Week 11 Tutorial Solutions

Question 1

Suppose we are determining the intensity of the pixel above (The pixel is shown by the dotted line box) with a value between 0 and 1. What value will the pixel have if sampling is performed by:

Sampling in the centre of the pixel 0
Sampling at the corners and taking the average 1/4
Double sampling and taking the average 2/9
Double sampling and using the following mask to perform a weighted average
```
1/16 1/16 1/16
1/16 1/2  1/16
1/16 1/16 1/16
```
1/16 + 1/16 = 1/8

Question 2:

The image below shows an eye ray from the camera hitting a glass sphere.

Draw the reflected and transmitted rays. Use Snell's law to compute refraction angles, assuming the speed of light in the sphere is 50% of the speed in air.

The normals are shown in green. The first collision creates reflected ray r1 and transmitted ray t1. If the angle of the incoming ray to the normal is 26.6 degrees, as shown, then the angle of the reflected ray is also 26.6 degrees. The angle of the transmitted ray can be computed by Snell's law to be 12.9 degrees to the normal as shown.
The transmitted ray will branch again as it exits the sphere. The angle of incoming ray t1 to the normal n2 also happens to be 12.9 degrees (due to the symmetry of the circle) so the angle of the outgoing transmitted ray t2 is 26.6 degrees to n2.

There is also an internally reflected ray r2.

Question 3:

We can rearrange the terms in Snell's Law to get:

What happens when this value is greater than one? Under what circumstances will this occur?

This is a case of total internal reflection. All the light is reflected internally and none is transmitted. Note that it can only occur when moving from a slow medium to a faster one (eg, from water to air).
In case you are interested, the general formulae for the amount of light reflected vs transmitted is the Fresnel equation . It depends on the polarisation of the light source. A simpler approximation is Schlick's Approximation .
Question 4:

Consider a cloud represented as the array of densities below:

The cloud has uniform colour C(x,y) = (1,1,1) and transparency alpha(x,y) equal to the density values above. The sky behind the cloud is blue C = (0,0,1).

Suppose we are looking up at the cloud from below. Assume the cloud is far enough away that we can render it with an orthographic camera (parallel rays) with little distortion.

Cast six equally spaced vertical rays through the cloud, taking samples distance 1 apart. Calculate the pixel values for each ray.

What happens to the result if we take samples at twice the frequency? or half? Is this behaviour correct? How could we adapt the algortihm from lectures to make it less sensitive to such changes?
An example for one ray, as illustrated, using back-to-front compositing:

What happens to the result if we take samples at twice the frequency? or half? Is this behaviour correct? How could we adapt the algortihm from lectures to make it less sensitive to such changes?
If the same method is applied with more samples along the ray, the transparency of the cloud is reduced because the algorithm as shown does not take sample density into account. If the cells are spaced every d units and sample points on the ray are r units apart then the alpha value should be:
Assignment Discuss any issues with the assignment.
Non-examinable Questions

How would you modify Bresenham's algorthm to draw lines of arbitrary thickness?
The Wu line algorithm is a method for drawing prefiltered anti-aliased lines. It does this by drawing pixels in pairs, one above and one below the line. The intensity of the pixel is based on its vertical distance from the line. If the pixel is distance d from the line, then its intensity is (1-d).
Use this algorithm to compute anti-aliased pixel values for the line from P(3,3) to Q(10,5) from Q1.
What are the trade-offs between this and a post-filtered version of Bresenham's algorithm?

Take a look at Eric Chan's method for computing anti-aliased lines using the GPU. What tradeoffs are involved in using this method?