An improved rendering technique for ray
tracing BeÂzier and B-spline surfaces
By Shyue-Wu Wang*, Zen-Chung Shih and Ruei-Chuan Chang
...
Both numerical and subdivision methods are widely used approaches for ray tracing parametric surfaces. However, the expense of ®nding the ray±surface intersection points is a major drawback. Thus, simpler and less memory-intensive strategies are needed to improve these methods without further complicating them. This work presents an ef®cient algorithm for enhancing the performance of both numerical and subdivision methods. The proposed technique can be extended to most applications based on these two methods. The computational time of both approaches is improved by 16±40%. Copyright # 2000 John Wiley & Sons, Ltd.
Received: 20 July 1999; Revised: 12 April 2000
KEY WORDS: ray tracing; Newton's method; BeÂzier clipping; parametric surfaces; BeÂzier surfaces
Introduction
Ray tracing parametric surfaces suffers from lengthy computation time, both in calculating the ray±surface intersection points and locating the closest intersection point. To improve performance, the time spent on both parts of the algorithm must be reduced as much as possible. However, researchers seldom consider redu-cing both factors simultaneously. In this paper, we present an ef®cient rendering technique for improving numerical methods and subdivision methods by redu-cing these two factors.
Newton's method is conventionally used to calculate the ray±surface intersection points, and researchers have always focused on how to locate the initial points ef®ciently. Toth1 performed interval analysis to ®nd
the initial points. Meanwhile, Lischinski and Gonczar-owski2later proposed an improved technique based on
Toth's results. Other researchers3±8 subdivided
sur-faces into patches and used various data structures to organize these patches. The initial points are obtained by traveling rays through the data structures and ®nding the intersections of rays and bounding volumes that enclose the patches. Although these methods3±8
can accelerate the calculation of the intersection point by selecting suitable initial points, the computational
cost of ®nding the initial points remains high. Joy and Bhetanabhotla9 ®rst adopted the ray coherence
prop-erty for selecting initial points. They took the closest intersection point of an adjoining ray previously calculated as the initial point, but faced the problems that an intersection point found through this method might not be the closest one, since not all of the ray±surface intersection points are checked.
Due to the basic process of subdivision methods,10±12
all the ray±surface intersection points are found by recursively subdividing the surfaces. The closest one is obtained after calculating all of them. That is, subdivi-sion methods do not attempt to reduce the time spent on selecting the closest intersection points.
The ray coherence property is important in accel-erating the calculation of the intersection points. For numerical methods, the closest intersection point of an adjoining ray previously calculated can be taken as the initial point for Newton's method. If an intersection point found in this manner can be veri®ed to be the closest, the computational cost of calculating the others can be eliminated.
Ray coherence can also be incorporated into sub-division methods to assist intersection calculation. During subdivision, it is preferable to subdivide the region containing the intersection point found by the previously traced ray. After an intersection point is obtained from this region, whether this point is the closest one is then veri®ed. In this case, the remaining subdividing processes can be avoided. Namely, the
*Correspondence to: Zen-Chung Shih, Department of Computer and Information Science, National Chiao Tung University,
THE JOURNAL OF VISUALIZATION AND COMPUTER ANIMATION J. Visual. Comput. Animat. 2000; 11: 209±219
closest intersection point can be located ef®ciently without ®nding the other points. Until now, no researchers have attempted to realize this idea to accelerate subdivision methods.
The following paragraphs describe a rendering technique that combines the ideas of improving both numerical methods and subdivision methods.
Given a surface that will be tested along a scan line, the ®rst ray that intersects the surface along the scan line is located. The closest intersection point is then located and marked. For the next ray, an intersection point is ®rst found by ray coherence and then whether this point is the closest one is veri®ed. If it is the closest point, it is not necessary to calculate all the other intersection points for this ray. Otherwise, all the other intersection points must be found and the closest one selected. This process continues for the succeeding rays along the scan line that intersect the surface.
This rendering technique is advantageous in that it not only reduces time spent on calculating the inter-section points but also locates the closest interinter-section points without ®nding all the other intersection points. The foundation of this rendering technique is how to verify whether an intersection point is the closest one. Originally, it was planned to follow up the operation of these two methods to modify the process of ®nding the intersection points without tremendous extra memory storage. With minor adjustment of these two methods to ®t the proposed rendering technique, the executive performance could be enhanced. Since numerical and subdivision methods have been applied widely in many practical directions, enhancing performance ef®ciency with the rendering technique proposed in this paper would be quite appropriate.
Since both numerical and subdivision methods have their own scheme to locate ray±surface intersections, it is dif®cult to create a general improvement scheme. To verify whether an intersection point is the closest, individual schemes are proposed for numerical and subdivision methods, respectively. Both schemes are based on the basic processes of numerical and subdivision methods, respectively. Two algorithms, a subdivision method and a numerical method, are modi®ed and implemented. Experimental results indi-cate that the improved algorithms can reduce total rendering time by 16±40%.
The rest of this paper is organized as follows. First, the improvement of numerical methods is described and applied to the algorithm proposed by Barth and StuÈrzlinger.4 Then, how to enhance the subdivision
method and apply the improvement to BeÂzier clipping
is discussed. Finally, experimental results and discus-sions are presented.
Numerical Methods
Numerical methods are a suitable and direct way to ®nd ray±surface intersection points. This section focuses on improving the algorithms3±8 that adopt
Newton's method but not utilize the ray coherence property. The algorithm proposed by Barth and StuÈrzlinger4 is taken as an example to demonstrate
how numerical algorithms can be improved with the novel rendering technique. Meanwhile, the problem of how to verify whether a found intersection point is the closest one should be solved. A scheme based on the subdivision strategy and veri®cation method is sug-gested, as originally proposed by Barth and StuÈrzlin-ger.4 The objective is to make the novel rendering
technique more ¯exible, suitable and easier to incorpo-rate into numerical methods, because many algo-rithms3,5,6are based on this algorithm.
The following discussion ®rst reviews Barth and StuÈrzlinger's algorithm. Since this algorithm can deal with both BeÂzier and B-spline surfaces, BeÂzier surfaces are selected as an example for explanation. Then, how to enable this algorithm to render an environment with the proposed technique is discussed.
Barth and StuÈrzlinger's Algorithm
Barth and StuÈrzlinger's algorithm consists of two steps. In the preprocessing step, the surface is subdivided adaptively into patches until each patch can be approxi-mated well enough using a plane parallelogram. The plane parallelogram is de®ned by two vectors,~v1and~v2,
as illustrated in Figure 1. In each step of the subdivision, a patch is subdivided into halves and the decision whether to halve along the u- or v-axis is based on the curvature in the u- and v-direction. During the subdivi-sion, these patches are arranged in a binary tree. The root encloses the entire surface and the leaves contain the almost plane parts of the surface.
To ef®ciently detect whether a patch is intersection free, a bounding volume called the enclosing paral-lelepiped is constructed for each patch stored in the binary tree. The enclosing parallelepiped of a patch is spanned by ~v1, ~v2, and ~v3where ~v3is obtained from the
cross-product of ~v1 and ~v2. Information of the
corre-sponding patch, including the enclosing
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epiped, is stored in each node of the binary tree, and is employed for the intersection tests or for calculating the initial point. For each leaf node, the parametric domain, and approximating parallelogram are also stored. The following Newton iteration works on the whole surface rather than merely the patch under consideration.
In the rendering step, a ray is traced through the binary tree of a tested surface. The traversal process starts from the root of the tree. If the ray strikes the enclosing parallelepiped, the traversal process con-tinues to the children and the same test is performed again. The above process locates all the leaf nodes whose parallelepipeds are intersected by the ray. For each node, an initial point is generated from the intersection between the ray and the corresponding parallelogram. If Newton's method converges, an intersection point is obtained. Barth and StuÈrzlinger considered that only one intersection point exists between the ray and the patch. If Newton's method fails to converge, a second iteration process is initiated with the entrance point into the parallelepiped as the initial point. Furthermore, a third iteration will be performed after an unsuccessful second iteration. If these three iteration processes fail to converge, the ray is considered not to intersect the patch.
Attention is also paid to the critical case in which a ray is nearly tangential to the part of the surface. Here, multiple intersections likely exist. This critical case is easily recognized because it occurs only when a ray passes through the parallelepiped approximately par-allel to the larger face, as shown in Figure 2. The following inequality is proposed to determine this critical case.
tan a >b
h 1
If equation (1) holds, the entrance point of the ray into the parallelepiped is taken as the initial point. Hence, the iteration is more likely to converge to the ®rst intersection. However, `equation (1) does not hold' is not a crucial condition for the patch being free of the intersections, or the located point being the closest one. Some artifacts may appear, but they are not easily noticeable in the anti-aliased picture.
Improvement of Barth and
StuÈrzlinger's Algorithm
Closely examining the basic process of this algorithm reveals that the primary function of the binary tree is to provide an ef®cient structure for ®nding the ray±patch pairs. In addition to providing the same capability, regular grids also offer a good mechanism for verifying whether an intersection point is the closet one. Unfortunately, the inherent property of the binary tree cannot provide such a mechanism. Regular grids are used to organize a surface instead of the corre-sponding binary tree. Details of the modi®ed algorithm are described as follows.
In the preprocessing step, each surface is subdivided according to the rules proposed by Barth and StuÈrzlin-ger. A regular grid associated with each surface is then constructed to organize these patches. The regular grid is created by applying the uniform space subdivision technique13 to the bounding volume of each surface.
The bounding volume used herein is an axis±aligned parallelepiped. Although the essential information of each ¯at patch is stored in the corresponding voxels, the binary tree does not need to be constructed.
In the rendering step, the primary rays are traced through the regular grids of the tested surfaces and the ray±surface intersection points are calculated accord-ing to the proposed renderaccord-ing technique. Given a surface that will be tested along a scan line, the ®rst ray that intersects the surface along the scan line is located. Then the closest intersection point is found and marked. The following ray±surface intersection points are calculated using ray-to-ray coherence along a scan line. Once an intersection point is found via ray Figure 2. The critical case that a ray is nearly tangential to
the part of the surface.
Figure 1. A part of a BeÂzier surface. The patch is de®ned over [u, u]r[v,v].
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coherence, a testing ray is traced from the intersection point toward the origin of the current traced ray. This testing ray is traced to verify whether the intersection point is the closest one by examining the intersection of the testing ray and the visited patches.
While the testing ray is traversing the regular grid, a list of patches whose enclosing parallelepipeds are hit by the testing ray is obtained. Then, equation (1) is checked for the testing ray and the patch containing the intersection point to be veri®ed. If this inequality does not hold, we regard that only one intersection of the testing ray and the patch exists according to the rule proposed by Barth and StuÈrzlinger. Otherwise, the entrance point of the testing ray into the parallelepiped is used as the initial point for iteration, and a closest intersection point is sought.
Next, if no other patches exist in the list, the veri®ed intersection point is regarded as the closest one. Otherwise, Newton's method is employed to ®nd the intersection points with the initial points obtained from the intersections of the testing ray and the intersected parallelepipeds. The detection process can either determine whether an intersection point is the closest or locate the real closest intersection point for the current traced ray.
The details of this process are illustrated with an example in Figure 3, where the dotted grids represent the regular grid and the curved segments represent the patches. Assume that P1is the intersection point found
by Newton's method without applying ray coherence and P2is the point found using P1as the initial point.
We conclude that P2 is the closest intersection point
because the testing ray TL4 only intersects with the
patch containing the point P2. Regardless, if P3is the
intersection point found with P2 as the initial point,
the closest intersection point P4 will be obtained,
because the testing ray TL5 will intersect with the
patch containing the point P4.
The above improvement has two advantages. First, this improvement reduces the time spent on both ®nding the initial points and locating the closest intersection point. Second, this improvement can easily be extended to the previous algorithms.3,5,6 For
example, to render triangular trimmed free-form surfaces, surfaces can be subdivided and the bounding volumes (tripipeds) for patches calculated using the methods proposed by StuÈrzlinger.6 Then, all patches
are stored in the corresponding regular grids. The improved rendering procedure mentioned above can be used to locate the ray±surface intersection points.
Subdivision Methods
Now we discuss the improvement of subdivision methods. We ®rst review the concept of subdivision methods. Most subdivision methods are based on a variation of the same idea. The domains that may contain intersection points are subdivided recursively and those without solutions are pruned. However, each method has a different means of ®nding the ray±surface intersection points. In this section, BeÂzier clipping is chosen as the method to be improved to illustrate how almost all the subdivision methods can adopt the proposed rendering technique to trace primary rays.
Compared with numerical methods, subdivision methods are not easily able to exploit ray coherence. The detailed processes of subdivision methods must be analyzed to achieve this goal. This section ®rst details the clipping process of BeÂzier clipping, and then discusses how to improve BeÂzier clipping.
Clipping Process of BeÂzier Clipping
The clipping process10 for ®nding the ray±surface
intersection points can be classi®ed into two types: straight-forward and bisection. For the straight-forward case, the u and v axes are clipped alternatively, each step achieves a considerable processing of void parts, and exits with `no intersections' or `one intersection'. In Figure 4, uj(vj) represents that the jth clipping stage is
Figure 3. An example of verifying the closest intersection point.
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performed along the u axis (v axis). For example, u1
denotes that the u axis is clipped at step 1 and v4 denotes that the v axis is clipped at step 4. In this
example, after six steps of clippings, the intersection point is obtained. Of course, this simple method cannot work if there is more than one intersection point in the patch.
For a bisection case, consider Figure 5, where Figure 5(a) illustrates the third clipping step (termed the bisection step). The clipping process fails to reduce the interval width of parameter u to a given threshold. Nishita et al.10 suggested a threshold value of 20% of
the interval width. Then the parametric space is split into halves and the clipping process is performed on each half. These two parts are denoted herein as ulow
and uhigh as illustrated in Figure 5(a). The clipping
process is ®rst performed on ulow. Then an intersection
point is found. On dealing with the uhigh part, the
clipping process also fails to reduce the interval width of parameter v to a given threshold, as illustrated in Figure 5(b). Hence, the uhighpart is again divided into
halves, uhighvlowand uhighvhigh. After recursively
follow-ing several steps, the other two intersection points are obtained, as presented in Figure 5(c).
An Improvement of BeÂzier Clipping
The characteristic feature of BeÂzier clipping, as intro-duced in the above section, is that the domains generated from the bisection steps are processed randomly and all the ray±surface intersection points are found to obtain the closest one. To improve BeÂzier clipping in accordance with the rendering technique, the aim herein is to clip the domains generated from the bisection steps in a reasonable order based on the ray coherence property. In this order, it is preferable to deal with the domain that contains or neighbors on the parameters of the closest intersection point found by the previously traced ray. After an intersection point is obtained from some domain, whether this point is the closest one will be veri®ed. If it is the closest, computations of clipping the remaining domains do not need to be performed. Namely, the closest inter-section point can be obtained without ®nding all the others. The following discussion ®rst describes a procedure for verifying whether an intersection point is the closest one. Details of the modi®ed algorithm are given as well.
In fact, while improving Barth and StuÈrzlinger's algorithm, a procedure for either verifying whether an intersection point is the closest one or locating the real closest one has already been proposed. In this paper, the aim herein is to enable the subdivision methods to solve the verifying problem in accordance with the algorithms themselves. Without introducing further complex methodology to help, the action that uses Figure 4. The clipping process of the straight-forward case
on ®nding the ray±surface intersection point.
Figure 5. The clipping process of a bisection case.
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Newton's method to locate the intersection points is removed from the procedure and make the resulting procedure answer `yes' or `no' only. Details of the modi®ed procedure are stated below.
On verifying whether an intersection point is the closest, a testing ray is traced through the point toward the viewpoint as usual. This process obtains a list of patches whose enclosing parallelepipeds are hit by the testing ray. The patch containing the intersection point is ®rst located and equation (1) is then checked for the testing ray and the patch. If this inequality holds, it is reported that this intersection point may not be the closest one, because more than one intersection is very likely to exist. Otherwise, the veri®cation process is continued for other patches of the list. If no other patches exist in the list, it is reported that this point is the closest. If other patches do exist in the list, it is reported that this point may not be the closest one.
The improved algorithm consists of two steps: the preprocessing step and rendering step. In the prepro-cessing step, the regular grid used to improve Barth and StuÈrzlinger's algorithm is naturally constructed for each surface. In the rendering step, the primary rays are traced in the scan line order according to the proposed rendering technique. Figure 6 illustrates the process in detail. The dashed lines denote shooting rays from the view point. The black dots indicate that the traced ray intersects the surface S and the closest intersection point is found. B denotes that the closest intersection point is found by direct BeÂzier clipping. B1
denotes that the clipping process is the straight-forward case and Bmrepresents that the clipping process is the
bisection case.
Given a surface that will be tested along a scan line, the ray (the 3rd ray in Figure 6) that intersects with the surface S along the scan line is ®rst located, and then the closest intersection point is found by direct BeÂzier clipping. For the succeeding rays that intersect with the surface S, if the ray±surface intersection is the straight-forward case, for example, the 4th, 5th, and 6th rays in Figure 6, BeÂzier clipping is applied to locate the intersection point directly.
For bisection cases, for example, the 7th or 8th rays in Figure 6, the bisection steps will produce some regions. First, the region that contains parameters of the closest intersection point found by the ray previously traced is dealt with; this region is called the prior region. Once the ®rst intersection point is found in this region, whether or not this point is the closest one can be veri®ed. If this point is the closest one, the computation is stopped. Otherwise, BeÂzier clipping is used to ®nd the remaining intersection points.
If no intersections could be obtained from the prior region, dealing with the region that adjoins the prior region is preferable. That is, the parameter domains of this region are beside those of the prior region because the intersection point obtained from the region tends to be the closest intersection point. This process is illustrated with an example shown in Figure 7. No intersection points are obtained from the prior region, hence the clipping process is applied to region R1and
the intersection point is found, indicated by a black dot. This intersection point is then veri®ed. If this point is the closest, the computation is stopped. Otherwise, BeÂzier clipping is used to ®nd the other intersection points.
Figure 6. The process of the modi®ed BeÂzier clipping on ®nding the ray±surface intersection points along a scan line. The dash lines represent shooting rays from the view point.
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During veri®cation, an alternative approach exists. That is, BeÂzier clipping can be used to locate the ray±patch intersection points directly rather than reporting that the intersection point is not the closest. In the present implementation it is found that the performance of applying this approach could accelerate restricted performance with the cost of extensive extra memory space to store patches. Hence, this approach is not adopted to improve BeÂzier clipping.
The major contributions of this improvement are that the number of subdividing processes can not only be reduced, but also that the closest intersection point can be located without ®nding all of the ray±surface intersection points.
Experimental Results
This section discusses the implementation results and compares them with previous approaches. The pro-posed algorithm and traditional algorithms are imple-mented in C programming language. All the results are run on the platform of a Pentium II 300 MHz CPU and 128 MB RAM under a Windows NT 4.0 environ-ment. Photographs for four different scenes were created, Figures 8±11, for performance comparison. The resolution of each image is 1024r1024 and all of the scenes displayed in this paper are generated using anti-aliasing (four rays per pixel) and a ray depth of two. The spectral sampling approach14 is also
employed to produce more realistic color in metals.
The implementation herein uses nine spectral samples. Moreover, the ¯at surfaces, such as a ¯oor or a ¯at mirror, displayed in this work are created by polygons rather than parametric surfaces.
Uniform space subdivision13 and simple min±max
bounding volume (axis-aligned parallelepiped) are used to reduce unnecessary ray±surface intersection tests. Tables 1 and 2 display the results from perform-ing the proposed approach and traditional algorithms on rendering all four distinct scenes. The data included are the sum of rendering a whole scene (not including the shading time) with 512r512 resolutions. Attention is only paid to the total execution time involved in ®nding ray±surface intersection points. The time units are in seconds.
Performance Comparison of Barth and
StuÈrzlinger's Algorithm
On implementing Barth and StuÈrzlinger's algorithm, the ray±surface intersection points on the two-dimensional space are calculated. First, a BeÂzier surface is projected according to two orthogonal planes representing the traced ray. The projection process is identical to that of BeÂzier clipping.10 Then
Newton's method is applied to solve the two-dimensional non-linear system. For B-spline surfaces, the method proposed by Yang8could be used for the
projection.
Table 1 presents the total rendering time and total number of intersection points found with the improved Barth and StuÈrzlinger's algorithm and original algo-rithm on tracing the primary rays. According to the Figure 7. An example of ®nding the intersections from the non-prior regions. (a) White dot represents the parameters of the closest intersection point found by the previous ray. Black and gray dots represent the parameters of the intersection points between the current ray and the surface. (b) We show the corresponding intersections between two adjoining rays and a surface.
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experimental results, the proposed method improved the performance by 16±40%. The more high-curvature surfaces that a scene contains, the more computation time is saved. This phenomenon occurs because high-curvature surfaces induce more multiple intersections between the ray and surface. For example, in Figures 9 and 10, most of the rays intersect the surfaces at two intersection points.
Performance Comparison of BeÂzier
Clipping
Table 2 summarizes experimental results of the improved BeÂzier clipping and original BeÂzier clipping Figure 9. Five toruses with 40 surfaces. Each torus is created
by 8 BeÂzier surfaces.
Figure 10. Twenty-four rings with 288 surfaces. Each ring is constructed by 12 BeÂzier surfaces.
Figure 11. A highly re¯ective and complex environment with 340 surfaces. The backdrop of the scene consists of curved mirrors. Two mirrors are perpendicular to the ¯oor and located on opposite sides of the ¯oor, which cannot be
seen in this rendering view. Figure 8. Newell's teapot with 32 surfaces.
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on rendering four distinct scenes for the primary ray. Notably, in the present implementation, BeÂzier clip-ping was modi®ed according to the schemes suggested by Campagna et al.15 According to the experimental
results, the proposed method improved the perfor-mance of BeÂzier clipping by 6±28%. The unsatisfactory performance in rendering Figure 8 is due to the body of the teapot. Most of the intersections between the body and traced rays are the straight-forward case. The execution ef®ciency can be obviously revealed from the number of intersection points found by the proposed method. The larger the number of intersection points implies a more ef®cient proposed method.
Comparing Tables 1 and 2 reveals two interesting results. The ®rst result is that Barth and StuÈrzlinger's algorithm ®nds the ray±surface intersection points faster than BeÂzier clipping. This phenomenon occurs because the computation cost of ®nding an intersection point of Newton's method is potentially lower than that of BeÂzier clipping. It must be emphasized that this paper aims to propose a rendering technique for improving these algorithms, not to compare the
difference and superiority of these algorithms. Each of these algorithms has its own advantages in ®nding the ray±surface intersection points.
The second result is that the total number of intersection points found by Barth and StuÈrzlinger's algorithm and BeÂzier clipping differ from each other, particularly on rendering Figures 9 and 10. This phenomenon occurs because if a ray is (or nearly) tangential to the part of a surface, BeÂzier clipping will locate many intersection points that are close to each other. This phenomenon occurs in many algorithms, such as the eigenvalue computation or Barth and StuÈrzlinger's algorithm3.
The Performance of Nonuniform
Sampling
According to the experimental results, the proposed rendering technique works well in the case in which an image plane is uniformly sampled. However, nonuni-form sampling also plays an important role in many
Barth and StuÈrzlinger's algorithm
Rendering time No. of intersection points
Figure No. of patches/No. of surfaces Originalalgorithm Improvedalgorithm Improvedperformance Originalalgorithm Improvedalgorithm
8 7594/32 10.7 8.57 20% 110169 100337
9 13872/40 36.8 21.9 40% 583972 302183
10 39042/288 16.7 13.3 20% 207517 157295
11 39358/340 31.5 26.5 16% 470133 260384
Table 1. Experimental results of the improved Barth and StuÈrzlinger's algorithm and original algorithm
BeÂzier clipping
Rendering time No. of intersection points
Figure No. of patches/No. of surfaces Originalalgorithm Improvedalgorithm Improvedperformance Originalalgorithm Improvedalgorithm
8 7594/32 32.9 30.8 6% 146910 102677
9 13872/40 128 101 21% 585952 329157
10 39042/288 66.5 50.3 24% 439421 198208
11 39358/340 145 105 28% 1170938 556281
Table 2. Experimental results of the improved BeÂzier clipping and original algorithm
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applications, such as anti-aliasing,18 parallel16 and
progressive17ray tracing. To demonstrate the
capabil-ity of the proposed rendering technique to deal with nonuniform sampling, these four scenes are rendered with a jittered sampling19 (four rays per pixel) In our
implementation, each pixel is subdivided into four subpixels and each sample point is located randomly within its subpixel. Table 3 displays the experimental results from the performance of the improved Barth and Sturzlinger's algorithm.
The above ®nding indicates that the performance of rendering the jittered sampling is only 2% less than that of rendering the uniform sampling. The same tests were also performed for the improved BeÂzier clipping, with similar results.
However, the rendering technique proposed herein may be inappropriate for distributed sampling, such as parallel and progressive ray tracing. For parallel ray tracing, the samples are selected adaptively from the image plane, which is based on the balanced load distribution. Furthermore, progressive ray tracing gen-erates the sample location in an order that images may be reconstructed from these samples. The adaptive stochastic sampling of the image plane that the algorithms adopted causes the proposed rendering technique to spend more time on ®nding the ray±sur-face intersection points. A similar problem is ray±sur-faced when dealing with the shadow rays, because shadow rays are lack of ray coherence. In our implementation, the improvement of performance for shadow rays is insigni®cant. Nevertheless, the shadow map19 is a
conventional means of accelerating shadow testing, and can also be applied to ray tracing parametric surfaces. The notion of constructing a shadow map is similar to that of rendering an image plane. Hence, the proposed approach is appropriate for generating shadow maps to accelerate shadow testing.
Conclusions
This paper presents a rendering technique for improv-ing both numerical methods and subdivision methods on rendering BeÂzier surfaces and B-spline surfaces. Our major contribution is that individual schemes are designed for both the numerical and subdivision methods to ®nd the ray±surface intersection points and locate the closest intersection points ef®ciently. Simplicity, reduced memory requirements, and en-hanced execution are the main characteristics of the proposed rendering technique. The results demon-strate that the improved algorithms can reduce total rendering time by 16±40%. The present implementa-tion only deals with the primary rays. However, the rendering technique can easily be extended to deal with secondary rays by applying the cone tracing20or
beam tracing21technique. In these techniques, rays are
traced in a group rather than scan line order.
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8 7594/32 33.9 34.7 x1%
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Table 3. Experimental results of the improved Barth and StuÈrzlinger's algorithm
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Authors' biographies:
Shyue-Wu Wang received the M.S. degree in Compu-ter Science from National Tsing Hua University, Taiwan, R.O.C., in 1992, and is currently a doctoral candidate at National Chiao-Tung University. His research interests include global illumination, real-time rendering, image-based rendering and geometric modeling.
Zen-Chung Shih was born on 10th February 1959, in Taipei, Taiwan, Republic of China. He received his B.S. degree in Computer Science from Chung-Yuan Chris-tian University in 1980, M.S. degree in 1982 and Ph.D. degree in 1985 in Computer Science from the National Tsing Hua University. Currently, he is a professor in the Department of Computer and Information Science at the National Chiao Tung University in Hsinchu. His current research interests include procedural texture synthesis, non-photorealistic rendering, global illumi-nation, and virtual reality.
Ruei-Chuan Chang was born on 30th January 1958, in Keelung, Taiwan, Republic of China. He received his B.S. degree in 1979, M.S. degree in 1981, and Ph.D. degree in 1984, all in Computer Science from the National Chiao Tung University. Currently he is a professor in the Department of Computer and Infor-mation Science at National Chiao Tung University in Hsinchu. He is also an associate research fellow at the Institute of Information Science, Academia Sinica, Taipei. His current research interests include design and analysis of algorithms, computer graphics, and system software.
IMPROVED RENDERING TECHNIQUE
