1. Week 5 - Friday
CS361

2. Last time What did we talk about last time? Quaternions
Vertex blending
Morphing

3. Questions?

4. Project 2

5. Morphing
Morphing is the technique for interpolating between two complete 3D models
It has two problems:
Vertex correspondence
What if there is not a 1 to 1 correspondence between vertices?
Interpolation
How do we combine the two models?

6. Morphing continued
We're going to ignore the correspondence problem (because it's really hard)
If there's a 1 to 1 correspondence, we use parameter s[0,1] to indicate where we are between the models and then find the new location m based on the two locations p0 and p1
Morph targets is another technique that adds in weighted poses to a neutral model

7. Projections
Finally, we deal with the issue of projecting the points into view space
Since we only have a 2D screen, we need to map everything to x and y coordinates
Like other transforms, we can accomplish projection with a 4 x 4 matrix
Unlike affine transforms, projection transforms can affect the w components in homogeneous notation

8. Orthographic projections
An orthographic projection maintains the property that parallel lines are still parallel after projection
The most basic orthographic projection matrix simply removes all the z values
This projection is not ideal because z values are lost
Things behind the camera are in front
z-buffer algorithms don't work

9. Canonical view volume
To maintain relative depths and allow for clipping, we usually set up a canonical view volume based on (l,r,b,t,n,f)
These letters simply refer to the six bounding planes of the cube
Left
Right
Bottom
Top
Near
Far
Here is the (OpenGL) matrix that translates all points and scales them into the canonical view volume

10. Stupid MonoGame
OpenGL normalizes to a canonical view volume from [-1,1] in x, [-1,1] in y, and [-1,1] in z
Just to be silly, MonoGame normalizes to a canonical view volume of [-1,1] in x, [-1,1] in y, and [0,1] in z
Thus, its projection matrix is:

11. Perspective projection
A perspective projection does not preserve parallel lines
Lines that are farther from the camera will appear smaller
Thus, a view frustum must be normalized to a canonical view volume
Because points actually move (in x and y) based on their z distance, there is a distorting term in the w row of the projection matrix

12. Perspective projection matrix
Here is the MonoGame projection matrix
It is different from the OpenGL again because it only uses [0,1] for z

13. MonoGame Projection Examples

14. Student Lecture: Light and Materials

15. Light

16. Light
[L]ight …travels so fast that it takes most races thousands of years to realise that it travels at all….
Douglas Adams
Light is one of the most complex phenomena in the universe
There are quantum effects, its dual wave/particle nature
We will constantly be approximating the effect of light, since figuring out its real effect is virtually impossible

17. Break it down We will consider three processes in lighting a scene
Emitting light
From the sun or light bulbs or whatever
Interaction of light
Light is absorbed by and scatters off objects in a scene
Detection by a sensor
A human eye (or a robot eye), camera, piece of film will sense the light
We have to give at least cursory attention to each process to get realistic rendering

18. Light sources
One of the easiest light sources to model are directional lights, such as the sun
With directional lights, all the light travels the same direction, which we can model with a light vector l
We assume that l is a unit vector
l is defined in the opposite direction the light is traveling

19. Intensity of illumination
Besides direction, we need to know the amount of light
Radiometry is the science of measuring light, and we'll talk more about it in two weeks
Irradiance is the light's power passing through a unit area surface perpendicular to l
Light can be colored by using RGB components

20. Surface irradiance Most light is not perpendicular to your surface
The surface irradiance is the perpendicular irradiance times cos θ, where θ is the angle between l and the surface normal n
This is why l is the opposite of the direction of light flowing (so that we don't have to negate it)
Also, we clamp the cos θ to [0,1] (no negative values)

21. Additive irradiance
Real light is coming from many different directions
The final effects of irradiance is additive
Just sum up all the individual light effects
Although we use RGB for light, there is not necessarily a maximum value
Light is perceived logarithmically by humans
High dynamic range displays and floating point color models can allow a better expression of light energy

22. Material

23. Material
Once we know how much and what direction of light we're dealing with, the material it hits impacts the final effect a great deal
These impacts are of two kinds:
Scattering
Absorption

24. Scattering Scattering is caused by an optical discontinuity
Difference in structure
Change in density
Scattering does not change the amount of light, only its direction
There are two types of scattering
Refraction (or transmission)
Reflection

25. Refraction
With refraction (or transmission) in (partially) transparent objects, the light continues to go through the object and may light other objects
There are light bending effects
Plus the Z-buffer algorithm doesn't work anymore
We won't deal with that now

26. Specular and diffuse components
Light that is reflected will have a different direction and color than light that was transmitted into the surface, partially absorbed, and scattered back out
We simplify by dividing into two terms
Specular term (the reflected light)
Diffuse term (the re-transmitted light)

27. Exitance Illumination reaching a surface is irradiance
Illumination leaving a surface is exitance (M)
Although our perception of light is logarithmic, light-matter interaction is linear:
Double the irradiance and you'll double the exitance
The ratio between exitance and irradiance is essentially the surface color that you see back
Surface color c = specular color + diffuse color

28. Modeling specular color
We will often assume that diffuse light has no directionality
Specular light, however, bounces off a surface and spreads out less if the surface is smoother
Color, texture, and the smoothness parameter are not absolute
We may change them depending on how far we are from the object

29. Sensors

30. Don’t believe your eyes
We are going to describe mathematical models of sensors
But how did humans investigate the nature of sensors in the first place?
Can you trust your own sensors?
Consider the following slide

32. Mach banding That slide is an example of Mach banding
In Mach banding, a lighter color on the edge of a darker color will appear to grow lighter as you get close to the border between them
The darker color will do the reverse
It's part of our brain's edge detection algorithm

33. Real sensors In general, sensors are made up of many tiny sensors
Rods and cones in the eye
Photodiodes attached to a CCD in a digital camera
Dye particles in traditional film
Typically, an aperture restricts the directions from which the light can come
Then, a lens focuses the light onto the sensor elements

34. Radiance
Irradiance sensors can't produce an image because they average over all directions
Lens + aperture = directionally specific
Consequently, the sensors measure radiance (L), the density of light per flow area AND incoming direction

35. Idealized sensors
In a rendering system, radiance is computed rather than measured
A radiance sample for each imaginary sensor element is made along a ray that goes through the point representing the sensor and point p, the center of projection for the perspective transform
The sample is computed by using a shading equation along the view ray v

36. Shading

37. Shading equations
After all this hoopla is done, we need a mathematical equation to say what the color (radiance) at a particular pixel is
There are many equations to use and people still do research on how to make them better
Remember, these are all rule of thumb approximations and are only distantly related to physical law

38. Lambertian shading Diffuse exitance Mdiff = cdiff  EL cos θ
Lambertian (diffuse) shading assumes that outgoing radiance is (linearly) proportional to irradiance
Because diffuse radiance is assumed to be the same in all directions, we divide by π (explained later)
Final Lambertian radiance Ldiff =

39. Upcoming

40. Next time…
Shading
Lambertian
Gouraud
Phong
Anti-aliasing

41. Reminders Keep working on Project 2, due Friday, March 17
Keep reading Chapter 5
Exam 1 is next Friday in class
Start reviewing everything up to today