# Central and Periodic Multi-Scale Discrete Radon Transforms
## 1 Introduction
### Conventional Discrete Radon Transform
Additionally, the shape of the output of 4 quadrants, joined together within a \(3N\times 4N\) rectangular memory footprint, where half of the space is wasted. interpolation will be required. In this basic quadrant, the lines to be considered are the discrete versions of lines of the form \(y=s\cdot x+d\), where \(s\) denotes the slope or ascent and \(d\) denotes the \(d\) displacement or intercept. The variety of names is due to the adoption of the algebraic slope-intercept formulation of the lines and, simultaneously, the adherence to previous authors' notations. Actually, in the context of discrete lines, it is more precise to denominate ascent, the parameter normally denoted as \(s\). Anyway, from that \(s\) parameter, the slope could be calculated as \(\frac{s}{N-1}\) and the angle of projection, relative to the positive direction of the \(x\)-axis, as \(\tan^{-1}(\frac{s}{N-1})\). From now on, the parameters on the Radon domain will be denoted as \(s\) and \(d\) and called slope and displacement. The slopes to be evaluated are those that generate ascents varying from 0 to \(N-1\) pixels on the vertical axis, whilst traversing \(N\) pixels on the horizontal axis. For each slope, there will be as much as \(2N-1\) displacements to be calculated. In case of the slope being maximum, all the \(2N-1\) displacements are different from zero, whilst, when slope is minimal, only \(N\) displacements will really touch the image. This can be better appreciated in Figure 2.  
These new algorithms have a goal to suppress the computations for the output at marginal displacements and, by doing so, the shape of the new DRTs output for the 4 quadrants will become a \(N\times 4N\) fully populated rectangle. In other words, the number of displacements considered for a single quadrant will be always \(N\), independent of the slope. Two alternative algorithms, with different uses and properties that overcome the aforementioned problem, have been developed, while still remaining of linearithmic complexity. The equivalent shearing for different slopes, shown in Figure 1, for conventional DRT is now shown for central DRT in Figure 3. The idea behind central DRT is to exclusively compute the \(N\) displacements around the projection of the center of the image. Therefore, the output of central DRT should be coincident with a portion (the central band on a light gray color on Figure 2 surrounded by yellow) of the output of the conventional DRT (surrounded by red). It can be seen that most of the input, when sheared and truncated to the central N displacements, will still be considered. In Figure 2, the values inside the red area, but outside the yellow area, those that are going to be discarded, are less than a 15% of the total. It will be demonstrated that this can be achieved with a saving of almost half of the computations and, in certain scenarios, this central band can substitute the whole output of the conventional DRT. The proposed transform will save computations in the  
Figure 2: Number of pixels to be summed together at different displacements, when calculating DRT for \(s=0\) (left) and \(s=N-1\) (middle). The number of pixels evaluated for the rest of slopes and displacements of two quadrants of a \(8\times 8\) image are shown on the right. forward transform; although the backward transform exists and converges, it will be much slower than the backward transform of conventional DRT, becoming practically unusable.