# CEng 477 Introduction to Computer Graphics Fall 2007

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1 Visible Surface Detection CEng 477 Introduction to Computer Graphics Fall 2007

2 Visible Surface Detection Visible surface detection or hidden surface removal. Realistic scenes: closer objects occludes the others. Classification: Object space methods Image space methods

3 Object Space Methods Algorithms to determine which parts of the shapes are to be rendered in 3D coordinates. Methods based on comparison of objects for their 3D positions and dimensions with respect to a viewing position. For N objects, may require N*N comparision operations. Efficient for small number of objects but difficult to implement. Depth sorting, area subdivision methods.

4 Image Space Methods Based on the pixels to be drawn on 2D. Try to determine which object should contribute to that pixel. Running time complexity is the number of pixels times number of objects. Space complexity is two times the number of pixels: One array of pixels for the frame buffer One array of pixels for the depth buffer Coherence properties of surfaces can be used. Depth buffer and ray casting methods.

5 Depth Cueing Hidden surfaces are not removed but displayed with different effects such as intensity, color, or shadow for giving hint for third dimension of the object. Simplest solution: use different colors intensities based on the dimensions of the shapes.

6 Back-Face Detection Back face detection of 3D polygon surface is easy Recall the polygon surface equation: Ax By Cz D 0 We need to also consider the viewing direction when determining whether a surface is back face or front face. The normal of the surface is given by: N = A, B, C

7 Back-Face Detection A polygon surface is a back face if: V view N 0 However, remember that after application of the viewing transformation we are looking down the negative z axis. Therefore a polygon is a back face if: 0,0, 1 N 0 or if C 0

8 Back-Face Detection We will also be unable to see surfaces with C=0. Therefore, we can identify a polygon surface as a back face if: C 0

9 Back-Face Detection Back face detection can identify all the hidden surfaces in a scene that contain non overlapping convex polyhedra. But we have to apply more tests that contain overlapping objects along the line of sight to determine which objects obscure which objects.

10 Depth-Buffer Method Also known as z buffer method. It is an image space approach Each surface is processed separately one pixel position at a time across the surface The depth values for a pixel are compared and the closest (smallest z) surface determines the color to be displayed in the frame buffer. Applied very efficiently on polygon surfaces Surfaces are processed in any order

11 Depth-Buffer Method

12 Depth-Buffer Method Two buffers are used Frame Buffer Depth Buffer The z coordinates (depth values) are usually normalized to the range [0,1]

13 Depth-Buffer Algorithm Initialize the depth buffer and frame buffer so that for all buffer positions (x,y), depthbuff (x,y) = 1.0, framebuff (x,y) =bgcolor Process each polygon in a scene, one at a time For each projected (x,y) pixel position of a polygon, calculate the depth z. If z < depthbuff (x,y), compute the surface color at that position and set depthbuff (x,y) = z, framebuff (x,y) = surfcol (x,y)

14 Calculating depth values efficiently We know the depth values at the vertices. How can we calculate the depth at any other point on the surface of the polygon. Using the polygon surface equation: Ax By D z= C

15 Calculating depth values efficiently For any scan line adjacent horizontal x positions or vertical y positions differ by 1 unit. The depth value of the next position (x+1,y) on the scan line can be obtained using z ' A x 1 By D = C =z A C

16 { x ' = x 1 m Calculating depth values efficiently For adjacent scan lines we can compute the x value using the slope of the projected line and the previous x value. z ' A/ m B =z C

17 Depth-Buffer Method Is able to handle cases such as View from the Right side

18 Z-Buffer and Transparency We may want to render transparent surfaces (alpha 1) with a z buffer However, we must render in back to front order Otherwise, we would have to store at least the first opaque polygon behind transparent one Partially transparent 3rd Front 1st or 2nd Opaque 2nd 3rd: Need depth of 1st and 2nd Opaque 1st 1st or 2nd Must recall this color and depth OK. No Problem Problematic Ordering

19 A-Buffer Method Extends the depth buffer algorithm so that each position in the buffer can reference a linked list of surfaces. More memory is required However, we can correctly compose different surface colors and handle transparent surfaces.

20 A-Buffer Method Each position in the A buffer has two fields: a depth field surface data field which can be either surface data or a pointer to a linked list of surfaces that contribute to that pixel position

21 Algorithm: Ray Casting Algorithm Cast ray from viewpoint through each pixel to find front most surface Viewer View Plane A B D C E It is like a variation of the depth buffer algorithm, in which we proceed pixel by pixel instead of proceeding surface by surface.

22 Object Space Methods

23 Depth Sorting Also known as painters algorithm. First draw the distant objects than the closer objects. Pixels of each object overwrites the previous objects

24 Depth-sort algorithm The idea here is to go back to front drawing all the objects into the frame buffer with nearer objects being drawn over top of objects that are further away. Simple algorithm: Sort all polygons based on their farthest z coordinate Resolve ambiguities Draw the polygons in order from back to front This algorithm would be very simple if the z coordinates of the polygons were guaranteed never to overlap. Unfortunately that is usually not the case, which means that step 2 can be somewhat complex.

25 Depth-sort algorithm First must determine z extent for each polygon z depth max z max z extent depth min x

26 Depth-sort algorithm Ambiguities arise when the z extents of two surfaces overlap. z surface 2 surface 1 x

27 Depth-sort algorithm y x

28 Depth-sort algorithm All polygons whose z extents overlap must be tested against each other. We start with the furthest polygon and call it P. Polygon P must be compared with every polygon Q whose z extent overlaps P's z extent. 5 comparisons are made. If any comparison is true then P can be written before Q. If at least one comparison is true for each of the Qs then P is drawn and the next polygon from the back is chosen as the new P.

29 Depth-sort algorithm 1. Do P and Q's x extents not overlap? 2. Do P and Q's y extents not overlap? 3. Is P entirely on the opposite side of Q's plane from the viewport? 4. Is Q entirely on the same side of P's plane as the viewport? 5. Do the projections of P and Q onto the (x,y) plane not overlap? If all 5 tests fail we quickly check to see if switching P and Q will work. Tests 1, 2, and 5 do not differentiate between P and Q but 3 and 4 do. So we rewrite 3 and 4 as: 3. Is Q entirely on the opposite side of P's plane from the viewport? 4. Is P entirely on the same side of Q's plane as the viewport?

30 Depth-sort algorithm x extents not overlap? z Q P if they do, test fails x

31 Depth-sort algorithm y extents not overlap? z Q P if they do, test fails y

32 Depth-sort algorithm Is P entirely behind the surface Q relative to the viewing position (i.e., behind Q s plane with respect to the viewport)? z P Q Test is true x

33 Depth-sort algorithm Is Q entirely in front of P's plane relative to the viewing position (i.e., the viewport)? z P Test is true Q x

34 Depth-sort algorithm Do the projections of P and Q onto the (x,y) plane not overlap? z y Q P Q P x Test is true hole in P x

35 If all tests fail Depth-sort algorithm then reverse P and Q in the list of surfaces sorted by maximum depth set a flag to say that the test has been performed once. If the tests fail a second time, then it is necessary to split the surfaces and repeat the algorithm on the 4 new split surfaces

36 Example: Depth-sort algorithm We end up processing with order Q2,P1,P2,Q1 z Q 2 P 1 P 2 Q 1 x

37 Binary Space Partitioning BSP tree: organize all of space (hence partition) into a binary tree Tree gives a rendering order: correctly traversing this tree enumerates objects from back to front Tree splits 3D world with planes The world is broken into convex cells Each cell is the intersection of all the half spaces of splitting planes on tree path to the cell Splitting planes can be arbitrarily oriented

38 Building BSP-Trees Choose a splitting polygon (arbitrary) Split its cell using the plane on which the splitting polygon lies May have to chop polygons in two (Clipping!) Continue until each cell contains only one polygon fragment (or object)

39 BSP-Tree Example C 4 A 3 B A + C B

40 Using a BSP-Tree Observation: Things on the opposite side of a splitting plane from the viewpoint cannot obscure things on the same side as the viewpoint Spliting plane This is a statement about rays: a ray must hit something on this side of the split plane before it hits the split plane and before it hits anything on the back side It is NOT a statement about distance things on the far side of the plane can be closer than things on the near side Gives a relative ordering of the polygons, not absolute in terms of depth or any other quantity

41 BSP Trees: Another example

42 BSP Trees: Another example

43 BSP Trees: Another example

44 BSP Trees: Another example

45 BSP Trees: Another example

46 Rendering BSP Trees renderbsp(bsptree *T) BSPtree *near, *far; if (T is a leaf node) { renderobject(t); return; } if (eye on left side of T >plane) near = T >left; far = T >right; else near = T >right; far = T >left; renderbsp(far); renderbsp(near);

47 BSP-Tree: Advantages One tree works for any viewing point

48 BSP-Tree: Disadvantages No bunnies were harmed in the example But what if a splitting plane passes through an object? Split the object; give half to each node: Ouch Worst case: can create up to O(n 3 ) objects!

49 Final Topics You are responsible of all the topics we have covered throughout the semester. However, you may expect more questions on the subjects covered after the midterm: Texture Mapping 3D Object Representations Illumination Visible Surface Detection

50 Texture Mapping Texture mapping process Specifying the texture image Texture coordinates: (s,t) texture space Magnification vs. minification: when do we need magnification/minification?

51 3D Object Representations Polygon representations Curved object representations: Natural Cubic Splines Hermite Curves Bezier Curves Forward difference calculations for cubic Scene graph representations CSG Octrees Fractals

52 Illumination Light sources Point, directional, spot lights Basic illumination model Ambient, diffuse, specular components Surface rendering methods Flat, Gouraud, and Phong shading

53 Visible Surface Detection Image Space versus Object Space methods Back face detection Depth buffer algorithm Depth sorting algorithm Binary Space Partitioning

54 Format of the Final Exam Expect questions of similar type as in the midtermexam

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