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Stable fitting of 2D curves and 3D surfaces by implicit polynomials
Amir Helzer1, Meir Barzohar, David Malah
1Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel. amir_helzer@hotmail.com
IEEE Transactions on Pattern Analysis and Machine Intelligence
|January 12, 2005
Summary
This study introduces new implicit polynomial (IP) fitting algorithms that are less sensitive to coefficient errors. These methods improve fitting accuracy and stability, especially for noisy 2D and 3D data.
Area of Science:
- Computer Vision
- Geometric Modeling
- Numerical Analysis
Background:
- Implicit polynomials (IPs) define curves and surfaces via polynomial equations.
- Fitting IPs to data is crucial in computer graphics and geometric modeling.
- Existing fitting algorithms can be sensitive to coefficient errors from numerical computations or quantization.
Purpose of the Study:
- To develop novel implicit polynomial fitting algorithms with reduced sensitivity to coefficient errors.
- To enhance the fitting tightness and stability of implicit polynomial representations.
- To improve the accuracy of fitting noisy 2D curves and 3D surfaces.
Main Methods:
- Analysis of the zero-set's sensitivity to coefficient perturbations.
- Development of two new algorithms based on minimizing error bounds and error variance.
- Comparison with established methods like the 3L and gradient-one algorithms.
Main Results:
- The proposed algorithms demonstrate reduced sensitivity to coefficient errors.
- Improved fitting tightness and stability, particularly for noisy data.
- Significant reduction in fitting errors compared to existing methods, especially for high-order polynomials and complex shapes.
Conclusions:
- The novel IP fitting algorithms offer superior performance over existing methods.
- These algorithms are robust in handling noisy data and complex geometric shapes.
- The findings have implications for accurate geometric data representation and analysis.