Please place your write-up in the folder outside my office door (or deposit an on-line version in the course directory). Some of the problems are taken from the exercises at the end of Chapter 10 "Reflectance Map: Photometric Stereo" from Horn's Robot Vision book. I handed out copies of this chapter in class today. There is an extra copy outside my door.
1. Exercise 5.1 from Computer Vision: A Modern Approach by Forsyth and Ponce. (10 pts)
2. Exercise 10-11 from Robot Vision by B. K. P. Horn. (25 pts)
3. Exercise 10-13 from Robot Vision by B. K. P. Horn. (25 pts)
4. Maxwell's Color Triangle (15 pts)
In color matching experiments, an additive mixture of three primary lights is used to reproduce a reference color. The basic facts about the additive mixture of lights were discovered by James Clerk Maxwell around 1855. In his matching experiments he used three primaries corresponding to orangish red (R at 700 nm), bluish violet (B at 436 nm), and an intermediate green (G at 546 nm). It is possible to represent these three colors as the vertices of an equilateral triangle with white at the center. Any specific color matching experiment consists of choose weights wi >= 0 subject to w1 + w2 + w3 = 1 such that T = w1 R + w2 B + w3 G, where T is the test light.
i) Identify the set of colors that can be matched successfully under these conditions. Explain the need for negative weights in achieving certain color matches. (Hint: What is the geometry of the set of colors that can be obtained by varying the weights?).
ii) Illustrate, using the color triangle, how adding an additional amount of a primary to the test light T can produce the same effect as a negative weight.
5. Finite Dimensional Linear Models. (25 pts)
Assume that the spectral radiance E(l) and the spectral albedo r(l) can each be described as the weighted (finite) sum of known basis functions. Similarly, assume that the spectral response functions for the sensor are known. In this case the sensor output at a pixel can be written pk = sumij ei rj gijk, where pk is the output of the kth receptor, ei and rj are the coefficients of the linear models for radiance and albedo, respectively, and the known constants gijk are obtained by integrating the basis functions over l. With respect to this scenario, demonstrate how color constancy can be achieved if an image region corresponding to a pure specular highlight can be identified. (i.e. Assume that a highlight has been found and derive the equations for color constancy in this case. Describe any necessary and sufficient conditions for the existence of a solution).
5. (Extra Credit +20) Exercise 6.5 from Computer Vision: A Modern Approach by Forsyth and Ponce.