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Rasterization Pipeline

Rasterization Pipeline. Rasterization Clipping. 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.

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Rasterization Pipeline

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  1. Rasterization Pipeline • Rasterization • Clipping

  2. 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

  3. 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

  4. Back-face culling • Done in screen space by hardware • Test sign of (xB-xA) (yC-yA) - (xC-xA) (yB-yA) C B A

  5. T-strips and fans • Reuse 2 vertices of last triangle strip fan Was important when hardware had only 3 registers Saves nearly 2/3 of vertex processing A vertex is still processed twice on average Less important in modern graphics boards

  6. Frustum culling • Build a tight sphere around each object • Center it at C=((xmax+xmin)/2, (ymax+ymin)/2, (zmax+zmin)/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 C E N

  7. N A B Simplification • Collapse edges Collapse A to B if |N•(AB)|<e Or use better error estimates

  8. 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

  9. Vertex clustering example

  10. 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

  11. v v’ Estimating edge-collapse error • 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)

  12. 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)=a11x2+a22y2+a33z2+2a23yz+2a13xz+2a12xy+2b1x+2b2y+2b3z+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:

  13. 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

  14. Q(v) measures flatness • Isosurfaces of Q • Are ellipsoids • Stretch in flat direction • Have eigenvalues proportional to the principal curvatures

  15. Texture map • Replace distant details with their images, pasted on simplified shapes or even no flat bill-boards

  16. 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();

  17. Occlusion culling • Is ball containing an object hidden behind the triangle?

  18. Occlusion culling in urban walkthrough • From Peter Wonka • Need only display a few facades--the rest is hidden

  19. 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

  20. 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)

  21. From-point visibility • Viewpoint/light ={a}(a single point) HIDDEN DETECTED CLEAR

  22. 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?

  23. 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

  24. 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 f c e d HIDDEN DETECTED CLEAR

  25. When does a triangle appear clockwise? • When do the vertices d, e, f, of a triangle appear clockwise when seen from a viewpoint a? f f e d d e clockwise counter-clockwise

  26. 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 • Procedures(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) f f e d d e clockwise counter-clockwise a a

  27. 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? f e d clockwise c

  28. 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 isin H whens(c,d,e,f) s(c,d,e,f) {RETURN cd•(cecf)>0} f e d clockwise c

  29. 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 c f e d a

  30. c f Clockwise from a e d a 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

  31. c e d Clockwise from a f 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?

  32. c e d Clockwise from a f 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||

  33. c f Clockwise from a e d a Is the moon hidden by a triangle? • Is the ball of center c and radius r hidden from a by trianglewith vertices d, e,and f?

  34. c f Clockwise from a e d a 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?

  35. 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? f f c c d e d e Hidden moon Hidden triangle

  36. When is a planar set A convex? q p 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

  37. What is the convex hull of a set A? The convex hull H(A) is the intersection of all convex sets that contain A.

  38. 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?

  39. 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

  40. 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.

  41. 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? S C D O

  42. 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.

  43. 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

  44. 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.

  45. n O D C 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 D n O C

  46. n O D C 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 n O D C

  47. c f Clockwise from a e d a Growing a hoop around the moon • Find the first intersection, i, betweenacand the mesh ofS • 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 appearin 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

  48. 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?)

  49. O b a A B O b a A B 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

  50. A B O b a (Jarek’s) Terminology and definitions? • HIDDEN: HO(A,B) • B is entirely hidden from A by O • ? • DETECTED: DO(A,B) • B can be detected from any point of A • ? • GUARDED: GO(A,B) • B can be guarded from a single point of A • ? • CLEAR: CO(A,B) • B can be guarded from any point of A • ? O A B A B O a b B A O

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