Feldkamp-Davis-Kress method (FDK) is an extension of the Filtered back-projection method to cone-beam geometry also known as the Radon Transform used in the conventional Fan Beam CT. Here we first show the workings of the Radon transform for 2D in the form of a convolution and then projection. When imaging an object made up of different materials, each of which have different attenuation coefficient ($\mu$), x-rays ($I$) are sent from a source parallel to a detector, such that the x-ray passes through the object and lands on the detector ( the resulting beam intensity is $I_0$). Each x-ray beam attenuates a certain amount depending on how much energy was aborted by the object. As shown in figure $$I=I_0exp(-\int_{path} \mu(x)dx) => \int_{path}\mu(x)dx = -ln(\frac{I}{I_0}) $$ The Radon transform of an image represented by the function $\mu(x,y)$ can be defined as a series of line integrals through $\mu(x,y)$ at different offsets from the origin. Which is mathematically defined as follows: $$R(r,\theta)=\iint_{-\infty}^{\infty} \mu(x,y) \delta(x cos\theta + y sin\theta-r) dx dy $$ Where $\theta$ is the angle of the line, and r is the perpendicular offset of the line. The resulting Radon transform is called a sinogram. In order to reconstruct the image from the sinogram, the inverse Radon transform is to sinogram. There are several methods to compute the inverse Radon transform but the one of interest in this study is the Filtered Back Projection method. This method is split into two phases, the projection and the filtration on the image. The projection phase is very similar to the Radon transform described above, except now the path integral are projected back onto a plane at their respective angles. This back-projection phase takes the form: $$f(x,y)=\int^\pi_0 R(x cos(\theta + y sin\theta, \theta)$$

Although the shape of the object is reconstructed, the reconstructed image is very noisy. To correct for this noise a high pass and Hanning filter is applied to the sinogram data in the frequency domain. Applying these filters significantly improves the quality of the reconstructed image. The FDK Methods extends the Radon transform to conical projections, and the transform can be given by the following: $$f_{FDK}(x,y,z)=\int^{2\pi}_0 \frac{R^2}{U(x,y,\beta)^2} \mu^F (\beta,a(x,y,\beta),b(x,y,z,\beta))\delta\beta$$ The detector position is given by: $$a(x,y,\beta)=R \frac{-x sin \beta +y cos \beta}{r + x cos \beta + y sin \beta}$$ $b(x,y,z,\beta)$ is given by: $$b(x,y,z,\beta)=z \frac{\beta}{R + x cos \beta + y sin \ beta}$$ The function $U(x,y,\beta)$ is the function of the distance between the source and the line parallel with the detector and is given by: $$U(x,y,\beta)=R+x cos\beta+y sin\beta$$