Ray Casting. 3D Rendering. Ray Casting. Ray Casting. Ray Casting. Ray Casting

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1 Rendering Ray asting The color of each pixel on the view plane depends on the radiance emanating from visible surfaces dam inkelstein rinceton University OS 6, Spring 00 Simplest method is ray casting Rays through view plane ye position iew plane Ray asting or each sample! onstruct ray from eye position through view plane! ind first surface intersected by ray through pixel! ompute color sample based on surface radiance Ray asting or each sample! onstruct ray from eye position through view plane! ind first surface intersected by ray through pixel! ompute color sample based on surface radiance Rays through view plane ye position Samples on view plane Ray asting Simple implementation: Image Rayast(amera camera, Scene scene, int width, int height) Image image = new Image(width, height); for (int i = 0; i < width; i++) for (int j = 0; j < height; j++) Ray ray = onstructraythroughixel(camera, i, j); Intersection hit = indintersection(ray, scene); image[i][j] = Getolor(hit); return image; Ray asting Simple implementation: Image Rayast(amera camera, Scene scene, int width, int height) Image image = new Image(width, height); for (int i = 0; i < width; i++) for (int j = 0; j < height; j++) Ray ray = onstructraythroughixel(camera, i, j); Intersection hit = indintersection(ray, scene); image[i][j] = Getolor(hit); return image;

2 onstructing Ray Through a ixel back right Up direction 0 Ray: = 0 + t towards iew lane onstructing Ray Through a ixel xample " = frustum half-angle d = distance to view plane right = towards x up = 0 + d*towards d*tan(")*right = 0 + d*towards + d*tan(")*right = + ((i + 0.) / width) * ( - ) = ( - 0 ) / " towards d right *d*tan(") Ray: = 0 + t Ray asting Simple implementation: Image Rayast(amera camera, Scene scene, int width, int height) Image image = new Image(width, height); for (int i = 0; i < width; i++) for (int j = 0; j < height; j++) Ray ray = onstructraythroughixel(camera, i, j); Intersection hit = indintersection(ray, scene); image[i][j] = Getolor(hit); return image; cceleration techniques! ounding volume hierarchies» S trees Ray-Sphere Intersection Ray-Sphere Intersection I Ray: = 0 + t Sphere: - O - r = 0 Ray: = 0 + t Sphere: - O - r = 0 Substituting for, we get: 0 + t - O - r = 0 lgebraic Method 0 r O Solve quadratic equation: at + bt + c = 0 where: a = b = ( 0 - O) c = r = 0 0 r O = 0 + t

3 Ray-Sphere Intersection II Ray-Sphere Intersection Ray: = 0 + t Sphere: - O - r = 0 L = O - 0 Geometric Method Need normal vector at intersection for lighting calculations N = ( - O) / - O t ca = L if (t ca < 0) return 0 d = L L - t ca if (d > r ) return 0 t hc = sqrt(r - d ) t = t ca - t hc and t ca + t hc 0 t ca L r t hc d r O 0 N r O = 0 + t» Triangle cceleration techniques! ounding volume hierarchies» S trees Ray-Triangle Intersection irst, intersect ray with plane Then, check if point is inside triangle 0 Ray-lane Intersection Ray-Triangle Intersection I Ray: = 0 + t lane: N + d = 0 Substituting for, we get: ( 0 + t) N + d = 0 Solution: t = -( 0 N + d) / ( N) = 0 + t lgebraic Method N heck if point is inside triangle algebraically or each side of triangle = T - = T - N = x Normalize N if (( - 0 ) N < 0) return LS; end T T N T 0 0

4 Ray-Triangle Intersection II heck if point is inside triangle parametrically ompute barycentric coordinates #, $: # = rea(t T ) / rea(t T T ) $ = rea(t T ) / rea(t T T ) rea(t T T ) = / (T-T) x (T-T) heck if point inside triangle. 0 % # % and 0 % $ % # + $ % T # T $ &#&$ T Other Ray-rimitive Intersections one, cylinder, ellipsoid:! Similar to sphere ox! Intersect front-facing planes, return closest onvex polygon! Same as triangle (check point-in-polygon algebraically) oncave polygon! Same plane intersection! More complex point-in-polygon test 0 ind intersection with front-most primitive in group Intersection indintersection(ray ray, Scene scene) min_t = infinity min_primitive = NULL or each primitive in scene t = Intersect(ray, primitive); if (t > 0 && t < min_t) then min_primitive = primitive min_t = t return Intersection(min_t, min_primitive)» cceleration techniques! ounding volume hierarchies» S trees ounding olumes heck for intersection with simple shape first ounding olumes heck for intersection with simple shape first

5 ounding olumes heck for intersection with simple shape first! If ray doesn t intersect bounding volume, then it doesn t intersect its contents ounding olumes heck for intersection with simple shape first! If ray doesn t intersect bounding volume, then it doesn t intersect its contents Still need to check for intersections with shape. ounding olume Hierarchies I uild hierarchy of bounding volumes! ounding volume of interior node contains all children ounding olume Hierarchies Use hierarchy to accelerate ray intersections! Intersect node contents only if hit bounding volume ounding olume Hierarchies III Sort hits & detect early termination indintersection(ray ray, Node node) // ind intersections with child node bounding volumes... // Sort intersections front to back... // rocess intersections (checking for early termination) min_t = infinity; for each intersected child i if (min_t < bv_t[i]) break; shape_t = indintersection(ray, child); if (shape_t < min_t) min_t = shape_t; return min_t;» cceleration techniques! ounding volume hierarchies» S trees

6 Uniform Grid onstruct uniform grid over scene! Index primitives according to overlaps with grid cells Uniform Grid Trace rays through grid cells! ast! Incremental Only check primitives in intersected grid cells Uniform Grid otential problem:! How choose suitable grid resolution? Too little benefit if grid is too coarse Too much cost if grid is too fine» cceleration techniques! ounding volume hierarchies» S trees Octree onstruct adaptive grid over scene! Recursively subdivide box-shaped cells into 8 octants! Index primitives by overlaps with cells Octree Trace rays through neighbor cells! ewer cells! More complex neighbor finding Generally fewer cells Trade-off fewer cells for more expensive traversal

7 » cceleration techniques! ounding volume hierarchies» S trees inary Space artition (S) Tree Recursively partition space by planes! very cell is a convex polyhedron inary Space artition (S) Tree Simple recursive algorithms! xample: point finding inary Space artition (S) Tree Trace rays by recursion on tree! S construction enables simple front-to-back traversal inary Space artition (S) Tree RayTreeIntersect(Ray ray, Node node, double min, double max) if (Node is a leaf) return intersection of closest primitive in cell, or NULL if none else dist = distance of the ray point to split plane of node near_child = child of node that contains the origin of Ray far_child = other child of node if the interval to look is on near side return RayTreeIntersect(ray, near_child, min, max) else if the interval to look is on far side return RayTreeIntersect(ray, far_child, min, max) else if the interval to look is on both side if (RayTreeIntersect(ray, near_child, min, dist)) return ; else return RayTreeIntersect(ray, far_child, dist, max) Other ccelerations Screen space coherence! heck last hit first! eam tracing! encil tracing! one tracing Memory coherence! Large scenes arallelism! Ray casting is embarassingly parallelizable etc.

8 cceleration Intersection acceleration techniques are important! ounding volume hierarchies General concepts! Sort objects spatially! Make trivial rejections quick! Utilize coherence when possible xpected time is sub-linear in number of primitives Summary Writing a simple ray casting renderer is easy! Generate rays! Intersection tests! Lighting calculations Image Rayast(amera camera, Scene scene, int width, int height) Image image = new Image(width, height); for (int i = 0; i < width; i++) for (int j = 0; j < height; j++) Ray ray = onstructraythroughixel(camera, i, j); Intersection hit = indintersection(ray, scene); image[i][j] = Getolor(hit); return image; Heckbert s business card ray tracer Next Time is Illumination! typedef structdouble x,y,zvec;vec U,black,amb=.0,.0,.0;struct sphere vec cen,color; double rad,kd,ks,kt,kl,ir*s,*best,sph[]=0.,6.,.,.,.,.,.9,.0,.,.8,0.,.7,-.,8.,-.,.,.,.,.,.7,.,0.,.0,.,.,8.,-.,.,.8,.8,.,.,.7,0.,0.,.,.,-6.,.,.,.8,.,7.,0.,0.,0.,.6,.,-.,-.,.,.8,.,.,.,0.,0.,0.,.,.,;yx;double u,b,tmin,sqrt(),tan();double vdot(,)vec,;return.x *.x+.y*.y+.z*.z;vec vcomb(a,,)double a;vec,;.x+=a*.x;.y+=a*.y;.z+=a*.z; return ;vec vunit()vec ;return vcomb(./sqrt( vdot(,)),,black);struct sphere*intersect (,)vec,;best=0;tmin=e0;s= sph+;while(s-->sph)b=vdot(,u=vcomb(-.,,s->cen)), u=b*b-vdot(u,u)+s->rad*s ->rad,u=u>0?sqrt(u):e,u=b-u>e-7?b-u:b+u,tmin=u>=e-7&& u<tmin?best=s,u: tmin;return best;vec trace(level,,)vec,;double d,eta,e;vec N,color; struct sphere*s,*l;if(!level--)return black;if(s=intersect(,));else return amb;color=amb;eta= s->ir;d= -vdot(,n=vunit(vcomb(-.,=vcomb(tmin,,),s->cen )));if(d<0)n=vcomb(-.,n,black), eta=/eta,d= -d;l=sph+;while(l-->sph)if((e=l ->kl*vdot(n,u=vunit(vcomb(-.,,l->cen))))>0&& intersect(,u)==l)color=vcomb(e,l->color,color);u=s->color;color.x*=u.x;color.y*=u.y;color.z *=U.z;e=-eta* eta*(-d*d);return vcomb(s->kt,e>0?trace(level,,vcomb(eta,,vcomb(eta*dsqrt (e),n,black))):black,vcomb(s->ks,trace(level,,vcomb(*d,n,)),vcomb(s->kd, color,vcomb (s->kl,u,black))));main()printf("%d %d\n",,);while(yx<*) U.x=yx%-/,U.z=/- yx++/,u.y=//tan(/.9906),u=vcomb(., trace(,black,vunit(u)),black),printf ("%.0f %.0f %.0f\n",U);/*minray!*/ Without Illumination With Illumination