How to See inside a French Milkman

Letters

Vol. 19 No. 16 · 21 August 1997

From Paul Taylor

Peter Campbell’s piece on medical imaging (LRB, 31 July) expressed a wish to know more about the mathematics of tomography. The problem is to reconstruct an image of a slice through a body from a set of readings, known as projections, taken as an X-ray source and detector are routed in a circle around the slice. If we know what is in a body we can calculate what will happen to an X-ray that passes through it. The problem here is the inverse – given the X-ray data, determine what is in the body through which it passed – and is, in the language of mathematics, ill-posed: it may not admit of a unique solution. There was, however, an approach to this kind of problem developed by an Austrian mathematician called Radon, 56 years before the inception of tomography.

The theorem deals with the Fourier transform of the image: the set of elementary wave-like patterns, each defined by a frequency, an amplitude and an orientation, which if superimposed would produce the image. The theorem states that the Fourier transform of the one-dimensional projection at each position of the detector forms one line through the two-dimensional Fourier transform of the image we seek to construct. So, if we have Fourier transforms of enough projections, we will have a set of lines from which we can assemble the Fourier transform of the image we require. There is a well-known equation for deriving an image from its Fourier transform. Rejigging this using the theorem, one can derive an equation which has two parts, corresponding to a two-stage process for image reconstruction: filtering and back-projection. The second stage is easy to explain: think of each X-ray as a jet of some kind of magic ink that can pass easily through air, less easily through tissue and hardly at all through bone, and imagine that the detector is a sponge which absorbs only the ink which passes in a straight line through the slice. Back-projection is equivalent to dragging the sponge across a sheet of paper in the direction corresponding to that travelled by the X-ray, so that all the ink in the sponge is transferred smoothly to the paper. If an exposure was taken for every possible line through the slice, the pattern of ink built up on the paper would be an (admittedly pretty ropey) image of the way ink was absorbed in the slice: which bits were bone, which were tissue and so on. The first stage can be thought of as a process which produces ‘filtered’ data suitable for back-projection, by combining the Fourier transform of the projection data with a mathematical function, the definition of which is neither perspicuous nor intuitive.

The calculations involved are not exactly the stuff of mental arithmetic and the rendering of the relevant acronym as Computer Assisted Tomography perhaps understates the contribution of the computer. The OED prefers Computerised Axial Tomography but the process is now universally known as CT. A much more exciting development in medical imaging acronyms is the addition of a lowercase ‘f’ before MRI. In MRI, as in CT, stacks of two-dimensional images form a three-dimensional representation of anatomy. Such scans can now be completed at a speed which allows the charting of fluctuations over time, enabling the reconstruction of images which are no longer representations of static anatomy but which visualise anatomical function. The promise of fMRI (functional Magnetic Resonance Imaging) is that we might be able not just to look inside a French milkman, but to see what he is thinking. Or at least the impact that his mental life is having on his cerebral blood flow. Such is our curiosity about this topic that it now attracts the kind of funding Peter Campbell worries might be lacking for medical imaging research.

Paul Taylor
UCL Medical School,