An adaptive and stable method for fitting implicit polynomial curves and surfaces

Representing 2D and 3D data sets with implicit polynomials (IPs) has been attractive because of its applicability to various computer vision issues. Therefore, many IP fitting methods have already been proposed. However, the existing fitting methods can be and need to be improved with respect to computational cost for deciding on the appropriate degree of the IP representation and to fitting accuracy, while still maintaining the stability of the fit. We propose a stable method for accurate fitting that automatically determines the moderate degree required. Our method increases the degree of IP until a satisfactory fitting result is obtained. The incrementability of QR decomposition with Gram-Schmidt orthogonalization gives our method computational efficiency. Furthermore, since the decomposition detects the instability element precisely, our method can selectively apply ridge regression-based constraints to that element only. As a result, our method achieves computational stability while maintaining fitting accuracy. Experimental results demonstrate the effectiveness of our method compared with prior methods.