indirect illuminationmisha/fall20/09.pdf · 2020. 9. 22. · summary. • ray casting (direct...

Indirect Illumination

Michael Kazhdan

(601.457/657)

HB Ch. 14.1, 14.2

FvDFH 16.1, 16.2

Surface Illumination Calculation• Multiple light source:

𝑁𝑁

𝐿𝐿1

𝐿𝐿2

𝑉𝑉

Viewer

𝐼𝐼 = 𝐼𝐼𝐸𝐸 + �𝐿𝐿

𝐾𝐾𝐴𝐴 ⋅ 𝐼𝐼𝐿𝐿𝐴𝐴 + 𝐾𝐾𝐷𝐷 ⋅ 𝑁𝑁, 𝐿𝐿 + 𝐾𝐾𝑆𝑆 ⋅ 𝑉𝑉,𝑅𝑅 𝑛𝑛 ⋅ 𝐼𝐼𝐿𝐿

Overview• Direct Illumination

Emission at light sources Direct light at surface points

• Global illumination Shadows Transmissions Inter-object reflections

Shadows• Shadow term tells if light sources are blocked

Cast ray towards each light source 𝐿𝐿𝑖𝑖 If the ray is blocked, do not consider the contribution of

the light.

Shadows• Shadow term tells if light sources are blocked

Cast ray towards each light source 𝐿𝐿𝑖𝑖 𝑆𝑆𝑖𝑖 = 0 if ray is blocked, 𝑆𝑆𝑖𝑖 = 1 otherwise

𝐿𝐿1

𝐿𝐿0


𝐾𝐾𝐴𝐴 ⋅ 𝐼𝐼𝐿𝐿𝐴𝐴 + 𝐾𝐾𝐷𝐷 ⋅ 𝑁𝑁, 𝐿𝐿 + 𝐾𝐾𝑆𝑆 ⋅ 𝑉𝑉,𝑅𝑅 𝑛𝑛 ⋅ 𝐼𝐼𝐿𝐿 ⋅ 𝑆𝑆𝐿𝐿

ShadowTerm

• Shadow term tells if light sources are blocked Cast ray towards each light source 𝐿𝐿𝑖𝑖 𝑆𝑆𝑖𝑖 = 0 if ray is blocked, 𝑆𝑆𝑖𝑖 = 1 otherwise

Shadows

𝑆𝑆0 = 1:𝐿𝐿0 contributes



ShadowTerm

𝐿𝐿1

𝐿𝐿0

• Shadow term tells if light sources are blocked Cast ray towards each light source 𝐿𝐿𝑖𝑖 𝑆𝑆𝑖𝑖 = 0 if ray is blocked, 𝑆𝑆𝑖𝑖 = 1 otherwise

Shadows



ShadowTerm

𝐿𝐿1

𝐿𝐿0

𝑆𝑆0 = 1:𝐿𝐿0 contributes

𝑆𝑆1 = 0:𝐿𝐿1 doesn’t contribute

Ray Casting• Trace primary rays from camera

Direct illumination from unblocked lights only

Recursive Ray Tracing• Also trace secondary rays from hit surfaces

Consider contributions from:1. Reflected Rays2. Refracted Rays

• Also trace secondary rays from hit surfaces Consider contributions from:

1. Reflected Rays2. Refracted Rays

Mirror Reflections

𝐿𝐿1

Recall:The specularity of a surface, 𝐾𝐾𝑆𝑆, measures how mirror-like it is.



𝐿𝐿0

• Also trace secondary rays from hit surfaces Consider contributions from:

1. Reflected Rays2. Refracted Rays

Mirror Reflections

𝐿𝐿1

𝐿𝐿0


𝐾𝐾𝐴𝐴 ⋅ 𝐼𝐼𝐿𝐿𝐴𝐴 + 𝐾𝐾𝐷𝐷 ⋅ 𝑁𝑁, 𝐿𝐿 + 𝐾𝐾𝑆𝑆 ⋅ 𝑉𝑉,𝑅𝑅 𝑛𝑛 ⋅ 𝐼𝐼𝐿𝐿 ⋅ 𝑆𝑆𝐿𝐿 + 𝐾𝐾𝑠𝑠 ⋅ 𝐼𝐼𝑅𝑅

Radiance for mirror

reflection ray

Mirror Reflections• Also trace secondary rays from hit surfaces


Transparent Refraction• Also trace secondary rays from hit surfaces


Radiance for refraction ray


𝐾𝐾𝐴𝐴 ⋅ 𝐼𝐼𝐿𝐿𝐴𝐴 + 𝐾𝐾𝐷𝐷 ⋅ 𝑁𝑁, 𝐿𝐿 + 𝐾𝐾𝑆𝑆 ⋅ 𝑉𝑉,𝑅𝑅 𝑛𝑛 ⋅ 𝐼𝐼𝐿𝐿 ⋅ 𝑆𝑆𝐿𝐿 + 𝐾𝐾𝑠𝑠 ⋅ 𝐼𝐼𝑅𝑅 + 𝐾𝐾𝑇𝑇 ⋅ 𝐼𝐼𝑇𝑇

𝐿𝐿1

𝐿𝐿0



𝐿𝐿1

𝐿𝐿0

Radiance for refraction ray



Transparency and Shadow• Problem:

If a surface is transparent, then rays to the light source may pass through the object

Over-shadowing

Transparency and Shadow• Problem:

If a surface is transparent, then rays to the light source may pass through the object

Need to modify the shadow term so that instead of representing a binary (0/1) value, it gives the fraction of light passing through.

⇒ Accumulate transparency values as the ray travels to the light source.



Transparency and Shadow• Accumulate transparency values as the ray

travels to the light source.

𝑆𝑆0 = 1:𝐿𝐿0 contributes fully



𝐿𝐿1

𝐿𝐿0



𝐿𝐿1

𝐿𝐿0


𝑆𝑆1 =?:





𝐾𝐾𝑇𝑇

𝐿𝐿1

𝐿𝐿0


𝑆𝑆1 = 1 ⋅ 𝐾𝐾𝑇𝑇 ⋅?:





𝐾𝐾𝑇𝑇

𝐿𝐿1

𝐿𝐿0


𝑆𝑆1 = 1 ⋅ 𝐾𝐾𝑇𝑇 ⋅ 𝐾𝐾𝑇𝑇 ⋅?:





𝐾𝐾𝑇𝑇

𝐿𝐿1

𝐿𝐿0


𝑆𝑆1 = 1 ⋅ 𝐾𝐾𝑇𝑇 ⋅ 𝐾𝐾𝑇𝑇:𝐿𝐿1 contributes partially





KT

𝐿𝐿1

𝐿𝐿0

For solid models we can use the distance, 𝑑𝑑, travelled through the

surface instead: 𝑆𝑆 = 𝑒𝑒−𝑑𝑑⋅𝐾𝐾𝑇𝑇


𝑆𝑆1 = 1 ⋅ 𝐾𝐾𝑇𝑇 ⋅ 𝐾𝐾𝑇𝑇:𝐿𝐿1 contributes partially



Transparent Refraction• When a light of light passes through a transparent

object, the ray of light can bend, (𝜃𝜃𝑖𝑖 ≠ 𝜃𝜃𝑟𝑟).

𝑁𝑁

𝐿𝐿

𝑇𝑇

𝜃𝜃𝑖𝑖

𝜃𝜃𝑟𝑟

Snell’s Law• The way that light bends is determined by the

indices of refraction of the internal and external materials 𝜂𝜂𝑖𝑖 and 𝜂𝜂𝑟𝑟:

The index of refraction of air is 𝜂𝜂 = 1.

𝑁𝑁

𝐿𝐿

𝑇𝑇

𝜃𝜃𝑖𝑖

𝜃𝜃𝑟𝑟

𝜂𝜂𝑟𝑟 ⋅ sin𝜃𝜃𝑟𝑟 = 𝜂𝜂𝑖𝑖 ⋅ sin𝜃𝜃𝑖𝑖


indices of refraction of the internal and external materials 𝜂𝜂𝑖𝑖 and 𝜂𝜂𝑟𝑟: 𝑁𝑁

𝐿𝐿

𝑇𝑇

𝜃𝜃𝑖𝑖

𝜃𝜃𝑟𝑟


Note (Critical Angle):

If 𝜂𝜂𝑖𝑖 > 𝜂𝜂𝑟𝑟 it is possible that 𝜂𝜂𝑖𝑖𝜂𝜂𝑟𝑟⋅ sin𝜃𝜃𝑖𝑖 > 1.

In this case:• There is no value of 𝜃𝜃𝑟𝑟 such that 𝜂𝜂𝑟𝑟 ⋅ sin𝜃𝜃𝑟𝑟 = 𝜂𝜂𝑖𝑖 ⋅ sin𝜃𝜃𝑖𝑖.• The light reflects off the surface and does not pass through.


indices of refraction of the internal and external materials 𝜂𝜂𝑖𝑖 and 𝜂𝜂𝑟𝑟: 𝑁𝑁

𝐿𝐿

𝑇𝑇

𝜃𝜃𝑖𝑖

𝜃𝜃𝑟𝑟

𝑇𝑇 =𝜂𝜂𝑖𝑖𝜂𝜂𝑟𝑟⋅ cos𝜃𝜃𝑖𝑖 − cos𝜃𝜃𝑟𝑟 ⋅ 𝑁𝑁 −

𝜂𝜂𝑖𝑖𝜂𝜂𝑟𝑟⋅ 𝐿𝐿




𝐿𝐿1

𝐿𝐿0


Snell’s Law and Caustics• Challenge:

If a surface is transparent, then rays to the light source may not travel along straight paths

⇒ For a given point of intersection, there may be multiple directions/paths a ray can travel to get to the light

⇒ Summing the intensity contribution from all of theses directions/paths we get bright caustics

This is difficult to address with ray-tracing

https://en.wikipedia.org/wiki/Caustic_(optics)

https://en.wikipedia.org/wiki/Caustic_(optics)

(Rough) Complexity Analysis• Simulating reflection and refraction in the ray

tracer, a ray splits into two at each bounce

𝐿𝐿1

𝐿𝐿0

CastReflectedRefracted

(Rough) Complexity Analysis• Simulating reflection and refraction in the ray

tracer, a ray splits into two at each bounce

⇒ Exponential growth in the number of rays as a function of the number of bounces

𝐿𝐿1

𝐿𝐿0

Note that we do not cast reflected rays from the interior of a shape.That is, if the dot-product of the surface normal

with the ray direction is positive

CastReflectedRefracted

Termination Criteria• How do we determine when to stop recursing?

If the ray bounces around too much If the contribution will be too small

Termination CriteriaPixel GetColor( scene , ray , ir , depth , cutOff ){

Pixel p(0,0,0)Ray reflect, refractIntersection hit = FindIntersection( ray , scene )if ( hit ){

p += GetSurfaceColor( hit.position )

reflect.direction = Reflect( ray.direction , hit.normal )reflect.position = hit.position + reflect.direction*εif( depth >0 && hit.kSpec>cutOff )

p += GetColor( scene , reflect , ir , depth-1 , cutOff/hit.kSpec )*hit.kSpec

refract.direction = Refract( ray.direction , hit.normal , ir , hit.ir )refract.position = hit.position + refract.direction*εif( depth >0 && hit.kTran>cutOff )

p += GetColor( scene , refract , hit.ir , depth-1 , cutOff/hit.kTran )*hit.kTran}return p

}

Pixel GetColor( scene , ray , ir , depth , cutOff ){



reflect.direction = Reflect( ray.direction , hit.normal )reflect.position = hit.position + reflect.direction*εif( depth >0 && hit.kSpec>cutOff )



p += GetColor( scene , refract , hit.ir , depth-1 , cutOff/hit.kTran ) *hit.kTran}return p

}

Termination Criteria






reflect.direction = Reflect( ray.direction , hit.normal )reflect.position = hit.position + reflect.direction*εif( depth>0 && hit.kSpec>cutOff )



p += GetColor( scene , refract , hit.ir , depth-1 , cutOff/hit.kTran ) *hit.kTran}return p

}




if( depth>0 && hit.kSpec>cutOff )

p += GetColor( scene , reflect , ir , depth-1 , cutOff/hit.kSpec )*hit.kSpec;

The first reflected ray should be cast if it cancontribute an intensity greater than the cut-off.

⇒ 𝐾𝐾𝑆𝑆1 > cut-off


𝐿𝐿1

𝐿𝐿0

𝐾𝐾𝑆𝑆1



The second reflected ray should be cast if it cancontribute an intensity greater than the cut-off.

⇒ 𝐾𝐾𝑆𝑆2 ⋅ 𝐾𝐾𝑆𝑆1 > cut-off

⇔ 𝐾𝐾𝑆𝑆2 > cut-off𝐾𝐾𝑆𝑆1


𝐿𝐿1

𝐿𝐿0

𝐾𝐾𝑆𝑆1

𝐾𝐾𝑆𝑆2



The second reflected ray should be cast if it cancontribute an intensity greater than the cut-off.

⇒ 𝐾𝐾𝑆𝑆2 ⋅ 𝐾𝐾𝑆𝑆1 > cut-off

⇔ 𝐾𝐾𝑆𝑆2 > cut-off𝐾𝐾𝑆𝑆1

⇒ The second reflected rayshould be cast if it can reflectat least cut-off

𝐾𝐾𝑆𝑆1 of the local intensity


𝐿𝐿1

𝐿𝐿0

𝐾𝐾𝑆𝑆1

𝐾𝐾𝑆𝑆2






refract.direction = Refract( ray.direction , hit.normal , ir , hit.ir )refract.position = hit.position + refract.direction*εif( depth>0 && hit.kTran>cutOff )


}











}


Why do we add a small amount of the direction to the position?

To ensure that the new ray does not hit its starting location!








}


Warning:In practice, cutOff is a scalar while hit.kTran/kSpec are RGB values.








}


Note:When we refract, we enter a new material and the associated index of refraction changes.

Illumination Examples• Ray tracing (rays to point light source)

Courtesy Henrik Wann Jensen

• Soft shadows (rays to area light source)

Illumination Examples


Illumination Examples• Caustics (rays from light source)


• Full Global Illumination

Illumination Examples


• GetColor calls itself recursively

Recursive Ray Tracing

Image RayTrace( Camera camera , Scene scene ,int width , int height ,int depth , float cutOff )

{Image image = new Image( width , height );for( int i=0 ; i<width ; i++ ) for( int j=0 ; j<height ; j++ ){

Ray ray = ConstructRayThroughPixel( camera , i , j );image[i][j] = GetColor( scene , ray , 1 , depth , cutOff );

}return image;

}

Summary• Ray casting (direct Illumination)

Use simple analytic approximations for light source emission and surface reflectance

• Recursive ray tracing (global illumination) Incorporate shadows, mirror reflections,

and pure refractions

More on global illumination later!

All of this is an approximationso that it is practical to compute

indirect illuminationmisha/fall20/09.pdf · 2020. 9. 22. · summary. • ray casting (direct...

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