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Gengsheng L Zeng1,2

  • 1Department of Computer Science, Utah Valley University, Orem, UT, 84058, USA. larry.zeng@uvu.edu.

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Summary
This summary is machine-generated.

This study explores extending frequency components for finite objects. An iterative algorithm showed slow convergence, suggesting the problem is too ill-posed for practical solutions.

Keywords:
Analytic continuationEntire functionImage reconstructionIterative projections onto convex sets algorithmLimited angle tomography

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Area of Science:

  • Signal Processing
  • Fourier Analysis
  • Complex Analysis

Background:

  • A finite support spatial-domain function's Fourier transform is an entire function.
  • Entire functions have Taylor series expansions that converge everywhere in the complex plane.
  • Analytic continuation theory implies unique object determination from limited frequency data.

Purpose of the Study:

  • To develop an iterative algorithm for extending measured frequency components into unmeasured regions.
  • To assess the practical solvability of reconstructing finite-sized objects from limited frequency data.

Main Methods:

  • Proposed an iterative algorithm to extrapolate frequency components.
  • Utilized computer simulations to evaluate the algorithm's performance.

Main Results:

  • The iterative algorithm demonstrated very slow convergence.
  • Simulations indicated significant challenges in extending measured frequency components.

Conclusions:

  • The problem of reconstructing finite-sized objects from limited frequency data is highly ill-posed.
  • Current methods, including the proposed algorithm, are not practically solvable due to slow convergence and ill-posedness.