Filtered back projection

Last revised by Andrew Murphy on 23 Mar 2023

Filtered back projection is an analytic reconstruction algorithm designed to overcome the limitations of conventional back projection; it applies a convolution filter to remove blurring. It was, up until recently the primary method in cross-sectional imaging reconstruction. 

It utilizes simultaneous equations of ray sums taken at differing angles of a sine wave to compute the values of attenuation coefficients within a cross section. 

It is achieved via an algorithm of 250,000 mathematical equations that can be solved by a high capacity computer. The attenuated profile or projection produced represented by the anatomy is stored in the memory of the computer, solved and reconstructed. Each pixel (picture element) corresponds to the voxel (volume element) of the image.

Simultaneous equations are used to ascertain the correct attenuation values in a ray sum. To represent it in an easy way, imagine a box with holes in each side, and there are four letters inside that correspond to A, B on upper part and C, D on the lower part, respectively.

Then inside the box has a coin that is located on A. The coin will correspond to "1" and "0" if not.

Looking in each holes simultaneously follows:

  • A+B = 1
  • B+D = 0
  • C+D = 0
  • A+C = 1

The solution to this will tell us that A = 1, and B, C, D will be 0.

This is the mathematical equation solved by computer simultaneously; the example earlier was just a four pixel, imagine them on 250,000.

This model is sound for single objects in an area, however, as we know, the human body has many objects with differing attenuation profiles, 4 points of reference will not suffice. To overcome this, measurements are recorded at multiple angles around the box (sinogram) using the method equation as above. 


Back projection has two distinctive limitations, noise and streak artifacts. It is due to the combination of these restrictions and the advancement of computers that iterative algorithms are slowly replacing the filtered back projection method of image reconstruction.

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