Implementation Issues

Transcription

1 Implementation Issues More from Interface point of view Y V U Eye N X Z World Coordinate System (WCS) Viewing Coordinate System (VCS)

2 View Coordinate System (VCS) Viewing coordinate system Position and orientation of the view plane Extent of the view plane (window) Position of the eye View Plane View Reference Point (VRP): the origin of VCS specified as (r x, r y, r z ) in WCS: center of the scene Normal to the view plane (n x, n y, n z )

3 View Coordinate System (VCS) View Plane Normal Direction (View Plane Normal VPN) n (n x,n y,n z ) User may provide normalized vector e.g. n x sin φ cos θ n y sin φ sin θ n z cos φ X Z φ r θ Y

4 View Coordinate System (VCS) View Plane Direction v v is a unit vector intuitively corresponding to up vector up vector is specified by the user in WCS up up (up.n)n v up / up Direction u n v up up u n x v ( Left Handed)

5 View Coordinate System (VCS) Window and Eye Window : left, right, bottom,top (w l,w r,w b,w t ) generally is centered at VRP (origin) w r w t Eye : e (e u,e v,e n ) Typically e (0,0,-E) n v u e w l w b

6 Transformation from WCS to VCS Y v r O (x, y) u ( x y ) ( a ( a u b) + r v b) M + r O X

7 Transformation from WCS to VCS Point object is represented as (a,b,c) in VCS (x,y,z) in WCS z y x z y x z y x n n n v v v u u u n v u M

8 [ ] [ ] [ ] T M r p M r p c b a r M c b a z y x p ) ( ) ( 1 + Transformation from WCS to VCS Conversion from one coordinate system to another Therefore a(p-r).u, b(p-r).v, c(p-r).n

9 [ ] [ ] [ ] T M r p M r p c b a r M c b a z y x p ) ( ) ( 1 + Conversion from one coordinate system to another z y x z y x z y x n n n v v v u u u n v u M (a,b,c) in VCS (x,y,z) in WCS Where 3D Viewing Interface (Revisit) Set up

10 Set up Steps 1. Define the view reference point (VRP) r 2. Obtain M defining u, v, n 3. Define the position of eye (in VCS) e ( e u, e v, e n ) typically ( 0, 0, -E ) corresponding point in WCS will be eye ( 0, 0, -E ) M + r 4. Define the window and the pixel location.

11 Set up W t v rows : 0 to MAXROW cols : 0 to MAXCOL (i,j) th pixel : (u i, v j, 0) W b n u i W l + i Δu v j W t j Δv e ( 0,0, -E) W l W r u Δu ( W r W l ) / MAXCOL Δv ( W t W b ) / MAXROW

12 Set up 5. Parametric equation of ray P i,j (t) eye + dir ij t eye ( 0, 0, -E ) M + r dir ij ( ( u i, v j, 0 ) e) M ( u i, v j, E ) M P ij (t) eye + dir ij t R(t) R o + R d t

13 Set up Find intersection with object (r i ) Find the normal at r i as r n Find intensity at I at the point (Illumination model) Find the pixel-color

14 Review Basic ray tracing (one level): ray casting Algorithm: For each pixel shoot a ray from eye (COP) Compute ray-object(s) intersection Obtain the closest intersection point (p) Compute normal at p Compute illumination (intensity) Set up: Obtain ray in WCS for intersection using transformations Transformation of objects: Equivalent transformation for the ray

15 D T 2 R 2 Eye B eye-ray A View Plane R 1 T 1 C F E

16 Eye C D R 1 T 1 R 2 T 2

17 Different Rays Eye ray (primary ray) Reflected ray Transmitted ray Shadow ray (secondary rays)

18 Reflected Ray Recall Reflection Vector N L θ i N θ r R L R 2 (L N) N -L R 2( L N) N L

19 Refracted Ray Snell s Law N I θ i sin sinθ θ i t η η t i θ t T

20 Refracted Ray Snell s Law I θ i N (cosθ i )N θ t M T I + (cosθi ) N M sinθi T (sinθt ) M (cos θt ) N sinθt ( I + (cos θi ) N) T (cosθt sinθi ηi T ( I + (cos θi ) N) (cos θt ) N η t ) N

21 L1 Eye Obj 3 Obj 1 L2 Obj 1 Obj 2

22 Eye Obj 3 L2 L1 Obj 1 Eye L1 L2 Obj 1 T R Obj 2 Obj 3 Obj 2

23 Eye Obj 3 L2 L1 Obj 1 Obj 2 L1 L2 Eye L1 L2 Obj 1 L1 T R L2 Obj 2 Obj 3 R T R

24 When to stop? When ray leaves the scene When the contribution to the overall intensity is small

25 Phong Illumination Model m i n i i s i i d a a n l s l d a a n l s l d a a total V R I k N L I k I k V R I k N L I k I k α I k θ I k I k reflection specular diffuse reflection reflection ambient I 1 ) ( ) ( ) ( ) ( cos cos Recursive Ray Tracing

26 Illumination When in shadow (single light source) I total ambient k I a a reflection

27 Illumination With reflection and transmission rays Itotal ( P) Ilocal (P) + krgi(pr ) + k tg (P ) t Global Illumination

28 T Whitted 1980

29 Other Example

30 Other features Ray tracing is an image based method (pixelization) Sampling aliasing Jagginess Moire patterns Anti-aliasing

31 Anti-alising Supersampling More number of rays per pixel Average the result