5.4 Reconstruction

Theory of filtered backprojection algorithm (fbp)

The FBP algorithm allows us to take the projections, PӨ(t), developed in the previous sections andreconstruct the original image, f(x,y).

A key idea is the Fourier Slice Theorem. It says that the Fourier Transform of a projection at an angle thetais equivalent to the values of the 2-dimensional Fourier Transform of the image evaluated along a radial line of the same angle.Knowing this fact, we are able to obtain the Fourier Transform of the image, F(u,v), from projections taken at multipleangles.

We start with 2-dimensional Inverse Fourier Transform:

Since we have projections for given angles, a change to polar coordinates is useful.

Using symmetry, this simplifies to:

Using the Fourier Slice Theorem, we substitute in the Fourier Transform of the projection,SӨ(ω).

With this formula, we are now able to reconstruct the original image. We now see that that the FPBalgorithm has certain benefits. We can begin reconstructing the image after the first projection has been calculated, since theimage is built up by summing over all the angles. This could increase speed and practicality for real time applications.