An open level set framework for image segmentation and restoration using the Mumford and Shah model

Abstract

In two dimensions, the Mumford and Shah functional for image segmentation and regularization¹⁵ has minimizers (<i>u</i>,<i>K</i>), where <i>u</i> is a piecewise-smooth approximation of the image data <i>f</i>, and <i>K</i> represents the set of discontinuities of <i>u</i> (a union of curves). Theoretically, the edge set <i>K</i> could include both closed and open curves. The current level set and piecewise-smooth Mumford-Shah based segmentation algorithms^{4, 23, 24} can only detect objects with closed edges, which are boundaries of open sets. We propose an efficient Mumford-Shah and level set based algorithm for segmenting images with edges which are made up of open curves or crack-tips. By adapting Smereka's open level set formulation²¹ to variational problems, we are able to extend the current piecewise-smooth and level-set based image segmentation methods, such as^{4, 23, 24} to the case of open curve segmentation. The algorithm retains many of the advantages of using level sets, such as well-defined boundaries and ability to change topology. We solve the resulting Euler-Lagrange equations by Sobolev <i>H</i>¹ gradient descent, avoiding instability and the need for additional regularization of the level set functions, while also accelerating convergence to the reconstructed image. Finally, we present the numerical implementation and experimental results on various noisy images.