Nt1310 Unit 1 Lab Report

Figure shows the intersection of line joining the camera center and image points ${\bf x}$ and ${\bf x'}$ which will be the 3D point ${\bf X}$.\\
\end{figure}

The ‘gold standard’ reconstruction algorithm minimizes the sum of squared errors between the measured and predicted image positions of the 3D point in all views in which it is visible, i.e.\\

\begin{equation} {\bf X=\textrm{arg min} \sum_{i} ||x_i-\hat{x_i}(P_i,X)||^2}
\end{equation}

Where ${\bf x_i}$ and ${\bf \hat{x_i}(P_i,X)}$ are the measured and predicted image positions in view $i$ under the assumption that image coordinate measurement noise is Gaussian-distributed, this approach gives the maximum likelihood solution for ${\bf X}$. Hartley and Sturm [3] describe a non-iterative

The required solution for the homogeneous 3D point ${\bf X}$ can be found by different methods each method is explained in Triangulation[9], in these thesis Iterative Linear Least Square method is used, it is easy to implement and gives fairly good result.

{\bf Iterative Linear Least Square}: The idea of this method is to change the weights of the linear equations adaptively so that the weighted equations correspond to the errors in the image coordinate measurements,\\
\begin{equation}
{\bf \varepsilon = uP_3^T X-P_1^T X}
\end{equation}
What we really want to minimize is the difference between measured image coordinates value ${\bf u}$ and the projections of ${\bf X}$ which is given by
\[{\bf \frac{P_1^TX}{P_3^TX}}\]
Specially we wish to minimize
\[{\bf \varepsilon ' =\frac{\varepsilon}{P_3^TX}}\]
This means that if the equation had been weighted by the factor ${\bf \frac{1}{w}}$ where $w=P_3^T X$ then the resulting error would have been precisely what really wanted to

Therefor proceed iteratively to adapts the weights we begin by setting $w_0=w_0'=1$ by solving the system of equations a solution ${\bf X_0}$ can be found. This is the precisely the solution found by the Linear Least Square method, from the ${\bf X_0}$ computes the weights. We Repeat this process several times at $i_{th}$ step multiply matrix ${\bf A}$ for the first view by ${\bf \frac{1}{w'}}$ where ${\bf w_i=P_3^T X_{i-1}}$ using the solution ${\bf x_{i-1}}$ found in the previous iteration. Within few iterations this process will converge in which case we will have ${\bf x_i=x_{i-1}}$ and so ${\bf w_i=P_3^T X_{i}}$. The error will be ${\bf {\varepsilon}_i = u- \frac{P_1^T X_i}{P_3^T X_i} }$ which is precisely the error in image measurements in equation