Deformable Medical Image Registration: A Survey <s> A. Continuous Optimization <s> A popular technique for nonrigid registration of medical images is based on the maximization of their mutual information, in combination with a deformation field parameterized by cubic B-splines. The coordinate mapping that relates the two images is found using an iterative optimization procedure. This work compares the performance of eight optimization methods: gradient descent (with two different step size selection algorithms), quasi-Newton, nonlinear conjugate gradient, Kiefer-Wolfowitz, simultaneous perturbation, Robbins-Monro, and evolution strategy. Special attention is paid to computation time reduction by using fewer voxels to calculate the cost function and its derivatives. The optimization methods are tested on manually deformed CT images of the heart, on follow-up CT chest scans, and on MR scans of the prostate acquired using a BFFE, T1, and T2 protocol. Registration accuracy is assessed by computing the overlap of segmented edges. Precision and convergence properties are studied by comparing deformation fields. The results show that the Robbins-Monro method is the best choice in most applications. With this approach, the computation time per iteration can be lowered approximately 500 times without affecting the rate of convergence by using a small subset of the image, randomly selected in every iteration, to compute the derivative of the mutual information. From the other methods the quasi-Newton and the nonlinear conjugate gradient method achieve a slightly higher precision, at the price of larger computation times. <s> BIB001 </s> Deformable Medical Image Registration: A Survey <s> A. Continuous Optimization <s> Whenever images taken at different times, from different viewpoints, and/or by different sensors need to be compared, merged, or integrated, image registration is required. Registration, also known as alignment, fusion, or warping, is the process of transforming data into a common reference frame. This book provides an overview of state-of-the-art registration techniques from theory to practice, plus numerous exercises designed to enhance readers understanding of the principles and mechanisms of the described techniques. It also provides, via a supplementary Web page, free access to FAIR.m, a package that is based on the MATLAB software environment, which enables readers to experiment with the proposed algorithms and explore the presented examples in more depth. Written from an interdisciplinary point of view, this book will appeal to mathematicians who want to learn more about image registration, medical imaging professionals who want to know more about and explore available imaging techniques, and computer scientists and engineers who want to understand the numerical schemes behind the techniques. The book is also appropriate for use as a course text at the advanced graduate level. Contents: FAIR Listings; FAIR Examples; List of Figures; List of Tables; Preface; Chapter 1: Introduction; Chapter 2: FAIR Concepts; Chapter 3: Image Interpolation; Chapter 4: Transforming Images by Parameterized Transformations; Chapter 5: Landmark-Based Registration; Chapter 6: Parametric Image Registration; Chapter 7: Distance Measures; Chapter 8: Regularization; Chapter 9: Nonparametric Image Registration; Chapter 10: Outlook; Bibliography; Symbols, Acronyms, Index. <s> BIB002
Continuous optimization methods are constrained to problems where the variables take real values and the objective function is differentiable. Image registration is a problem where the application of continuous optimization methods has been studied. Continuous optimization methods estimate the optimal parameters following an update rule of the following form: BIB002 where denotes the vector of parameters of the transformation, indexes the number of iteration, is the step size or gain factor, and defines the search direction. The search direction is calculated by taking into account both the matching and the regularization term. Therefore, it should be written as . Nonetheless, we prefer the use of in order to reduce the clutter of unnecessary notation. There are various ways to define the previous parameters. For example, the step size may be constant, decrease with each iteration or such that it minimizes the objective function along the search direction (exact or inexact line search). The search direction can be specified by exploiting only first-order information or, for example, by also taking into consideration second-order information. It is the choice of these parameters that distinguishes different methods. Commonly used methods include (see also Fig. 3 ): 1) gradient descent (GD), 2) conjugate gradient (CG), 3) Powell's conjugate directions, 4) Quasi-Newton (QN), 5) Levenberg-Marquardt (LM), and 6) stochastic gradient descent. Klein et al. BIB001 reported a study comparing optimization strategies in image registration using mutual information as similarity metric and cubic -spline FFDs as deformation model.