Cameras and Triangles. CS6965 Fall 2011

Transcription

1 Cameras and Triangles CS6965

2 Camera models The camera maps pixels to rays What kind of camera models might we want? 2

3 Camera models Typical: Orthographic Pinhole (perspective) Advanced: Depth of field (thin lens approximation) Sophisticated lenses ( A realistic camera model for computer graphics, Kolh, Mitchell, Hanrahan) Fish-eye lens Arbitrary distortions 3

4 Camera models Map pixel coordinates -1 to 1 Pay careful attention to pixel centers Can be combined with ray generation Non-square images Longest dimension is -1 to 1, shorter is smaller (still centered at 0) Or camera knows about aspect ratio 1 (-.75, -.75) (-.25,.25) 4

5 Orthographic projection Film is just a rectangle in space Rays are parallel (same direction) 5

6 Orthographic projection Specify with center (P) and two vectors (u, v) O = P + xu + yv V = u v u, v : image size u = aspect ratio v square image: u v = 0 6 u v P O V

7 Pinhole camera Most common model for ray tracing Image is projected upside down onto image plane 7

8 Pinhole camera Easier to think about right side up Focal point is also called the eye point 8

9 Pinhole camera Parameters: E : Eye point (focal point) C : Lookat point Up : Up vector θ: Field of view Up E θ C 9

10 Pinhole camera Top View C Construction: L = C E (look or gaze direction) L L n = L u tmp = Ln Up = utmp Ln v tmp Up L v θ θ 2 u 10 E

11 Pinhole camera Top View C How do we get the lengths of u/v? tan θ 2 = u L n u = tan θ 2 u = u tmp tan θ 2 u tmp Up L n v θ θ 2 u 11 E

12 Pinhole camera Top View C What about v? aspect ratio = a = u v u = tan θ 2 Up L n v θ 2 u v = tan θ 2 a v = v tmp v tmp tan θ 2 a 12 E θ

13 Pinhole camera Top View Finally O = E V = L n + xu + yv v L n V u 13 E

14 Camera Implementation ulen is computed using a tan() Just pass in the value (precompute) ulen = tan(hfov/2.*pi/180.); // for degrees Normalize ray directions 14

15 Data structures: Camera Base class: Method to compute ray for image x,y [-1,1] Pinhole camera: Position Lookat point or gaze direction Up vector Aspect ratio Derived values: u, v, normalized gaze dir 15

16 Camera void makeray(ray& ray, float x, float y) const; 16

17 Group Iterate over all objects, calling intersect For now we can stick with what we have (list) More, better groups later 17

18 HitRecord bool hit(float t, const Primitive& hit_prim, const Material& hit_matl) Use like: hitrecord.hit(t, this, mymatl) Bool return value for optimizations (and more uses later) 18

19 Ray-plane intersection To find the intersection of a ray with a plane, determine where both equations are satisfied at the same time: NiP + d = 0 and P = O + tv Ni( O + tv ) + d = 0 NiO + tni V + d = 0 tni V = ( d + Ni O ) ( ) t = d + Ni O NiV 19

20 If Ray-plane intersection NiV = 0(or close to it) then the ray is parallel to the plane The parameter t defines the point where the ray intersects the plane To determine the point of intersection, just plug it back into the ray equation O + t V ( ) 20

21 Ray-disk intersection Center: C Radius: r Normal: N Plane equation: N P d = 0 d = N C intersect plane with ray: t = d O N V N Determine if point is inside circle: P = O + tv D = P C P D C r if (D D < r 2 ) hit 21

22 Other planar objects A number of other planar object can be handled in the same way: Ring (two circles) Parallelogram (specified with point and 2 vectors) Triangle 22

23 Triangle Several ways to do ray-triangle intersections: Similar to disc (Dot products to perform inside/outside test) Similar to disc (Project point to XY, XZ or YZ plane for inside/ outside test) Plücker coordinates Barycentric coordinates 23

24 Barycentric coordinates 0 b 1,b 2,b 3 1 P 1 b 1 + b 2 + b 3 = 1 P = b1 P 1 + b2 P 2 + b3 P 3 = b 1 P 1 + b2 P 2 + ( 1 b1 b 2 )P 3 P 2 b 3 b 2 P b 1 P 3 24

25 Barycentric coordinates 0 b 1,b 2,b 3 1 b 1 + b 2 + b 3 = 1 P = b1 P 1 + b2 P 2 + b3 P 3 = b 1 P 1 + b2 P b1 b 2 P 1 ( )P 3 O + tv = b 1 P 1 + b2 P 2 + ( 1 b1 b 2 )P 3 (3 equations, 3 unknowns) P 2 b 3 b 2 P b 1 P 3 25

26 Ray-triangle intersection O + tv = b 1 P 1 + b2 P 2 + ( 1 b1 b 2 )P 3 tv + b ( 1 P 1 P3 ) + b 2 P 2 P3 = P1 P3 e 1 ( ) e 2 = P2 P3 s = O P3 ( ) ( ) ( ) = ( O P ) 3 V x e 1x e 2 x t s x V y e 1y e 2 y b 1 = s y V z e 1z e 2z b 2 s z Use Cramer's rule to solve system 26

27 Solution s x e 1x e 2 x s y e 1y e 2 y V x s x e 2 x V y s y e 2 y V x e 1x s x V y e 1y s y t = s z e 1z e 2z,b 1 = V z s z e 2z,b 2 = V z e 1z s z V x e 1x e 2 x V x e 1x e 2 x V x e 1x e 2 x V y e 1y e 2 y V y e 1y e 2 y V y e 1y e 2 y V z e 1z e 2z V z e 1z e 2z We will also use this property: A T = A V z e 1z e 2z 27

28 Remember this slide? Scalar triple product A (BxC) = B (CxA) = C (AxB) Can also be written as the determinant: Ax Ay Az Bx By Bz Cx Cy Cz This one is pretty cool, but we won t use it often. 28

29 Improved solution denom = V x V y V z e 1x e 1y e 1z e 2 x e 2 y e 2z = V e 1 e2 denom = e 1 ( V e2 ) ( ) t = e 2 s e 1 denom ( ) b 1 = s V e 2 denom ( ) ( ) = e 1 b 2 = V s e 1 denom 29 e2 V ( ) = e 1 V e2 ( )

30 Ray-triangle intersection e 1 = p1 p3 e 2 = p2 p3 r 1 = V e2 denom = e 1 r1 if (Abs(denom) < epsilon) miss, return; 1 invdenom = denom s = O p3 b 1 = ( s r 1 )invdenom if (b 1 < 0 b 1 > 1) miss, return; r 2 = s e1 b 2 = V ( r 2 )invdenom if (b 2 < 0 b 1 + b 2 > 1) miss, return; t = ( e 2 r2 )invdenom hit,save b1/b2 for normal/texture interpolation 30

31 Add / sub / mult Compare Divide e 1 = p1 p3 3 e 2 = p2 p3 3 r 1 = V e2 9 denom = e 1 r1 5 if (Abs(denom) < epsilon) miss, return; 2 1 invdenom = denom s = O p3 3 b 1 = ( s r 1 )invdenom 6 if (b 1 < 0 b 1 > 1) miss, return; 2 r 2 = s e1 9 b 2 = V ( r 2 )invdenom 6 if (b 2 < 0 b 1 + b 2 > 1) miss, return; 1 2 t = ( e 2 r2 )invdenom 6 hit,save b1/b2 for normal/texture interpolation 2 total 20 / 29 / 45 / 51 2 / 4 / 6 / 8 0 /1 /1 /1 0 / 0 / 0 /

32 Normal interpolation true normal: e 1 e2 Smooth shading: normal specified at each vertex: N = b 1 N 1 + b2 N 2 + ( 1 b1 b ) 2 N 3 32

33 HitRecord Add a scratchpad to your hitrecord: char data[maxsize]; template<typename T> getscratchpad() { assert(sizeof(t) <= MAXSIZE); return reinterpret_cast<t*>(data); } 33

34 Scratchpad Use it like this: struct TriangleCoords{double b1, b2}; if(hitrecord.hit(t, this, matl)){ // I am now the closest TriangleCoords* tsave =hitrecord.getscratchpad<trianglecoords>(); tsave->b1=b1; tsave->b2=b2; } 34

35 Normal computation Triangle::normal( ) { TriangleCoords* tsave =hitrecord.getscratchpad<trianglecoords>(); N=tsave->b1*N1+tsave->b2*N2 +(1-tsave->b1-tsave->b2)*N3; N.normalize(); return N; } 35

36 Boxes Axis aligned boxes Parallelepiped 12 triangles? 6 planes with squares? 36

37 Ray-box intersection N P d = 0 t = d N O N V x plane: N = t = d O x V x [ ] P 1 P 2 V d 1 = P 1x,d 2 = P 2 x t x1 = P 1x O x V x,t x2 = P 2 x O x V x O t x1 t x2 Same for y, z planes 37

38 Intersection of intervals Intersection occurs where x interval overlaps y interval x interval: min(t x1,t x2 ) t max(t x1,t x2 ) y interval: min(t y1,t y2 ) t max(t y1,t y2 ) intersection: max(min x,min y ) t min(max x, max y ) y interval P 2 P 1 O x interval 38

39 Better box 39

40 Other cool intersections Extrusions Surfaces of revolution Swept spheres (Center, Radius vary) Metaballs Isosurface Torus Cones/Cylinders Superquadric Spline surfaces 40

41 Improved Sphere Intersection 41

42 Ray-sphere intersection Ray: P = O + tv Sphere: ( P C ) ( P C ) ( ) V + ( O C ) ( O C ) r 2 Solution: t 2 V V + 2t O C Vector OC = O C a = V V b = 2OC V c = OC OC r 2 roots : b ± b2 4ac 2a 42

43 Ray-sphere intersection OC = O C a = V V b = 2OC V c = OC OC r 2 disc = b 2 4ac if (disc > 0) root = disc denom = 2a b + root t 1 = ( ) denom b root t 2 = ( ) denom (revisited) 43

44 Counting operations Op Add/Sub/Mult Compare Divide OC = O C 3 a = V V 5 b = 2OC V 6 c = OC OC r 2 7 disc = b 2 4ac 5 if (disc > 0) 1 root = disc 1 denom = 2a 1 b + root t 1 = denom b root t 2 = denom Total 26 / 30 1 / 5 0 / 2 0 /1 44

45 First optimization There are a lot of 2s in there - can we get rid of them? Vector OC = O C a = V V b ʹ = OC V b = 2OC V = 2bʹ c = OC OC r 2 roots : 2 b ʹ ± 4bʹ 2 4ac 2a = b ʹ ± bʹ 2 ac a 45

46 Counting operations Op Add/Sub/Mult Compare Divide OC = O C 3 a = V V 5 b ʹ = OC V 5 c = OC OC r 2 7 disc = bʹ 2 ac 4 if (disc > 0) 1 root = disc 1 denom = a ( ) t 1 = b ʹ denom t 2 = bʹ denom Total 24 / 27 1 / 5 0 / 2 0 /1 46

47 Unit vectors What if V = 1? (unit direction) Vector OC = O C a = V V = 1 b ʹ = OC V c = OC OC r 2 roots : b ʹ ± bʹ 2 c 47

48 Counting operations Op Add/Sub/Mult Compare Divide OC = O C 3 b ʹ = OC V 5 c = OC OC r 2 7 disc = bʹ 2 c 2 if (disc > 0) 1 root = disc 1 ( ) 2 2 ( ) 1 2 t 1 = b ʹ + root t 2 = bʹ root Total 17 / 20 1 / /1 48

49 A different derivation Determine if ray is inside of sphere V V O C r ( C O ) ( C O ) > r 2 ( C O ) ( C O ) < r 2 O C r 49

50 Closest approach Find the closest approach of the ray to the sphere center V t ca O V t ca O C t ca = C O ( ) V C 50

51 Distance to surface V O t ca t hc r ( C O ) ( C O ) D C t 2 hc = r 2 D 2 D 2 = ( C O ) ( C O ) 2 t ca t 2 hc = r 2 ( C O ) ( C O ) 2 + t ca 51

52 Finally t ca t hc O t C origin outside sphere: t = t ca origin inside sphere: t = t ca + 2 t hc 2 t hc 52

53 Ray-sphere intersection Vector CO = C O = CO CO 2 L co t ca = CO V ( ) 2 if L co < r 2 2 t hc else = r 2 2 L co root: t ca + 2 t hc 2 + t ca if(tca<0) behind ray, no roots else 2 t hc 2 if(t hc else = r 2 2 L co 2 + t ca <0) misses, no roots root: t ca 2 t hc 53

54 Add / Sub / Mult Compare Divide Vector CO = C O 3 2 L co = CO CO 5 t ca = CO V 5 2 t hc ( ) if L co < r 2 = r L co + t ca root: t ca + else 2 t hc if(tca<0) behind ray, no roots 1 2 if(t hc 2 t hc else = r L co + t ca 3 <0) misses, no roots 1 else root: t ca 2 t hc Total 18 /14 /17 /18 2 / 2 / 3 / 4 1 / 0 / 0 /1 54

55 Results ops vs ops Important early exit (sphere behind ray with no sqrt) Use this insight to do the same in the algebraic version Use CO instead of OC (saves 1 op) Early exit with c<0 Exercise left to reader 55

56 Program 2 Due sometime Assigned sometime 56

57 End Slides 57