Please place your write-up in the folder outside my office door. 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. Extra copies are available from the TA.
Grading: This is the undergraduate level version. The answers from the students in 4495 will determine the mean for this assignment track. Any students from 7495 who do this version of the assignment will be graded on a much tougher curve (i.e. we will assume that you are going to get everything right). Each question below is worth 3% of the course grade, for a total of 15%.
1. Exercise 5.5 from Computer Vision: A Modern Approach by Forsyth and Ponce.
A few points about Gaussian spheres may be useful for the next problem. We have discussed this sphere implicitly in class without defining it. The Gaussian sphere is a 3-D sphere of radius 1. It gives a unique parameterization of surface normals. Each possible normal vector for a surface patch is a point on the Gaussian sphere (since that point is defined by a unit vector with two degrees-of-freedom of orientation). The geometry of Lambert's Law is simpler on the Gaussian sphere than it is in the (p,q) gradient space. This is illustrated in Figure 10-24 in the handout from Horn's book and is explored in the question below.
2. Exercise 10-7 from Robot Vision by B. K. P. Horn. Note that while Figure 10-24 gives the geometry of the situation, you are being asked to derive equations for points on the Gaussian sphere using the definition of a Lambertian surface.
3. Exercise 10-11 from Robot Vision by B. K. P. Horn.
4. Greg Thoenen has written a java applet for a color matching game using the RGB color space. This game is analogous to the color matching experiments we discussed in class. Your goal is to find the right mixture of R, G, B primaries to reproduce the test color. A test color is chosen at random each time you load the page.
i) Play the color matching game. Once you have obtained a successful match, print out the screen and turn it in with your solutions. The Prnt Scrn key under Windows will capture the entire display to the clipboard. You can paste it into any imaging program such as Paint (Start->Programs->Accessories->Paint under Windows 2000) and print it. There are also many free screen capture programs from sites such as AnalogX.
ii) Describe any difficulties you encountered during matching. What strategy did you use to obtain a correct match? Can you interpret your strategy in terms of a particular way of searching in RGB?
5. Finite dimensional linear color models describe the color imaging process as the integration of smooth signals formed by a finite set of basis functions over l.
i) In class we discussed several scenarios (e.g. specular highlights, average reflectance) in which simplified versions of this model could be used to solve the color constancy problem. Characterize the general situation for color constancy with respect to this model. What has to be known? What has to be true for there to be a unique solution for color constancy?
ii) Suppose the scene contains a surface patch whose reflectance function is known to be linear with a unit slope. For simplicity we consider a 1-D scanline through the image. Let x denote the coordinate along the scanline. We assume that the scanline intersects the surface patch between known coordinates xs and xe. The reflectance function has the form r(l,x) = [Sjfj(l)](x - xs) + Sjbjfj(l) for all pixels xs < x < xe inside the surface patch, where the bj are the unknown offsets of the linear reflectance function. The patch with linear reflectance uses the same finite basis functions fj(l) as the rest of the surfaces in the scene. As before, we assume that the constants gijk are known, and that they encapsulate all of the available information about the basis functions (i.e. do not assume that you have access to the fj(l) directly). Describe a color constancy algorithm for this situation.
6. (Extra Credit +3%) Exercise 10-13 from Robot Vision by B. K. P. Horn.