Lecture Hidden Surface Removal and Rasterization Taku Komura 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. 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) Visibility of primitives A scene primitive can be invisible for reasons: Primitive lies outside field of view Primitive is back-facing Primitive is occluded by one or more objects nearer the viewer wrong visibility correct visibility Definitions: Visible surface algorithms. 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 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. Back face culling. Or if N. V > 0 we are viewing the back face so polygon is obscured. 6
Painters algorithm (object space). BSP (Binary Space Partitioning) Tree. Draw surfaces in back to order nearer polygons paint over farther ones. Supports transparency. Key issue is order determination. Doesn t always work see image at right. One of class of list-priority algorithms returns ordered list of polygon fragments for specified view point (static pre-processing stage). Divide scene into (relative to normal) and back half-spaces. Choose a polygon from each side split scene again. Recursively divide each side until each node contains only polygon. View of scene from above 7 a b a b Divide scene into (relative to normal) and back half-spaces. Divide scene into (relative to normal) and back half-spaces. Choose a polygon from each side split scene again. back Choose a polygon from each side split scene again. back Recursively divide each side until each node contains only polygon. a b Recursively divide each side until each node contains only polygon. a b Lecture 9 9/0/0079 Displaying a BSP tree. Divide scene into (relative to normal) and back half-spaces. Choose a polygon from each side split scene again. Recursively divide each side until each node contains only polygon. a a b back b 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 half-space, draw polygons behind root first, then the root polygon, then polygons in. 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. Lecture 9 9/0/007
BSP performance measure 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. Lecture 9 9/0/007 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 & Algorithms, pp 79 8. September 99. Lecture 9 9/0/007 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. 6 7 8
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 9 0 Filling in Triangles Scan line algorithm Filling in the triangle by drawing horizontal lines from top to bottom Triangle Rasterization Consider a D triangle with vertices p0, p, p. Let p be any point in the plane. We can always find a, b, c such that Barycentric coordinates Checking whether a pixel is inside / outside the triangle We will have if and only if p is inside the triangle. We call the barycentric coordinates of p. Computing the baricentric coordinates of the interior pixels The triangle is composed of points p0 (x0,y0), p (x, y), p(x,y) (α,β,γ) : barycentric coordinates Only if 0<α,β,γ<, (x,y) is inside the triangle Depth can be computed by αz0 + βz +γz Can do the same thing for color, normals, textures 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
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,], we plot it (and compute the depth) 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 6 Z-buffer performance Brute-force image-space algorithm scores best for complex scenes easy to implement and is very general. Storage overhead: O() Time to resolve visibility to screen precision: O(n) n: number of polygons Ex. Architectural scenes Here there can be an enormous amount of occlusion 7 Lecture 9 9/0/007 8 Occlusion at various levels Portal Culling (object-space) D F Model scene as a graph: Nodes: Cells (or rooms) Edges: Portals (or doors) C A B E G Graph gives us: Potentially visible set Lecture 9 9/0/007 9. Render the room. If portal to the next room is visible, render the connected room in the portal region. Repeat the process along the scene graph C A B D E 9/0/007 0
Summary References for hidden surface removal 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 Foley et al. Chapter, all of it. Introductory text, Chapter, all of it Baricentric coordinates www.cs.caltech.edu/courses/cs7/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 6