Chirality and Reconstruction with Relative Poses

From the decomposition of the Essential matrix, up to a multiplicative factor, there are therefore 4 possible configurations (the two rotation matrices and the associated translation vectors) that can be recombined to obtain the original Essential matrix once again. To determine which decomposition is the correct one, the only method is to find the configuration that reconstructs the majority of the three-dimensional points appropriately, or more simply, the configuration that results in the majority of points having the camera coordinate $z$ being positive.

Let $\left( \mathbf{R}, \mathbf{t} \right)$ be a decomposition of the Essential matrix, and let $\mathbf{m}_1$ and $\mathbf{m}_2$ be the camera coordinates of two corresponding points. Define $\tilde { \mathbf{m}_1 } = \left( \tilde{u}_1, \tilde{v}_1, 1 \right)$ and $\tilde { \mathbf{m}_2 } = \left( \tilde{u}_2, \tilde{v}_2, 1 \right)$ such that

$$\tilde { \mathbf{m}_1 } = 1/z_1 \mathbf{m}_1 \qquad \tilde { \mathbf{m}_2 } = 1/z_2 \mathbf{m}_2$$ (9.73)

represents the normalized camera coordinates of a pair of corresponding points, that is, It seems that you have provided a placeholder for a mathematical block but did not include any specific content to translate. Please provide the text or equations you would like translated, and I will be happy to assist you!

The objective is to derive the coordinates $z_1$ and $z_2$ through which, by evaluating their positivity, it can be inferred that the corresponding points are frontal with respect to the observer, thus allowing us to deduce the correctness of the decomposition of the Essential matrix. Utilizing the formalism (9.73), the equation (9.38) becomes

$$z_2 \tilde { \mathbf{m}_2 } = z_1 \mathbf{R} \tilde { \mathbf{m}_1 } + \mathbf{t}$$ (9.74)

. By solving the equation with $z_1$ as the unknown, we ultimately obtain

$$z_1 = \dfrac{t_x - \tilde{u}_2 t_z}{ \left( \tilde{u}_2 \ma... ...f{r}_3 - \mathbf{r}_2 \right) \cdot \tilde { \mathbf{m}_1 } }$$ (9.75)

having indicated with $\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3$ the 3 rows of the matrix $\mathbf{R}$. The first equation uses the coordinate $\tilde{u}_2$ to derive $z_1$, while the second utilizes the coordinate $\tilde{v}_2$. In this way, the coordinate $\mathbf{m}_1$ is obtained, from which $\mathbf{m}_2$ can be immediately derived through equation (9.38), particularly

$$z_2 = \mathbf{r}_3 \cdot \mathbf{m}_1 + t_z$$ (9.76)

to assess the frontality of the other element of the pair.

It is worth noting that the solution to the problem could be obtained by directly solving the system (9.74) as if it were an overdetermined linear system with 2 unknowns and 3 equations (a similar approach to what has been discussed in section 9.3.1).

In both cases, an algebraic quantity is optimized, and therefore it will not be the maximum likelihood estimate of the three-dimensional point: unlike the algorithms discussed in section 9.3.1, this approach is indeed somewhat unsuitable for deriving precise world coordinates but is sufficient to verify that the choice of decomposition is correct.

It should always be noted that since the vector $\mathbf{t}$ is extracted from the Essential matrix, it is known up to a multiplicative factor; thus, the estimated points are known up to a multiplicative factor as well.

It is worth noting that this discussion is clearly generic and can be applied to the case of three-dimensional reconstruction when the relative positioning between sensors is known.