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Inverse Abel transform

Inverse Abel transform. Kaarel Piip 2 nd year PhD student Institute of Physics. Outline. Idea of the Abel transform Analytical formula for the inverse transform Application to experimental problems Data processing and numerical algorithms Con c lusions. Abel transform.

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Inverse Abel transform

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  1. Inverse Abel transform Kaarel Piip 2nd year PhD student Institute of Physics

  2. Outline • Idea of the Abel transform • Analytical formula for the inverse transform • Application to experimental problems • Data processing and numerical algorithms • Conclusions Kaarel Piip ISC 02.04.2014

  3. Abel transform • It is named after Norwegian mathemtician Niels Henrik Abel (1802-1829) • Abel transform occures naturally in the case of silindrical symmetry. • One typical case is the optical emission (light) of sylindrically symmetric object: galaxy, plasma column etc. • We do not see the real radial distribution f(r), we see the Abel transform F(y) of this distribution. Kaarel Piip ISC 02.04.2014

  4. Inverse Abel Transform • Assuming that f(r) drops to zero more quickly than 1/r the Abel inverse transform has the analytical form. To check just apply the Abel transform to this formula. • It is worth mentioning that the solution is unique (assuming no absorbtion): knowing F(y) the f(r) could be calculated. Kaarel Piip ISC 02.04.2014

  5. Applycation for experimental results • Experimental results are discrete (usually pixels). • There is experimental noise. • Images are usually not perfectly symmetrical. • For the integral there is a singularity at y=r. • Simple numerical calculations for dF/dy lead to large errors. • It is not possible to calculate till infinity. Kaarel Piip ISC 02.04.2014

  6. Data processing and numerical algorithms • Typically the experimental data is smoothed and sometimes mirroring is used to ensure the silindrical symmetry. • Basically there are two schemes to calculate the Abel inverse: • Fit the experimental data with some good analytical fuction (Gaussian, Lorentzian etc) and calculate the integral analytically. • Use numerical methods to calculate the Abel inverse. In this presentation we concentrate on this. Kaarel Piip ISC 02.04.2014

  7. Numerical algorithms I Direct method • Easy to implement and understand. • Needs smooth data and is sensitive to noise near the symmetry axis. Kaarel Piip ISC 02.04.2014

  8. Numerical algorithms II Nestor-Olsen method • Easy to implement and understand. • Converges better than the direct method. • Does not include division with small floats. Kaarel Piip ISC 02.04.2014

  9. Numerical algorithms III Fourier-Hankel method • Converges better. • Could be used to remove noise. • Includes double sum: calculation time is longer. Kaarel Piip ISC 02.04.2014

  10. Conclusions • Inverse Abel transform is associated with noticeable difficulties for numerical implementation. • Altough direct schemes could be applied, they are not accurate. • Integral transformations and special functions offer a method for more accurate algorithms. • In addition to mathematical problems there are several experimental problems: noise, asymmetry etc. Kaarel Piip ISC 02.04.2014

  11. Thank you! Kaarel Piip ISC 02.04.2014

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