Equivalent convolution strategy for the evolution computation in parametric active contour model

Abstract

Parametric active contour model is an efficient approach for image segmentation. However, the high cost of evolution computation has restricted their potential applications to contour segmentation with long perimeter. Extensive algorithm debugging and analysis indicate that the inverse matrix calculation and the matrix multiplication are the two major reasons. In this paper, a novel simple and efficient algorithm for evolution computation is proposed. Motivated by the relationship between the eigenvalues and the entries in the circular Toeplitz matrix, each entry expression of inverse matrix is firstly derived through mathematical deduction, and then, the matrix multiplication is simplified into a more efficient convolution operation. Experimental results show that the proposed algorithm can significantly improve the computational speed by one to two orders of magnitude and is even more efficient for contour extraction with large perimeter.

Appendix

$$\begin{aligned} \begin{aligned}\sum _{j=0}^{n-1} \kappa _{j}^{-1} &=\frac{2}{\sqrt{-\varDelta }} \textrm{Im}\left(\frac{\zeta }{1-\zeta ^{2}}\frac{\sum _{j=0}^{n-1}\zeta ^j+\sum _{j=0}^{n-1}\zeta ^{n-j}}{1-\zeta ^{n}}\right)\\ {}&=\frac{2}{\sqrt{-\varDelta }} \textrm{Im}\left(\frac{\zeta }{(1-\zeta )^{2}}\right) =\frac{2}{\sqrt{-\varDelta }} \textrm{Im}\left(\frac{1}{(1-\zeta )(\frac{1}{\zeta }-1)}\right)\\& =\frac{2}{\sqrt{-\varDelta }} \textrm{Im}\frac{1}{(1+\mu +\sqrt{\mu ^2-1})\left(\frac{1}{-\mu -\sqrt{\mu ^2-1}}-1\right)}\\& =-\frac{1}{\sqrt{-\varDelta }} \textrm{Im}\frac{1}{1+\mu } =-\frac{1}{\sqrt{-\varDelta }} \textrm{Im}\frac{1}{1+\frac{-(\alpha +4\beta )+\sqrt{\alpha ^2-4\beta }}{4\beta }}\\& =\frac{1}{\sqrt{-\varDelta }} \textrm{Im}(\alpha +\sqrt{-\varDelta }\text {i})=1 \end{aligned} \end{aligned}$$