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Figure 4. A perspective view of a building. The walls of the building do not appear to be vertical The ray casting algorithm divides the viewing plane into four areas, and treats the areas differently. The lines that divide the areas pass through COP’ and their slopes

Figure 4 A perspective view of a building. The walls of the building do not appear to be vertical The ray casting algorithm divides the viewing plane into four areas, and treats the areas differently. The lines that divide the areas pass through COP’ and their slopes

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nn ae We use perspective projection and arbitrary viewing parameters for the viewing environment. The (u,v,w) coordinate system is right-handed. The matrix of the viewing transformation is
nn ae We use perspective projection and arbitrary viewing parameters for the viewing environment. The (u,v,w) coordinate system is right-handed. The matrix of the viewing transformation is
Figure 1. A ray plane and a ray line. The ray plane is perpendicular to the DTM base plane and contains COP. The ray line is the intersection of the ray plane and the projection plane. Two rays that intersect the ray line are shown. They are projected onto the same line on the DTM base plane More effective rendering is possible if vertical ray coherence is exploited in the ray casting of DTMs. If two rays are projected onto an identical line on the DTM base plane, by vertical ray coherence, the ray which is higher hits the surface of the earth at a position further away from the viewing position. Two terms — ray plane and ray line — are defined in this paper so that we may further discuss the conditions in which vertical ray coherence can be used. The ray plane passing through the point P is shown in Figure 1. A ray plane
Figure 1. A ray plane and a ray line. The ray plane is perpendicular to the DTM base plane and contains COP. The ray line is the intersection of the ray plane and the projection plane. Two rays that intersect the ray line are shown. They are projected onto the same line on the DTM base plane More effective rendering is possible if vertical ray coherence is exploited in the ray casting of DTMs. If two rays are projected onto an identical line on the DTM base plane, by vertical ray coherence, the ray which is higher hits the surface of the earth at a position further away from the viewing position. Two terms — ray plane and ray line — are defined in this paper so that we may further discuss the conditions in which vertical ray coherence can be used. The ray plane passing through the point P is shown in Figure 1. A ray plane
Figure 2. A ray plane. The ray plane in Figure 1 is shown in detail with the elevation values. Pixels are generated along the ray line, from Q which is nearest to COP’. Note that the ray line must not be vertical in the projection plane. The DTM is scanned from the point nearest to COP”. If we know that the ray passing through P hits the terrain at H, we may scan the DTM from K to find the hit of the ray passing through P The ray plane in Figure 1 is reproduced in Figure 2 to explain vertical ray coherence. Since the rays passing through P and P’ intersect the same ray line, they are projected onto the same line on the DTM base plane. Since P’ is higher than P, the ray passing through P’ has a greater w value at any (u,v) on the projected line. To determine where the ray passing through P hits the terrain surface, DTM is scanned along the projected line starting at a position nearest to COP”, and the w co-ordinate of the ray and height (©) are compared at each point. Suppose the
Figure 2. A ray plane. The ray plane in Figure 1 is shown in detail with the elevation values. Pixels are generated along the ray line, from Q which is nearest to COP’. Note that the ray line must not be vertical in the projection plane. The DTM is scanned from the point nearest to COP”. If we know that the ray passing through P hits the terrain at H, we may scan the DTM from K to find the hit of the ray passing through P The ray plane in Figure 1 is reproduced in Figure 2 to explain vertical ray coherence. Since the rays passing through P and P’ intersect the same ray line, they are projected onto the same line on the DTM base plane. Since P’ is higher than P, the ray passing through P’ has a greater w value at any (u,v) on the projected line. To determine where the ray passing through P hits the terrain surface, DTM is scanned along the projected line starting at a position nearest to COP”, and the w co-ordinate of the ray and height (©) are compared at each point. Suppose the
Figure 3. Ray lines are not vertical. Three ray planes and corresponding ray lines are shown. Since the ray planes are not parallel, in turn the ray lines are not parallel. Thus the ray lines are not always vertical in the projection plane By using the concepts of ray planes and ray lines presented above, we may obtain an efficient ray casting algorithm which exploits vertical ray coherence. To use vertical ray coherence in rendering, we should process pixels in ray line order. Because the ray lines are neither vertical nor parallel to each other in the projection plane, this is quite an unnatural order of processing. Furthermore, tessellating the entire window with non-parallel ray lines is quite a challenging task. Figure 3 shows
Figure 3. Ray lines are not vertical. Three ray planes and corresponding ray lines are shown. Since the ray planes are not parallel, in turn the ray lines are not parallel. Thus the ray lines are not always vertical in the projection plane By using the concepts of ray planes and ray lines presented above, we may obtain an efficient ray casting algorithm which exploits vertical ray coherence. To use vertical ray coherence in rendering, we should process pixels in ray line order. Because the ray lines are neither vertical nor parallel to each other in the projection plane, this is quite an unnatural order of processing. Furthermore, tessellating the entire window with non-parallel ray lines is quite a challenging task. Figure 3 shows
Figure 5. The generation of ray lines. Since the ray lines are not parallel, more ray lines are generated as it gets further from COP’. Level 0 and level 1 ray lines are marked with solid circles and solid squares at the starting pixel, respectively. Pixels covered by each level of ray lines are shaded in light gray and dark gray, respectively As shown in Figure 5, values for the pixels covered by the ray lines passing hrough the bottom pixels of the window are detemined by ray casting. These ray ines are called level 0 ray lines, and marked with small solid circles at the centres of the starting pixels. The distance d depends on the field of view and the angle between the DTM base plane and the projection plane: the further the projection plane is from COP, the narrower the field of view and the greater d; the steeper he projection plane, the further COP’ from COP and the greater d. We can verify his in Figure 3. Note that the distance d is selected arbitrarily for convenience so hat level 0 ray lines may start at the bottom of the window. In fact any integer smaller than the distance between COP’ and the bottom of the window will work. Ray casting for the pixels covered by the same ray line can be processed efficiently,
Figure 5. The generation of ray lines. Since the ray lines are not parallel, more ray lines are generated as it gets further from COP’. Level 0 and level 1 ray lines are marked with solid circles and solid squares at the starting pixel, respectively. Pixels covered by each level of ray lines are shaded in light gray and dark gray, respectively As shown in Figure 5, values for the pixels covered by the ray lines passing hrough the bottom pixels of the window are detemined by ray casting. These ray ines are called level 0 ray lines, and marked with small solid circles at the centres of the starting pixels. The distance d depends on the field of view and the angle between the DTM base plane and the projection plane: the further the projection plane is from COP, the narrower the field of view and the greater d; the steeper he projection plane, the further COP’ from COP and the greater d. We can verify his in Figure 3. Note that the distance d is selected arbitrarily for convenience so hat level 0 ray lines may start at the bottom of the window. In fact any integer smaller than the distance between COP’ and the bottom of the window will work. Ray casting for the pixels covered by the same ray line can be processed efficiently,
Figure 7. An image generated by the algorithm. A 512 X 512 image generated from a 901 X 901 DTM. It is generated in 0.4 seconds on a Silicon Graphics Indigo2 and 0.6 seconds on a Sun Sparc 10
Figure 7. An image generated by the algorithm. A 512 X 512 image generated from a 901 X 901 DTM. It is generated in 0.4 seconds on a Silicon Graphics Indigo2 and 0.6 seconds on a Sun Sparc 10
Figure 8. Other views of the same area as Figure 7. The same area viewed from different viewpoints
Figure 8. Other views of the same area as Figure 7. The same area viewed from different viewpoints
Figure 10. Average rendering time with various viewing directions. The view plane normal i: (cos@,sin@,1), where 0 labels the x axis I IDR EE There are several approximations in the algorithm: interpolation of pixels not covered by lines; integer co-ordinates of COP’; and slight displacement of pixel co- ordinates from the ray plane. These approximations do not seem to cause any significant problem on a frame in our experience. However, if this algorithm is used to produce an animated sequence, all of the approximations may contribute to a significant amount of temporal aliasing, especially crawling of object boundaries. We do not have a good solution for temporal aliasing due to integer co-ordinates of COP’, but effects of other approximations may be alleviated by simple modification of the algorithm. Instead of selecting a pixel with the nearest x co-ordinate at each
Figure 10. Average rendering time with various viewing directions. The view plane normal i: (cos@,sin@,1), where 0 labels the x axis I IDR EE There are several approximations in the algorithm: interpolation of pixels not covered by lines; integer co-ordinates of COP’; and slight displacement of pixel co- ordinates from the ray plane. These approximations do not seem to cause any significant problem on a frame in our experience. However, if this algorithm is used to produce an animated sequence, all of the approximations may contribute to a significant amount of temporal aliasing, especially crawling of object boundaries. We do not have a good solution for temporal aliasing due to integer co-ordinates of COP’, but effects of other approximations may be alleviated by simple modification of the algorithm. Instead of selecting a pixel with the nearest x co-ordinate at each
Related topics:
Cognitive ScienceComputer ScienceVisualizationDigital Terrain ModelsPerspective Projection
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