Project 3

3-1: Camera Projection Matrix

Projection matrix M has 9 variables but combining intrinsic and extrinsic matrices leads to 8 DoF. I set the last parameter of M to 1 and solve for M in an iterative code below: 

nps = size(Points_2D,1);
A = zeros(nps*2,11);
A(1:2:nps*2,1:3) = Points_3D;
A(1:2:nps*2,4) = 1;
A(1:2:nps*2,9:11) = -Points_3D.*repmat(Points_2D(:,1),[1,3]);
A(2:2:nps*2,5:7) = Points_3D;
A(2:2:nps*2,8) = 1;
A(2:2:nps*2,9:11) = -Points_3D.*repmat(Points_2D(:,2),[1,3]);
b = zeros(nps*2,1);
b(1:2:nps*2) = Points_2D(:,1);
b(2:2:nps*2) = Points_2D(:,2);
M = A\b;
M = [M;1];
M = reshape(M,[],3)';
Above code results in: 

>> proj3_part1

  The projection matrix is:

  -2.0466    1.1874    0.3889  243.7330

  -0.4569   -0.3020    2.1472  165.9325

  -0.0022   -0.0011    0.0006    1.0000

The total residual is: <15.6217>
The estimated location of camera is: <303.0967, 307.1842, 30.4223>
Given in the project by the following relation: "M_norm_A = -0.5970* M" the center is: 

Center =

   -1.5126   -2.3517    0.2827
Figures below show the projections and center of the camera respctively.


3-2: Fundamental Matrix Estimation (Normalization for extra credit)

This subsection is the same as part 1, but I ran SVM twice and set the smallest singular value to zero to calculate the fundemetal matrix. Epipolar lines are shown for the two images given 20 points.

Without normalization 1

Without normalization 2
I also opt for the normalization using average least square. IT perfects the results as truely guessed. And finaly the results with the normalization:

 


With normalization 1

With normalization 2

Part 3-3: Fundamental Matrix with RANSAC (Normalization is included)

I uses VL_feat SIFT to find 800+ interest point. RANSAC will lead to calculate the best option or solution for fundamental matrix. I used 8 points in RANSAC. Figure below show the results in Mount rushmore.

Image 1 without normalization

And here is the correspondence:


Corespondence without normalization
Same results with the normalization:

Image 1 with normalization

Image 2 with normalization

Correspondence with normalization