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Interconversion between truncated Cartesian and polar expansions of images
Wooram Park1, Gregory S Chirikjian
1Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218 USA. wpark7@jhu.edu
Summary
This study introduces a novel algorithm for lossless data conversion between Cartesian and polar coordinates using truncated expansions. The method ensures numerical stability and is applied to image data and polar-Cartesian interpolation.
Area of Science:
- Applied Mathematics
- Numerical Analysis
- Signal Processing
Background:
- Data representation in different coordinate systems is crucial for various scientific and engineering applications.
- Direct conversion between coordinate systems can lead to numerical instability.
- Efficiently handling sampled data from 2-D functions requires robust conversion methods.
Purpose of the Study:
- To develop a lossless algorithm for converting data between Cartesian and polar coordinates.
- To address the numerical instability issues in direct coefficient conversion.
- To enable efficient application to discrete image data and solve interpolation problems.
Main Methods:
- Utilizing Laguerre functions and Fourier basis for polar coordinates.
- Employing Hermite functions for Cartesian coordinates.
- Deriving relationships between truncated expansion coefficients for lossless conversion.
- Implementing resampling techniques to avoid direct coefficient conversion and enhance stability.
Main Results:
- A novel algorithm for lossless Cartesian-polar coordinate data conversion is proposed.
- The algorithm demonstrates improved numerical stability by using resampling.
- A method for optimally fitting truncated expansions to discrete image data is presented.
- The algorithm is successfully applied to the polar-Cartesian interpolation problem.
Conclusions:
- The proposed algorithm provides an effective and numerically stable method for lossless data conversion between Cartesian and polar coordinates.
- The technique is adaptable for discrete image data processing and interpolation tasks.
- This work contributes to more efficient and accurate data manipulation in scientific computing.
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