From nonlinear optimization to convex optimization through firefly algorithm and indirect approach with applications to CAD/CAM

Immunological Approach for Full NURBS Reconstruction of Outline Curves from Noisy Data Points in Medical Imaging

Curve reconstruction from data points is an important issue for advanced medical imaging techniques, such as computer tomography (CT) and magnetic resonance imaging (MRI). The most powerful fitting functions for this purpose are the NURBS (non-uniform rational B-splines). Solving the general reconstruction problem with NURBS requires computing all free variables of the problem (data parameters, breakpoints, control points, and their weights). This leads to a very difficult non-convex, nonlinear, high-dimensional, multimodal, and continuous optimization problem. Previous methods simplify the problem by guessing the values for some variables and computing only the remaining ones. As a result, unavoidable approximations errors are introduced. In this paper, we describe the first method in the literature to solve the full NURBS curve reconstruction problem in all its generality. Our method is based on a combination of two techniques: an immunological approach to perform data parameterization, breakpoint placement, and weight calculation, and least squares minimization to compute the control points. This procedure is repeated iteratively (until no further improvement is achieved) for higher accuracy. The method has been applied to reconstruct some outline curves from MRI brain images with satisfactory results. Comparative work shows that our method outperforms the previous related approaches in the literature for all instances in our benchmark.

Cuckoo search with Lévy flights for weighted Bayesian energy functional optimization in global-support curve data fitting

The problem of data fitting is very important in many theoretical and applied fields. In this paper, we consider the problem of optimizing a weighted Bayesian energy functional for data fitting by using global-support approximating curves. By global-support curves we mean curves expressed as a linear combination of basis functions whose support is the whole domain of the problem, as opposed to other common approaches in CAD/CAM and computer graphics driven by piecewise functions (such as B-splines and NURBS) that provide local control of the shape of the curve. Our method applies a powerful nature-inspired metaheuristic algorithm called cuckoo search, introduced recently to solve optimization problems. A major advantage of this method is its simplicity: cuckoo search requires only two parameters, many fewer than other metaheuristic approaches, so the parameter tuning becomes a very simple task. The paper shows that this new approach can be successfully used to solve our optimization problem. To check the performance of our approach, it has been applied to five illustrative examples of different types, including open and closed 2D and 3D curves that exhibit challenging features, such as cusps and self-intersections. Our results show that the method performs pretty well, being able to solve our minimization problem in an astonishingly straightforward way.