1. Computer Graphics Inf4/MSc 1 Computer Graphics Lecture 5 Hidden Surface Removal and Rasterization Taku Komura

2. Computer Graphics Inf4/MSc 2 Hidden surface removal Drawing polygonal faces on screen consumes CPU cycles – Illumination We cannot see every surface in scene – We don ’ t want to waste time rendering primitives which don ’ t contribute to the final image.

3. Computer Graphics Inf4/MSc 3 Visibility (hidden surface removal)‏ A correct rendering requires correct visibility calculations Correct visibility –when multiple opaque polygons cover the same screen space, only the closest one is visible (remove the other hidden surfaces)‏ –wrong visibility correct visibility

4. Computer Graphics Inf4/MSc 4 Visibility of primitives A scene primitive can be invisible for 3 reasons: –Primitive lies outside field of view –Primitive is back-facing –Primitive is occluded by one or more objects nearer the viewer

5. Computer Graphics Inf4/MSc 5 Visible surface algorithms. Definitions: Object space techniques: applied before vertices are mapped to pixels Back face culling, Painter’s algorithm, BSP trees Image space techniques: applied while the vertices are rasterized Z-buffering

6. Computer Graphics Inf4/MSc 6 Back face culling. The vertices of polyhedra are oriented in an anticlockwise manner when viewed from outside – surface normal N points out. Project a polygon. –Test z component of surface normal. If negative – cull, since normal points away from viewer. –Or if N. V > 0 we are viewing the back face so polygon is obscured.

7. Computer Graphics Inf4/MSc 7 Painters algorithm (object space). Draw surfaces in back to front order – nearer polygons “paint” over farther ones. Supports transparency. Key issue is order determination. Doesn’t always work – see image at right.

8. Computer Graphics Inf4/MSc BSP (Binary Space Partitioning) Tree. One of class of “list-priority” algorithms – returns ordered list of polygon fragments for specified view point (static pre-processing stage). Choose polygon arbitrarily Divide scene into front (relative to normal) and back half-spaces. Split any polygon lying on both sides. Choose a polygon from each side – split scene again. Recursively divide each side until each node contains only 1 polygon. 3 4 1 2 5 View of scene from above

9. Computer Graphics Inf4/MSc 19/10/2007 Lecture 9 9 BSP Tree. Choose polygon arbitrarily Divide scene into front (relative to normal) and back half-spaces. Split any polygon lying on both sides. Choose a polygon from each side – split scene again. Recursively divide each side until each node contains only 1 polygon. 3 3 4 1 2 5 5a 5b 1 2 5a 4 5b back front

10. Computer Graphics Inf4/MSc BSP Tree. Choose polygon arbitrarily Divide scene into front (relative to normal) and back half-spaces. Split any polygon lying on both sides. Choose a polygon from each side – split scene again. Recursively divide each side until each node contains only 1 polygon. 3 3 4 1 2 5 5a 5b 4 5b back front 2 15a front

11. Computer Graphics Inf4/MSc BSP Tree. Choose polygon arbitrarily Divide scene into front (relative to normal) and back half-spaces. Split any polygon lying on both sides. Choose a polygon from each side – split scene again. Recursively divide each side until each node contains only 1 polygon. 3 3 4 1 2 5 5a 5b back front 2 15a front 5b 4

12. Computer Graphics Inf4/MSc 19/10/2007 Lecture 9 12 Displaying a BSP tree. Once we have the regions – need priority list BSP tree can be traversed to yield a correct priority list for an arbitrary viewpoint. Start at root polygon. –If viewer is in front half-space, draw polygons behind root first, then the root polygon, then polygons in front. –If polygon is on edge – either can be used. –Recursively descend the tree. If eye is in rear half-space for a polygon – then can back face cull.

13. Computer Graphics Inf4/MSc 19/10/2007 Lecture 9 13 BSP Tree. A lot of computation required at start. –Try to split polygons along good dividing plane –Intersecting polygon splitting may be costly Cheap to check visibility once tree is set up. Can be used to generate correct visibility for arbitrary views.  Efficient when objects don’t change very often in the scene.

14. Computer Graphics Inf4/MSc 19/10/2007 Lecture 9 14 BSP performance measure Tree construction and traversal (object-space ordering algorithm – good for relatively few static primitives, precise) Overdraw: maximum Front-to-back traversal is more efficient Record which region has been filled in already Terminate when all regions of the screen is filled in S. Chen and D. Gordon. “Front-to-Back Display of BSP Trees.” IEEE Computer Graphics & Algorithms, pp 79–85. September 1991.

15. Computer Graphics Inf4/MSc 15 Z-buffering : image space approach Basic Z-buffer idea: rasterize every input polygon For every pixel in the polygon interior, calculate its corresponding z value (by interpolation)‏ Track depth values of closest polygon (smallest z) so far Paint the pixel with the color of the polygon whose z value is the closest to the eye.

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20. Computer Graphics Inf4/MSc 20 Implementation. Initialise frame buffer to background colour. Initialise depth buffer to z = max. value for far clipping plane For each triangle –Calculate value for z for each pixel inside –Update both frame and depth buffer

21. Computer Graphics Inf4/MSc Filling in Triangles Scan line algorithm –Filling in the triangle by drawing horizontal lines from top to bottom Barycentric coordinates –Checking whether a pixel is inside / outside the triangle

22. Computer Graphics Inf4/MSc 22 Triangle Rasterization Consider a 2D triangle with vertices p0, p1, p2. Let p be any point in the plane. We can always find a, b, c such that We will have if and only if p is inside the triangle. We call the barycentric coordinates of p.

23. Computer Graphics Inf4/MSc 23 Computing the baricentric coordinates of the interior pixels (α,β,γ) : barycentric coordinates Only if 0<α,β,γ<1, (x,y) is inside the triangle Depth can be computed by αZ 0 + βZ 1 +γZ 2 Can do the same thing for color, normals, textures The triangle is composed of 3 points p 0 (x 0,y 0 ), p1 (x1, y1), p2(x2,y2)‏

24. Computer Graphics Inf4/MSc 24 Bounding box of the triangle First, identify a rectangular region on the canvas that contains all of the pixels in the triangle (excluding those that lie outside the canvas). Calculate a tight bounding box for a triangle: simply calculate pixel coordinates for each vertex, and find the minimum/maximum for each axis

25. Computer Graphics Inf4/MSc 25 Scanning inside the triangle Once we've identified the bounding box, we loop over each pixel in the box. For each pixel, we first compute the corresponding (x, y) coordinates in the canonical view volume Next we convert these into barycentric coordinates for the triangle being drawn. Only if the barycentric coordinates are within the range of [0,1], we plot it (and compute the depth)‏

26. Computer Graphics Inf4/MSc 26 Why is z-buffering so popular ? Advantage Simple to implement in hardware. –Memory for z-buffer is now not expensive Diversity of primitives – not just polygons. Unlimited scene complexity Don’t need to calculate object-object intersections. Disadvantage Extra memory and bandwidth Waste time drawing hidden objects Z-precision errors May have to use point sampling

27. Computer Graphics Inf4/MSc 27 Z-buffer performance Brute-force image-space algorithm scores best for complex scenes – easy to implement and is very general. Storage overhead: O(1)‏ Time to resolve visibility to screen precision: O(n)‏ –n: number of polygons

28. Computer Graphics Inf4/MSc 19/10/2007 Lecture 9 28 Ex. Architectural scenes Here there can be an enormous amount of occlusion

29. Computer Graphics Inf4/MSc 19/10/2007 Lecture 9 29 Occlusion at various levels

30. Computer Graphics Inf4/MSc 19/10/200730 Portal Culling (object-space) Model scene as a graph: Nodes: Cells (or rooms) Edges: Portals (or doors) Graph gives us: Potentially visible set 1.Render the room 2.If portal to the next room is visible, render the connected room in the portal region 3.Repeat the process along the scene graph A B C D E F G A B DCE

31. Computer Graphics Inf4/MSc Summary Z-buffer is easy to implement on hardware and is an important tool We need to combine it with an object-based method especially when there are too many polygons BSP trees, portal culling

32. Computer Graphics Inf4/MSc 32 References for hidden surface removal Foley et al. Chapter 15, all of it. Introductory text, Chapter 13, all of it Baricentric coordinates www.cs.caltech.edu/courses/cs171/barycent ric.pdf Or equivalents in other texts, look out for: –(as well as the topics covered today)‏ –Depth sort – Newell, Newell & Sancha –Scan-line algorithms