A Convex-Nonconvex variational method for the additive decomposition of functions on surfaces

We present a Convex-NonConvex variational approach for the additive decomposition of noisy scalar f ields defined over triangulated surfaces into piecewise constant and smooth components. The energy functional to be minimized is defined by the weighted sum of three terms, namely an L2 fidelity term for the noise component, a Tikhonov regularization term for the smooth component and a Total Variation (TV)-like non-convex term for the piecewise constant component. The last term is parametrized such that the free scalar parameter allows to tune its degree of non- convexity and, hence, to separate the piecewise constant component more effectively than by using a classical convex TV regularizer without renouncing to convexity of the total energy functional. A method is also presented for selecting the two regularization parameters. The unique solution of the proposed variational model is determined by means of an efficient ADMM-based minimization algorithm. Numerical experiments show a nearly perfect separation of the different components

A method to improve the computational efficiency of the Chan-Vese model for the segmentation of ultrasound images

Purpose Advanced image segmentation techniques like the Chan-Vese (CV) models transform the segmentation problem into a minimization problem which is then solved using the gradient descent (GD) optimization algorithm. This study explores whether the computational efficiency of CV can be improved when GD is replaced by a different optimization method. Methods Two GD variants from the literature (Nesterov accelerated, Barzilai-Borwein) and a newly developed hybrid variant of GD were used to improve the computational efficiency of CV by making GD insensitive to local minima. One more variant of GD from the literature (projected GD) was used to address the issue of maintaining the constraint on boundary evolution in CV which also increases computational cost. A novel modified projected GD (Barzilai-Borwein projected GD) was also used to overcome both problems at the same time. The effect of optimization method selection on processing time and the quality of the output was assessed for 25 musculoskeletal ultrasound images (five anatomical areas). Results The Barzilai-Borwein projected GD method was able to significantly reduce computational time (average(±std.dev.) reduction 95.82 % (±3.60 %)) with the least structural distortion in the delineated output relative to the conventional GD (average(±std.dev.) structural similarity index: 0.91(±0.06)). Conclusion The use of an appropriate optimization method can substantially improve the computational efficiency of CV models. This can open the way for real-time delimitation of anatomical structures to aid the interpretation of clinical ultrasound. Further research on the effect of the optimization method on the accuracy of segmentation is needed

Convex non-convex image segmentation

A convex non-convex variational model is proposed for multiphase image segmentation. We consider a specially designed non-convex regularization term which adapts spatially to the image structures for a better control of the segmentation boundary and an easy handling of the intensity inhomogeneities. The nonlinear optimization problem is efficiently solved by an alternating directions methods of multipliers procedure. We provide a convergence analysis and perform numerical experiments on several images, showing the effectiveness of this procedure