Say you perform a dual-energy x-ray CT scan using a high and low energy spectrum on three different materials. Since you're using two different energy levels, there will be two sinograms. But how would you reconstruct the main image of some $3$-material phantom from two separate sinograms?
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$\begingroup$ If the answer is really unknown then the corresponding patent will be worth a lot more than a 100 points of reputation here. $\endgroup$– my2ctsCommented Jun 22, 2018 at 18:20
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For a scan along a particular line we have $$\int_{x_0}^{x_1} A_i(x)dx = \ln\frac{I_{1,i}}{I_{0,i}},$$ where $A_i(x)$, $I_{0,i}$, and $I_{1,i}$ are the attenuation coefficient, initial intensity, and final intensity at energy $E_i$, respectively, and $i=1,2$. (This is Beer's law.) We will roughly have $A_2=\alpha A_1$, where $\alpha$ is some dimensionless constant relating the attenuation coefficients at energies $E_1$ and $E_2$. Thus, we should be able to pool the data consisting of $\alpha\ln(I_{1,1}/I_{0,1})$ and $\ln(I_{1,2}/I_{0,2})$. Then we perform an inverse Radon transform on the pooled data. One should be able to estimate $\alpha$ from the data.
Another grossly simple method that I would apply first would be to use an image processing algorithm to combine the inverse Radon transforms of the two sinograms. (Roughly, this assumes $\alpha = 1$.)
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$\begingroup$ How would you pool the data into one image for reconstruction? And what image processing algorithm combines the inverse Randon transform of two sinograms? $\endgroup$– Oliver GCommented Jun 25, 2018 at 11:34
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$\begingroup$ @OliverG: (1) Use the two discrete sets of data to create a "fitted" sinogram. Apply the inverse Radon transform to this. (2) I am no expert on image processing but I believe that GIMP, for example, can be used to merge images. $\endgroup$ Commented Jun 25, 2018 at 14:56
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$\begingroup$ Do you know of any specific ways to create a "fitted" sinogram from two sinograms? I would like to test out some methods. $\endgroup$– Oliver GCommented Jun 25, 2018 at 18:52
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$\begingroup$ @OliverG: How you implement this depends on the data and the computational power available. If the data set isn't too large I would first try to fit the data from the two sinograms to a high degree polynomial. $\endgroup$ Commented Jun 25, 2018 at 21:12
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1$\begingroup$ @OliverG: However, it appears that the images have the same $t$s and $\theta$s, which simplifies things. Suppose $R_{1ij}$ and $R_{2ij}$ represent the $ij$th elements of sinograms 1 and 2, respectively. Then $\alpha = \sum_{ij}R_{2ij}/\sum_{ij}R_{1ij}$. Let $R_{ij} = (\alpha R_{1ij} + R_{2ij})/2$. Apply the inverse Radon transform to $R$. You may also wish to weight one sinogram relative to the other, in which case you might try $R'_{ij} = (1-p)\alpha R_{1ij} + p R_{2ij}$, where $0\le p\le 1$. $\endgroup$ Commented Jun 26, 2018 at 16:17