I am reading a paper titled "Real-Time Correlative Scan Matching" which explains a scan-matching algorithm used to obtain pose-pose constraints in pose-graph Simultaneous Localization And Mapping (SLAM). The paper mentions that their "method is probabilistically motivated, making it easy to incorporate a probabilistic prior and to compute a covariance estimate."
On page 4, section III.F "Computing Covariance", the method to compute the covariance of a range of robot pose values $x_i$ is given as: Once the value of the cost function has been evaluated over a range of values of $x_i$, a multivariate Gaussian distribution $(j)$ can be fit to the data. Let $x_i^{(j)}$ be the $j$th evaluation of $x_i$:
$\sum_{K} = x_i^{(j)}(x_i^{(j)})^T \cdot p(x_i^{(j)} | x_{i-1}, u, m, z)$
$\sum_{j} u = x_i^{(j)} \cdot p(x_i^{(j)} | x_{i-1}, u, m, z)$
$\sum_{s} = p(x_i^{(j)} | x_{i-1}, u, m, z)$
$\mathbf{\Sigma}_x = (1/s)K - (1/s^2)uu^T$
I am trying to understand the following:
What is meant by "cost function". So far I think cost function means the likelihood probability distribution i.e. if one sample is $p(x_i | ...)$ then the distribution from which this sample is drawn is the cost function.
The above equations show how covariance can be computed for a robot pose $x_i$ but I need to figure out how I can get the covariance of features/landmarks from the robot's map $m$ to allow information theory-based autonomous exploration as described in the paper "Decentralized active information acquisition: Theory and application to multi-robot SLAM"
Is there a way I can use the data and theory above to achieve goal (2)?