Radon Transform - Inversion Formulas

Inversion Formulas

Explicit and computationally efficient inversion formulas for the Radon transform and its dual are available. The Radon transform in n dimensions can be inverted by the formula

where

and the power of the Laplacian (−Δ)(n−1)/2 is defined as a pseudodifferential operator if necessary by the Fourier transform

For computational purposes, the power of the Laplacian is commuted with the dual transform R* to give

c_nf = \begin{cases} R^*\frac{d^{n-1}}{ds^{n-1}}Rf & n \rm{\ odd}\\ R^*H_s\frac{d^{n-1}}{ds^{n-1}}Rf & n \rm{\ even} \end{cases}

where Hs is the Hilbert transform with respect to the s variable. In two dimensions, the operator Hsd/ds appears in image processing as a ramp filter. One can prove directly from the Fourier slice theorem and change of variables for integration that for a compactly supported continuous function ƒ of two variables

f =\frac{1}{2}R^{*}H_s\frac{d}{ds}Rf.

Thus in an image processing context the original image ƒ can be recovered from the 'sinogram' data Rƒ by applying a ramp filter (in the variable) and then back-projecting. As the filtering step can be performed efficiently (for example using digital signal processing techniques) and the back projection step is simply an accumulation of values in the pixels of the image, this results in a highly efficient, and hence widely used, algorithm.

Explicitly, the inversion formula obtained by the latter method is

if n is odd, and

if n is even.

The dual transform can also be inverted by an analogous formula:

Read more about this topic:  Radon Transform

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