2. 1 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Rasterization Pipeline  Rasterization  Clipping

3. 2 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Rasterization costs  Per vertex –Lighting calculations –Perspective transformation –Slopes calculation  Per pixel –Interpolations of z, c (and leading trailing edges, amortized) –Read, compare, right to update the z-buffer and frame buffer

4. 3 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Acceleration techniques  Reduce per pixel cost –Reduce resolution –Back-face culling  Reduce per vertex cost –T-strips and fans –Frustum culling –Simplification –Texture map  Reduce both –Occlusion culling

5. 4 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Back-face culling  Done in screen space by hardware  Test sign of (x B -x A ) (y C -y A ) - (x C -x A ) (y B -y A ) A B C

6. 5 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac T-strips and fans  Reuse 2 vertices of last triangle fan strip Was important when hardware had only 3 registers Less important in modern graphics boards Saves nearly 2/3 of vertex processing A vertex is still processed twice on average

7. 6 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Frustum culling  Build a tight sphere around each object –Center it at C=((x max +x min )/2, (y max +y min )/2, (z max +z min )/2 ) –Compute radius r as min||CV|| for all vertices V  Test the sphere against each half-space of the frustum –If N(EC)<-r then sphere is outside N E C

8. 7 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Simplification  Collapse edges N AB Collapse A to B if |N(AB)|<eOr use better error estimates

9. 8 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Vertex clustering (Rossignac-Borrel)  Subdivide box around object into grid of cells  Coalesce vertices in each cell into one “attractor”  Remove degenerate triangles –More than one vertex in a cell –Not needed for dangling edge or vertex

10. 9 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Vertex clustering example

11. 10 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Hausdorff distance between T-meshes  Expensive to compute, –because it can occur away from vertices and edges! A B The point of B that is furthest away from A is closest to 3 points on A that are inside faces

12. 11 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac  What is the error produced by edge collapses?  Volume of the solid bounded by old and new triangles –requires computing their intersection  Exact Hausdorff distance between old and new triangles –expensive to compute precisely  Distance between v and planes of all triangles incident on collapsed vertices –one dot product per plane (expensive, but guaranteed error bound)  Quadratic distance between v and planes old triangles –evaluate one quadratic form representing all the planes (cheap but not a bound) –leads to closed form solution for computing the optimal V’ (in least square sense) Estimating edge-collapse error v’v

13. 12 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Measuring Error with Quadrics (Garland)  Given plane P, and point v, we can define a quadric distance Q(v) = D(v,P) 2 –P goes through point p and has normal N –D(v,P) = pv  N –Q(p) = (pv  N) 2, which is a quadratic polynomial in the coordinatges of v –If v=(x,y,z), Q(p)=a 11 x2+a 22 y2+a 33 z2+2a 23 yz+2a 13 xz+2a 12 xy+2b 1 x+2b 2 y+2b 3 z+c  Q may be written represented by 3x3 matrix A,a vector b,and a scalar c  Q(v) may be evaluated as v  (Av)+2b  v+c –or in matrix form as:

14. 13 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Optimal position for v  Pick v to minimize average distance to planes Q(v) –set dQ(v)/dv = 0, which yields: v = -A -1 b

15. 14 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Q(v) measures flatness  Isosurfaces of Q –Are ellipsoids –Stretch in flat direction –Have eigenvalues proportional to the principal curvatures

16. 15 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Texture map  Replace distant details with their images, pasted on simplified shapes or even no flat bill-boards

17. 16 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Texture mapping textureMode(NORMALIZED); PImage maya = loadImage(maya.jpg"); beginShape(QUADS); texture(maya); vertex(-1, 1, 1, 1, 1); vertex( 1, 1, 1, 0, 1); vertex( 1, -1, 1, 0, 0 ); vertex(-1, -1, 1, 1, 0); … endShape();

18. 17 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Occlusion culling  Is ball containing an object hidden behind the triangle?

19. 18 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Occlusion culling in urban walkthrough  From Peter Wonka  Need only display a few facades--the rest is hidden

20. 19 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Shadows / visibility duality  Point light source –Shadow (volume): set of points that do not see the light  From-point visibility –Occluded space: set of points hidden from the viewpoint  Area light source (assume flat for simplicity) –Umbra: set of points that do not see any light –Penumbra: set of points that see a portion of the light source –Clearing: set of points that see the entire light source  From-cell visibility (cell C) –Hidden space: set of points hidden from all points in C –Detectable space: set of points seen from at least one point in C –Exposed space: set of points visible from all points in C

21. 20 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac From-point visibility  What is visible from a given viewpoint –Not from a given area  Usually computed at run-time for each frame –Should be fast –Or done in background before the user starts looking around  May be pre-computed for a set of important viewpoints –Can be slower –User will be invited to jump to these (street crossings) –Or to navigate along pre-computed paths (videos)  May be useful for pre-computing from-cell visibility –Which identifies all objects B that are invisible when the viewpoint is in a cell A –The cell may be reduced to a trajectory (taxi metaphor)

22. 21 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac From-point visibility  Viewpoint/light = {a} (a single point) HIDDENDETECTEDCLEAR

23. 22 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Zen break  The moon is a perfect ball  It is full and perfectly clear  You see it through your window  You wonder about the meaning of life  You trace its outline on the glass Is is a circle? Can you prove it?

24. 23 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Nothing is perfect  In general, the outline of the moon is an ellipse  Proof: –The pencil of rays from your eye to the moon is a cone –The intersection of that cone with the window if a conic –If the moon is clear and visible through the window: it is an ellipse –When the window is orthogonal to the axis of the cone: it is a circle

25. 24 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Cherchez la lune  The moon is a ball B(c,r) of center c and radius r  The occluder is a triangle with vertices d, e, and f  Assume viewer is at point a  Write the geometric formulae for testing whether the moon is hidden behind the triangle HIDDENDETECTEDCLEAR de f c

26. 25 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac When does a triangle appear clockwise?  When do the vertices d, e, f, of a triangle appear clockwise when seen from a viewpoint a? clockwise de f counter-clockwise ed f

27. 26 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac A triangle appears clockwise when  The vertices d, e, f, of a triangle appear clockwise when seen from a viewpoint a when s(a,d,e,f) is true  Procedure s(a,d,e,f) {RETURN ad(ae  af)>0} –ad(ae  af) is a 3x3 determinant –It is called the mixed product (or the triple product) clockwise de f counter-clockwise ed f a a

28. 27 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Testing a point against a half-space?  Consider the plane through points d, e, and f –It splits space in 2 half-spaces  Consider the half-space H of all points that see d, e, f clockwise  How to test whether point c is in H? clockwise de f c

29. 28 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Testing a point against a half-space  Consider the plane through points d, e, and f  Consider the half-space H of all points that see d, e, f clockwise  A point c is in H when s(c,d,e,f) s(c,d,e,f) {RETURN cd(ce  cf)>0} clockwise de f c

30. 29 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Is a point hidden by a triangle?  Is point c hidden from viewpoint a by triangle (d, e, f)? –Write down all the geometric test you need to perform d e f a c

31. 30 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Test whether a point is hidden by a triangle  To test whether c is hidden from a by triangle d, e, f do –IF NOT s(a,d,e,f) THEN swap d and e –IF NOT s(a,d,e,c) THEN RETURN FALSE –IF NOT s(a,e,f,c) THEN RETURN FALSE –IF NOT s(a,f,d,c) THEN RETURN FALSE –IF s(c,d,e,f) THEN RETURN FALSE –RETURN TRUE Clockwise from a d e f a c

32. 31 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Testing a ball against a half-space?  Consider the half-space H of all points that see (d, e, f) clockwise  When is a ball of center c and radius r in H? Clockwise from a d e f c

33. 32 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Testing a ball against a half-space  Consider the half-space H of all points that see (d, e, f) clockwise  A ball of center c and radius r is in H when: ec (ed  ef) > r ||ed  ef|| Clockwise from a d e f c

34. 33 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Is the moon hidden by a triangle?  Is the ball of center c and radius r hidden from a by triangle with vertices d, e, and f? Clockwise from a d e f a c

35. 34 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Is the moon hidden by a triangle?  To test whether the ball of center c and radius r hidden from a by triangle with vertices d, e, and f do: –IF NOT s(a,d,e,f) THEN swap d and e –IF ac (ad  ae) < r || ad  ae || THEN RETURN FALSE –IF ac (ae  af) < r || ae  af || THEN RETURN FALSE –IF ac (af  ad) < r || af  ad || THEN RETURN FALSE –IF ec (ed  ef) < r || ed  ef || THEN RETURN FALSE –RETURN TRUE  Explain what each line tests  Are these test correct?  Are they sufficient? Clockwise from a d e f a c

36. 35 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Is the moon hiding a triangle?  Triangle d, e, f is hidden from viewpoint a by a ball of center c and radius r when its 3 vertices are (hidden). –Proof? –How to test it? Hidden moon d f c Hidden triangle d f c e e

37. 36 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac When is a planar set A convex? seg(p,q) is the line segment from p to q A is convex  (  p  A  q  A  seg(p,q)  A ) p q p q p q

38. 37 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is the convex hull of a set A? The convex hull H(A) is the intersection of all convex sets that contain A.

39. 38 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Is a convex occluder hiding a polyhedron?  A polyhedron P is hidden by a convex occluder O if all of its vertices are –Proof? –Is a subdivision surface hidden if its control vertices are?  We do not need to test all of the vertices –Only those of a container (minimax box, bounding ball, convex hull) Proof? –Only those on the silhouette of a convex container Proof?  What if the occluder is not convex? –Can it still work?

40. 39 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Strategies for non-convex occluders? What if O is not convex? –replace O by an interior ball Not easy to compute a (nearly) largest ball inside O The average of the vertices of O may be outside of O –replace O by an interior axis-aligned box Not easy to compute a largest box inside –replace O by a convex set it contains O may be too thin Andujar, Saona Navazo CAD, 32 (13), 2000

41. 40 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac The Hoop principle The occluder does not have to be convex, provided that a portion of its surface appears convex to the viewer and that the object is behind it.

42. 41 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Hoop from non-convex occluders  Pick a candidate occluder S –large triangle mesh –close to the viewer  Select a set D of triangles of S whose border C is a hoop: –single loop –that has a convex projection on the screen  Build a shield O –on plane behind D –bounded by projection of C  O is a convex occluder –anything hidden by O –is hidden by D –and hence by S –sure? O C D S

43. 42 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac When does a 3D polyloop appear convex? Each pair of consecutive edges defines a linear half-space from where their common vertex appears convex. The portion of space from which a loop C of edges appears convex is the intersection of these half-spaces or of their complements.

44. 43 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Two strategies for building hoops  Grow a patch D on the surface S, or  Grow a curve C on the boundary of interior octree cells –simplify it, ensuring that the simplified border lies inside the solid

45. 44 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Testing against a hoop  Given a hoop C, we can easily test whether the viewpoint or a whole cell sees it as convex. –Test whether the viewpoint (or whether all the vertices of the cell,) lie in the intersections of all half-spaces.  But this is not sufficient if we want to construct a shield from it. –Why not? We must ensure that the shield O is behind the portion D of the original surface S that is bounded by C.

46. 45 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Ensuring that O is behind D Store with the hoop the thickness (depth of projection of D on the normal to C) Compute normal using sum of cross-products Use it to place the shield O sufficiently behind the hoop C so that it does not intersect D C D n O C D n O

47. 46 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Ensuring that D occludes O Store with the hoop the thickness (range of projection of D on the normal n to C) Ensure that the viewer is not in that range … or shoot a ray from viewer to O and check parity of hits with D C D n O C D n O

48. 47 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Growing a hoop around the moon  Find the first intersection, i, between ac and the mesh of S  Consider the triangle (d, e, f) of S that contains i  If the triangle hides the moon, we are done (return hidden)  Otherwise grow it by adding neighboring triangles –Let C be the boundary of this patch D of triangles –Identify bad vertices of C that appear in front of the moon –Add triangles to make these vertices interior –Ensuring that C remains a simple loop –Stop when C has no bad vertices (return hidden) –or when you can’t grow (return not hidden)  Test the surface patch D bounded by C –Count intersections with ac –If odd then return hidden Clockwise from a d e f a c

49. 48 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Cell-to-cell visibility Now, let us look at the problem of pre-computing whether any portion of a cell A is visible from a viewpoint in cell B. Why? (Applications?)

50. 49 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Notation and assumptions  Two sets, A and B  Two points a  A, b  B  O is the occluder  A, B, and O are disjoint  ab is the line segment from a to b  a–b  O  ab = , a sees b (ignore self-occlusion)  a | b  O  ab ≠ , a is occluded from by O a b BA O a b BA O

51. 50 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac (Jarek’s) Terminology and definitions?  HIDDEN: H O (A,B) –B is entirely hidden from A by O –?  DETECTED: D O (A,B) –B can be detected from any point of A –?  GUARDED: G O (A,B) –B can be guarded from a single point of A –?  CLEAR: C O (A,B) –B can be guarded from any point of A –? a b BA O B A B A O a b B A O O

52. 51 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac  HIDDEN: H O (A,B) –B is entirely hidden from A –H O (A,B)   a,  b: a | b  DETECTED: D O (A,B) –B can be detected from any point of A –D O (A,B)   a,  b: a–b  GUARDED: G O (A,B) –B can be guarded from some point of A –G O (A,B)   a,  b: a–b  CLEAR: C O (A,B) –B can be guarded from any point of A –C O (A,B)   a,  b: a–b Definitions a b BA O B A O B A O a b B A O

53. 52 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Properties (and proofs?)  C O (A,B)  C O (B,A) and H O (A,B)  H O (B,A)  C O (A,B)  G O (A,B) and G O (B,A)  G O (A,B)  D O (B,A) proof ? a b BA O H O (A,B)   a,  b: a | b B A O D O (A,B)   a,  b: a–b B A O G O (A,B)   a,  b: a–b a b B A O C O (A,B)   a,  b: a–b

54. 53 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Properties and proofs  C O (A,B)  C O (B,A) and H O (A,B)  H O (B,A)  C O (A,B)  G O (A,B) and C O (A,B)  G O (B,A)  G O (A,B)  D O (B,A) proof G O (A,B)   a,  b: a–b   b,  a: b–a  D O (B,A) a b BA O H O (A,B)   a,  b: a | b B A O D O (A,B)   a,  b: a–b B A O G O (A,B)   a,  b: a–b a b B A O C O (A,B)   a,  b: a–b

55. 54 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Counter-example?  Draw O such that none of the 8 relations hold a b BA O H O (A,B)   a,  b: a | b B A O D O (A,B)   a,  b: a–b B A O G O (A,B)   a,  b: a–b a b B A O C O (A,B)   a,  b: a–b B A O? H O (A,B), G O (A,B), D O (A,B), C O (A,B), H O (B,A), G O (B,A), D O (B,A), C O (B,A) are all false

56. 55 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Counter-example?  Draw O such that none of the 8 relations hold a b BA O H O (A,B)   a,  b: a | b B A O D O (A,B)   a,  b: a–b B A O G O (A,B)   a,  b: a–b a b B A O C O (A,B)   a,  b: a–b H O (A,B), G O (A,B), D O (A,B), C O (A,B), H O (B,A), G O (B,A), D O (B,A), C O (B,A) are all false a b B A O

57. 56 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Cell-to-cell visibility (C2CV)  Viewpoint is in cell A  Small objects in cell B need not be rendered if cell A is occluded from cell B by some surface S: H S (A,B)?

58. 57 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Terminology  Assume that surface S, cell A, and cell B are disjoint?  R = convex hull of A  B (contains all rays from A to B)  T = tube (boundary of R not on A or B)  C connected component of S  T

59. 58 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac The Occlusion Theorem The surface S occludes cell A from cell B, if and only if: 1. 1.There exists a connected component, C, of S  T that is bounding a portion D of S and has no boundary in the interior of R. 2.Given an arbitrary (non-singular) ray, ab, from a  A to b  B, ab has an odd number of intersections with D  C.

60. 59 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Proof

61. 60 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Various cases

62. 61 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac ShieldTester for triangle meshes Obtain an edge e AB of T from a vertex of A to a vertex of B IF e AB  S =  THEN RETURN (“visible”) FOR EACH triangle x of S that is stabbed by e AB DO { Start a WALK that traces a connected component C of S  T IF the WALK did not run into a boundary of S THEN { INVADE the portion of S to the left of the triangles crossed by the WALK along C IF the invaded portion has no hole THEN IF the number of triangles that were stabbed by e AB and visited by the WALK and the INVADE is odd THEN RETURN (“occluded”) } RETURN (“undecided”)

63. 62 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Walking one step

64. 63 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Turning left on S

65. 64 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Invade process

66. 65 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Hot edges

67. 66 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Results

68. 67 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Is the moon hidden by a tree?  Can’t grow a patch of edge-connected triangles –Leaves/branches are small and disjoint  Use approximate shadow fusion –Render tree(s) –Mark area of rendered pixels –Construct big rectangle inside it (how?) –Let M= max(z-buffer inside rectangle) –Push back rectangle to z=M –Use it as occluder for other objects (behind the tree)  Accuracy depends on screen resolution –Can be improved by multiple passes (zoomed 1/4 of image each)  Use bounding spheres around objects to test them –Conservative part –Use a hierarchy of bounding sphere (hierarchical z-buffer)

69. 68 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Occluded volumes  Assume that we have an area light source or a cell containing the viewpoint that is a polyhedron  Assume that all occluders are polyhedra  Can we compute the hidden volume (umbra)? Difficult because it is bounded by curved surfaces

70. 69 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What you must remember  How to build a rasterizer  How to accelerate it

71. 70 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Examples of questions for exams  When is it no-longer useful to simplify the model?  What are the 7 acceleration techniques?  Which ones reduce per-vertex cost?  How much savings should you expect from each technique?