Chapter 6 Shading

Models of received radiation

Model of received radiation

Assuming one distant light source, the intensity that surface element j receives is: i received
 n j
 s for n j
 s

0
0 otherwise n j is the surface normal (a unit vector) s is the illumination direction n j
 s is the cos of the angle between n j product) when n j and s (dot and s are unit vectors

Model of received radiation

Assuming one distant light source, the intensity that surface element j receives is: i received
 n j
 s for n j
 s

0
Why the restriction
(n j o s) > 0?
0 otherwise n j is the surface normal (a unit vector) s is the illumination direction n j
 s is the cos of the angle between n j product) when n j and s (dot and s are unit vectors

Dot product
 a and b are two vectors.

It is useful to be able to calculate the angle between a and b.

That’s where the dot product comes in.

The length of a vector:
(unit vectors have a length=1)

Dot product
Calculating the dot product:
Converting the dot product to an angle:

Dot product

Calculating the dot product:
– What is the value of cos 90 degrees?
– What 2 systems are used to represent angles?
 Degrees and …?
– What representation is used in Java? C/C++?
Calculator?

Back to our model of received radiation

Extending our model

Let’s extend our model to include the viewpoint.

Type of surfaces/surface reflection:
1. Diffuse (AKA Lambertian) – relatively viewpoint independent
2. Specular – very viewpoint dependent

Diffuse reflection
AKA Lambertian (after Johann Heinrich
Lambert (August 26, 1728 – September
25, 1777), a Swiss mathematician, physicist, and astronomer.)

Diffuse reflection
 independent of viewpoint, V
– Light reaching a surface element is reflected evenly in all directions of the hemisphere centered at that surface element.

Diffuse reflection

Albedo
– amount that is diffusely reflected
– Ratio (fraction) of total reflected light to total received light (by a surface element).
– Low for dark surfaces (0.04 for charcoal); high for light surfaces (0.9 for fresh snow).


Typically for
“rough” surfaces.

i diffuse

Diffuse reflection k j
 i received for n j
 v
0 otherwise where
 j is the particular surface element,

 k j is the surface albedo, n j is the surface normal (a unit vector), and
 v is the viewpoint.

0

Diffuse reflection

Specular reflection

Mirror-like / smooth / polished surfaces.

Distributes energy in a narrow cone about the ray of reflection.

Surfaces have a “shininess”  associated with them.
– Values of 100 or more for shiny surfaces.

Specular reflection

Defn. specular reflection = mirror-like reflection.
– Wavelength of reflected light is similar to the source and is independent of surface color.

Specular reflection

Defn., highlight = bright spot caused by the specular reflection of a light source.
– Indicates that the object is wavy, metallic, or glassy.

Specular reflection i specular where
R


R  V
 
R is the reflected ray, and

2 N

N  
S


S

Specular reflection i specular where
R


R  V
 
R is the reflected ray

2 N

N  
S


S

i specular

Specular reflection

R  V
 
R

2 N

N  
S


S where
R is the reflected ray
V is the viewpoint
N is the surface normal
S is the ray of received illumination
(See http://mathworld.wolfram.com/Reflection.html
for derivation of vector R.)

Specular vs. diffuse
Specular
– Wavelength of reflected light is similar to the source and is independent of surface color.
– Viewpoint dependent.

Darkening with distance

The intensity of light energy reaching a surface decreases with the distance of that surface from the light source.

Mercury receives more light from the sun than the earth.

Complications

Surface models of real objects typically have both specular and diffuse components.
– An apple has both specular and diffuse reflective components.

Complications

There are typically many light sources and many inter-surface reflections (referred to as
light).

Phong shading model
Three components:
1. ambient
2. diffuse
3. specular

Phong shading model

Flat vs. Phong Shading

Phong shading model

For each light source (there may be many), m:
– the components i m,s and i m,d
, are the intensities (often as RGB values) of the specular and diffuse components of the light sources, respectively.

Phong shading model

A single i a lighting.
term controls the ambient

It is sometimes computed as a sum of contributions from the light sources.

Phong shading model
For each light source, m,
– L m is the direction vector from the point on the surface toward each light source,
– N is the normal at this point of the surface,
– R m is the direction that a perfectly reflected ray of light (represented as a vector) would take from this point of the surface, and
– V is the direction towards the viewer.

Phong shading model
Then the shade value for each surface point
I p is calculated using this equation, which is the Phong reflection model:
I p
 k a i a
  m
 lights k
 d

L m
 N
 i m , d
 k s

R m
 V
  i m , s
 k a
: ambient reflection constant, the ratio of reflection of the ambient term present in all points in the scene rendered

Phong shading model
I p
 k a i a
  m
 lights k
 d

L m
 N
 i m , d
 k s

R m
 V
  i m , s
 k d
: diffuse reflection constant, the ratio of reflection of the diffuse term of incoming light (Lambertian reflectance)

I p
 k a i a
Phong shading model
  m
 lights k
 d

L m
 N
 i m , d
 k s

R m
 V
  i m , s
 k s
: specular reflection constant, the ratio of reflection of the specular term of incoming light
 : is a shininess constant for this material, which decides how "evenly" light is reflected from a shiny spot, and is very large for most surfaces, on the order of 50, getting larger the more mirror-like they are.

Phong shading model

One last feature of the Phong shading model (for smooth shading):
– Vectors are assigned at each polygonal vertex, and shading is interpolated across the surface of the polygon.

Summary



Shading is complicated!
We can model surfaces of objects as (coplanar) polygonal patches.
– Normal vector is very important.
– Each has its own color, specular, and diffuse
(Lambertian) characteristics.
We can have multiple light sources in a scene.
– Each may have its own: intensity, location, color, and direction.

The Phong model includes 3 components: specular, diffuse, and ambient.