# Inverse Multiscale Discrete Radon Transform by Filtered Backprojection
## 2 Proposed Method
### Characterization of the Impulse Responses
#### 2.2.1 Time Variant Impulse Responses
We will analyze the case \(N=8\) for simplicity, and therefore the boundaries will be: \(s_{2},s_{1},s_{2},s_{0},s_{2},s_{1},s_{2}\). It is shown in Figure 6 how the lines of slopes 0 to 7 are distributed by height for three possible bit boundaries as they travel to their right. Those bit boundaries would be the ones that would cross the lines depending if the origin point was at position 0, 1 or 2, in a problem of size \(N=8\). In the case depicted at the bottom--corresponding to starting from position 0--the first bit boundary, \(s_{2}\), breaks the group composed of the lines of slopes 0 to 7 into two groups: the lines with slopes 0 to 3 which do not ascend at that boundary, and the lines of slopes 4 to 7, that ascend because they have a 1 in the bit 2 of their slopes. The rest of the diagram can be interpreted in the same way. It can be seen that if we join the cells that contain each number from 0 to 7 with line segments we would obtain the 7 discrete lines of the DRT:  
Figure 5: Bits boundaries from position \(N/2\) to \(2N+N/2\) in the \(4N\) extended domain. lines that ascend in a 'broken' way but in an orderly fashion and that finish after traveling N positions to the right, having ascended what its slope determines: 0 to 7. The shape of the impulse response will be determined by the number of lines that occupy each height at a certain distance, in this case, moving away from the starting point to the right (separated by square brackets) and ascending from height 0 (separated by commas inside the square brackets): \([8]\), \([4,4]\), \([2,4,2]\), \([2,2,2,2]\), \([1,2,2,2,1]\), \([1,2,1,1,2,1]\), \([1,1,1,2,1,1]\), \([1,1,1,1,1,1,1,1]\). For each distance \(u\), we have \(u\) values that add to N which are palindromes. Considering the intermediate case, which simulates an impulse at position 1 (extended N+1), is equivalent to thinking of the boundaries for case 0 shifting to the left, with the disappearance of a \(s_{2}\) and the appearance of a \(s_{0}\) from the right. The impulse response will change with respect to the previous starting position, and we obtain: \([8]\), \([4,4]\), \([2,4,2]\), \([1,3,3,1]\), \([1,2,2,2,1]\), \([1,1,2,2,1,1]\), \([1,1,1,2,1,1,1]\), \([1,0,2,1,1,2,0,1]\). However, in the next position, 2 (extended \(N+2\)), although the bit boundaries change with respect to both cases 0 and 1, the impulse response is the same as in case 0. This is because the roles of boundaries \(s_{0}\) and \(s_{1}\) are interchanged with respect to that case, but given that the path that each line follows does not matter but the number of lines crossing at each height, the final result in terms of impulse response is the same. With the proposed adaptation from original slopes to extended ones, which replicates the two least significant bits of the slopes, we have managed to make the bits \(\tilde{s}_{1}\) and \(\tilde{s}_{0}\) equal to \(s_{0}\), and therefore a periodicity appears after \(N/4\) displacements of the impulse starting position. After N/4 shifts, the bit boundaries repeat, only that swapping the roles of bits \(s_{0}\) and \(s_{1}\). Therefore, the number of different impulse responses that we need to study will be \(N/4\). They will have size \((2\cdot N-1)\times(2\cdot N-1)\), in 2 dimensions: this is, the central value will be surrounded by \(N-1\) values on each side; and characterize half the impulse response (only in one dimension). They form a kind of 'butterfly' which is different from zero only in the positions closer to the horizontal axis than it is to the vertical one. Examples of this sort of 'butterflies' will be shown later. Figure 6: Depiction of how slopes ascend for different bits boundaries, for size \(N=8\). The total impulse response at position \((x,y)\) will be the sum of the horizontal half-impulse response \(x\%(N/4)\) with the horizontal transposed half-impulse response \(y\%(N/4)\), see Figure 7, with (%) representing the remainder of integer division.