Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <s> An iterative algorithm is given for solving a system Ax=k of n linear equations in n unknowns. The solution is given in n steps. It is shown that this method is a special case of a very general method which also includes Gaussian elimination. These general algorithms are essentially algorithms for finding an n dimensional ellipsoid. Connections are made with the theory of orthogonal polynomials and continued fractions. <s> BIB001 </s> Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <s> Abstract THE conjugate gradient method was first described in [1, 2] for solving sets of linear algebraic equations. The method, being iterative in form, has all the merits of iterative methods, and enables a set of linear equations to be solved (or what amounts to the same thing, the minimum of a quadratic functional in finite-dimensional space to be found) after a finite number of steps. The method was later extended to the case of Hilbert space [3–5], and to the case of non-quadratic functionals [6, 7]. The present paper proves the convergence of the method as applied to non-quadratic functionals, describes its extension to constrained problems, considers means for further accelerating the convergence, and describes experience in the practical application of the method for solving a variety of extremal problems. <s> BIB002 </s> Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <s> In this paper a nonmonotone steplength selection rule for Newton’s method is proposed, which can be viewed as a generalization of Armijo’s rule. Numerical results are reported which indicate that the proposed technique may allow a considerable saving both in the number of line searches and in the number of function evaluations. <s> BIB003 </s> Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <s> In this paper the authors present a new approach for the nonrigid registration of contrast-enhanced breast MRI. A hierarchical transformation model of the motion of the breast has been developed. The global motion of the breast is modeled by an affine transformation while the local breast motion is described by a free-form deformation (FFD) based on B-splines. Normalized mutual information is used as a voxel-based similarity measure which is insensitive to intensity changes as a result of the contrast enhancement. Registration is achieved by minimizing a cost function, which represents a combination of the cost associated with the smoothness of the transformation and the cost associated with the image similarity. The algorithm has been applied to the fully automated registration of three-dimensional (3-D) breast MRI in volunteers and patients. In particular, the authors have compared the results of the proposed nonrigid registration algorithm to those obtained using rigid and affine registration techniques. The results clearly indicate that the nonrigid registration algorithm is much better able to recover the motion and deformation of the breast than rigid or affine registration algorithms. <s> BIB004 </s> Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <s> Two new consistent image registration algorithms are presented: one is based on matching corresponding landmarks and the other is based on matching both landmark and intensity information. The consistent landmark and intensity registration algorithm produces good correspondences between images near landmark locations by matching corresponding landmarks and away from landmark locations by matching the image intensities. In contrast to similar unidirectional algorithms, these new consistent algorithms jointly estimate the forward and reverse transformation between two images while minimizing the inverse consistency error-the error between the forward (reverse) transformation and the inverse of the the reverse (forward) transformation. This reduces the ambiguous correspondence between the forward and reverse transformations associated with large inverse consistency errors. In both algorithms a thin-plate spline (TPS) model is used to regularize the estimated transformations. Two-dimensional (2-D) examples are presented that show the inverse consistency error produced by the traditional unidirectional landmark TPS algorithm can be relatively large and that this error is minimized using the consistent landmark algorithm. Results using 2-D magnetic resonance imaging data are presented that demonstrate that using landmark and intensity information together produce better correspondence between medical images than using either landmarks or intensity information alone. <s> BIB005 </s> Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <s> A variational method for nonrigid registration of multimodal image data is presented. A suitable deformation will be determined via the minimization of a morphological, i.e., contrast invariant, matching functional along with an appropriate regularization energy. The aim is to correlate the morphologies of a template and a reference image under the deformation. Mathematically, the morphology of images can be described by the entity of level sets of the image and hence by its Gauss map. A class of morphological matching functionals is presented which measure the defect of the template Gauss map in the deformed state with respect to the deformed Gauss map of the reference image. The problem is regularized by considering a nonlinear elastic regularization energy. Existence of homeomorphic, minimizing deformation is proved under assumptions on the class of admissible deformations. With respect to actual medical applications, suitable generalizations of the matching energies and the boundary conditions are pre... <s> BIB006 </s> Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <s> In this paper, we present a linear setting for statistical analysis of shape and an optimization approach based on a recent derivation of a conservation of momentum law for the geodesics of diffeomorphic flow. Once a template is fixed, the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point along the geodesic is completely determined by the momentum at the origin through geodesic shooting equations. The space of initial momentum provides a linear representation of the nonlinear diffeomorphic shape space in which linear statistical analysis can be applied. Specializing to the landmark matching problem of Computational Anatomy, we derive an algorithm for solving the variational problem with respect to the initial momentum and demonstrate principal component analysis (PCA) in this setting with three-dimensional face and hippocampus databases. <s> BIB007 </s> Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <s> This paper constructs metrics on the space of images {\cal I} defined as orbits under group actions {\cal G}. The groups studied include the finite dimensional matrix groups and their products, as well as the infinite dimensional diffeomorphisms examined in Trouve (1999, Quaterly of Applied Math.) and Dupuis et al. (1998). Quaterly of Applied Math.). Left-invariant metrics are defined on the product {\cal G} \times {\cal I} thus allowing the generation of transformations of the background geometry as well as the image values. Examples of the application of such metrics are presented for rigid object matching with and without signature variation, curves and volume matching, and structural generation in which image values are changed supporting notions such as tissue creation in carrying one image to another. <s> BIB008 </s> Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <s> This paper presents a methodology and algorithm for generating diffeomorphisms of the sphere onto itself, given the displacements of a finite set of template landmarks. Deformation maps are constructed by integration of velocity fields that minimize a quadratic smoothness energy under the specified landmark constraints. We present additional formulations of this problem which incorporate a given error variance in the positions of the landmarks. Finally, some experimental results are presented. This work has application in brain mapping, where surface data is typically mapped to the sphere as a common coordinate system. <s> BIB009 </s> Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <s> This paper proposes a method to match diffusion tensor magnetic resonance images (DT-MRIs) through the large deformation diffeomorphic metric mapping of vector fields, focusing on the fiber orientations, considered as unit vector fields on the image volume. We study a suitable action of diffeomorphisms on such vector fields, and provide an extension of the Large Deformation Diffeomorphic Metric Mapping framework to this type of dataset, resulting in optimizing for geodesics on the space of diffeomorphisms connecting two images. Existence of the minimizers under smoothness assumptions on the compared vector fields is proved, and coarse to fine hierarchical strategies are detailed, to reduce both ambiguities and computation load. This is illustrated by numerical experiments on DT-MRI heart images. <s> BIB010 </s> Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <s> We present a novel, variational and statistical approach for shape registration. Shapes of interest are implicitly embedded in a higher-dimensional space of distance transforms. In this implicit embedding space, registration is formulated in a hierarchical manner: the mutual information criterion supports various transformation models and is optimized to perform global registration; then, a B-spline-based incremental free form deformations (IFFD) model is used to minimize a sum-of-squared-differences (SSD) measure and further recover a dense local nonrigid registration field. The key advantage of such framework is twofold: 1) it naturally deals with shapes of arbitrary dimension (2D, 3D, or higher) and arbitrary topology (multiple parts, closed/open) and 2) it preserves shape topology during local deformation and produces local registration fields that are smooth, continuous, and establish one-to-one correspondences. Its invariance to initial conditions is evaluated through empirical validation, and various hard 2D/3D geometric shape registration examples are used to show its robustness to noise, severe occlusion, and missing parts. We demonstrate the power of the proposed framework using two applications: one for statistical modeling of anatomical structures, another for 3D face scan registration and expression tracking. We also compare the performance of our algorithm with that of several other well-known shape registration algorithms <s> BIB011 </s> Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <s> This paper reviews the development of dierent versions of nonlinear conjugate gradient methods, with special attention given to global convergence properties. <s> BIB012 </s> Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <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> BIB013 </s> Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <s> In order to compare anatomical and functional brain imaging data across subjects, the images must first be registered to a common coordinate system in which anatomical features are aligned. Intensity-based volume registration methods can align subcortical structures well, but the variability in sulcal folding patterns typically results in misalignment of the cortical surface. Conversely, surface-based registration using sulcal features can produce excellent cortical alignment but the mapping between brains is restricted to the cortical surface. Here we describe a method for volumetric registration that also produces an accurate one-to-one point correspondence between cortical surfaces. This is achieved by first parameterizing and aligning the cortical surfaces using sulcal landmarks. We then use a constrained harmonic mapping to extend this surface correspondence to the entire cortical volume. Finally, this mapping is refined using an intensity-based warp. We demonstrate the utility of the method by applying it to T1-weighted magnetic resonance images (MRIs). We evaluate the performance of our proposed method relative to existing methods that use only intensity information; for this comparison we compute the intersubject alignment of expert-labeled subcortical structures after registration. <s> BIB014 </s> Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <s> In this paper, we propose a novel method for the registration of volumetric images of the brain that optimizes the alignment of both cortical and subcortical structures. In order to achieve this, relevant geometrical information is extracted from a surface-based morph and diffused into the volume using the Navier operator of elasticity, resulting in a volumetric warp that aligns cortical folding patterns. This warp field is then refined with an intensity driven optical flow procedure that registers noncortical regions, while preserving the cortical alignment. The result is a combined surface and volume morph (CVS) that accurately registers both cortical and subcortical regions, establishing a single coordinate system suitable for the entire brain. <s> BIB015 </s> Deformable Medical Image Registration: A Survey <s> 1) Gradient Descent Methods: <s> Previous contributions to both the research and open source software communities detailed a generalization of a fast scalar field fitting technique for cubic B-splines based on the work originally proposed by Lee . One advantage of our proposed generalized B-spline fitting approach is its immediate application to a class of nonrigid registration techniques frequently employed in medical image analysis. Specifically, these registration techniques fall under the rubric of free-form deformation (FFD) approaches in which the object to be registered is embedded within a B-spline object. The deformation of the B-spline object describes the transformation of the image registration solution. Representative of this class of techniques, and often cited within the relevant community, is the formulation of Rueckert who employed cubic splines with normalized mutual information to study breast deformation. Similar techniques from various groups provided incremental novelty in the form of disparate explicit regularization terms, as well as the employment of various image metrics and tailored optimization methods. For several algorithms, the underlying gradient-based optimization retained the essential characteristics of Rueckert's original contribution. The contribution which we provide in this paper is two-fold: 1) the observation that the generic FFD framework is intrinsically susceptible to problematic energy topographies and 2) that the standard gradient used in FFD image registration can be modified to a well-understood preconditioned form which substantially improves performance. This is demonstrated with theoretical discussion and comparative evaluation experimentation. <s> BIB016
An approach to optimize the objective function is by following the direction that decreases the energy, or its negative gradient. In other words, the direction is given as . Klein et al. BIB013 studied two variants of gradient descent. The first employed a function of the step size that decayed with each iteration, while the second was based on the inexact line search algorithm of Moré and Thuente  . Other line strategies include keeping the step size fixed, monotone line search BIB003 , line search and golden section search  . Gradient descent has been used to solve various registration problems. In the LDDMM framework, usually posed in a variational setting, gradient descent is often used to solve the problem [67], BIB007 , BIB010 . Johnson and Christensen's consistent registration approach BIB005 as well as Rueckert et al.'s FFD registration algorithm BIB004 were also based on a gradient descent optimization scheme. Without trying to give a full account of all registration methods that employ gradient descent, let us also cite two more variational approaches BIB006 , BIB011 . 2) Conjugate Gradient Methods: Techniques that have better convergence rates than gradient descent have also been tested. Conjugate gradient descent methods try to exploit the knowledge conveyed by the previous gradients and propose a search direction that does not follow the new gradient but is conjugate to the previous direction. Thus, the direction now is given as , where usually denotes a linear combination, . Different ways to define the weighting factor . Among the well-known formulas for , one may cite the Fletcher-Reeves  , the Polak-Ribière  , the Polak-Ribière-Polyak BIB002 and the Hestenes-Stiefel BIB001 . For a review on CG methods, the interested reader is referred to the work of Hager and Zhang BIB012 . Some examples of registration methods that use conjugate gradient descent as an optimizer are BIB008 , BIB009 , BIB015 , BIB014 . An interesting approach tailored for FFD image registration using a preconditioned gradient scheme was presented in BIB016 . Tustison et al. BIB016 observed that problematic energy topologies appear in the standard gradient schemes for FFD image registration. This is caused by the nature of the uniform -splines that leads to disproportionate weighting of the control points. The authors proposed an approach to account for this fact by normalizing the gradient based on the spline basis functions.