Spatial redistribution of irregularly-spaced Pareto fronts for more intuitive navigation and solution selection

A multi-objective optimization approach is o.en followed by an a posteriori decision-making process, during which the most appropriate solution of the Pareto set is selected by a professional in the .eld. Conventional visualization methods do not correct for Pareto fronts with irregularly-spaced solutions. However, achieving a uniform spread of solutions can make the decision-making process more intuitive when decision tools such as sliders, which represent the preference for each objective, are used. We propose a method that maps anm-dimensional Pareto front to an (m-1)-simplex and spreads out points to achieve a more uniform distribution of these points in the simplex while maintaining the local neighborhood structure of the solutions as much as possible. .is set of points can then more intuitively be navigated due to the more uniform distribution. We test our approach on a set of non-uniformly spaced 3D Pareto fronts of a real-world problem: deformable image registration of medical images. The results of these experiments are visualized as points in a triangle, showing that we indeed achieve a representation of the Pareto front with a near-uniform distribution of points where these are still positioned as expected, i.e., according to their quality in each of the objectives of interest

A first step toward uncovering the truth about weight tuning in deformable image registration

Deformable image registration is currently predominantly solved by optimizing a weighted linear combination of objectives. Successfully tuning the weights associated with these objectives is not trivial, leading to trial-and-error approaches. Such an approach assumes an intuitive interplay between weights, optimization objectives, and target registration errors. However, it is not known whether this always holds for existing registration methods. To investigate the interplay between weights, optimization objectives, and registration errors, we employ multi-objective optimization. Here, objectives of interest are optimized simultaneously, causing a set of multiple optimal solutions to exist, called the optimal Pareto front. Our medical application is in breast cancer and includes the challenging prone-supine registration problem. In total, we studied the interplay in three different ways. First, we ran many random linear combinations of objectives using the well-known registration software elastix. Second, since the optimization algorithms used in registration are typically of a local-search nature, final solutions may not always form a Pareto front. We therefore employed a multi-objective evolutionary algorithm that finds weights that correspond to registration outcomes that do form a Pareto front. Third, we examined how the interplay differs if a true multi-objective (i.e., weight-free) image registration method is used. Results indicate that a trial-and-error weight-adaptation approach can be successful for the easy prone to prone breast image registration case, due to the absence of many local optima. With increasing problem difficulty the use of more advanced approaches can be of value in finding and selecting the optimal registration outcomes

Diversifying Multi-Objective Gradient Techniques and their Role in Hybrid Multi-Objective Evolutionary Algorithms for Deformable Medical Image Registration

Gradient methods and their value in single-objective, real-valued optimization are well-established. As such, they play a key role in tackling real-world, hard optimization problems such as deformable image registration (DIR). A key question is to which extent gradient techniques can also play a role in a multi-objective approach to DIR. We therefore aim to exploit gradient information within an evolutionary-algorithm-based multi-objective optimization framework for DIR. Although an analytical description of the multi-objective gradient (the set of all Pareto-optimal improving directions) is available, it is nontrivial how to best choose the most appropriate direction per solution because these directions are not necessarily uniformly distributed in objective space. To address this, we employ a Monte-Carlo method to obtain a discrete, spatially-uniformly distributed approximation of the set of Pareto-optimal improving directions. We then apply a diversification technique in which each solution is associated with a unique direction from this set based on its multi- as well as single-objective rank. To assess its utility, we compare a state-of-the-art multi-objective evolutionary algorithm with three different hybrid versions thereof on several benchmark problems and two medical DIR problems. Results show that the diversification strategy successfully leads to unbiased improvement, helping an adaptive hybrid scheme solve all problems, but the evolutionary algorithm remains the most powerful optimization method, providing the best balance between proximity and diversity

On the usefulness of gradient information in multi-objective deformable image registration using a B-spline-based dual-dynamic transformation model: comparison of three optimization algorithms

The use of gradient information is well-known to be highly useful in single-objective optimization-based image registration methods. However, its usefulness has not yet been investigated for deformable image registration from a multi-objective optimization perspective. To this end, within a previously introduced multi-objective optimization framework, we use a smooth B-spline-based dual-dynamic transformation model that allows us to derive gradient information analytically, while still being able to account for large deformations. Within the multi-objective framework, we previously employed a powerful evolutionary algorithm (EA) that computes and advances multiple outcomes at once, resulting in a set of solutions (a so-called Pareto front) that represents efficient trade-offs between the objectives. With the addition of the B-spline-based transformation model, we studied the usefulness of gradient information in multiobjective deformable image registration using three different optimization algorithms: the (gradient-less) EA, a gradientonly algorithm, and a hybridization of these two. We evaluated the algorithms to register highly deformed images: 2D MRI slices of the breast in prone and supine positions. Results demonstrate that gradient-based multi-objective optimization significantly speeds up optimization in the initial stages of optimization. However, allowing sufficient computational resources, better results could still be obtained with the EA. Ultimately, the hybrid EA found the best overall approximation of the optimal Pareto front, further indicating that adding gradient-based optimization for multiobjective optimization-based deformable image registration can indeed be beneficial