Edge enhancement using symmetric B-spline basis on closed periodic zone

Aiming at the weakness of lacking complete theoretical explanation for derivative operators and ignoring keeping natural transition for enhanced edges in the existing time domain edge enhancement methods, this paper establishes a new theory that explains derivative operators from standpoint of B-spline interpolation and presents a novel edge enhancement method using non-linear transforming first derivative of given signal or image. First we shift nai¿ve B-spline basis to establish symmetric B-spline basis, next we use orthogonality properties of complex exponentials to establish orthogonal B-spline basis on closed periodic zone and derive parallel computing formula for coefficients of orthogonal B-spline basis; we further use relation between coefficients of orthogonal B-spline basis and coefficients of symmetric B-spline basis to achieve parallel serial computing formulas for interpolation coefficients of symmetric B-spline basis. At last, we derive first and second derivate operators in terms of interpolation coefficients of symmetric B-spline basis and present a novel method of edge enhancement using nonlinear transforming first derivative of given signal or image. Experiment results show that, the new theory established in this paper can explain derivative operators theoretically from standpoint of B-spline interpolation. The derivative operators derived in this paper have no phase deviation and the novel method of edge enhancement enhances edges in image with natural transition and sharp visual quality.