cs 775 advanced computer graphicsparagc/teaching/2015/cs775/lectures/07_… · cs775: lecture 7...

CS 775Advanced Computer Graphics

Lecture 7 : Ray Tracing 2

CS775: Lecture 7 Parag Chaudhuri, 2015

Ray Tracing

● Transforming normals

q

q '

sc t

s 'c ' t 'M

M−1

We also need the normal at the point of intersection. How does the normal transform when the object undergoes an affine transformation?

Object Space

World SpaceN

N '


Ray Tracing

● Transforming normals

Incorrect

Correct


● If is an vector in the tangent plane then after the transformation it becomes

● The correct transformation to be applied to the normal should keep it perpendicular to the tangent vector.

Ray Tracing

● Transforming normals – think about the tangents instead

N

V T

N '

V 'T

N '

V 'T

Incorrect Incorrect

V 'T=M.V T

V T


Ray Tracing

● Transforming normals – think about the tangents instead

N

V T

N '

V 'T

N '

V 'T

Incorrect Incorrect

N T V T=0⇒ N T

M−1 M V T=0⇒N T . M−1

M.V T =0⇒N T . M−1

V 'T=0⇒ N 'T V 'T=0

⇒ N '=NT . M−1T=M−1

T . N


Ray Tracing

● Ray Casting – Single Step Ray Tracing

– For every pixel in the image● Shoot a ray ● Find closest intersection with object● Find normal at the point of intersection● Compute illumination at point of

intersection● Assign pixel color


Ray Tracing

● Recursive Ray Tracing

Viewer

ViewPlane

A

B

C

D

E

F


Ray Tracing


A

BC

D

EF

Eye

D

DC

C F

E

Eye

● Primary Rays - from the Eye

● Secondary Rays – Reflection, Refraction

● Shadow Rays


Ray Tracing

● Reflected Ray

– Angle of Incidence = Angle of Reflection

I

I

N

R

−I . N N

−I . N NR=I−2I . N N

I R

Incident Ray

Reflected Ray

Surface Normal

I

R

N


Ray Tracing

● Transmitted Ray

I sinI=T sin T

I

T

N

M

M sinI

I

T

−N

−N cosI

T

I

I

T=

sinT

sin I

=R

I=M sinI−N cosI

⇒ M= IN cosI /sinI

T =−N cosTM sinT

=−N cosT IN cosI sinT

sin I=R

I R cosI−cosT N

=RI R cosI−1− R

2 1−cos2 I N

=RI R −N .I −1− R

2 1−−N .I 2 N

Normalize the result!

Snell-Descartes Law

- Incident Ray

So the transmitted ray is given by:


Ray Tracing

● Transmitted Ray

I sinI=T sin T

I

T

N

M

M sinI

I

T

−N

−N cosI

T

I

I

T=

sinT

sin I

=R

I=M sinI−N cosI

⇒ M= IN cosI /sinI

T =−N cosTM sinT

=−N cosT IN cosI sinT

sin I=R

I R cosI−cosT N

=RI R cosI−1− R

2 1−cos2 I N

=RI R −N .I −1− R

2 1−−N .I 2 N

Snell-Descartes Law

So the transmitted ray is given by:

What happens when the square root is imaginary?

- Incident Ray


Ray Tracing

● Transmitted RayN

T=m

I=1

Entering and Leaving transmissive material is different – check dot product with normal.

N

I=m

T=1

I

T

I

T


Ray Tracing

● Total Internal Reflection

Image from Color and light in nature by David K. Lynch, William Charles Livingston


Ray Tracing

Whitted, T. "An improved illumination model for shaded display”, Communications of the ACM, 23(6):343-349, 1980.


Ray Tracing

Enright, D., Marschner, S. and Fedkiw, R., "Animation and Rendering of Complex Water Surfaces", SIGGRAPH 2002, ACM TOG 21, 736-744 (2002).


Ray Tracing

● Illumination : The Phong Model

– For a single light source total illumination at any point is given by:

I=k a I ak d I dk s I s

where

is the contribution due to ambient reflection

is the contribution due to diffuse reflection

is the contribution due to specular reflection

k a I a

k d I d

k s I s


Ray Tracing

● Components of the Phong Model

● Ambient Illumination:

– Represents the reflection of all indirect illumination.

– Has the same value everywhere.

– Is an approximation to computing Global Illumination.

I a

From http://en.wikipedia.org/wiki/Global_illumination (14/08/2009)

http://en.wikipedia.org/wiki/Global_illumination


Ray Tracing


● Diffuse Illumination:

– Assumes Ideal Diffuse Surface – that reflects light equally in all direction.

– Surface is very rough at microscopic level. For e.g., Chalk and Clay.

I d=I L cosL


Ray Tracing



– Reflects light according to Lambert's Cosine Law

I d=I L cosL

I d=I L cosL

=I L L . N

: vector to the light source

: intensity of the light source

: surface normal

L

N

I L

=I L L . N N−L

≡


Ray Tracing



– Reflects light according to Lambert's Cosine Law

I d=I L cosL

I d=I L cosL

=I L L . N N−L

≡

If and are in opposite directions then the dot product is negative. Use to get the correct value.

If is distance to the light source and is its true intensity then a distance based attenuation is modelled by an inverse square falloff, i.e.,

L Nmax L . N , 0

r I t

I L=I t /r2


Ray Tracing


● Specular Illumination:

– Ideal specular surface reflects only along one direction.

– Reflected intensity is view dependent – Mostly it is along the reflected ray but as we move away some of the reflection is slightly offset from the reflected ray due to microscopic surface irregularites.

I s=I L cosnv=I L R . V

n

NL R

L

NL R

L VV


Ray Tracing


● Specular Illumination:– is called the coefficient of shininess and

I s=I L cosnv=I L R . V

n

n

cosn

I L=I t /r2


Ray Tracing

● The Phong Illumination Model

I=k a I ak d I dk s I s

– are material constants defining the amount of light that is reflected as ambient, diffuse and specular. They may be defined in as three values with R, G, B components.

k a , k d , k s

http://en.wikipedia.org/wiki/Phong_shading


Ray Tracing

● The Blinn-Phong Illumination Model

H=LV

∥LV∥

NL R

H

v=

=

⇒v−= or v=2I s=I L cosn 'ϕ=I L( H⃗ . N⃗ )

n

V

v


● Local Illumination Model

Global Illumination Model

● Reflected and transmitted components may also be attenuated based on distance the ray travels.

Ray Tracing

I local=k a I a ∑1≤i≤m

kd I dik s I si

I global=I localkr I reflectedk t I transmitted


● Surface Material Properties

● Colour – For each object there can be a

– Diffuse colour, Specular colour, Reflected colour and Transmitted colour

– Remember differently coloured light is at different wavelength so:

● Accounting for shadows:

Ray Tracing

I =k a C d I a ∑1≤i≤m

k d Cd I dik s C s I sik r C r I rk t C t I t

I =k a C d I a ∑1≤i≤m

Si k d Cd I dik s C s I sik r C r I rk t C t I t


Ray Tracing


A

B C

D

EF

Eye

D

Eye

L1

L2


Ray Tracing


A

B C

D

EF

Eye

D

Eye

L1

L2

C D

L1

L2

L1

L2


Ray Tracing


A

B C

D

EF

Eye

D

Eye

L1

L2

C D L1

L2

L1

L2

C

E

FL1

L2

● Stop

– When a ray leaves the scene

– Contributed intensity is too less

L1

L2

L1

L2


Ray Tracing


A

B C

D

EF

Eye

D

Eye

L1

L2

C D L1

L2

L1

L2

C

E

FL1

L2

● Complexity?

L1

L2

L1

L2


Ray Tracing


A

B C

D

EF

Eye

D

Eye

L1

L2

C D L1

L2

L1

L2

C

E

FL1

L2

● Complexity = h x w x Nobjects

x interesection cost x depth of recursion x N

shadow_rays x ...

L1

L2

L1

L2


Ray Tracing

● Aliasing – Discrete samples of a continuous world


Ray Tracing

● Anti-aliasing – Shoot more rays per pixel - super sample!


Ray Tracing

● Sampling strategies for anti-aliasing

Regular super sampling

Adaptive super sampling

Stochastic or jittered super sampling

● Aliasing vs. Noise

● Aggregating the samples.


Ray Tracing


Distributed Ray Tracing

http://www.cs.utexas.edu/~fussell/

http://web.cs.wpi.edu/~matt/courses/cs563/talks/dist_ray/dist.html


The Rendering EquationLo x , , , t =L ex , , , t ∫

f r x , ' , , , t L i x , ' , , t −⋅nd '

Lo x , , , t is the total amount of light of wavelength , directed outward along direction at time t , from a particular position x

Le x , , , t is the emitted light.

L i x , ' , , t is the light of wavelength , coming inward toward xfrom direction ' at time t

f r x , ' , , , t is the bidirectional reflectance distribution function (BRDF),i.e., the proportion of light reflected from ' toat position x , time t ,and at wavelength

∫

... d ' is the integral over a sphere of inward directions

−⋅n is the attenuation of incident light due to incident angle


The Rendering EquationLo x , , , t =L ex , , , t ∫

f r x , ' , , , t L i x , ' , , t −⋅nd '

● Is this enough?● BTDF - Refraction, BSDF – Sub surface

scattering● Phosphoresence● Diffraction● Atmospheric Scattering

cs 775 advanced computer graphicsparagc/teaching/2015/cs775/lectures/07_… · cs775: lecture 7...

Documents