In the last post [Link], we were able to reconstruct our image. However, the result was encouraging but a little bit disappointing. Fortunately, this can be improved by allowing the computation of Fourier transforms with higher precision.
According to the formula [Wikipedia], the 1D Discrete Fourier transform of a signal composed of N points is calculated with a spectral precision δF = ƒ_{s} / N with frequencies in the range of [-ƒ_{s}/2;+ƒ_{s}/2] where ƒ_{s} is the sampling frequency.
One way to compute the Fourier transform with a higher precision (a.k.a involving more frequencies) is to add zero values to the input data. This technique is called zero padding.
In ImageJ, the zero padding is simply done by using the Image > Adjust > Canvas Size... command and by modifying the width (choose a power of 2).
Note: Be sure to check the 'Zero Fill' checkbox.In our test image, the sinogram has a width of 256, then the next power of 2 is 512. Thus, I pad it into a larger image of size 512x180 as shown in Fig. 1. Thus, the Fourier transforms are now computed with 512 values.
Fig. 1: Padding of the sinogram |
Fig.2: 2D reconstruction from sinogram of Fig.1. The image must be cropped to its original size (here, 256 x 256). |
Fig. 3: Final 2D reconstruction |
<< Previous: Part II - 2D reconstruction
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