Math 2.0

Integral transforms, operational calculus Radon transform and tomography Problem

Tomographic reconstruction

Builds on Radon transformInverse problemWell-posed problem

Problem

Let ff be an unknown function on ℝn\R^n, n≥2n \ge 2, with compact support. Given the Radon transform RfR f, or the X-ray transform of ff, on a set of lines or hyperplanes, possibly with measurement errors, recover ff. In practice the data are values Rf(θj,sk)R f(\theta_j, s_k) at finitely many directions θj\theta_j and offsets sks_k. The task is then to compute a good approximation of ff.

Remarks

With complete and exact data in ℝn\R^n, the problem has a unique solution, given by the Radon inversion formula; Radon gave the inversion in the plane in 1917. The problem is an inverse problem that is not well-posed in L2L^2: the inversion contains the filter ∣σ∣n−1\abs{\sigma}^{n-1}, which amplifies high-frequency noise, so some regularization is needed. With finitely many directions, uniqueness fails: for any finite set of directions there is a nonzero smooth function with compact support whose projections in these directions all vanish. The main algorithms are filtered back-projection, Fourier reconstruction based on the projection-slice theorem, and iterative algebraic methods such as the Kaczmarz method. Limited-angle data and interior data lead to harder versions of the problem, with weaker stability.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Inverse problem, concept, no evidence levelInverse problemconceptRadon transform, concept, no evidence levelRadon transformconceptWell-posed problem, concept, no evidence levelWell-posed problemconceptTomographic reconstruction, problem, no evidence levelTomographic reconstru…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Inverse problem concept
  2. Radon transform concept
  3. Well-posed problem concept
  4. Tomographic reconstruction problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Tomographic_reconstruction
About this page
TagHK1TJFXNG9
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Kindproblem
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MSC44A12, 92C55, 65R32
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AboutRadon transform
UsesRadon transform, Inverse problem, Well-posed problem
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