CS 563 Advanced Topics in Computer Graphics. by Emmanuel Agu

Transcription

1 CS 563 Advanced Topics in Computer Graphics by Emmanuel Agu

2 Parsing: uses lex and yacc: core/pbrtlex.l and core/pbrtparse.y After parsing, a scene object is created (core/scene.*) Rendering: Scene::Render() is invoked. PBRT Flow (generates sample positions for eye rays and integrators)

3 PBRT Architecture

4 Geometric classes Chapter : Representation and operations for the basic math: points, vectors and rays. core/geometry.* and core/transform.* Chapter 3 (Shapes): Actual scene geometry such as triangles and spheres. Chapter 4: Acceleration structures (uniform grid, kd-tree, BVH, etc)

5 Coordinate system Points, vectors and normals: 3 floating-point coordinate values: x, y, z defined under a coordinate system. A coordinate system defined by: Origin + frame Handedness? y z PBRT uses right hand coordinate system Y OpenGL uses right hand coordinate system x +z x

6 Diff. b/w points = vector v = Q P Vector-Point Relationship Sum of point and vector = point v + P = Q P v Q

7 Vector Operations Define vectors a = ( a1, a, a3 ) b = b b, ) ( 1, b3 Then vector addition: a + b = ( a 1 + b1, a + b, a3 + b3 ) and scalar, s a a+b b

8 Vector Operations Scaling vector by a scalar as = ( a s, as, a3 1 s ) Note vector subtraction: a b = ( a1 + ( b1 ), a + ( b ), a3 + ( b3 )) a a-b a.5a b

9 Magnitude of a Vector Magnitude of a a = a 1 + a... + an Normalizing a vector (unit vector) â = a a = vector magnitude Note magnitude of normalized vector = 1. i.e a1 + a... + an = 1

10 Vectors class Vector { public: <Vector Public Methods> float x, y, z; } (no need to use selector and mutator)

11 Dot and cross product Dot(v, u) AbsDot(v, u) Cross(v, u) (v, u, v u) form a coordinate system cosθ u v u v = sinθ u v u v = v u? ( ) ( ) ( ) z y y x z z x x z y y z z y x u v u v u v u v u v u v u v u v u v = = =

12 Normalization PBRT vector methods Length(v)- returns length of vector, v LengthSquared(v) (returns length of v) Normalize(v) returns a vector, does not normalize in place

13 Coordinate system from a vector Construct a local coordinate system from a vector. inline void CoordinateSystem(const Vector &v1, Vector *v, Vector *v3) V1 normalized already. Construct v: perpendicular vector of v1 by Zero out 1 component of v1 Swap other components V1 x v = v3: 3 rd vector

14 Points are different from vectors explicit Vector(const Point &p); You have to convert a point to a vector explicitly (no accidents, know what you are doing). Vector v=p; Vector v=vector(p); Points

15 Vector v; Point p, q, r; float a; q q=p+v; q=p-v; v=q-p; v Operations for points r=p+q; a*p; p/a; p (This is only for the operationa p+ß q.) PBRT supports: Distance(p,q); DistanceSquared(p,q);

16 Normals A surface normal (or just normal) is a vector that is perpendicular to a surface at a particular position.

17 Different than vectors sometimes Particularly when applying transformations. Implementation similar to Vector, except Normal cannot be added to a point Cannot take the cross product of two normals. Normal is not necessarily normalized. Conversion between Vector and Normal must be explicit Normals

18 o class Ray { public: <Ray Public Methods> Point o; Vector d; mutable float mint, maxt; float time; }; d (for motion blur) mint maxt r( t) = o + td t Rays (They may be changed even if Ray is const.) Initialized as RAY_EPSILON to avoid self intersection. Ray r(o, d); Point p=r(t);

19 Ray differentials Used to estimate projected area for a small part of a scene Used for texture antialiasing. class RayDifferential : public Ray { public: <RayDifferential Methods> bool hasdifferentials; Ray rx, ry; };

20 Bounding boxes Avoid intersection tests inside a volume if ray doesn t hit bounding volume. Benefits depends on: Expense of testing volume vs objects inside Tightness of the bounding volume. Popular bounding volumes: sphere, axis-aligned bounding box (AABB), oriented bounding box (OBB). y z x

21 Bounding boxes class BBox { public: <BBox Public Methods> Point pmin, pmax; } BBox::pMax Point p,q; BBox b; float delta; BBox(p,q) // no order for p, q BBox::pMin Union(b,p) Given point & Bbox, return new larger bounding box containing point (bbox) and Bbox. Union(b, b)

22 Point p,q; BBox b; b.expand(delta): Expand old bounding box by factor delta Bounding boxes pmax + delta pmin - delta

23 Point p,q; BBox b; Bounding boxes b.overlaps(b): do two bounding boxes overlap each other in x,y,z Returns boolean. True (overlaps) or false (does not overlap) b b

24 Point p,q; BBox b; Bounding boxes b.inside(p): Is point p inside bounding box? Returns boolean (true or false) Volume(b): Returns volume of bounding volume (x * y * z) pmax pmin

25 Point p,q; BBox b; Bounding boxes b.maximumextent()(which bounding box axis is the longest; useful for building kd-tree) b.boundingsphere(c, r) (returns center and radius of bounding sphere) Example: generate random ray which intersects scene geometry pmax pmin

26 class Transform {... private: Reference<Matrix4x4> m, minv; } save space, but can t be modified after construction Transformations Transform stores element of 4x4 matrix Also computes and stores matrix inverse, minv (avoid repeatedly computing inverse)

27 Transformations Translate(Vector(dx,dy,dz)) Scale(sx,sy,sz) RotateX(a) = ),, ( dz dy dx dz dy dx T = 1 ),, ( sy sy sx sz sy sx S = 1 cos sin sin cos 1 ) ( θ θ θ θ θ R x T 1 ) ( ) ( θ θ x x R R = Question: How does x-roll matrix above differ based on axes handedness?

28 Rotate(a, Vector(1,1,1)) Rotation around an arbitrary axis a? v v

29 Rotate(a, Vector(1,1,1)) Rotation around an arbitrary axis a v p = a(v a) v p? v 1 v 1 = v p = 1 v = v1 v v a v v = p + v 1 cosθ + v sinθ '

30 Caller specifies: camera (eye position), Look at point Up vector LookAt Transformation Want to compute 4x4 transform matrix that converts from world space to eye space y P v u n world x z

31 LookAt(Point &pos, Point look, Vector &up) up look dir Look-at pos Vector dir=normalize(look-pos); Vector right=cross(dir, Normalize(up)); Vector newup=cross(right,dir); pos 1

32 Point: q=t(p), T(p,&q) Applying transformations use homogeneous coordinates implicitly Vector: u=t(v), T(u, &v) Normal: treated differently than vectors because of anisotropic transformations n t = n T t = Transform should keep its inverse For orthonormal matrix, S=M ( n' ) T t' = ( ) T Sn Mt = n T Point: (p, 1) Vector: (v, ) T S Mt = T S M = I T S = M

33 Transform Bbox? Applying transformations transform its 8 corners and expand to include all 8 points.

34 Differential geometry DifferentialGeometry: a self-contained representation for a particular point on a surface so that all the other operations in pbrt can be executed without referring to the original shape. Contains Position Surface normal Parameterization Parametric derivatives Derivatives of normals Pointer to shape

35 Ray-Surface Intersection

36 Ray: Ray-Plane Intersection r r r t r P = O+t D O v D t < N r P r Plane: r r r ( P P ) N = ax+ by+ cz+ d = P r Solve for intersection Substitute ray equation into plane equation r r r r r r r ( P P ) N= ( O+ td P ) N = r r r ( O P ) N t = r r D N

37 Sphere A sphere of radius r at the origin Implicit: x +y +z -r = Parametric: f(?,? ) x=rsin? cos? y=rsin? sin? z=rcos?

38 Sphere

39 Algebraic solution Perform in object space, WorldToObject(r, &ray) Assume that ray is normalized for a while r z y x = + + ( ) ( ) ( ) r td o td o td o z z y y x x = = + + C Bt At z y x d d d A + + = ) ( z z y y x x o d o d o d B + + = r o o o C z y x + + = Step 1

40 Algebraic solution t = B B A 4AC t 1 = B + B A 4AC Step If (B -4AC<) then the ray misses the sphere. B -4AC=? Step 3 Calculate t and test if t < Step 4 Calculate t 1 and test if t 1 <

41 Cylinder φ = uφ max x = r cosφ y = r sinφ z = z + v min ( zmax zmin ) First consider sides Later consider cap/base

42 Cylinder Implicit equation for cylinder x + y r = Substituting in ray equation Giving ( ) ( ) ox + td x + oy + td y = At + Bt + C A = + d x d y = B = ( d xox + d yoy ) C = ox + oy r Solve for t r

43 Cylinder

44 References/Shamelessly stolen Pat Hanrahan, CS 348B, Spring 5 class slides Yung-Yu Chuang, Image Synthesis, class slides, National Taiwan University, Fall 5 Kutulakos K, CSC 53H: Visual Modeling, course slides UIUC CS 319, Advanced Computer Graphics Course slides