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  1. CS 431/636 Advanced Rendering Techniques Dr. David Breen University Crossings 149 Tuesday 6PM  8:50PM Presentation 2 4/7/09

  2. Start Up • Any questions from last time? • Go over sampling image plane? • Or intersection algorithms?

  3. Slide Credits • Leonard McMillan, Seth Teller, Fredo Durand, Barb Cutler - MIT • G. Drew Kessler, Larry Hodges - Georgia Institute of Technology • John Hart - University of Illinois • Rick Parent - Ohio State University

  4. More Geometry & Intersections

  5. Ray/Plane Intersection Ray is defined by R(t) = Ro + Rd*t where t  0 Ro = Origin of ray at (xo, yo, zo) Rd = Direction of ray [xd, yd, zd] (unit vector) Plane is defined by [A, B, C, D] Ax + By + Cz + D = 0 for a point in the plane Normal Vector, N = [A, B, C] (unit vector) A2 + B2 + C2 = 1 D = - N • P0 (P0 - point in plane)

  6. Ray/Plane (cont.) Substitute the ray equation into the plane equation: A(xo + xdt) + B(yo + ydt) + C(zo +zdt) + D = 0 Solve for t: t = -(Axo + Byo + Czo + D) / (Axd + Byd + Czd) t = -(N • Ro - N • P0 ) / (N • Rd)

  7. N • Rd = 0 N • Rd > 0 Ro Ro t < 0 t > 0 What Can Happen?

  8. Ray/Plane Summary Intersection point: (xi, yi, zi) = (xo + xdti, yo + ydti, zo + zdti) • Calculate N • Rd and compare it to zero. • Calculate ti and compare it to zero. • Compute intersection point. • Flip normal if N • Rd is positive

  9. Ray-Parallelepiped Intersection • Axis-aligned • From (X1, Y1, Z1) to (X2, Y2, Z2) • Ray P(t)=Ro+Rdt y=Y2 y=Y1 x=X1 x=X2 Rd Ro

  10. y=Y2 y=Y1 x=X1 x=X2 Rd Ro Naïve ray-box Intersection • Use 6 plane equations • Compute all 6 intersection • Check that points are inside boxAx+By+Cz+D ≤ 0

  11. y=Y2 y=Y1 x=X1 x=X2 Rd Ro Factoring out computation • Pairs of planes have the same normal • Normals have only one non-0 component • Do computations one dimension at a time • Maintain tnear and tfar (closest and farthest so far)

  12. Test if parallel • If Rdx = 0, then ray is parallel • If Rox < X1 or Rox > x2 return false y=Y2 y=Y1 Rd x=X1 x=X2 Ro

  13. t2 y=Y2 t1 y=Y1 x=X1 x=X2 Rd Ro If not parallel • Calculate intersection distance t1 and t2 • t1 = (X1-Rox)/Rdx • t2 = (X2-Rox)/Rdx

  14. t2x t1x tfar tnear y=Y2 t2y tfar Rd Ro y=Y1 t1y x=X1 x=X2 Test 1 • Maintain tnear and tfar • If t1 > t2, swap • if t1 > tnear, tnear = t1 • if t2 < tfar, tfar = t2 • If tnear > tfar, box is missed

  15. Test 2 • If tfar < 0, box is behind Rd Ro y=Y2 tfar t2x t1x y=Y1 x=X1 x=X2

  16. y=Y2 tfar tnear y=Y1 t1y Rd x=X1 x=X2 Ro Algorithm recap • Do for all 3 axes • Calculate intersection distance t1 and t2 • Maintain tnear and tfar • If tnear > tfar, box is missed • If tfar < 0, box is behind • If box survived tests, return intersection at tnear • If tnear is negative, return tfar

  17. Ray/Ellipsoid Intersection Ray/Cylinder Intersection Ellipsoid's surface is defined by the set of points {(xs, ys, zs)} satisfying the equation: (xs/ a)2 + (ys/ b)2 + (zs/ c)2 - 1 = 0 Cylinder's surface is defined by the set of points {(xs, ys, zs)} satisfying the equation: (xs)2 + (ys)2 - r2 = 0 -z0 ≤ zs ≤ z0 Centers at origin

  18. Ray/Ellipsoid Intersection Ray/Cylinder Intersection • Substitute ray equation into surface equations • This is a quadratic equation in t: • At2 + Bt + C = 0 • Analyze as before • Solve for t with quadratic formula • Plug t back into ray equation - Done • Well,… not exactly

  19. Ray/Cylinder Intersection • Is intersection point Pi between -Z0 and Z0? • If not, Pi is not valid • Also need to do intersection test with z = -Z0 , Z0 plane • If (Pix)2 + (Piy)2 ≤ r2, you’ve intersected a “cap” • Which valid intersection is closer?

  20. Superquadrics

  21. Bezier Patch • Patch of order (n, m) can be defined in terms of a set of (n + 1)(m + 1) control points Pi+1,j+1 for integer indices i = 0 to n, j = 0 to m.

  22. Tesselate Patches and Superquadrics

  23. Tesselate Patches and Superquadrics

  24. Utah Teapot • Modeled by 32 Bézier Patches • Control points available at http://www.holmes3d.net/graphics/teapot

  25. SMF Triangle Meshes v -1 -1 -1 v 1 -1 -1 v -1 1 -1 v 1 1 -1 v -1 -1 1 v 1 -1 1 v -1 1 1 v 1 1 1 f 1 3 4 f 1 4 2 f 5 6 8 f 5 8 7 f 1 2 6 f 1 6 5 f 3 7 8 f 3 8 4 f 1 5 7 f 1 7 3 f 2 4 8 f 2 8 6 vertices triangles Draw data structure

  26. Triangle Meshes (.iv)

  27. Transformations & Hierarchical Models

  28. a e i m b f j n c g k o d h l p x y z w x' y' z' w' = p' = M p Homogeneous Coordinates • Add an extra dimension • in 2D, we use 3 x 3 matrices • In 3D, we use 4 x 4 matrices • Each point has an extra value, w

  29. a e i 0 b f j 0 c g k 0 d h l 1 x y z 1 x' y' z' 1 = Homogeneous Coordinates • Most of the time w = 1, and we can ignore it

  30. 1 0 0 0 0 1 0 0 0 0 1 0 tx ty tz 1 x y z 1 x' y' z' 1 x' y' z' = Translate(c,0,0) Translate (tx, ty, tz) y p' p • Why bother with the extra dimension?Because now translations can be encoded in the matrix! x c

  31. sx 0 0 0 0 sy 0 0 0 0 sz 0 0 0 0 1 x y z 1 x' y' z' 1 = Scale(s,s,s) Scale (sx, sy, sz) p' y p q' • Isotropic (uniform) scaling: sx = sy = sz • You only have to implement uniform scaling q x

  32. ZRotate() Rotation y p' θ • About z axis p x z cos  sin  0 0 -sin  cos  0 0 0 0 1 0 0 0 0 1 x y z 1 x' y' z' 1 =

  33. 0 cos  sin  0 0 -sin  cos  0 0 0 0 1 x y z 1 x' y' z' 1 1 0 0 0 = cos  0 -sin  0 0 1 0 0 sin  0 cos  0 0 0 0 1 x y z 1 x' y' z' 1 = Rotation • About x axis: • About y axis:

  34. kxkx(1-c)+c kykx(1-c)+kzs kzkx(1-c)-kys 0 kykx(1-c)-kzs kyky(1-c)+c kzky(1-c)+kxs 0 kxkz(1-c)+kys kykz(1-c)-kxs kzkz(1-c)+c 0 0 0 0 1 x y z 1 x' y' z' 1 = Rotate(k, ) Rotation y θ k • About (kx, ky, kz), an arbitrary unit vector (Rodrigues Formula) x z where c = cos  & s = sin 

  35. 0 0 0 0 0 0 1 1 1 How are transforms combined? Scale then Translate (5,3) (2,2) Scale(2,2) Translate(3,1) (1,1) (3,1) (0,0) (0,0) Use matrix multiplication: p' = T ( S p ) = ((TS) p) 0 1 0 2 0 2 1 0 3 1 2 0 0 0 2 0 3 1 TS = = Caution: matrix multiplication is NOT commutative!

  36. Non-commutative Composition Scale then Translate: p' = T ( S p ) = TS p (5,3) (2,2) Scale(2,2) Translate(3,1) (1,1) (3,1) (0,0) (0,0) Translate then Scale: p' = S ( T p ) = ST p (8,4) (4,2) Translate(3,1) Scale(2,2) (6,2) (1,1) (3,1) (0,0)

  37. 0 0 0 0 0 0 1 1 1 Non-commutative Composition Scale then Translate: p' = T ( S p ) = TS p 0 1 0 0 2 0 0 2 0 1 0 0 3 1 1 2 0 0 0 0 1 2 0 0 3 1 1 TS = = Translate then Scale: p' = S ( T p ) = ST p 0 2 0 1 0 2 2 0 0 0 1 0 3 1 2 0 6 2 ST = =

  38. Transformations in Ray Tracing

  39. Transformations in Modeling • Position objects in a scene • Change the shape of objects • Create multiple copies of objects • Projection for virtual cameras • Animations

  40. Scene Description Scene Materials Camera Lights Background Objects

  41. Simple Scene Description File Camera { center 0 0 10 direction 0 0 -1 up 0 1 0 } Lights { numLights 1 DirectionalLight { direction -0.5 -0.5 -1 color 1 1 1 } } Background { color 0.2 0 0.6 } Materials { numMaterials <n> <MATERIALS> } Group { numObjects <n> <OBJECTS> }

  42. Hierarchical Models • Logical organization of scene

  43. Simple Example with Groups Group { numObjects 3 Group { numObjects 3 Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { numObjects 2 Group { Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { Box { <BOX PARAMS> } Sphere { <SPHERE PARAMS> } Sphere { <SPHERE PARAMS> } } } Plane { <PLANE PARAMS> } }

  44. Adding Materials Group { numObjects 3 Group { numObjects 3 Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { numObjects 2 Group { Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { Box { <BOX PARAMS> } Sphere { <SPHERE PARAMS> } Sphere { <SPHERE PARAMS> } } } Plane { <PLANE PARAMS> } }

  45. Adding Transformations

  46. Using Transformations • Position the logical groupings of objects within the scene • Transformation in group

  47. Directed Acyclic Graph is more efficient and useful

  48. Processing Model Transformattions • Goal • Get everything into world coordinates • Traverse graph/tree in depth-first order • Concatenate transformations • Can store intermediate transformations • Apply/associate final transformation to primitive at leaf node • What about cylinders, superquadrics, etc.? • Transform ray!

  49. r minor r major (x,y) r = 1 (0,0) World Space Object Space Transform the Ray • Map the ray from World Space to Object Space pWS = M pOS pOS = M-1 pWS