This work deals with fitting 2D and 3D implicit polynomials (IPs) to 2D curves and 3D surfaces, respectively. The zero-set of the polynomial is determined by the IP coefficients and describes the data. The polynomial fitting algorithms proposed in this paper aim at reducing the sensitivity of the polynomial to coefficient errors. Errors in coefficient values may be the result of numerical calculations, when solving the fitting problem or due to coefficient quantization. It is demonstrated that the effect of reducing this sensitivity also improves the fitting tightness and stability of the proposed two algorithms in fitting noisy data, as compared to existing algorithms like the well-known 3L and gradient-one algorithms. The development of the proposed algorithms is based on an analysis of the sensitivity of the zero-set to small coefficient changes and on minimizing a bound on the maximal error for one algorithm and minimizing the error variance for the second. Simulation results show that the proposed algorithms provide a significant reduction in fitting errors, particularly when fitting noisy data of complex shapes with high order polynomials, as compared to the performance obtained by the abovementioned existing algorithms.

Presents an approach to contour representation and coding. It consists of an improved fitting of high-degree (4/sup th/ to 18/sup th/) implicit polynomials (IPs) to the contour which is robust to coefficient quantization. The proposed approach to solve the fitting problem is a modification of the 3L linear solution developed by Lei et al. (1997) and is more robust to noise and to coefficient quantization. We use an analytic approach to limit the maximal fitting error between each data point and the zero-set generated by the quantized polynomial coefficients. We than show that consideration of the quantization error (which led to a specific sensitivity criterion) also brought about a significant improvement in fitting IPs to noisy data, as compared to the 3L algorithm.