Analytic continuation and incomplete data tomography.

Area of Science:

• Medical imaging
• Applied mathematics
• Signal processing

Background:

• Medical imaging objects possess compact support, meaning their Fourier transforms are entire functions.
• Entire functions can theoretically be reconstructed from limited data using methods like power series expansions.
• Power series expansions necessitate calculating all derivative orders, which is infeasible for discretely sampled data.

Purpose of the Study:

• To propose an alternative method for analytic continuation of entire functions derived from medical imaging data.
• To overcome the limitations of power series expansions in reconstructing functions from discrete samples.

Main Methods:

• The study proposes an analytic continuation method utilizing the Nyquist-Shannon sampling theorem.
• This novel approach involves solving a system of linear equations.
• The method does not require the computation of function derivatives.

Main Results:

• Computer simulations with noiseless data were conducted to validate the proposed method.
• The results demonstrate the feasibility of analytic continuation using the Nyquist-Shannon sampling theorem.
• The study highlights that analytic continuation is an extremely ill-conditioned problem.

Conclusions:

• The proposed method offers a viable alternative for analytic continuation in medical imaging.
• Solving linear equations based on the sampling theorem circumvents the need for derivative calculations.
• Despite its effectiveness, the inherent ill-conditioned nature of analytic continuation remains a significant challenge.