Computed tomography (CT) generates images of a patient's density. Data is collected by sending x-ray radiation through the patient. The ratio of transmitted to incident radiation represents the line integral of the patient's density, (since radiation travels in straight lines through the patient. CT data is subject to consistency conditions, both integral (Helgason-Ludwig) and differential (Fritz John). CT systems measure characteristic data for John's ultrahyperbolic equation

u_{x1,x1} + u_{x2,x2} = u_{y1,y1} + u_{y2,y2}

By solving the characteristic boundary value problem for John's equation we can compute unmeasured CT data within the characteristic cone from measured data on the boundary. Computing additional data give us tremendous flexibility in choice of reconstruction algorithm, (but cannot provide additional information about the imaging object). We will discuss - and hopefully improve - numerical methods for solving John's equation for third-generation and open-gantry systems. Practically, total flop count is not as important as run-time (data flow is very important!) Also, reconstruction algorithms do not require measuring all lines through the object for a mathematically exact reconstruction. Therefore, we need only compute some of the missing views accurately. Optimizing image quality and total reconstruction time is an open problem.

References:

Dym & McKean, "Fourier Series and Integrals", especially the section on the Radon transform.

"Practical Cone Beam Algorithm," by Feldkamp, Davis and Kress, JOSA vol 1, no 6 gives the nuts and bolts of what we'll probably use for our next-generation systems.

John's "The ultrahyperbolic Differential Equation with Four independent variables" Duke Math J, 1938, pp.300 - 322 gives the equation.

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