Digital Image Synthesis g g y Yung-Yu Chuang

(1)

Acceleration

Digital Image Synthesis g g y Yung-Yu Chuang

10/8/2009 10/8/2009

with slides by Mario Costa Sousa, Gordon Stoll and Pat Hanrahan

Classes

• Primitive (in core/primitive.*)

G t i P i iti – GeometricPrimitive – InstancePrimitive

A t

– Aggregate

• Two types of accelerators are provided (in accelerators/*.cpp)

– GridAccel – KdTreeAccel

Hierarchy

P i iti Primitive

I t A t

Geometric Primitive

Instance Primitive

Aggregate

T

Material Shape

Primitive

class Primitive : public ReferenceCounted {

<Primitive interface>

}

class InstancePrimitive : public Primitive {

…

Reference<Primitive> instance;

} }

(2)

Interface

BBox WorldBound();

bool CanIntersect();

geometry bool CanIntersect();

bool Intersect(const Ray &r,

Intersection *in);

// update maxt Intersection *in);

bool IntersectP(const Ray &r);

void Refine(vector<Reference<Primitive>>

&refined);

id F ll R fi ( t <R f <P i iti >>

void FullyRefine(vector<Reference<Primitive>>

&refined);

A Li ht *G tA Li ht() t i l

AreaLight *GetAreaLight();

BSDF *GetBSDF(const DifferentialGeometry &dg, ) material const Transform &WorldToObject);

Intersection

• primitive stores the actual intersecting primitive hence Primitive >GetAreaLight and primitive, hence Primitive->GetAreaLight and GetBSDF can only be called for

GeometricPrimitive GeometricPrimitive

More information than DifferentialGeometry;

struct Intersection {

y;

also contains material information

<Intersection interface>

DifferentialGeometry dg;

const Primitive *primitive;

Transform WorldToObject;

};

GeometricPrimitive

• represents a single shape

h ld f t h d it i l

• holds a reference to a Shape and its Material, and a pointer to an AreaLight

Reference<Shape> shape;

Reference<Material> material; // BRDF AreaLight *areaLight; // emittance

• Most operations are forwarded to shape

Object instancing

61 unique plant models, 1.1M triangles, 300MBq p , g , 4000 individual plants, 19.5M triangles

(3)

InstancePrimitive

R f <P i iti > i t

Reference<Primitive> instance;

Transform InstanceToWorld, WorldToInstance;

Ray ray = WorldToInstance(r);

if (!instance->Intersect(ray, isect)) return false;

r.maxt = ray.maxt;

isect->WorldToObject = isect->WorldToObject

*WorldToInstance;

Aggregates

• Acceleration is a heart component of a ray tracer because ray/scene intersection accounts tracer because ray/scene intersection accounts for the majority of execution time

G l d th b f / i iti

• Goal: reduce the number of ray/primitive

intersections by quick simultaneous rejection of f i iti d th f t th t b groups of primitives and the fact that nearby intersections are likely to be found first

• Two main approaches: spatial subdivision, object subdivision

• No clear winner

Acceleration techniques Bounding volume hierarchy

(4)

Bounding volume hierarchy

1) Find bounding box of objects

Bounding volume hierarchy

1) Find bounding box of objects 2) S li bj i

2) Split objects into two groups

Bounding volume hierarchy

2) Split objects into two groups 3) Recurse

Bounding volume hierarchy

(5)

Bounding volume hierarchy

Where to split?

• At midpoint

• Sort and put half of the objects on each side

• Sort, and put half of the objects on each side

• Use modeling hierarchy

BVH traversal

• If hit parent, then check all children

(6)

BVH traversal

• Don't return intersection immediately because the other subvolumes may have a closer

the other subvolumes may have a closer intersection

Bounding volume hierarchy

Bounding volume hierarchy Space subdivision approaches

Quadtree (2D) Unifrom grid Quadtree (2D)

Octree (3D)

Unifrom grid

(7)

Space subdivision approaches

KD tree BSP tree

Uniform grid

P

Preprocess scene 1. Find bounding box

Uniform grid

P

Preprocess scene 1. Find bounding box 2 Determine grid resolution 2. Determine grid resolution

(8)

Uniform grid

P

Preprocess scene 1. Find bounding box 2 Determine grid resolution 2. Determine grid resolution 3. Place object in cell if its

bounding box overlaps the cell

Uniform grid

P

Preprocess scene 1. Find bounding box 2 Determine grid resolution 2. Determine grid resolution 3. Place object in cell if its

bounding box overlaps the cell 4. Check that object overlaps cell

(expensive!)

Uniform grid traversal

P

Preprocess scene Traverse grid

3D li 3D DDA 3D line = 3D-DDA (Digital Differential Analyzer)

Analyzer)

1 2

x x

y m  y  b

mx y  

1

2

x

x 

1

  1

 i

i

x

b mx

y

_i_₁



_i_₁



^naive

m

y

_i_1



_i



_DDA

octree

Octree

(9)

K-d tree

A

Leaf nodes correspond to unique regions in space

K-d tree

A

B

A

K-d tree

A

B B

B

A

K-d tree

C A

B B

B

A

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K-d tree

C A

B B

C B

C

A

K-d tree

C A

D BB

C B

C

A

K-d tree

C A

D BB

C B

C

A

D

K-d tree

A

D B

B CC CC

D A

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K-d tree traversal

A

D B

B CC CC

D

A

BSP tree

6 55

9 7

9 10 8

1 2

11

4 2

3

BSP tree

6 5

5 11

inside outside 9

7

inside ones

outside ones 9

10 8 1

2

11

4 2

3

BSP tree

6 55 11

2 5

9 6 7

3 4

6 7 8 9

10 8 1

2

9 10 11 11

4 2

3

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BSP tree

6 5 1

9 5

9b 5

1

9 10 8

7

9a

6 7

11b 8

9b 1

2

11

9a 10 11a 11a

11b

4

2 11a

3

BSP tree

6 5

9 9b

2 1

5

1

9 10 8

7

2 3

5

6 8

9a

11b 1

2

11a 11 4 7 9b

4 2

9a 11b

3 10

11a

BSP tree traversal

6 5

9 5

9b 2

1 5

1

9 10 8

7

2 3

5

6 8

9a

11b 9a

11b 1

2

11a 11 4 7 9b

4 2

i t

9a 11b

3 point

10 11a

BSP tree traversal

6 5

9 5

9b 2

1 5

1

9 10 8

7

2 3

5

6 8

9a

11b 9a

11b 1

2

11a 11 4 7 9b

4 2

i t

9a 11b

3 point

10 11a

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BSP tree traversal

6 5

9 9b

2 1

5

1

9 10 8

7

2 3

5

6 8

9a

11b 9a

11b 1

2

11a 11 4 7 9b

2

point

9a 11b 4

point 3 10

11a

Ray-Box intersections

• Both GridAccel and KdTreeAccel require it Q i k j i d i i

• Quick rejection, use enter and exit point to traverse the hierarchy

• AABB is the intersection of three slabs

Ray-Box intersections

1 D_x

O

x

t x 



¹

t

1

D

x

t

₁

 O

x

₁



x=x₀ x=x₁

Ray-Box intersections

bool BBox::IntersectP(const Ray &ray,

float *hitt0, float *hitt1) {

float t0 = ray.mint, t1 = ray.maxt;

for (int i = 0; i < 3; ++i) {

float invRayDir = 1.f / ray.d[i];

float tNear = (pMin[i] - ray.o[i]) * invRayDir;

float tFar = (pMax[i] - ray.o[i]) * invRayDir;(p [ ] y [ ]) y ; if (tNear > tFar) swap(tNear, tFar);

t0 = tNear > t0 ? tNear : t0;

i i

t0 = tNear > t0 ? tNear : t0;

t1 = tFar < t1 ? tFar : t1;

if (t0 > t1) return false;

}

segment intersection intersection is empty

}

if (hitt0) *hitt0 = t0;

if (hitt1) *hitt1 = t1;

return true;

}

(14)

Grid accelerator

• Uniform grid

Teapot in a stadium problem

• Not adaptive to distribution of primitives.

Have to determine the number of voxels

• Have to determine the number of voxels.

(problem with too many or too few)

GridAccel

Class GridAccel:public Aggregate {

<GridAccel methods>

u_int nMailboxes;

MailboxPrim *mailboxes;

int NVoxels[3];

BBox bounds;

Vector Width, InvWidth;

V l ** l

Voxel **voxels;

ObjectArena<Voxel> voxelArena;

t ti i t M ilb Id static int curMailboxId;

}

mailbox

struct MailboxPrim {

Reference<Primitive> primitive;

Int lastMailboxId;

}

(15)

GridAccel

GridAccel(vector<Reference<Primitive> > &p, bool forRefined bool refineImmediately) bool forRefined, bool refineImmediately) : gridForRefined(forRefined) {

// Initialize with primitives for grid // Initialize with primitives for grid vector<Reference<Primitive> > prims;

if (refineImmediately) if (refineImmediately)

for (u_int i = 0; i < p.size(); ++i) [i] >F ll R fi ( i )

p[i]->FullyRefine(prims);

else i

prims = p;

GridAccel

// Initialize mailboxes for grid nMailboxes = prims size();

nMailboxes = prims.size();

mailboxes

= (MailboxPrim*)AllocAligned(

nMailboxes * sizeof(MailboxPrim));

for (u int i 0; i < nMailboxes; ++i) for (u_int i = 0; i < nMailboxes; ++i)

new (&mailboxes[i])MailboxPrim(prims[i]);

Determine number of voxels

• Too many voxels → slow traverse, large

memory consumption (bad cache performance) memory consumption (bad cache performance)

• Too few voxels → too many primitives in a l

voxel

• Let the axis with the largest extent have

3 N³

partitions (N:number of primitives)

Vector delta = bounds.pMax - bounds.pMin;

int maxAxis=bounds.MaximumExtent();

float invMaxWidth=1.f/delta[maxAxis];

float cubeRoot=3.f*powf(float(prims.size()),1.f/3.f);

float voxelsPerUnitDist=cubeRoot * invMaxWidth;

Calculate voxel size and allocate voxels

for (int axis=0; axis<3; ++axis) {

NVoxels[axis]=Round2Int(delta[axis]*voxelsPerUnitDist);

NVoxels[axis]=Clamp(NVoxels[axis], 1, 64);

}

Width[axis]=delta[axis]/NVoxels[axis];

InvWidth[axis]=

(Width[axis]==0.f)?0.f:1.f/Width[axis];

}

int nVoxels = NVoxels[0] * NVoxels[1] * NVoxels[2];

voxels=(Voxel **)AllocAligned(nVoxels*sizeof(Voxel *));

memset(voxels 0 nVoxels * sizeof(Voxel *));

memset(voxels, 0, nVoxels * sizeof(Voxel *));

(16)

Conversion between voxel and position

int PosToVoxel(const Point &P, int axis) { int v=Float2Int(

(P[axis]-bounds.pMin[axis])*InvWidth[axis]);

return Clamp(v, 0, NVoxels[axis]-1);

}

float VoxelToPos(int p, int axis) const { return bounds pMin[axis]+p*Width[axis];

return bounds.pMin[axis]+p Width[axis];

}

Point VoxelToPos(int x, int y, int z) const { return bounds pMin+

return bounds.pMin+

Vector(x*Width[0], y*Width[1], z*Width[2]);

}

inline int Offset(int x, int y, int z) {

return z*NVoxels[0]*NVoxels[1] + y*NVoxels[0] + x;

}

Add primitives into voxels

for (u_int i=0; i<prims.size(); ++i) {

<Find voxel extent of primitive>

<Add primitive to overlapping voxels>

} }

<Find voxel extent of primitive>

BBox pb = prims[i]->WorldBound();

int vmin[3] vmax[3];

int vmin[3], vmax[3];

for (int axis = 0; axis < 3; ++axis) { vmin[axis] = PosToVoxel(pb pMin axis);

vmin[axis] = PosToVoxel(pb.pMin, axis);

vmax[axis] = PosToVoxel(pb.pMax, axis);

} }

<Add primitive to overlapping voxels>

for (int z = vmin[2]; z <= vmax[2]; ++z) for (int y = vmin[1]; y <= vmax[1]; ++y)

for (int x = vmin[0]; x <= vmax[0]; ++x) { int offset = Offset(x, y, z);

if (!voxels[offset]) { voxels[offset]

= new (voxelArena)Voxel(&mailboxes[i]);

} } else {

// Add primitive to already-allocated voxel voxels[offset]->AddPrimitive(&mailboxes[i]);

voxels[offset] >AddPrimitive(&mailboxes[i]);

} }

(17)

Voxel structure

struct Voxel {

<Voxel methods>

union {

MailboxPrim *onePrimitive;

MailboxPrim **primitives;

};

u_int allCanIntersect:1;

i t P i iti 31 u_int nPrimitives:31;

}

Packed into 64 bits

AddPrimitive

void AddPrimitive(MailboxPrim *prim) { if (nPrimitives == 1) {

MailboxPrim **p = new MailboxPrim *[2];

p[0] = onePrimitive;

primitives = p;

}

else if (IsPowerOf2(nPrimitives)) { int nAlloc = 2 * nPrimitives;;

MailboxPrim **p = new MailboxPrim *[nAlloc];

for (u_int i = 0; i < nPrimitives; ++i) p[i] = primitives[i];

p[i] = primitives[i];

delete[] primitives;

primitives = p;

} }

primitives[nPrimitives] = prim;

++nPrimitives;

} }

GridAccel traversal

bool GridAccel::Intersect(

Ray &ray Intersection *isect) { Ray &ray, Intersection *isect) {

<Check ray against overall grid bounds>

<Get ray mailbox id>

<Set up 3D DDA for ray>

<Walk ray through voxel grid>

}

Check against overall bound

float rayT;

if (bounds Inside(ray(ray mint))) if (bounds.Inside(ray(ray.mint)))

rayT = ray.mint;

else if (!bounds IntersectP(ray &rayT)) else if (!bounds.IntersectP(ray, &rayT))

return false;

Point gridIntersect ray(rayT);

Point gridIntersect = ray(rayT);

(18)

Set up 3D DDA (Digital Differential Analyzer)

• Similar to Bresenhm’s line drawing algorithm

Set up 3D DDA (Digital Differential Analyzer)

blue values changes along the traversal

NextCrossingT[1] Out

voxel

g g

voxel index

DeltaT[0]

rayT

Step[0]=1 DeltaT[0]

NextCrossingT[0]

Pos

DeltaT: the distance change when voxel changes 1 in that direction

Set up 3D DDA

Pos[axis]=PosToVoxel(gridIntersect, axis);

if (ray.d[axis]>=0) {

NextCrossingT[axis] = rayT+

(VoxelToPos(Pos[axis]+1 axis)-gridIntersect[axis]) (VoxelToPos(Pos[axis]+1,axis)-gridIntersect[axis]) /ray.d[axis];

DeltaT[axis] = Width[axis] / ray.d[axis];

Step[axis] = 1;

Out[axis] = NVoxels[axis]; 1

Out[axis] NVoxels[axis];

} else { ...

D_x

Step[axis] = -1;

Out[axis] = -1;

} }

} Width[0]

Walk through grid

for (;;) {

*voxel=voxels[Offset(Pos[0] Pos[1] Pos[2])];

*voxel=voxels[Offset(Pos[0],Pos[1],Pos[2])];

if (voxel != NULL) hitSomething |=

hitSomething |=

voxel->Intersect(ray,isect,rayId);

<Advance to next voxel>

}

t hitS thi

return hitSomething;

Do not return; cut tmax instead Do not return; cut tmax instead Return when entering a voxel

that is beyond the closest found intersection.

(19)

Advance to next voxel

int bits=((NextCrossingT[0]<NextCrossingT[1])<<2) + ((NextCrossingT[0]<NextCrossingT[2])<<1) + ((NextCrossingT[1]<NextCrossingT[2]));

const int cmpToAxis[8] = { 2, 1, 2, 1, 2, 2, 0, 0 };

int stepAxis=cmpToAxis[bits];

if (ray.maxt < NextCrossingT[stepAxis]) break;

Pos[stepAxis]+=Step[stepAxis];

Pos[stepAxis]+ Step[stepAxis];

if (Pos[stepAxis] == Out[stepAxis]) break;

NextCrossingT[stepAxis] += DeltaT[stepAxis];

conditions

x<y x<z y<z

0 0 0 x≥y≥z 2

0 0 1 x≥z>y 1

0 1 0 -

0 1 1 z>x≥y 1 1 0 0 y>x≥z 2 1 0 0 y>x≥z 2

1 0 1 -

1 1 0 y≥z>x 0 1 1 0 y≥z>x 0 1 1 1 z>y>x 0

KD-Tree accelerator

• Non-uniform space subdivision (for example, kd tree and octree) is better than uniform grid kd-tree and octree) is better than uniform grid if the scene is irregularly distributed.

Spatial hierarchies

A A

A

Letters correspond to planes (A) Point Location by recursive search

(20)

Spatial hierarchies

A A

B B

A

Letters correspond to planes (A, B) Point Location by recursive search

Spatial hierarchies

A

D

A

B

B C

C

D

A

Letters correspond to planes (A, B, C, D) Point Location by recursive search

Variations

octree

kd tree octree bsp tree

kd-tree bsp-tree

“Hack” kd-tree building

• Split axis

R d bi l t t t

– Round-robin; largest extent

• Split location

– Middle of extent; median of geometry (balanced tree)

• Termination

– Target # of primitives, limited tree depth

• All of these techniques stink.

(21)

Building good kd-trees

• What split do we really want?

Cl Id th th t k t i h – Clever Idea: the one that makes ray tracing cheap – Write down an expression of cost and minimize it

G d t ti i ti

– Greedy cost optimization

• What is the cost of tracing a ray through a cell?

Cost(cell) = C_trav + Prob(hit L) * Cost(L) + Prob(hit R) * Cost(R)

Splitting with cost in mind

Split in the middle

To get through this part of empty space, you need to test all triangles on the right.

• Makes the L & R probabilities equal P tt ti t th L & R t

• Pays no attention to the L & R costs

Split at the median

• Makes the L & R costs equal

P tt ti t th L & R b biliti

• Pays no attention to the L & R probabilities

(22)

Cost-optimized split

Since Cost(R) is much higher, make it as small as possible

• Automatically and rapidly isolates complexity

P d l h k f t

• Produces large chunks of empty space

Building good kd-trees

• Need the probabilities

T t t b ti l t f

– Turns out to be proportional to surface area

• Need the child cell costs

– Simple triangle count works great (very rough approx.)

ll “b ” – Empty cell “boost”

Cost(cell) = C_trav + Prob(hit L) * Cost(L) + Prob(hit R) * Cost(R)

= C_trav + SA(L) * TriCount(L) + SA(R) * TriCount(R)

i th ti f th t t t t th t t C_trav is the ratio of the cost to traverse to the cost to intersect

C_trav= 1:80 in pbrt (found by experiments)

Surface area heuristic

2n splits;

must coincides with object boundary. Why?

S

a

S

_b

a b

a

p

a

 S

^b

b

p S

 S

Termination criteria

• When should we stop splitting?

B d d th li it b f t i l – Bad: depth limit, number of triangles – Good: when split does not help any more.

h h ld f

• Threshold of cost improvement

– Stretch over multiple levels

– For example, if cost does not go down after three splits in a row, terminate

• Threshold of cell size

– Absolute probability SA(node)/SA(scene) small

(23)

Basic building algorithm

1. Pick an axis, or optimize across all three 2 B ild f did li l i ( 2. Build a set of candidate split locations (cost

extrema must be at bbox vertices) 3. Sort or bin the triangles

4. Sweep to incrementally track L/R counts, cost p y , 5. Output position of minimum cost split

Running time: T ( N )  N log N  2 T ( N / 2 ) Running time:

N N

N T

N T N N N T

log

2

) (

) 2 / ( 2 log )

(







• Characteristics of highly optimized tree

– very deep, very small leaves, big empty cells

Ray traversal algorithm

• Recursive inorder traversal

tmax t*

* t

t

tmin

*

t t tmin t* tmax

* t

* min

t t

tmaxt min max min

Intersect(L,tmin,tmax) Intersect(L,tmin,t*)Intersect(R,tmin,tmax) Intersect(R,t*,tmax)( , , )

a video for kdtree

Tree representation

8-byte (reduced from 16-byte, 20% gain)

struct KdAccelNode { interior

struct KdAccelNode { ...

union {

u_int flags; // Both float split; // Interior u int nPrims; // Leaf

leaf u_int nPrims; // Leaf

};

union {

n

union {

u_int aboveChild; // Interior MailboxPrim *onePrimitive; // Leaf MailboxPrim **primitives; // Leaf };

} }

Tree representation

1 8 23

S E M

flags

2 n

Flag: 0,1,2 (interior x, y, z) 3 (leaf)

(24)

KdTreeAccel construction

• Recursive top-down algorithm d h

8 13l ( )

• max depth =

81.3log(N)

If (nPrims <= maxPrims || depth==0) {

<create leaf>

}

Interior node

• Choose split axis position

M d i t – Medpoint – Medium cut

A h i ti – Area heuristic

• Create leaf if no good splits were found

• Classify primitives with respect to split

Choose split axis position

cost of no split: 

^N

k i k t

1

) (

cost of split:

^{k 1}^

 





 ^B ^N^A

k k i A N

k k i B

t P t b P t a

t

1 1

) ( )

(

assumptions:

1. t

_i_i

is the same for all primitives p

2. t

_i

: t

_t

= 80 : 1 (determined by experiments, main factor for the performance)

cost of split:

cost of no split:

t_iN

) )(

1

( _e _B _B _A _A

i

t t b p N p N

t   

p )

cost of split:

t i⁽ e⁾⁽pB B pA A⁾

s

B

A B

p ( | ) 

B C

A

s

A

p ( | )

B C

A

Choose split axis position

Start from the axis with maximum extent, sort all edge events and process them in order all edge events and process them in order

A C

B

C

a₀ b₀ a₁ b₁ c₀ c₁

(25)

Choose split axis position

If there is no split along this axis, try other axes.

When all fail, create a leaf.

KdTreeAccel traversal

t_max

ToDo stack t_plane

t_min

far near

KdTreeAccel traversal

ToDo stack t_max

t_min

far near

(26)

KdTreeAccel traversal

t_max

far

ToDo stack t_plane

t_min

near

KdTreeAccel traversal

ToDo stack t_max

t_min

KdTreeAccel traversal

t

t_min

ToDo stack t_max

KdTreeAccel traversal

bool KdTreeAccel::Intersect

(const Ray &ray, Intersection *isect) (const Ray &ray, Intersection isect) {

if (!bounds.IntersectP(ray, &tmin, &tmax))( ( y, , )) return false;

KdAccelNode *node=&nodes[0];

while (node!=NULL) {

if (ray.maxt<tmin) break;

if (!node->IsLeaf()) <Interior>

else <Leaf>

}

} ToDo stack

(max depth)

(27)

Leaf node

1. Check whether ray intersects primitive(s) inside the node; update ray’s maxt

inside the node; update ray s maxt 2. Grab next node from ToDo queue

Interior node

1. Determine near and far (by testing which side O is)

O is)

below above below above

node+1 &(nodes[node->aboveChild])

2. Determine whether we can skip a node

node+1 &(nodes[node >aboveChild])

t t_plane t_plane tt_min

t_max t_min

t_max

near far near far

tt_plane

Acceleration techniques Best efficiency scheme

(28)

References

• J. Goldsmith and J. Salmon, Automatic Creation of Object Hierarchies for Ray Tracing IEEE CG&A 1987 Object Hierarchies for Ray Tracing, IEEE CG&A, 1987.

• Brian Smits, Efficiency Issues for Ray Tracing, Journal of Graphics Tools, 1998. p ,

• K. Klimaszewski and T. Sederberg, Faster Ray Tracing K. Klimaszewski and T. Sederberg, Faster Ray Tracing Using Adaptive Grids, IEEE CG&A Jan/Feb 1999.

• Whang et. al., Octree-R: An Adaptive Octree for efficient ray tracing, IEEE TVCG 1(4), 1995.

• A. Glassner, Space subdivision for fast ray tracing. IEEE CG&A, 4(10), 1984