Fast CSG Voxelization by Frame Buffer Pixel Mapping
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1 Fast CSG Voxelization by Frame Buffer Pixel Mapping Shiaofen Fang 1 Duoduo Liao Department of Computer and Information Science Indiana University Purdue University Indianapolis ABSTRACT This paper describes a fast algorithm for the volume conversion and rendering of CSG models constructed from both geometric and volumetric primitives Using 3D texture mapping and frame buffer pixel operations, the algorithm can interactively generate a binary volume of the CSG model The result can be used for volume rendering and other applications Boolean operations are implicitly computed by a Point-Classification Map, and implemented by a hardware assisted frame buffer pixel map The algorithm can be applied to any regions of interest of the model, thus provides a multi-resolution rendering solution through dynamic voxelization of the viewing regions Since no pre-processing is required for any change of the CSG tree, it can be used as an effective rendering tool in a volumetric CSG modeling environment Keywords voxelization, CSG modeling, volume rendering, 3D texture mapping 1 INTRODUCTION The Constructive Solid Geometry (CSG) representation allows users to define complex 3D solid objects by hierarchically combining simple geometric primitives using Boolean operations and affine transformations [14, 9] It is a very popular and powerful solid modeling scheme, and is particularly suitable for interactive object manipulations and design Traditionally, CSG primitives are defined by simple analytic objects, such as cubes, cylinders and spheres Some recent CSG algorithms can also support primitives that are general solid models defined by their boundary surfaces Using voxel-based volume representations, a further extension can include objects extracted from volume data sets using intensity thresholding These volume data sets may come from various types scanning of real objects, such as CT, MRI, and microscopy images, or from the sampling of implicit or procedural functions Such extended CSG models are sometimes called Volumetric CSG models [7, 6], and are very useful in applications such as medical imaging, W Michigan St, SL 280, Indianapolis, IN sfang@csiupuiedu surgical simulation, and amorphous phenomenon modeling For the simplicity of presentation, in this paper, we will still refer to this extended model as CSG model Due to the lack of explicit representation of surface boundaries, CSG display is not directly supported by standard graphics systems Although several interactive CSG rendering algorithms have previously been developed [8, 13, 17, 20], they cannot be directly applied when volume data sets are involved In principle, there are two potential solutions for the rendering of volumetric CSG models The first is to explicitly extract an iso-surface from the volume data set, and convert the model into a regular geometric CSG model, which can then be displayed by an existing CSG rendering algorithm This method is, however, very slow and difficult for two reasons: (a) the iso-surface extraction process is time-consuming, and (b) the extracted iso-surface normally contains an extremely large number of tiny polygons, which are difficult and expensive to display and operate The second solution is to first convert the CSG model into a voxel-based volume representation, and then display the resulting volume using a standard volume rendering algorithm This solution, however, requires a fast volume conversion (voxelization) process for volumetric CSG models The voxelization problem for general curve and surface objects have been extensively studied [5, 4, 10, 11, 16] Voxelization algorithms for more complex models have also received more research attention recently Conceptually, CSG voxelization is a set membership classification problem with respect to the CSG object for all sampling points in a volume space Early CSG voxelization methods can date back to the spatial enumeration representation by point classification [12] More recent work includes the Beam Oriented method [15], the volume sampling method [18], the point sampling algorithm [1], the octree-based algorithm [7], and the distance volume algorithm [2] A common problem of these algorithms is that the voxelization is slow and does not support interactive modeling applications In [6], we described a hardware accelerated CSG voxelization algorithm that uses combinations of frame buffer blending functions to carry out the CSG Boolean operations Although that algorithm is reasonably fast for small scale interactive applications, its performance have been limited by the need of generating an intermediate texture object for each Boolean operation node, and by the hardware restrictions in blending function combinations In this paper, we present a new CSG voxelization and rendering algorithm that employs a Point-Classification Map (PCM) for Boolean operations based on a frame buffer color encoding scheme The result is a binary voxelization of the CSG model, which can be used for volume rendering and other applications The algorithm is clearly faster than the one given in [6] To overcome the volume resolu-
2 tion limitations, this algorithm can also be applied to any regions of interest of the model This provides a natural multi-resolution rendering solution for the rendering of different scales of the CSG scene Since the dynamic changes of the CSG tree do not require any pre-processing time in this algorithm, it is particularly suitable for the interactive volumetric CSG modeling applications In the following, we will first discuss, in Section 2, some basic concepts and definitions regarding CSG modeling using volume data sets The Point-Classification Map (PCM) method will be described in Section 3 Section 4 presents the binary CSG voxelization algorithm The CSG volume rendering process using the voxelization result is described in Section 5 Section 6 discussed some implementation issues and results We conclude the paper in Section 7 with some further remarks and future work T Teapot U - T F CT data T sphere Figure 1: A CSG model using a CT data set T: affine transformation; F: thresholding operation 2 THE CSG MODEL Let a scalar volume V be represented by an intensity function IèP è, where P is sampled over a 3D regular grid in a volume space A solid object can be extracted from a volume by thresholding with an intensity interval ës; të (s é t) More specifically, a point P belongs to the object if and only if IèP è 2 ës; të This process may also be defined as a thresholding operation F s;t : V! R 3,ie F ës;të èv è=fp 2 R 3 : IèP è 2 ës; tëg Multiple intensity intervals may also be used in the thresholding operation The result is the union of the objects defined by the individual intensity intervals: èv è=fp 2 R3 : FSës i ;t i ë IèP è 2 ë ës i;t iëg p bits 0 i p n 2-1 n=p+q bits q i*2 +j i j C C1 op q bits 0 j q 2-1 Figure 2: Combining two PCMs C2 In general, a CSG model can be constructed by the following two types of primitives: 1 A general solid model defined by its boundary surfaces 2 A solid object defined by a thresholding operation from a volume data set The thresholding operation provides additional flexibilities through the changes of the thresholding intervals to generate objects from different layers of the volume This can be considered part of the CSG editing process, and is an essential feature for the designing of volume based objects It should be mentioned that although volume data sets are used, the Boolean operations are only carried out with the (binary) solid objects extracted from the volume data An example of such CSG models is given in Figure 1 The binary voxelization of a CSG model is the process of generating a binary volume of the CSG model within a given volume domain (region of interest) Only 0 and 1 intensity values are generated in binary voxelization, with 1 representing an inside point and 0 outside point The result from the binary voxelization can be directly rendered, as a solid object, by 3D texture mapping It can also serve as a volume space mask for the volume rendering of the CSG model using the original data sets If the binary volume is considered as a spatial enumeration representation of the CSG object, it can also be used in many other solid modeling applications such as integral property computation, finite element analysis and layered manufacturing 3 POINT-CLASSIFICATION MAP An essential component of a CSG voxelization process is the membership classification of points in a volume space with respect to the 3D object represented by the CSG model For a given CSG tree, The final CSG classification of a point is uniquely determined by its classifications with respect to all the CSG primitives This relationship is defined by a Point-Classification Map (PCM) that maps a complete primitive classification result to an in/out CSG classification If the CSG tree has n leaf nodes (primitives), numbered from 0 to n, 1, ann-bit binary number can be used to encode the classification results of a point with respect to all n primitives We call this binary number the classification index A1 at the jth bit of the index indicates that the point is inside the jth primitive, and a 0 indicates outside Thus, for a given CSG tree, the classification index uniquely determines the final CSG classification Since there are only 2 n different possible classification indices, a look-up table of 2 n entries (ie the Point-Classification Map) will be sufficient to describe all possible point classification cases The PCM of a CSG tree can be easily constructed by the following recursive procedure
3 PCM PROCEDURE CSG_PCM (CSG_NODE N) IF (N is a leaf node) RETURN the PCM of the primitive; ELSE PCM_left = CSG_PCM (Nleft); PCM_right = CSG_PCM (Nright); PCM_combined = combine (PCM_left, PCM_right, Nop); RETURN PCM_combined; IF This procedure recursively computes a PCM table for a binary node by combining the PCM tables of its two child nodes, as shown in Figure 2 The index domain of the combined PCM is the Cartesian product of the index domains of the two sub-pcms The combined classification result is simply a binary Boolean operation (given by the CSG node) of the classifications in the two sub-pcms, ie c = c 1 éopé c 2 In practical implementation, a common PCM memory space can be shared by all levels of recursive calls to avoid unnecessary memory resource consumption 4 CSG VOXELIZATION In this section, we present a hardware accelerated CSG voxelization algorithm using certain hardware features in 3D graphics systems The required graphics system features include the polygon based graphics engine, frame buffer pixel map and logical operations, and 3D texture mapping In the following, we will first describe a basic algorithm that does not consider the frame buffer depth limitation In fact, it assumes that each pixel of the frame buffer has as many bits as the number of primitives in the CSG tree This basic algorithm will then be modified to accommodate the frame buffer limitation 41 A Basic Algorithm A cubic volume space is first defined over the CSG model The algorithm proceeds slice-by-slice in a front-to-back order by moving the Z-plane, a plane parallel to the projection plane, along the viewing direction to generate slices for all primitives (as shown in Figure 3) For each slice, the algorithm defines the viewing volume of the system as the thin space between two adjacent Z-planes inside the volume boundary, and then renders each primitive within this viewing volume When the algorithm moves from Z-plane to Z-plane, slices of the primitives are displayed and composited onto the frame buffer in a front-to-back order Using proper color coding of the the primitives, the algorithm can directly generate slices of the point classification indices in the frame buffer These classification indices can then be mapped, by the PCM, to form a binary volume of the CSG model The PCM can be implemented by the frame buffer pixel map defined in OpenGL, which is basically a color look-up table for frame buffer pixels The distance between adjacent Z-planes determines the resolution of the volume in the Z direction The resolutions in the X and Y directions are determined by the size of the display window If the CSG tree has n primitives, n distinct colors are assigned to different primitives so that the color code of the jth primitive is a binary number, with the jth bit set to 1 and all other bits set to 0 For a spatial point P,thecolorof P with respect to the jth primitive, C jèp è, is defined as the color of the jth primitive if P is inside the primitive, and 0 otherwise Now, if we combine C 1èP è;c 2èP è; æææ;c nèp è using a logical operation OR or XOR, the result, CèP è, is exactly the classification index of point P Thus, for each Z-plane, the algorithm generates a slice for each primitive, and then composites the slices from the primitives into one single slice of classification indices in the frame buffer using appropriate frame buffer pixel functions This composition process is carried out differently for the two different types of primitives: æ For a primitive defined by its surface boundaries, since only the boundary surfaces are drawn, we need to have a way to determine the interior points The idea is similar to the solid voxelization algorithm described in [4, 6], and is based on the principle that when a ray is shot from a pixel towards the jth primitive in the viewing direction, it has to enter the primitive object first (jth color bit becoming 1) and stay there (keeping the jth bit 1) until it exits the object (changing the jth bit to 0) This can be done by drawing the boundary surfaces of each primitive with a logical XOR operation The clipping hardware of the graphics engine will ensure that only the parts of the surfaces within the thin viewing volume are displayed When a slice is complete, the frame buffer will not be cleared, ie the frame buffer content of the slice will be used for blending operations with the subsequent slices This way, the XOR operation will automatically set the jth color bit to 1 for all interior points, and 0 for all outside points Since the pixel colors on the slice generated by the jth primitive has 0 s at all bit positions except the jth, the XOR operation for the jthprimitivewillhave noeffect tothe classifications of other primitives æ If a primitive, say the ith primitive, is defined by a volume data set, its Z-plane can be rendered by 3D texture mapping using the texture object defined by the volume data set A color look-up table may be used to define a transfer function that maps the intensity values within the threshold interval of the ith primitive to the primitive s color and other intensity values to 0 Apparently, multi-interval thresholding operations may also be similarly defined Before drawing the Z-plane, however, the ith bit needs to be cleared for all pixels on the frame buffer This cannot be done by simply clearing the entire frame buffer since the bits for all geometric primitives need to be retained for the XOR operation We can, however, draw the Z-plane, as a solid color polygon using a color that is the binary complement of the ith primitive s color with a logical AND frame buffer operation This basically set the ith bit to 0 for all pixels in the frame buffer while keeping all other bits unchanged Once the classification indices are generated in the frame buffer for each slice, the Point-Classification Map can be applied to the frame buffer image to generate the final CSG classifications The result (for each slice) will then be copied to the texture memory as one slice of the final volume Using the frame buffer pixel map, the above two steps can be combined into one The pixel map is a hardware implemented look-up table that maps the color of each pixel in current frame buffer to a new color Once defined, pixel map can be invoked by any frame buffer image operations, including the copy operation from frame buffer to texture memory In other words, the PCM is automatically applied during the process of copying the frame buffer image to the texture memory This process will be repeated for each Z-plane, leading to a complete binary volume representation in the texture memory The following procedure summarizes this algorithm:
4 PROCEDURE CSG_Voxelization (CSG_NODE root) Generate a cumulative transformation matrix for each primitive; Define texture objects for all volume primitives; Assign color codes to all primitives; pcm = CSG_PCM(root); Clear frame buffer; FOR (each Z_plane) Define the thin viewing volume; FOR (each primitive) Define system transformation matrix; IF (geometric primitive) Set the XOR logical operation; Set the primitive s color; Render the primitive; ELSE Clear the frame buffer s color bit for this primitive; Bind 3D texture; Set color look-up table; Set OR logical operation; Draw Z-plane by 3D texture mapping; IF Define frame buffer pixel map using pcm; Copy frame buffer to texture memory; 42 Region-Based Voxelization In order for this algorithm to work correctly, the slicing process has to start from the outside of the object In other words, the object needs to be completely contained by the volume space Consequently, the volume space may need to be made very large Since the size (resolution) of the volume is limited by the system memory and the texture memory, large spatial region can lead to low resolution voxel representations of the object details But in order for this algorithm to work correctly, the slicing process has to start To overcome this problem, the voxelization process should be applied on-the-fly to only a region of interest (eg the viewing region) of the scene With a fixed sized volume representation, region-based dynamic voxelization leads to a natural multi-resolution volume rendering approach with desired level of details for different viewing regions This region-based voxelization process, however, requires a small modification of the basic voxelization algorithm The basic algorithm assumes that the slicing process starts from the outside of all primitives But when the Z-plane starts from the front face of a subregion of the spatial domain, the XOR operation may not work correctly for all pixels since the parity is no longer guaranteed for every pixel If we define the 0th slice as the space between the first Z-plane of the region and the z =,1 plane, as shown in Figure 3, this 0th slice can be used to represent and store information about the starting status of each pixel To generate the content of the 0th slice, we can draw all the geometric primitives once using the 0th slice as the viewing volume, with the XOR logical operation set for the frame buffer The resulting frame buffer image will then be carried over to the voxelization process to ensure proper parities for all the pixels Y 0th slice Z Figure 3: 43 A Multi-Pass Approach volume space viewing volume for the ith slice There are two frame buffer limitations that restrict the implementation of this algorithm: 1 The bit depth of the frame buffer is normally limited to 8 to 32 bits The common RGBA frame buffer has 8 bits for each component of the RGB color, and another 8 bits for the alpha channel 2 The frame buffer pixel map used to implement the PCM can only have separate look-up tables for individual RGBA components Therefore, we can have four look-up tables of 256 (8 bits) entries, but not any cross combinations Thus, in general, the basic voxelization algorithm can only handle a CSG tree with eight primitives, ie using only one color component of the frame buffer bits and the corresponding pixel map To overcome this limitation, a multi-pass approach can be employed for large CSG trees The idea is to repeatedly apply the basic volume conversion algorithm to subtrees of the CSG tree At each step, a subtree with at most 8 primitives is chosen and voxelized into a texture object in the 3D texture memory This texture object is then used to replace the entire subtree as a new volume primitive Repeating this process will lead to an iterative trimming of the CSG tree, which eventually will be small enough to be directly voxelized This multi-pass algorithm will have some speed penalty from the extra traffic between the frame buffer and the texture memory But from our experience, the overhead is fairly small This is mainly because that reading from frame buffer to texture memory is a very fast and direct cache operation that does not take up main memory cycles Another drawback is that it puts more pressure on the texture memory requirement, since several intermediate volumes may need to be stored in the texture memory simultaneously, potentially causing more frequent texture memory swapping, and consequently a performance penalty One way to reduce the number of intermediate volumes is to always choose the subtree with the most number of volume primitives, so that the old volumes can be replaced by the newly generated volume Nevertheless, for large CSG trees, the texture memory swapping between the multiple passes is inevitable But as long as the texture memory can hold all the texture objects simultaneously for each pass (at most eight texture objects), texture memory swapping is not a serious problem since it only occurs once for each pass X
5 5 CSG VOLUME RERING The result from the above voxelization process is a binary volume stored in the system s texture memory It can be directly rendered by 3D texture mapping [3, 19], or written to the main memory for other applications A common problem in the direct rendering of binary volumes is that it often suffers severe aliasing problem due to the lack of gray level intensity data for smooth surface boundaries The problem can, however, be alleviated by a multi-pass 3D texture mapping approach Since 3D texture mapping uses trilinear interpolations for sample points around the surface boundaries of the object, the interpolated results will have intensity values between the 0 and 1 binary values This essentially generates some gray level intensity data around the surfaces, and has a similar effect as an anti-aliasing process in surface rendering For smoother results, this process can be iteratively repeated by writing each texture mapped slice (without blending) from frame buffer back to the texture memory as a slice of a refined volume In general, more iterations generate smoother surfaces, with higher rendering cost But our experiments show that two or three iterations are normally sufficient for reasonably smooth surfaces For CSG models involving volume data sets, it is also possible to generate blended images using the intensity values in the original data sets This is done by using the binary voxelization result as a 3D mask that extracts, slice-by-slice, the volumetric portion of the model from a union of the rendering results of the volumetric primitives For each slice, all the volumetric primitives are first individually rendered and combined in the frame buffer using a logical OR operation (or any other desired blending function) The part of this slice that is within the spatial region of the CSG model is then extracted by drawing the same slice from the mask (by 3D texture mapping) using a logical AND frame buffer operation To avoid part of the binary volume being removed by the AND operation, a white background needs to be used in drawing the slices for the volumetric primitives Since the primitives can be rendered using their original volume data, better blending results can be achieved 6 IMPLEMENTATION The volume conversion and rendering algorithm described in this paper has been implemented on an SGI Onyx-2 workstation using a single processor Reality Engine and 64 MB texture memory For large CSG trees, the multi-pass volume conversion algorithm is used with 8 primitives processed in each pass The program is written in C and OpenGL 11 In the examples shown in Figure 4, each model is converted into a binary volume Among the primitives used, a CT head and a knot object are volume data sets The rest are all synthetic geometric primitives A zoom-in result of (f) is given in (i) All the images except (d)(e)(h) are generated directly from the binary voxelization results using a two-pass texture mapping approach as described in Section 5 Images in (d)(e)(h) are generated using the 3D mask approach Timing data for these examples are shown in the following table The timings are affected mainly by three factors: (a) the complexity of the scene, (b) the size of the frame buffer window, and (c) the number of slices The scene complexity determines how fast the graphics system can render the scene during the slicing process The window size determines the size of the slices in the resulting volume, which (along with the number of slices) in turn affects the voxelization speed due to the large number of image movements between the frame buffer and the texture memory sample image a b c d e voxelization time (sec) number of primitives sample image f g h i voxelization time (sec) number of primitives CONCLUSIONS A fast CSG voxelization and rendering algorithm is described in this paper Since volume techniques are used, volume data sets can be naturally included in the CSG construction process Such CSG models allow solid modeling and CAD/CAM applications to input objects from sampled or scanned data sets in the design and manufacturing processes, and are also potentially useful for other applications involving both geometric objects and volume data sets, such as surgical simulation and biomedical data exploration Our algorithm makes use of commonly available graphics hardware features, and is able to provide interactive feedbacks for the modification and manipulation of general CSG models Thus, it can be an effective tool in interactive CSG modeling and volume graphics environments Several future works are being planned The current algorithm is only able to use one color channel of the frame buffer memory We would like to investigate an approach that can make full use of all 32 frame buffer bits in the point classification process for better performance with large CSG trees The algorithm has so far only been tested on relatively small CSG trees The applicability and performance of this algorithm on very large scale CSG problems (eg thousands of primitives) still need to be tested and analyzed REFERENCES [1] D E Breen Constructive cubes: CSG evaluation for display using discrete 3D scalar data sets In Eurographics 91, pages , 1991 [2] David E Breen, Sean Mauch, and Ross T Whitaker 3d scan conversion of CSG models into distance volumes In Proc IEEE/ACM symposium on Volume Visualization, pages 7 14, 1998 [3] Brian Cabral, Nancy Cam, and Jim Foran Accelerated volume rendering and tomographic reconstruction using texture mapping hardware In Proc 1994 Symposium on Volume Visualization, pages 91 98, October 1994 [4] Hongsheng Chen and Shiaofen Fang Fast voxelization of 3D synthetic objects ACM Journal of Graphics Tools, 3(4):33 45, 1999 [5] D Cohen and A Kaufman Scan-conversion algorithms for linear and quadratic objects In A Kaufman, editor, Volume Visualization, pages , 1991 [6] Shiaofen Fang and Hongsheng Chen Hardware accelerated voxelization In Volume Graphics, Chapter 20, pages Springer-Verlag, March 2000 [7] Shiaofen Fang and R Srinivasan Volumetric CSG a modelbased volume visualization approach In Proc 6th International Conference in Central Europe on Computer Graphics and Visualization, pages 88 95, 1998 [8] J Goldfeather, J Molnar, S Turk, and H Fuchs Near real-
6 time CSG rendering using tree normalization and geometric pruning IEEE Computer Graphics and Application, 9:20 28, 1989 [9] Christoph M Hoffmann Geometric and Solid Modeling: An Introduction Morgan Kaufmann Publishers, 1989 [10] Arie Kaufman Efficient algorithms for 3D scan-conversion of parametric curves, surfaces, and volumes In SIGGRAPH 87, volume 21, pages , July 1987 [11] Arie Kaufman and Eyal Shimony 3D scan-conversion algorithms for voxel-based graphics In Proceedings of 1986 Workshop on Interactive 3D Graphics, pages 45 75, October 1986 [12] YT Lee and A A G Requicha Algorithms for computing the volume and other integral properties of solids Communications of the ACM, 25(9): , 1982 [13] Ari Rappoport and S Spitz Interactive Boolean operations for conceptual design of 3D solids Computer Graphics, SIG- GRAPH 97, pages , August 1997 [14] A A G Requicha Representation for rigid solids: Theory, methods and systems Computing Surveys, 12(4): , December 1980 [15] Naeem Shareef and Roni Yagel Rapid previewing via volume-based solid modeling In Solid Modeling 95, pages , May 1995 [16] Milos Sramek and Arie Kaufman Object voxelization by filtering In Proc IEEE/ACM symposium on Volume Visualization, pages , 1998 [17] W C Thibault and B F Naylor Set operations on polyhedra using binary space partitioning trees SIGGRAPH 87, pages , 1987 [18] Sidney Wang and Arie Kaufman Volume-sampled 3D modeling IEEE Computer Graphics and Application, 14:26 32, September 1994 [19] R Westermann and E Thomas Efficiently using graphics hardware in volume rendering applications In SIGGRAPH 98, pages , 1998 [20] T F Wiegand Interactive rendering of CSG models Computer Graphics Forum, 15(4): , 1996
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