sweeping defined with real functions. We managed to solve several problems that could not find general solution in one model before. There, we were able to define a complex swept with any arbitrary CSG object moving by any complex trajectory with self-intersections. In this paper, we apply the same model to volume data. In Figure 6, we cut volumetric head with a CSG cutter that was implemented as subtraction a swept solid created by moving cutter from the head. We used trilinear interpolation to get a real function defining the head and we did not identify any specific problems here compared to the previously studied set-theoretic operations over CSG and swept object. 4.2 Sweeping by a volumetric object Figure 4. Volumetric head is applied settheoretic operations and unified with CSG primitives. Figure 5. Metamorphosis between a volumetric head and a union of three analytically defined objects. Continuing our research on sweeping by an arbitrary functionally defined solid, we expand our consideration to sweeping by a volumetric object. To achieve this goal, we defined a generator, being a volumetric object, with real functions as it is described in the previous paragraph. Here, we met some problem that lies in the nature of volumetric data used. These data, normally, do not have a distance property that is very important for the implementation of our method of sweeping. To achieve it, we set up this distance property artificially so that values of function outside the object obey some simple rule of quadric behavior defined as follows: New_value = Old_value*(1-x*x-y*y-z*z). Figure 7. Sweeping by a volumetric head. This rule, applied after the trilinear interpolation is made, does provide a distance property for any point outside the volume. After that, the method described in [8] is applied. The results of sweeping by a volumetric head is shown in Figure 7. Figure 8. Hair styling. Hair strands are unified with a volumetric head. In Figure 8, the hair before and after styling are shown. 4.3 Hypertexturing Elsewhere in [9, 10], we present our approach to hypertexturing. We define the hypertexture with a real function f ( x , y , z ) ≥ 0 and then unify it with some functionally defined object where the texture should be applied. With this method, we could define moss, corrosion, snow, fur, and hair. We extended this approach to volume data and here present the improved hair modelling technique that allows us to grow and style solid hair on a volumetric head. The hair is thought as a set of functionally defined generalized solid cylinders. Their union gives the hair that can be then unified with any other solid as it is done in CSG. The process of hair growing and styling is done as follows. We select some point in 3D that is supposed to be an origin for the local coordinate system associated with the hair. Then, for any arbitrary point we define its direction cosines α , β , γ . Next, we find the direction cosines for the single hair strand nearest to this point. The hair strands are defined as long solid cylinders passing through the nodes of a pseudo-random spherical parametric grid. After the direction cosines of the nearest cylinder-strand are found, we just check if the tested point belongs to this cylinder. The radius of the hair strand and the angular distance between the strands are defined by the user. Styling and cutting hair are achieved with nonlinear coordinate transformations and cutting operations done with subtraction of different CSGs or swept volumes. For hair behavior we used pseudo-physical laws based on different formulae defining distance properties for the given head. Figure 9. Volumetric head and hair unified with a pillar. Figure 9 presents another example of intermixing volumetric and analytically defined geometric objects to form a solid object with curved surfaces. In this example we use volume data of a voxel head with simulated hair in synthetic carving on a pillar. 4.4 Feature-based sculpting and separation Feature-based sculpting provides deformation defined by displacements of control points linked to the object features. Two sets of points Q and D are given. Points Q are defined for the non deformed object. Points D result from Q when the deformation is applied to the initial object. Control points can be freely chosen by the user since the algorithm does not use any regular grid. It can be thought as a nonlinear 3D space mapping defined in a finite set of points [11]. The problem is to find a smooth mapping function that approximately describes the spatial transformation. We define the inverse mapping needed to transform our object as Q = U(D)+D, where the volume spline U(D) generates displacements of the initial points Q. We use a volume spline interpolating scattered data as mentioned above. - calculate a mapping function and apply it to the reference half-space; - perform a set-theoretic subtraction of a deformed reference half-space from the initial object. Figure 10. Volumetric head deformed using space mapping. Figure 10 shows the volumetric head deformed using space mapping. Only four control points have been used for the deformation. Figure 12. Separating the volumetric head with a curved half-space. The separated 3D object is shown in Figure 12. 5. Conclusions Figure 11. Combined mapping in 3D space: spatial and temporal transformations. Previously considered example (see Figure 5) of primitive metamorphosis between objects having different genus shows that it is a difficult problem to reach visual effects of smooth transition between the shapes because they are not homeomorphic. In Figure 11, we illustrate the 3D application of combined mapping, that is spatial and temporal transformations. The problem is to obtain the visually smooth transformation between the initial ellipsoidal shape (genus 0) and the final shape (genus 1) in presence of obstacles. A defining function for a final shape of a block with hole was obtained using set-theoretic operations on 3D primitives. To solve the problem we use a combination of the linear interpolation or temporal transformation of two shapes and nonlinear space mapping discussed above. We define the space mapping by displacement of control points in the local coordinate system of the moving object. The same nonlinear space mapping approach can be applied to design a curved half-space separating two parts of a volumetric object. Control points for partition or separation of 3D geometric object are placed on or near the surface and the following scheme is used: - specify a defining function for the object; - define one-to-one projections of control points onto a reference half-space, for example a plane; An approach to volume modelling combining representations by voxel data and by real functions of three variables is presented. Questions of conversion between two representations are discussed. We illustrate our approach by several advanced operations on a volumetric object. The traditional areas such as solid modelling and volume graphics also can benefit from this approach: Solid modelling. A new source of shapes appears. Instead of voxelization of exactly represented objects such as spheres, tori and others (as discussed in [24]), a solid modeller keeps as high accuracy as possible while dealing with volumetric objects. 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