Image analysis in medical imaging: recent advances in selected examples

Medical imaging has developed into one of the most important fields within scientific imaging due to the rapid and continuing progress in computerised medical image visualisation and advances in analysis methods and computer-aided diagnosis. Several research applications are selected to illustrate the advances in image analysis algorithms and visualisation. Recent results, including previously unpublished data, are presented to illustrate the challenges and ongoing developments

Solutions of Inequality Constrained Spline Optimization Problems with the Active Set Method

We solve the problem of finding a near-interpolant curve, subject to constraints, which minimizes the bending energy of the curve. Using B-splines as our tools, we give a brief overview of spline properties and develop several different cases of inequality constrained optimization problems of this type. In particular, we develop the active set method and use it to solve these problems, emphasizing the fact that this algorithm will converge to a solution in finite iterations. Our solution will solve an open problem regarding near-interpolant spline curves. Furthermore, we supplement this with an iterative technique for better choosing data sites so as to further minimize the bending energy of the spline curve, offering an easy solution to the problem of free data sites

On smoothing problems with one additional equality condition

Two problems of approximation in Hilbert spaces are considered with one additional equality condition: the smoothing problem with a weight and the smoothing problem with an obstacle. This condition is a generalization of the equality, which appears in the problem of approximation of a histogram in a natural way. We characterize the solutions of these smoothing problems and investigate the connection between them. First published online: 14 Oct 201

On the problems of smoothing and near-interpolation

Abstract. In the first part of this paper we apply a saddle point theorem from convex analysis to show that various constrained minimization problems are equivalent to the problem of smoothing by spline functions. In particular, we show that near-interpolants are smoothing splines with weights that arise as Lagrange multipliers corresponding to the constraints in the problem of near-interpolation. In the second part of this paper we apply certain fixed point iterations to compute these weights. A similar iteration is applied to the computation of the smoothing parameter in the problem of smoothing. 1