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Unformatted text preview: CS221 Problem Set #3 1 CS 221, Autumn 2007 Problem Set #3: Vision and Bayes Nets Due by 9:30am on Thursday, November 15. Please see the course information page on the class website for late homework submission instructions. SCPD students can also fax their solutions to (650) 7251449 with a filled out route form 1 as the cover page. We will not accept solutions by email or courier. 1 Written part (100 points) NOTE: These questions require thought, but do not require long answers. Please try to be as concise as possible. 1. [14+3 points] Hough transforms In class, we studied the Hough transform method for efficiently finding straight lines among a large number of points. In this problem, we will extend the same ideas to look for shapes other than straight lines. (a) [7 points] Suppose we are given m points { ( X 1 ,Y 1 ) , ( X 2 ,Y 2 ) ,..., ( X m ,Y m ) } in two dimensions. We know that the data lies approximately on some small number of circles in two dimensions. Devise a Hough transform algorithm to efficiently find these circles. You can assume bounds on the location of the center of the circles, and their radii. If you are discretizing variables, you do not need to specify the number or size of the “buckets” (discretization intervals) used to discretize your variables. (b) [3 extra credit points] Show that with a perspective camera, a 3D circle will appear as an ellipse on the 2D image plane. (Note: By “3D circle” we are referring to a ring or a disc or a hoop, not a sphere. In other words, all points on a 3D circle lie in a plane and are equidistant from a point called the centre of the circle. You can assume that the 3D circle does not intersect the image plane. You can ignore the case when the projection is just a straight line segment.) (c) [7 points] Because of the result from (1b), it is often more useful to look for ellipses rather than circles in an image. For example, the rim of a mug is circular, but will look like an ellipse in an image. Devise a Hough transform algorithm to efficiently find ellipses passing through m given points { ( X 1 ,Y 1 ) , ( X 2 ,Y 2 ) ,..., ( X m ,Y m ) } in two dimensions. (Note: There are several possible ways to parametrize an ellipse in 2D analytically, and you should feel free to use whichever formulation you like best. Please feel free to ask us if you don’t know what an ellipse is.) As in part (1a), you do not need to specify the number or size of your “buckets” (discretization intervals), and you can assume known bounds on the parameters of the ellipses. 1 Available from http://scpd.stanford.edu/scpd/students/routing.htm CS221 Problem Set #3 2 2. [16 points] Stereopsis In class, we studied the formation of images in the perspective camera. In this problem, we will consider stereopsis, where we have two cameras C and C ′ and we attempt to estimate the 3D depth (or distance from the image plane) of various points from their images in the two cameras....
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 Winter '09
 KOLLER,NG
 Artificial Intelligence, Line segment, Stereopsis

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