Trying to implement a loss function read from a journal-article in python

Computer science undergrad here. I am trying to understand Eqn 12 from this paper so that I can implement it in python code. In this paper, the NN model takes a blurred image as input and outputs a sharp (deblurred) image and the kernel that can produce the same blurred image after multiplying with the sharp image.

Here -

• $\widetilde{K_t}$ = kernel predicted matrix
• $K_t^{train}$ = ground truth kernel (for training) matrix
• $\widetilde{X_t}$ = predicted sharp image matrix
• $X_t^{train}$ = actual sharp image matrix

The equation here is as follows:

This equation takes matrice $\widetilde{K_t}$, $K_t^{train}$, $\widetilde{X_t}$,$X_t^{train}$ and a list of weights as input and tries to find the weighted mean squared error loss $\widetilde{K_t}$ and $K_t^{train}$ and between $\widetilde{X_t}$ and $X_t^{train}$. This is what I can understand, probably I am wrong.

But the part that I don't understand is where $K_t$ and $X_t$ are taking inputs like this $\widetilde{K_t}(\{w^l, b^l, ...\})$ and $\widetilde{X_t}(\{w^l, b^l, ...\}, \eta)$. Here $w^l$, $b^l$, etc are weights with which $\widetilde{K_t}$ and $\widetilde{X_t}$ are to be multiplied with (I think).

Also, I don't understand where $K_t^{train}$ and $X_t^{train}$ are being passed through a translational operator, which is described as

Tτ {·} is the translation operator in 2D that performs a shift by $\ τ ∈ R^2$

Does this mean, $X_t^{train} = X_t^{train} + (X_t^{train})^{2}$ ?

Also what is the difference between $\tau_{-\tau}$ and the $\tau_{\tau}$ operator used in equation 12.

Let me rephrase all my questions regarding the equation:

1. If $\widetilde{X_t}$ and $\widetilde{K_t}$ are matrices, how can they take input? Moreover, what do the curly braces ($\{ \}$) mean? For example here : $\widetilde{X_t}(\{w^l, b^l, ...\}, \eta)$?
2. What do the operators $\tau_{-\tau}\{.\}$ and the $\tau_{\tau}\{.\}$ do to any matrix and what is their difference?
3. I want to be able to understand the equation enough so that I can implement this in python

Thank you so much for your time.