Shading 3D Models

Shading 3D Models
An Introduction

Computer Science Department

bernstdh@jmu.edu

Motivation

An Observation:
- We commonly approximate "curved" objects with a series of "flat" polygons (e.g., triangles)
An Implication:
- When we light objects they appear to be faceted
A Problematic Aspect of Human Vision:
- This is made worse because, at dark-light borders, exciting and inhibiting photoreceptors do not cancel each other out

Motivation (cont.)

One Resolution:
- Use curved objects
Another Resolution:
- Modify the way flat surfaces are filled (which is usually called shading)

Gouraud Shading

Approach:
- Specify normals (that, hopefully, correspond to the normals of the curved object) at vertices
- Determine the intensity at each vertex
- Use a linear interpolation of the vertex intensities at each pixel
Visualization:
Notation:
- The normal at a point, $\bs{q}$ , is denoted by $\bs{n_{q}}$
- The lighting intensity at a point, $\bs{q}$ , is denoted by $I_{q}$

Gouraud Shading (cont.)

The Intial Calculations:
- Calculate $I_{a}$ , $I_{b}$ , and $I_{c}$ using $\bs{n_{a}}$ , $\bs{n_{b}}$ , and $\bs{n_{c}}$
Initial Interpolations:
- $I_{f} = (1 - \phi) I_{a} + \phi I_{b}$
- $I_{t} = (1 - \tau) I_{b} + \tau I_{c}$
For Each Point on the Scan Line:
- $I_{p} = (1 - \pi) I_{f} + \pi I_{t}$
- where $\pi = \frac{||\bs{p} - \bs{f}||} {||\bs{t} - \bs{f}||}$
- Note: $\pi = 0$ when $\bs{p} = \bs{f}$ in which case $I_{p} = I_{f}$ . Similarly, $\pi = 1$ when $\bs{p} = \bs{t}$ in which case $I_{p} = I_{t}$ .

Phong Shading

Approach:
- Specify normals (that, hopefully, correspond to the normals of the curved object) at vertices
- Use a linear interpolation of the vertex normals at each pixel
The Intial Interpolations:
- $\bs{n_{f}} = (1 - \phi) \bs{n_{a}} + \phi \bs{n_{b}}$
- $\bs{n_{t}} = (1 - \tau) \bs{n_{b}} + \tau \bs{n_{c}}$
For Each Point on the Scan Line:
- $\bs{n_{p}} = (1 - \pi) \bs{n_{f}} + \pi \bs{n_{t}}$
- where $\pi = \frac{||\bs{p} - \bs{f}||} {||\bs{t} - \bs{f}||}$

Phong Shading (cont.)

An Observation:
- A convex combination of two vectors with unit length has less than unit length
Visualization:
Implications:
- The "interpolated normals" must be re-normalized
- Failure to do so results in a slow algorithm for Gouraud shading [see Duff (1979)]

Shading Languages

The Idea:
- Specify shading rules in a special-purpose language
Some Examples:
- Assembly Language
- C-like Language [e.g., OpenGL]
- Shade Trees [e.g., Cook (1984)]
- Declarative Languages [e.g., RenderMan]

There's Always More to Learn