lets assume we have the pixel value of some interest points as well as their covariance matrices that defined as follows: $$ \mathbf x^{\prime} = \begin{bmatrix} x^{\prime}\\ y^{\prime}\\ \end{bmatrix} , \mathbf \Sigma_{x\prime x\prime} = \begin{bmatrix} \sigma^2_{x^{\prime}} & \sigma_{x^{\prime} y^{\prime}}\\ \sigma_{x^{\prime} y^{\prime}} & \sigma^2_{y^{\prime}} \\ \end{bmatrix} $$ after that, pixel points convert to homogenous coordinate from Euclidean coordinate and are projected to the normalised camera coordinate by $K^{-1}$, where K represents the camera matrix.
$$
\mathbf x = K^{-1} \mathbf x^{\prime} = \begin{bmatrix}
x\\
y\\
1\\
\end{bmatrix}
$$
in the next step, we compute the covariance matrices for each new $\mathbf x$ point.
$$
\mathbf \Sigma_{\mathbf x \mathbf x} = J_{\pi} \mathbf \Sigma_{x^{\prime} x^{\prime}} J_{\pi}^{T} =
\begin{bmatrix}
\sigma^2_{x} & \sigma_{x y} & 0\\
\sigma_{x y} & \sigma^2_{y} & 0\\
0 & 0 & 0\\
\end{bmatrix}
$$
where $J_{\pi}$ is the jacobian of forward projection.
We also know the rank of matrix $\mathbf \Sigma_{\mathbf x \mathbf x}$ is 2 and it's singular and invertable.
Next, we compute the normalised point $\mathbf x$ and we called it $\mathbf v$.
$$
\mathbf v =
\begin{bmatrix}
v_x\\
v_y\\
v_z\\
\end{bmatrix} = \frac {\mathbf x}{\Vert \mathbf x \Vert},
\mathbf \Sigma_{\mathbf v \mathbf v} =
\begin{bmatrix}
\sigma^2_{v_x} & \sigma_{v_{xv}} & \sigma_{v_{xz}} \\
\sigma_{v_{xz}} & \sigma^2_{v_y} & \sigma_{v_{yz}} \\
\sigma_{v_{zx}} & \sigma_{v_{zy}} & \sigma^2_{v_z}\\
\end{bmatrix}
$$
where
$$
\mathbf \Sigma_{\mathbf v \mathbf v} = J \mathbf \Sigma_{\mathbf x \mathbf x} J^{T}, J = \frac{1}{\Vert \mathbf x \Vert}(\mathbf I_{3} - \mathbf v \mathbf v^{T})
$$
We also know the $\mathbf \Sigma_{\mathbf v \mathbf v}$ remains invertable. From now, I can compute the rest procedure and update $\mathbf \Sigma_{\mathbf v \mathbf v}$.
My main question is, after updating the $\mathbf \Sigma_{\mathbf v \mathbf v}$, how can I update the $\mathbf \Sigma_{\mathbf x \mathbf x}$ and $\mathbf \Sigma_{\mathbf x^{\prime} \mathbf x^{\prime}}$ sequentially. In fact, the main difficult part of this job is extracting the $\mathbf \Sigma_{\mathbf x \mathbf x}$ from updated $\mathbf \Sigma_{\mathbf v \mathbf v}$ while we should keep the rank of $\mathbf \Sigma_{\mathbf x \mathbf x}$, 2 and set the last column and row of that $0$ (similar to formula).
It is noticeable, all equations, which I mentioned here, come from MLPnP reference and here, I just summarised the equations 2-6.