An approximate method for Abel inversion using Chebyshev polynomials

• We have applied Chebyshev operational matrix of integration for Abel’s inversion.
• The proposed approach is simpler compare to others existing ones.
• Only small size operational matrix is required to provide the solution at high accuracy.

Many problems in physics like reconstruction of the radially distributed emissivity from the line-of-sight projected intensity, the 3-D image reconstruction from cone-beam projections in computerized tomography, etc. lead naturally, in the case of radial symmetry, to the study of Abel’s type integral equation. In this paper, a new stable algorithm based on shifted Chebyshev polynomial approximation is presented and analyzed. First, Chebyshev operational matrix of integration P is constructed and then it is used to reduce the integral equation to a system of algebraic equation which can be solved easily. The method is quite accurate and stable even though the approximations are performed using polynomials of degree up to 5. Some test examples from the plasma diagnostics are illustrated to demonstrate the effectiveness and stability of the proposed method.