Radon transform#Dual transform

Filtered back projection ( FBP also for Filtered Back Projection ) is a system based on the Radon transform method of image reconstruction, which is primarily used in computed tomography. The process has the great advantage that it is fast, as it requires little processing power. However, errors of the imaging system can not be corrected. In SPECT as in PET, it was now replaced by the iterative reconstruction process.

Example

An example illustrates the algorithm:

Imagine an illuminated front aquarium in which to map the contours of the fish at a location behind the screen. Performs one light source and screen at equal angular increments around the aquarium around, you get many projections of the fish on the screen. To determine afterwards the location of fish in the aquarium from these data, one takes each of the recorded projections and projected it on the volume of the aquarium back (hence the name of the method ). It is clear that in this case, the depth information is not considered, that is finally the deep image of the fish on the canvas is smeared in the projection direction over the image. This error, however, can be effectively suppressed by applying a suitable filter in the image reprojection.

The point spread function of the unfiltered back-projection is where the amount in the spatial domain is; This means that if the object to be imaged is only one point having the coordinates ( delta function ), then the unfiltered backprojection image obtained as a signal at the location, which is proportional to. The " filtering" mathematically corresponds to a convolution. Using the convolution theorem can be deployed by transforming the back-projected image in the Fourier space, with, the amount in Fourier space multiplied, and is then again transformed back into the spatial domain. According to the sampling theorem, the filter can be cut off at a certain spatial frequency. Also note that data from a computed tomography be discreet and not as mathematically actually needed continuously. Therefore, the CT is not " exactly " and there is not the ideal filter, as for example, averaged between the discrete points ( Shepp -Logan filter) or averaged ( Ram -Lak filter) can be. Depending on which filter to use, it is either more contrast, but noisy or less contrast, but reduced noise.