1. Review 2: Illumination, Shading, Texturing and Anti-aliasing
Jian Huang, CS594, Spring 2002

2. Illumination Vs. Shading
Illumination (lighting) model: determine the color of a surface point by simulating some light attributes.
Shading model: applies the illumination models at a set of points and colors the whole image.

3. Local illumination
Only consider the light, the observer position, and the object material properties

4. Basic Illumination Model
Simple and fast method for calculating surface intensity at a given point
Lighting calculating are based on:
The background lighting conditions
The light source specification: color, position
Optical properties of surfaces:
Glossy OR matte
Opaque OR transparent (control refection and absorption)

5. Ambient light (background light)
The light that is the result from the light reflecting off other surfaces in the environment
A general level of brightness for a scene that is independent of the light positions or surface directions -> ambient light
Has no direction
Each light source has an ambient light contribution, Ia
For a given surface, we can specify how much ambient light the surface can reflect using an ambient reflection coefficient : Ka (0 < Ka < 1)

6. Ambient Light
So the amount of light that the surface reflect is therefore
Iamb = Ka * Ia

7. Diffuse Light
The illumination that a surface receives from a light source and reflects equally in all directions
This type of reflection is called Lambertian Reflection (thus, Lambertian surfaces)
The brightness of the surface is indepenent of the observer position (since the light is reflected in all direction equally)

8. Lambert’s Law
How much light the surface receives from a light source depends on the angle between its angle and the vector from the surface point to the light (light vector)
Lambert’s law: the radiant energy ’Id’ from a small surface da for a given light source is:
Id = IL * cos(q)
IL : the intensity of the light source
is the angle between the surface
normal (N) and light vector (L)

9. The Diffuse Component The total diffuse reflection = ambient + diffuse
Surface’s material property: assuming that the surface can reflect Kd (0<Kd<1), diffuse reflection coefficient) amount of diffuse light:
Idiff = Kd * IL * cos(q)
If N and L are normalized, cos(q) = N*L
Idiff = Kd * IL * (N*L)
The total diffuse reflection = ambient + diffuse
Idiff = Ka * Ia + Kd * IL * (N*L)

10. Examples
Sphere diffusely lighted from various angles !

11. Specular Light
These are the bright spots on objects (such as polished metal, apple ...)
Light reflected from the surface unequally to all directions.
The result of near total reflection of the incident light in a concentrated region around the specular reflection angle

12. Phong’s Model for Specular
How much reflection light you can see depends on where you are

13. Phong Illumination Curves
Specular exponents are much larger than 1;
Values of 100 are not uncommon.
: glossiness, rate of falloff

14. Specular Highlights
Shiny surfaces change appearance when viewpoint is changed
Specularities are caused by microscopically smooth surfaces.
A mirror is a perfect specular reflector

15. Phong Illumination
Moving Light
Change n

16. Putting It All Together
Single Light (white light source)

17. Flat Shading

18. Smooth Shading Need to have per-vertex normals Gouraud Shading
Interpolate color across triangles
Fast, supported by most of the graphics accelerator cards
Phong Shading
Interpolate normals across triangles
More accurate, but slow. Not widely supported by hardware

19. Gouraud Shading Normals are computed at the polygon vertices
If we only have per-face normals, the normal at each vertex is the average of the normals of its adjacent faces
Intensity interpolation: linearly interpolate the pixel intensity (color) across a polygon surface

20. Gouraud Shading

21. Phong Shading Model
Gouraud shading does not properly handle specular highlights, specially when the n parameter is large (small highlight).
Reason: colors are interpolated.
Solution: (Phong Shading Model)
1. Compute averaged normal at vertices.
2. Interpolate normals along edges and scan-lines. (component by component)
3. Compute per-pixel illumination.

22. Phong Shading

23. What Dreams May Come

24. How to efficiently add graphics detail?
Solution - (its really a cheat!!)
How?
MAP surface detail from a predefined (easy to model) table (“texture”) to a simple polygon

25. Texture Mapping Problem #1 Fitting a square peg in a round hole
We deal with non-linear transformations
Which parts map where?

26. Texture Mapping Problem #2 Mapping from a pixel to a “texel”
Aliasing is a huge problem!

27. What is a Texture? Given the (texture/image index) (u,v), want:
F(u,v) ==> a continuous reconstruction
= { R(u,v), G(u,v), B(u,v) }
= { I(u,v) }
= { index(u,v) }
= { alpha(u,v) }
= { normals(u,v) }
= { surface_height(u,v) }
= ...

28. RGB Textures
Places an image on the object
“Typical” texture mapping

29. Opacity Textures
A binary mask, really redefines the geometry.

30. Bump Mapping
This modifies the surface normals.
More on this later.

31. Displacement Mapping
Modifies the surface position in the direction of the surface normal.

32. Texture and Texel Each pixel in a texture map is called a Texel
Each Texel is associated with a (u,v) 2D texture coordinate
The range of u, v is [0.0,1.0]

33. (u,v) tuple
For any (u,v) in the range of (0-1, 0-1), we can find the corresponding value in the texture using some interpolation

34. Two-Stage Mapping Model the mapping: (x,y,z) -> (u,v)
Do the mapping

35. Image space scan For each y
For each x
compute u(x,y) and v(x,y)
copy texture(u,v) to image(x,y)
Samples the warped texture at the appropriate image pixels.
inverse mapping

36. Image space scan Problems: Finding the inverse mapping
Use one of the analytical mappings
Bi-linear or triangle inverse mapping
May miss parts of the texture map
Texture
Image

37. Inverse Mapping
Need to transform back to world space to do the interpolation
Orientation in 3D image space
Foreshortening
(.8,1)
(.5,1)
(.5,.7)
(.1,.6)
(.6,.2)

38. Texture space scan For each v
For each u
compute x(u,v) and y(u,v)
copy texture(u,v) to image(x,y)
Places each texture sample to the mapped image pixel.
Forward mapping

39. Texture space scan Problems: May not fill image Forward mapping needed

40. Texture Mapping Mapping to a 3D Plane Simple Affine transformation
rotate
scale
translate y z v x u

41. Texture Mapping Mapping to a Cylinder u -> q v -> z
Rotate, translate and scale in the uv-plane
u -> q
v -> z
x = r cos(q), y = r sin(q) v u

42. Texture Mapping Mapping to Sphere Impossible!!!!
Severe distortion at the poles
u -> q
v -> f
x = r sin(q) cos(f)
y = r sin(q) sin(f)
z = r cos(q)

43. Sampling
What we have in computer graphics is a point sampling of our scene, or:
I(x) = f(x)•ST(x)
What we would like is more of an integration across the pixel (or larger area):
I(x) = f(x) h(x)
What should h(x) be?

44. Sampling Theorem The Shannon Sampling Theorem
A band-limited signal f(x), with a cutoff frequency of l, that is sampled with a sampling spacing of T may be perfectly reconstructed from the discrete values f[nT] by convolution with the sinc(x) function, provided:
l is called the Nyquist limit.

45. Sampling and Anti-aliasing
If don’t have Nyquist rate, then aliasing artifacts.

46. Two possible solutions
So far we just mapped one point, results in bad aliasing (resampling problems)
Two possible solutions:
Super-sampling: not very good (slow!)
Low-pass filtering: popular approaches – mipmaps, SAT

47. Quality considerations
Pixel area maps to “weird” (warped) shape in texture space v ys
pixel xs u

48. Quality considerations
We need to:
Calculate (or approximate) the integral of the texture function under this area
Approximate:
Convolve with a wide filter around the center of this area
Calculate the integral for a similar (but simpler) area.

49. Quality considerations
the area is typically approxiated by a rectangular region (found to be good enough for most applications)
filter is typically a box/averaging filter - other possibilities
how can we pre-compute this?

50. Summed Area Table (SAT)
Determining the rectangle:
Find bounding box and calculate its aspect ratio
pixel u v xs ys

51. Summed Area Table (SAT)
Determine the rectangle with the same aspect ratio as the bounding box and the same area as the pixel mapping.
pixel u v xs ys

52. Summed Area Table (SAT)
Center this rectangle around the bounding box center.
Formula:
Area = aspect_ratio*x*x
Solve for x – the width of the rectangle
Other derivations are also possible using the aspects of the diagonals, …

53. Summed Area Table (SAT)
Calculating the color
We want the average of the texel colors within this rectangle
(u4,v4) u v
(u3,v3) + - - +
(u2,v2)
(u1,v1) + - + -

54. Summed Area Table (SAT)
To get the average, we need to divide by the number of texels falling in the rectangle.
Color = SAT(u3,v3)-SAT(u4,v4)-SAT(u2,v2)+SAT(u1,v1)
Color = Color / ( (u3-u1)*(v3-v1) )
This implies that the values for each texel may be very large:
For 8-bit colors, we could have a maximum SAT value of 255*nx*ny
32-bit pixels would handle a 4kx4k texture with 8-bit values.
RGB images imply 12-bytes per pixel.