# Inverse Multiscale Discrete Radon Transform by Filtered Backprojection
## 2 Proposed Method  
### Extending Backprojection  
It must be emphasized that the input data to our inverse transform will be the unaltered data that the conventional direct DRT provides. This data will be modified as little as possible: in fact, it would be more accurate, instead of using the term modify, to talk about rearranging those data to make them fit into a DTR four times larger.  
We advanced previously that our goal is to make the backprojected lines grow up to three times the size of the input image. Therefore, any point in the original domain will have, in the extended domain, at least \((N-1)\) pixels of distance in straight line up to the new borders -being N the number of pixels of the input image-. This condition will make it possible to later deconvolve using 2D kernels of size \((2\cdot N-1)\times(2\cdot N-1)\).  
Although our goal is to have at the lines backprojected to at least three times the initial size, actually we will make them grow to four times their original length. Because the algorithm works bitwise, we will grow by two bits the size of the image to keep it a power of two (that is 4\(\times\) bigger per dimension), then we will simply ignore \(N\) elements at each dimension (becoming 3\(\times\) bigger per dimension), as shown in Figure 3.  
The initial domain of the image is shown in the inner square in Figure 3. Four lines of slope 0, 1, 2 and 3 are drawn through the upper right corner of said square. Marked in green are the points that intersect the left edge of the original image-square. Two squares that are three and four times bigger than the original, are also drawn. Marked in blue are the intersection of the same lines of slopes 0 to 3 in the left edge of these squares.  
To the right, top and bottom, are depicted the relative positions of transformed data of slopes s = 0 to 3. Green and blue determining if they pertain to the original or the extended domain, respectively. In the original DRT the lines with consecutive values of \(s\) are also deployed at consecutive values of decreasing \(d\). In the case of having a DRT of side four times bigger, these data would not belong to adjacent slopes, but multiples of four, and the intersections would not happen adjacently neither, but they would fall into multiples of two.  
Figure 2: Depiction of extension of backprojected lines to a domain three times larger than input image size. See text for details.  
This is the basic idea behind the extension of a DRT to four times its original size. It is not necessary to modify the direct transform nor its backprojection algorithms, but to reorder the data. The formula that allows that a backprojected point leaves exactly the same footprint inside the original domain, and as symmetrical as possible outside of it, is slightly more complex than what was described in the introduction as the basic idea, in order to deal with the annoyances of a three-fold extension in a bitwise algorithm. More details on Section 2.2.1. The actual reordering formula for each quadrant being:  
\[\mathfrak{R}_{4}f(4N+d-2\cdot(s+(s\%2)),\ 4\cdot s+3\cdot(s\%2))\leftarrow \mathfrak{R}f(d,s)\]  
\(s\%2\) represents the remainder of the integer division between the slope and \(2\), that is the least significant bit of the slope: \(s_{0}\). The reordering of the data has an \(O(N)\) complexity and an example can be observed in Figure 4.