## CSCE 5013 - Homework 6

Due Date - December 8, 2011 at 5:00 PM

### 1. Problem Statement:

The goal of this assignment is to gain an understanding of the
mathematical framework for stereo vision. In particular, your task
is to work out the equations for calculating the depth of objects
in the scene given different camera setups. Since this is a written
assignment you will NOT be required to implement or test these equations.

### 2. Two camera stereo:

To start with, draw a diagram to show how two
pin-hole cameras pointing in the same direction would view a scene.
The first camera has its focal point at location (0,0,0) and the second
camera is located at (d,0,0). They are both pointing in the direction
of the positive z-axis with their image planes at z = -f.

- Assume that the point (a,b,c) is visible in both cameras. Where
will this point be projected in both cameras? What is the disparity?
- Assume that the pixels (x1,y1) in camera1 and (x2,y2) in camera2
correspond to the point in the scene. What assumption can we make
about y1 and y2? What is the formula to calculate the z coordinate of
this point?

### 3. Moving camera stereo:

Draw a diagram to illustrate a pin-hole camera moving back and forth
along a rod taking multiple images of a scene. The initial position of the
focal point is (0,0,0). As the camera moves along the x-axis at uniform
speed it takes N pictures when the focal point is at (0,0,0), (d,0,0),
(2d,0,0) ... (Nd,0,0). As before, the camera is pointing in the direction
of the positive z-axis with the image plane is z = -f.

- Assume that the point (a,b,c) is visible in all images.
What is the disparity between images taken at (0,0,0) and (d,0,0)?
What is the disparity between images taken at (d,0,0) and (2d,0,0)?
Finally what is the disparity between images taken at (0,0,0) and (Nd,0,0)?
- Based on the information above, what is the relationship
between the distance between camera positions and how easy/hard it
is to search for corresponding points in a pair of images?
- Describe how you would combine information from 3 or more images
to obtain better depth estimates than the basic 2 camera setup.
Be specific about the formula you would use to calculate the z coordinate
of a point that appears at known locations in each of your images.
For example, pixel (x1,y) in image1, (x2,y) in image2, and (x3,y) in image3.

### 4. Project Submission:

There is no need to submit this project online. Instead bring a hard
copy to class or put it in my mailbox in the department office.