# How can I infer the cost function from Kruppa's simplified equations

The following equations are Kruppa's simplified equations used in camera autocalibration. My objective here is to infer the cost function(Error Function) from this equations, So I can minimize the cost function through Levenberg-Marquardt nonlinear optimization algorithm.

(I already did the part of optimization, and only need the cost function in mathematical form so I can implement it).

Can anybody explain how the mathematical cost function will look like in simple language?

$$W^{-1} = K K^{T}$$,

where $$W^{-1}$$ is the dual of $$IAC$$, $$K$$ is the camera intrinsics matrix

$$r,s$$ are the first, and the second singular values of $$S$$

$$u_1, u_2, u_3$$ are the column-vectors of $$U$$,

$$v_1, v_2, v_3$$ are the column-vectors of $$V$$,

$$USV = SVD(Fundemental\quad Matrix\__{ij})$$,

and $$i,j$$ are two different images from moving camera.

$$\rho_{1}/ \phi_{1} = \rho_{2}/ \phi_{2} = \rho_{2}/ \phi_{2}$$ where $$\rho$$ is the numerator of each term of the equations, and $$\phi$$ is the denominator of each term, now the equation(30) in the paper proposed that: $$\rho_{1}* \phi_{2} - \phi_{1}* \rho_{2} = 0$$ and $$\rho_{1}* \phi_{3} - \phi_{1}* \rho_{3} = 0$$ now taking the Frobenius norm $$\parallel.\parallel_{fro}$$ for the two equations yields two values, Thus the cost function is the sum of this values.