Homework - 8

In this homework, we reconstruct the 3D structure of an image using the point correspondences obtained by edge detection and NCC measure. There are a series of steps involved to reconstruct the 3D structure which will be discussed in the next section. Algorithm: Although its possible to reconstruct the 3D structure using multiple images, for the sake of simplicity, I am using two images taken using my mobile camera. (a) Estimate the fundamental matrix (F) using the correspondences inputted by the user manually. Let the correspondences be (are the elements of the fundamental matrix. We require 8 or more such correspondences to compute F. We build a matrix A with rows as the left hand elements of above expression for each user inputted correspondence. Note: i. Its required to do some normalization before doing the above step to ensure a better estimate of F for various scales of input image. We find the normalization matrices T 1 , T 2 which transform the points (x i , , x i) to (ˆ x i , ˆ x i). ii. When the points are normalized, after the computation of F, it has to be denormalized. F = T T 2 F T 1 iii. The rank of the Fundamental matrix should be 2 which can be ensured by doing an Singular Value Decomposition and then making the last eigen value zero. F = UDV T (b) Compute the epipoles (e, e) using the Fundamental matrix (F). They are the right and left null vectors of the Fundamental matrix and can be calculated using Singular Value Decomposition.