## 3 FRBs in the Era of SKAO/ngVLA
### Radon and Hough Transform  
Finding far-away and dim FRBs requires an algorithm that maximises the signal-to-noise ratio of the pulse by integrating the incoming dynamic spectrum along the possible quadratic dispersion curves. Such operations happen over sets of curves form a family of mathematical transformations that include, for example, the Radon transform [44] and Hough transform [45]. Recently, experimental attempts have been made to test the feasibility of such algorithms [46; 47]. A key step here is a transformation of the dynamic spectrum from a \(t-\
u\) space to \(t-\frac{1}{\
u^{2}}\) space. This transformation of the parameter space turns the dispersion framework into a linear problem where the intensity \(I\) can be characterised as \(I(d,t)\), where \(d\) is some function of the DM, and \(t\) is the time measure with a certain reference. Then, any search with a number of DM trials can be defined as a family of linear functions in the \(d-t\) parameter space, at which point finding a FRB becomes a peak detection problem using these transforms. The advantage here is that finding peaks is more robust and insensitive to any RFI that does not satisfy the \(\
u^{-2}\) dependence. Below, we briefly describe the Radon transform.  
Mathematically, the Radon transform is an integral transform that takes any function and projects it in a 2D space of lines in a plane. This output consists of the line integrals along the paths in a certain direction. Since it is a projection transform, it has found a wide variety of medical applications. Intriguingly, one can also use a Radon transform to convert the received dynamic spectrum from a radio telescope to a projection space where a dispersed FRB can be approximated as a line with some arbitrary slope. We do note that this conversion would require resampling and interpolation of the data which would also consume computational resources. These steps can be made faster with the use of fast interpolation algorithms like the nearest neighbour algorithm. A Radon transform is highly sensitive to any linear features in the parameters space. This means that one can transform the dynamic spectrum (function of time \(t\) and frequency \(\
u\)) to another parameter space (a function of time \(t\) and \(\
u^{2}\)) where any dispersed signal would be a linear function. The position and angle of that function then produce a singular peak, as shown in Figure 7. These Radon transforms can then be used to search for astrophysical signals or could be served to a neural network to classify FRBs.  
Figure 6: **(Left panel)** the radio transient phase space showing the spectral duration (product of the duration and frequency of emission) of coherent emitters compared to the specific luminosity (luminosity integrated over a finite radio band) of the coherent emission with two shaded regions that are mostly unexplored. The parallel lines denote the minimum specific luminosity for a given brightness temperature as a function of spectral duration. **(Right panel)** the fraction of recovered S/N as a function of boxcar width. The shaded region shows the typical range of boxcar widths used in conventional FRB searches.