The history of mathematics in biomedical imaging illustrates how mathematicians and speciality scientists can make rapid progress when they work in teams. Most of the early work on modern medical image reconstruction was developed very slowly by individuals working independently. At first, success was stifled by the lack of mathematical input, but later on, partnerships between mathematicians and medical scientists resulted in immediate successes.

The mathematical formulations underpinning the three-dimensional image reconstruction techniques now known as X-ray computer-assisted tomography (X-ray CT, also known as CAT scan), positron emission tomography (PET), single photon emission tomography (SPECT), and magnetic resonance imaging (MRI) were laid by Johann Radon in 1917, but the Radon transform was not discovered until 60 years later. The first success in reconstruction tomography involving elegant mathematical applications was that of physicist and radioastronomer Ronald Bracewell, who in 1956 used the Fourier projection (the central slice theorem) as the basis for reconstructing the regions of microwave radiation emitted from the Sun disk. The connection between Radon's mathematics and Bracewell's early work was not made until 20 years later, in the mid-1970s. The development of medical reconstruction tomography proceeded independently of Bracewell's contributions.

Medical computed tomography began to be developed in the early 1960s and proceeded slowly because there was little mathematical input. The earliest X-ray CT demonstration was by a neurologist, William Oldendorf, who in 1961 single-handedly engineered an X-ray reconstruction of the transverse section of an object consisting of iron and aluminum nails. Although an inventive experimental study, it utilized a crude method of simple back projection. The patented invention that resulted was deemed impractical because it required lengthy analysis. Oldendorf worked without the input of a mathematician and without any knowledge of the work of Radon or Bracewell. In 1963, David Kuhl, a physician, and Roy Edwards, an engineer, invented a method of imaging radionuclide distributions. They even performed clinical studies in patients 9 years before the first patient X-ray tomogram. Since the mathematics needed for an accurate mapping had not been incorporated into their method and computer operating systems in 1963 were unable to quickly perform even simple back projection, the resolution of Kuhl's scanner was only as good as that obtained with existing methods of radionuclide imaging.

A crucial mathematical contribution to the reconstruction problem was made in 1963 and 1964 by the physicist/mathematician Allan Cormack. His contributions were directly motivated by two problems. First, medical radiotherapy treatment required the ability to determine the body's attenuation coefficient distributions so that externally applied radiation could be targeted at the tumor. Second, a mathematical algorithm was needed for reconstructing the three-dimensional distribution of radionuclide concentrations from data collected by a PET instrument developed in 1962.

Independent of the above developments, Godfrey Hounsfield, a computer engineer and industrial researcher, invented the first practical device for performing X-ray computer-assisted tomography on human beings. In 1967, oblivious of the earlier mathematics of Radon, Bracewell, or Cormack and of the instruments developed by Oldendorf and Kuhl, Hounsfield used an X-ray source and X-ray detector with a test bed for obtaining projections through a cadaver brain. Nine hours of data acquisition and two hours of computation were required to obtain a single two-dimensional plane from multiple one-dimensional profiles or projections. Hounsfield used a simple arithmetic reconstruction technique that was merely an iterative estimation method of solving a series of simultaneous equations (i.e., each equation represented a line integral of attenuation coefficients through the head). This method was entirely independent of and different from the method published 8 years earlier by Cormack.