Ray Tracing 2. Trace Harder. COMP 575/770 Spring 2013

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1 Ray Tracing 2 Trace Harder COMP 575/770 Spring 2013

2 Ray-Triangle Intersection Core operation while ray tracing triangle meshes Given: Ray with origin o and direction d Triangle with vertices v 0, v 1, and v 2 Figure out the intersection point

3 Ray-Triangle Intersection Any point on the ray is given by: p = o + td Any point on the triangle is given by: p = v 0 + β v 1 v 0 + γ v 2 v 0 Barycentric coordinates!

4 Ray-Triangle Intersection One equation for each coordinate: In matrix form: x o + tx d = x a + β x b x a + γ x c x a y o + ty d = y a + β y b y a + γ y c y a z o + tz d = z a + β z b z a + γ(z c z a ) x a x b x a x c x d y a y b y a y c y d z a z b z a z c z d β γ t = x a x e y a y e z a z e Solve using Cramer s rule

5 Ray-Triangle Intersection A faster approach: Moller-Trumbore intersection algorithm eleration/fast%20minimumstorage%20raytriangle%20intersection.pdf

6 Ray-Triangle Intersection In general, two parts to any intersection test: Determining t (ray-plane test) Determining β, γ (barycentric test) Ray-plane test finds a potential hit point Barycentric test checks that a hit point exists Does the ordering matter?

7 Ray-Triangle Intersection Eye Rays Need to find the closest hit If t is greater than the current closest hit, bail Ray-plane test before barycentric test Shadow Rays Need to find any hit in an interval If the ray can t possible intersect, bail Barycentric test before ray-plane test

8 Triangle Meshes Test each ray with each triangle? Too slow! 1M pixels (rays), 1000 triangles = 1 billion tests! Rasterization is much better at this Why?

9 Triangle Meshes Ray Tracing: for each pixel for each triangle test (ray, triangle) Rasterization: for each triangle for each pixel test (pixel, triangle)

10 Triangle Meshes Ray Tracing: for each pixel for each triangle test (ray, triangle) Rasterization: for each triangle for each pixel potentially covered test (pixel, triangle)

11 Triangle Meshes Ray Tracing: for each pixel for each triangle potentially intersecting test (ray, triangle) Rasterization: for each triangle for each pixel potentially covered test (pixel, triangle)

12 Acceleration Structures A special data structure to accelerate ray tracing Used to quickly determine which triangles might intersect a ray Usually a hierarchical tree structure Object hierarchies Spatial hierarchies

13 Outline Ray-Triangle Intersection Bounding Volume Hierarchy Ray-Box Intersection kd Trees Efficient Ray Tracing

14 Bounding Volume Hierarchies Hierarchical clustering of objects (triangles) Each cluster represented as a bounding volume

15 Bounding Volume Hierarchies

16 Bounding Volume Hierarchies Typically binary trees Each internal node splits a set of triangles into two Each node stores its bounding volume Each leaf typically contains one triangle Variations are possible

17 Bounding Volumes A BVH can be built using any kind of bounding volume Typical choices: Spheres Axis-Aligned Bounding Boxes (AABBs) Oriented Bounding Boxes (OBBs) Discrete Oriented Polytopes (DOPs) Need to test whether a ray passes through a bounding volume

18 Ray-AABB Intersection An AABB is a region bounded by 3 slabs 2D example shown here for convenience

19 Ray-AABB Intersection AABB intersection expressed as 3 slab intersection tests 2 tests in 2D Need to find interval of t in which ray is in AABB

20 Ray-AABB Intersection Consider a box with min. coordinates (x 0, y 0, z 0 ) and max. coordinates x 1, y 1, z 1 The x-slab goes from x = x 0 to x = x 1 Which will the ray encounter first? Depends on which way the ray is traveling If d x > 0, x min = x 0, x max = x 1, otherwise the other way around

21 Ray-AABB Intersection Ray intersection with x = x min : o x + td x = x min Similarly for intersection with x = x max Ray passes through the x-slab in the interval [t xmin, t xmax ] t xmin = x min o x d x t xmax = x max o x d x

22 Ray-AABB Intersection Repeat process for y-slab (and z-slab) Ray is in AABB when it s in both (all three) slabs Need to find the intersection of intervals

23 Ray-AABB Intersection Ray intersects AABB if t min t max

24 BVH Traversal Recursive algorithm: at current node: if ray doesn t intersect current node s bounding volume, bail if current node is leaf: check for intersection with stored triangle else (if current node is internal node): recurse on children Start with root node Still have to test against multiple triangles Hopefully, bounding volume tests eliminate most triangles

25 Outline Ray-Triangle Intersection Bounding Volume Hierarchy Ray-Box Intersection kd Trees Efficient Ray Tracing

26 kd Trees Or, binary spatial partitioning (BSP) trees Instead of a hierarchy of objects, a hierarchy of spatial regions Allows front-to-back spatial sorting Stop as soon as a ray hits something

27 kd Trees Binary tree of spatial regions At each node, the spatial region is split using an axis-aligned plane x y y

28 kd Trees

29 kd Trees

30 kd Trees

31 kd Trees

32 kd Tree Traversal Recursive algorithm: at current node: if current node is leaf: intersect with all stored triangles if ray hits a stored triangle, stop else (if current node is internal node): find intersection of ray with splitting plane if ray intersects both children: recurse on near side, then far side otherwise: recurse on side it intersects

33 kd Tree Construction Given: An AABB describing a region of space ( cell ) List of triangles in the cell Core operation: Pick an axis-aligned plane to split the cell Assign triangles to each sub-cell Some triangles may be in both Recurse Subject to some termination criteria

34 kd Tree Construction What axis to split along? Where to place splitting plane? When to terminate recursion?

35 kd Tree Construction What axis to split along? Round-robin? Largest extent? Where to place splitting plane? Middle of extent? Median of geometry? When to terminate recursion? 1 triangle left in cell? Max. tree depth?

36 Outline Ray-Triangle Intersection Bounding Volume Hierarchy Ray-Box Intersection kd Trees Efficient Ray Tracing

37 Efficient Ray Tracing Many techniques developed to speed up ray tracing Sophisticated tree construction algorithms Compact, cache-friendly memory layout Parallel programming on multi-core CPUs SIMD instructions for exploiting coherence Replace recursion with iteration Optimize inner loops

38 BVH vs kd Tree BVH: Pros: Can be updated for dynamic geometry Cons: Slower traversal, takes more memory kd Tree: Pros: Faster traversal, takes less memory Cons: Static scenes only

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