Anmol Kalia

Camera Calibration

This required solving a simple system of linear equations. The first step was to build the system of equations. This was followed by solving using svd to find the appropraite matrix.

Example of code with highlighting

This is the MATLAB code I used to solve this problem.

x=zeros(size(Points_2D,1),4);
oneMat=ones(size(Points_2D,1),1);
Points_3D=[Points_3D,oneMat];
uMat=repmat(Points_2D(:,1),1,4);
vMat=repmat(Points_2D(:,2),1,4);
A=[Points_3D,x,-uMat.*Points_3D];
B=[x,Points_3D,-vMat.*Points_3D];
Mat=[A;B]
[U,S,V]=svd(Mat);
M=V(:,end);
M=reshape(M,[],3)';

Result of Part 1

The projection matrix is:

-0.4583,	0.2947,	0.0140,	-0.0040
0.0509,	0.0546,	0.5411,	0.0524
-0.1090,	-0.1783,	0.0443,	-0.5968

The total residual is: 0.0445
The estimated location of camera is: -1.5127, -2.3517, 0.2826

Fundamental Matrix Estimaton

This was again solving a set of linear equations and then using SVD

Result of Part 2

-0.0000	0.0000	-0.0019
0.0000	0.0000	0.0172
-0.0009	-0.0264	0.9995

RANSAC and Fundamental Matrix Estimation

In order to accomplish this I ran 1000 iterations of RANSAC to find the best possible fundamental matrix by using 9 randomly chosed points to build a fundamental matrix and the finding the fundamental matrix with the maximum number of inliers.

I also used normlized points to get a better fundamental matrix and thus, better epipolar lines.

Results in a table

The first image in each image set is the image with non normalized points. The second image has normalized points.

Mount Rushmore

Notre Dame

Episcopal Gaudi

Name	Features in image 1	Features in image 2	Matches	Inliers	Inliers with Normalization	threshold
Mount Rushmore	5583	5249	835	124	282	0.01
Notre Dame	768	1633	266	266	180	0.1
Episcopal Gaudi	1289	1035	126	96	42	0.1

Conclusions