On the Accuracy of Dense Fisheye Stereo

Author: Johannes Schneider

Year: 2016

Notes:

generative probabilistic model for stereo matching
Bayesian approach that builds a prior over the disparity space with a set of triangulated points
MAP to compute disparity on the epipolar line

Uses the equi distance model i.e. the same as us
The rectification model projects the epipolar planes to the same image row in both images
We note the ray of a pixel $[x_1, x_2, x_3]$, the rectified pixel $\mathbf{x}^$ $has the following coordinates: $$ \mathbf{x}^ = \begin{bmatrix} \text{atan2} \left ( x_1, \sqrt{x_2^2+x_3^2} \right ) \ \text{atan2} (x_2, x_3) \end{bmatrix} $$
propagation of uncertainty on rectified image with jacobians
3D point cloud using the apical angle
the apical angle (see figure 3) is linked to the disparity on the epipolar rectified image by $\gamma_{\psi} = d / c$
the depth is finally computed with the sin law and the baseline

approximates that accuracy of the pixel coordinates increases with $\phi$
build an empirical stochastic variance model with orthogonal planes
improve the stochastic variance model by: $$ \Sigma_{ll} = \hat{\sigma}_1^2 I_2 + \hat{\sigma}_2^2 diag[\phi^{2p}] $$
the observation was the normal of three planes that are orthogonals: the orthogonality can be used to confirm the stochastic variance model

for variance, use SGM and ELAS to produce dense PC on 30 stereo images and compute the 3 normal directions of the planes with RANSAC
the residuals of each inliers of the 3 planes are computed to update the covariance factors $\hat{\sigma}_1$ and $\hat{\sigma}_2$
$p$ is set to 8 as it matches better observations
the orthogonality of planes is more precise with the improved stochastic model than with the classical one
for $\phi > 40°$ the precision drops significantly
ELAS and SGM performs similarly but ELAS slighly faster and more precise