A problem in computer vision and 3d reconstruction is getting the camera's intrinsics parameters. A common solution is to use an object in which one knows the measurements of the shape before hand, such as a checker board. The issue with this method is that it must be done every time the camera parameter's change such as focal length and magnification.
I'm trying to implement the camera self-calibration discussed in A Simple Technique for Self-Calibration. The essential matrix is constrained by its two singular values. This can be used to recover the camera's intrinsics without doing a manual calibration (i.e., with a checkerboard). I am a little confused by how the cost function can be minimized. Here's what I understand so far:
essential matrix $$E=K_2^TFK_1$$
intrinsic matrix $$K=\begin{bmatrix}\alpha_x & s & u_0 \\ 0 & \epsilon\alpha_x & v_0 \\ 0 & 0 & 1\end{bmatrix}$$
- $\alpha_x$ product of focal length and magnification factor [solve]
- $\epsilon$ aspect ratio [assume provided, I guess from camera or EXIF data?]
- $u_0 v_0$ are the coordinates of the principal point [assume 0, 0]
- $s$ skew [assume 0]
cost function $$C(K_i,i=1..n)=\sum_{ij}^n(\sigma1_{ij}-\sigma2_{ij})/\sigma1_{ij}$$ the $\sigma$s are the singular values of $K_j^TF_{ij}K_j$
Question: How is this cost function being minimized?