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Unformatted text preview: ECE 181b Homework 4 Two View Geometry May 5, 2006 In this homework you will explore the geometry of two views (focusing on the epipolar constraint and the fundamental matrix) using the tools of projective geometry. Henceforth we will adopt the following conventions (READ CAREFULLY): • Boldface letters indicate points or vectors. • For vectors/points in P 3 we use capital letters, Sans font (e.g. X ), for vectors/points in P 2 we use lower case letters, Sans font (e.g. x ). • For vectors/points in R 3 we use capital letters, normal font (e.g. X ), for vectors/points in R 2 we use lower case letters, normal font (e.g. x ). • All the quantities related to the second camera are identified using the superscript prime ( ). • We will assume that the world coordinate system coincides with the coordinate system of the first camera. • For sake of convenience we will adopt the coordinate system convention described at http://vision.ece.ucsb.edu/ ~ zuliani/Code/lattice.png . Caveat. To receive full credit your answers must be clearly justified. Make sure to include the relevant steps that were required to obtain the numerical answer. 1 Calculating the Fundamental Matrix In this section we will study the basic steps to build a matlab function to estimate the fundamental matrix. The final goal is to obtain an estimate of the fundamental matrix F starting from a set of point correspondences ( x i , x i ) such that for each pair: x T i F x i = 0 (1) 1 Question 1 Let: F = f 1 f 4 f 7 f 2 f 5 f 8 f 3 f 6 f 9 and let f = f 1 . . . f 9 T ∈ R 9 . Show that the epipolar constraint (1) can be written as the following inner product: a ( x i , x i ) f = 0 (2) where a ( x i , x i ) is a row vector that depends only on the coordinates of the points x i and x i . Suggestion. Write explicitly (1) and collect the terms that multiply the components of F to form the row vector a ( x i , x i ) . Recall what was done to estimate an homography... Answer 1 If we let x i = x i y i 1 T and x i = x i y i 1 T , by expanding (1) we obtain: ( x i x i ) f 1 + ( y i x i ) f 2 + x i f 3 + ( x i y i ) f 4 + ( y i y i ) f 5 + y i f 6 + x i f 7 + y i f 8 + f 9 = 0 that, in vector form can be written as: x i x i y i x i x i x i y i y i y i y i x i y i 1 f Hence we conclude that: a ( x i , x i ) = x i x i y i x i x i x i y i y i y i y i x i y i 1 Question 2 Answer the following questions, providing an adequate justification. • How many degrees of freedom has the fundamental matrix F ?...
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- Fall '08
- Yi, Fundamental matrix, Epipolar geometry, Image rectification