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  2. Radon transform - Wikipedia

    en.wikipedia.org/wiki/Radon_transform

    Radon transform. Maps f on the (x, y)-domain to Rf on the (α, s)-domain.. In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line.

  3. Tomographic reconstruction - Wikipedia

    en.wikipedia.org/wiki/Tomographic_reconstruction

    In theory, the inverse Radon transformation would yield the original image. The projection-slice theorem tells us that if we had an infinite number of one-dimensional projections of an object taken at an infinite number of angles, we could perfectly reconstruct the original object, f ( x , y ) {\displaystyle f(x,y)} .

  4. Surface wave inversion - Wikipedia

    en.wikipedia.org/wiki/Surface_wave_inversion

    The fifth is a high-resolution Linear Radon transformation performed by Luo et al. (2008). [10] In performing a wave-field transformation, a slant stack is done, followed by a Fourier transform . The way in which a Fourier transform changes x-t data into x-ω (ω is angular frequency) data shows why phase velocity dominates surface wave ...

  5. Projection-slice theorem - Wikipedia

    en.wikipedia.org/wiki/Projection-slice_theorem

    Take a two-dimensional function f(r), project (e.g. using the Radon transform) it onto a (one-dimensional) line, and do a Fourier transform of that projection. Take that same function, but do a two-dimensional Fourier transform first, and then slice it through its origin, which is parallel to the projection line. In operator terms, if

  6. Mellin inversion theorem - Wikipedia

    en.wikipedia.org/wiki/Mellin_inversion_theorem

    Then is recoverable via the inverse Mellin transform from its Mellin transform . These results can be obtained by relating the Mellin transform to the Fourier transform by a change of variables and then applying an appropriate version of the Fourier inversion theorem. [1]

  7. Abel transform - Wikipedia

    en.wikipedia.org/wiki/Abel_transform

    A geometrical interpretation of the Abel transform in two dimensions. An observer (I) looks along a line parallel to the x axis a distance y above the origin. What the observer sees is the projection (i.e. the integral) of the circularly symmetric function f(r) along the line of sight.

  8. Vincenty's formulae - Wikipedia

    en.wikipedia.org/wiki/Vincenty's_formulae

    Vincenty suggested a method of accelerating the convergence in such cases (Rapp, 1993). An example of a failure of the inverse method to converge is (Φ 1, L 1) = (0°, 0°) and (Φ 2, L 2) = (0.5°, 179.7°) for the WGS84 ellipsoid. In an unpublished report, Vincenty (1975b) gave an alternative iterative scheme to handle such cases.

  9. Inverse problem - Wikipedia

    en.wikipedia.org/wiki/Inverse_problem

    An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field.