Parametric Level Set Methods for Inverse Problems

In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set parameters. We show that using the appropriate form of parameterizing the level set function results a significantly lower dimensional problem, which bypasses many difficulties with traditional level set methods, such as regularization, re-initialization and use of signed distance function. Moreover, we show that from a computational point of view, low order representation of the problem paves the path for easier use of Newton and quasi-Newton methods. Specifically for the purposes of this paper, we parameterize the level set function in terms of adaptive compactly supported radial basis functions, which used in the proposed manner provides flexibility in presenting a larger class of shapes with fewer terms. Also they provide a "narrow-banding" advantage which can further reduce the number of active unknowns at each step of the evolution. The performance of the proposed approach is examined in three examples of inverse problems, i.e., electrical resistance tomography, X-ray computed tomography and diffuse optical tomography

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

Segmentation of images with low-contrast edges

A vast amount of the current research in medical image analysis has aimed to develop improved techniques of image segmentation. Of the existing approaches, active contour methods have proven effective by incorporating edge or region information from the image into a level set formulation. However, complications arise in images containing regions of low-contrast due to noise, occlusions, or partial volume effects, which are often unavoidable in practical applications. Incorporating prior shape information into the segmentation framework provides a more accurate and robust solution by constraining the evolving contour to resemble a target shape. Two methods are presented to incorporate a shape prior into existing active contour segmentation methods, including the edge-based geodesic active contours model and a fast update implementation of the region-based Chan-Vese model. Applying these methods to synthetic and real images demonstrates that an improved result can be obtained for images containing low-contrast edge regions

Discrete gradient methods for solving variational image regularisation models

Discrete gradient methods are well-known methods of geometric numerical integration, which preserve the dissipation of gradient systems. In this paper we show that this property of discrete gradient methods can be interesting in the context of variational models for image processing, that is where the processed image is computed as a minimiser of an energy functional. Numerical schemes for computing minimisers of such energies are desired to inherit the dissipative property of the gradient system associated to the energy and consequently guarantee a monotonic decrease of the energy along iterations, avoiding situations in which more computational work might lead to less optimal solutions. Under appropriate smoothness assumptions on the energy functional we prove that discrete gradient methods guarantee a monotonic decrease of the energy towards stationary states, and we promote their use in image processing by exhibiting experiments with convex and non-convex variational models for image deblurring, denoising, and inpainting