Analytic 2D Reconstruction Properties of the Xray Transform

In this section, we solve the inverse 2D x-ray transform. A closed-form solution of the integral equation, Eq. (3) is first derived assuming a continuous sampling of the sinogram variables over (s, 0) e [—Rp,Rp ] ^ [0, n]. An approximation to this exact solution will then be written in terms of the discrete data samples (defined by Eq. (5)), leading to the standard filtered-backprojection algorithm (FBP). We refer for this section to the comprehensive books by Natterer [2,4], Kak and Slaney [5], Barrett and Swindell [6], and Barrett and Myers [7]. First, two properties of Eq. (3) should be stressed:

• The problem is invariant for translations in the sense that the x-ray transform of a translated image ft(x,y) = f(x - tx,y - ty) is (Xft)(s, 0) = (Xf)(s - tx cos 0 - ty sin 0, 0). Translating the image simply shifts each sino-gram row.

• The problem is invariant for rotations in the sense that the x-ray transform of a rotated image fg(x, y) = f(x cos 0 - y sin Q, x sin 0 + y cos 0) is (Xf0)(s, 0) = (Xf)(s, 0+ 0).

These two invariances, and also the algorithms described in the next sections, are valid only when the scanner measures all line integrals crossing the support of the image (the disc of radius RF), so that the sinogram is sampled over the complete range (s, 0) e [- Rf, Rf] x [0, n]. When this condition is not satisfied, the problem is called an incomplete data problem (among many references, see [2] Ch. VI, [4, 8, 9]). This happens in particular with hexagonal or octagonal scanners such as the Siemens/CPS HHRT, where the gaps between adjacent flat panel detectors cause unmeasured diagonal bands in the sinogram [10]. Before applying the FBP algorithm presented below, the incompletely measured sinograms must first be com pleted by estimating the missing LOR data. When the gaps in the sinogram are not too wide, simple interpolation can be used, but more sophisticated techniques have been proposed [11,12]. An alternative is to apply iterative reconstruction techniques, which are less sensitive to the specific geometry. We note, however, that the use of iterative methods does not provide a solution for the missing data problem. Rather it simplifies the introduction of prior knowledge which can partially compensate for the missing data.

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