Deformable Medical Image Registration: A Survey <s> 7) Stochastic Gradient Descent Methods: <s> A new information-theoretic approach is presented for finding the registration of volumetric medical images of differing modalities. Registration is achieved by adjustment of the relative position and orientation until the mutual information between the images is maximized. In our derivation of the registration procedure, few assumptions are made about the nature of the imaging process. As a result the algorithms are quite general and can foreseeably be used with a wide variety of imaging devices. This approach works directly with image data; no pre-processing or segmentation is required. This technique is, however, more flexible and robust than other intensity-based techniques like correlation. Additionally, it has an efficient implementation that is based on stochastic approximation. Experiments are presented that demonstrate the approach registering magnetic resonance (MR) images with computed tomography (CT) images, and with positron-emission tomography (PET) images. Surgical applications of the registration method are described. <s> BIB001 </s> Deformable Medical Image Registration: A Survey <s> 7) Stochastic Gradient Descent Methods: <s> A new information-theoretic approach is presented for finding the pose of an object in an image. The technique does not require information about the surface properties of the object, besides its shape, and is robust with respect to variations of illumination. In our derivation few assumptions are made about the nature of the imaging process. As a result the algorithms are quite general and may foreseeably be used in a wide variety of imaging situations. ::: ::: Experiments are presented that demonstrate the approach registering magnetic resonance (MR) images, aligning a complex 3D object model to real scenes including clutter and occlusion, tracking a human head in a video sequence and aligning a view-based 2D object model to real images. ::: ::: The method is based on a formulation of the mutual information between the model and the image. As applied here the technique is intensity-based, rather than feature-based. It works well in domains where edge or gradient-magnitude based methods have difficulty, yet it is more robust than traditional correlation. Additionally, it has an efficient implementation that is based on stochastic approximation. <s> BIB002 </s> Deformable Medical Image Registration: A Survey <s> 7) Stochastic Gradient Descent Methods: <s> Image registration is one of today's challenging image processing problems, particularly in medical imaging. Since the problem is ill posed, one may like to add additional information about distortions. This applies, for example, to the registration of time series of contrast-enhanced images, where variations of substructures are not related to patient motion but to contrast uptake. Here, one may only be interested in registrations which do not alter the volume of any substructure. In this paper, we discuss image registration techniques with a focus on volume preserving constraints. These constraints can reduce the non-uniqueness of the registration problem significantly. Our implementation is based on a constrained optimization formulation. Upon discretization, we obtain a large, discrete, highly nonlinear optimization problem and the necessary conditions for the solution form a discretized nonlinear partial differential equation. To solve the problem we use a variant of the sequential quadratic programming method. Finally, we present results on synthetic as well as on real-life data. <s> BIB003 </s> Deformable Medical Image Registration: A Survey <s> 7) Stochastic Gradient Descent Methods: <s> The goal of image registration is twofold. One goal is to enforce a certain similarity of two images by geometrically transforming one of the images. The second goal is to keep this transformation meaningful or regular. There exists a large amount of approaches aiming for regularity. Most of those are based on certain regularization techniques, others use so-called regridding options. ::: ::: Here, we present a mathematically sound formulation that explicitly controls the deformation in terms of the determinant of the Jacobian of the transformation. In contrast to similar work, we use pointwise inequality constraints, i.e., the volume is controlled voxel by voxel and not by integral measures. This approach guaranties grid regularity and prevent folding. ::: ::: As it turns out, the discretization of the volume constraint inequality is not straightforward. Therefore, we present a new type of discretization enabling the detection of twists in a pixel or a voxel. Such detection is crucial since a twists indicates that a transformation is physically meaningless. ::: ::: To solve the large-scale inequality constrained optimization problem, we present a numerical approach based on an interior point method. We finally present some numerical examples that demonstrate the advantage of including inequality constraints explicitly. <s> BIB004 </s> Deformable Medical Image Registration: A Survey <s> 7) Stochastic 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> BIB005 </s> Deformable Medical Image Registration: A Survey <s> 7) Stochastic Gradient Descent Methods: <s> In this work, we extend a previously demonstrated entropy based groupwise registration method to include a free-form deformation model based on B-splines. We provide an efficient implementation using stochastic gradient descents in a multi-resolution setting. We demonstrate the method in application to a set of 50 MRI brain scans and compare the results to a pairwise approach using segmentation labels to evaluate the quality of alignment. Our results indicate that increasing the complexity of the deformation model improves registration accuracy significantly, especially at cortical regions. <s> BIB006 </s> Deformable Medical Image Registration: A Survey <s> 7) Stochastic Gradient Descent Methods: <s> We present a stochastic gradient descent optimisation method for image registration with adaptive step size prediction. The method is based on the theoretical work by Plakhov and Cruz (J. Math. Sci. 120(1):964---973, 2004). Our main methodological contribution is the derivation of an image-driven mechanism to select proper values for the most important free parameters of the method. The selection mechanism employs general characteristics of the cost functions that commonly occur in intensity-based image registration. Also, the theoretical convergence conditions of the optimisation method are taken into account. The proposed adaptive stochastic gradient descent (ASGD) method is compared to a standard, non-adaptive Robbins-Monro (RM) algorithm. Both ASGD and RM employ a stochastic subsampling technique to accelerate the optimisation process. Registration experiments were performed on 3D CT and MR data of the head, lungs, and prostate, using various similarity measures and transformation models. The results indicate that ASGD is robust to these variations in the registration framework and is less sensitive to the settings of the user-defined parameters than RM. The main disadvantage of RM is the need for a predetermined step size function. The ASGD method provides a solution for that issue. <s> BIB007 </s> Deformable Medical Image Registration: A Survey <s> 7) Stochastic Gradient Descent Methods: <s> Nonrigid image registration methods using intensity-based similarity metrics are becoming increasingly common tools to estimate many types of deformations. Nonrigid warps can be very flexible with a large number of parameters and gradient optimization schemes are widely used to estimate them. However, for large datasets, the computation of the gradient of the similarity metric with respect to these many parameters becomes very time consuming. Using a small random subset of image voxels to approximate the gradient can reduce computation time. This work focuses on the use of importance sampling to reduce the variance of this gradient approximation. The proposed importance sampling framework is based on an edge-dependent adaptive sampling distribution designed for use with intensity-based registration algorithms. We compare the performance of registration based on stochastic approximations with and without importance sampling to that using deterministic gradient descent. Empirical results, on simulated magnetic resonance brain data and real computed tomography inhale-exhale lung data from eight subjects, show that a combination of stochastic approximation methods and importance sampling accelerates the registration process while preserving accuracy. <s> BIB008 </s> Deformable Medical Image Registration: A Survey <s> 7) Stochastic Gradient Descent Methods: <s> Radiation therapy for cervical cancer can benefit from image registration in several ways, for example by studying the motion of organs, or by (partially) automating the delineation of the target volume and other structures of interest. In this paper, the registration of cervical data is addressed using mutual information (MI) of not only image intensity, but also features that describe local image structure. Three aspects of the registration are addressed to make this approach feasible. First, instead of relying on a histogram-based estimation of mutual information, which poses problems for a larger number of features, a graph-based implementation of alpha-mutual information (alpha-MI) is employed. Second, the analytical derivative of alpha-MI is derived. This makes it possible to use a stochastic gradient descent method to solve the registration problem, which is substantially faster than nonderivative-based methods. Third, the feature space is reduced by means of a principal component analysis, which also decreases the registration time. The proposed technique is compared to a standard approach, based on the mutual information of image intensity only. Experiments are performed on 93 T2-weighted MR clinical data sets acquired from 19 patients with cervical cancer. Several characteristics of the proposed algorithm are studied on a subset of 19 image pairs (one pair per patient). On the remaining data (36 image pairs, one or two pairs per patient) the median overlap is shown to improve significantly compared to standard MI from 0.85 to 0.86 for the clinical target volume (CTV, p = 2 x 10(-2)), from 0.75 to 0.81 for the bladder (p = 8 x 10(-6)), and from 0.76 to 0.77 for the rectum (p = 2 x 10(-4)). The registration error is improved at important tissue interfaces, such as that of the bladder with the CTV, and the interface of the rectum with the uterus and cervix. <s> BIB009 </s> Deformable Medical Image Registration: A Survey <s> 7) Stochastic Gradient Descent Methods: <s> It is implicitly assumed in most non-rigid registration methods that either corresponding structures can be found in both images, or that the regularization associated with the class of transformations chosen produce pertinent deformations elsewhere. However, when the images to register have a different contrast and resolution (e.g. in vivo / post mortem), or to minimize the deformation of some specific tissues (e.g. tumor, bones), it is necessary to have local control on the displacement field in these regions. We propose in this work an original registration method based on cubic B-spline transformations which allows to bound the variation of volume induced by the estimated transformation (variation of the determinant of the Jacobian of the transformation relatively to 1) in a predefined region. Lagrange multipliers are used to perform the constrained minimization of mutual information. We tested our method on artificial images with different tolerances on the volume variation and on one real 3D image: it performed efficiently while respecting the constraints. <s> BIB010
The aforementioned techniques cover the deterministic gradient methods that are used most often to solve the optimization problems that arise when tackling image registration. In medical image registration, the computation of the derivative information can be computationally demanding because of the great dimensionality of both the data and the search space. Thus, to alleviate the computational burden, researchers have investigated the use of stochastic gradient approaches. Their update rule is based on an approximation of the gradient, or . The variants of the stochastic gradient approach differ with respect to how the gradient is approximated. In BIB005 , three approaches were discussed. The first one, referred to as Kiefer-Wolfowitz (KW), approximates the gradient by a finite difference scheme. The second one, known as Simultaneous Perturbation (SP), estimates the gradient by perturbing it not along the basis axis but instead along a random perturbation vector whose elements are independent and symmetrically Bernoulli distributed. The last method was proposed by Robbins and Monro (RM). It is more general, in the sense that it only assumes that an approximation of the gradient exists. This method uses a step-size that decreases with time in order to decrease the inaccuracy. BIB005 estimated the gradient by using a subset of the image voxels sampled uniformly. The conclusion of BIB005 is that the RM method performs best. The RM method was extended in two subsequent works by employing adaptive image-driven strategies. Klein et al. BIB007 presented an adaptive step mechanism, while Bhagalia et al. BIB008 proposed an edge-driven importance sampling to improve the gradient approximation. Stochastic gradient descent schemes have been applied in image registration settings that employ lower degrees of freedom deformation models (e.g., global linear or cubic -spline FFDs). For some applications of stochastic gradient see BIB006 , BIB001 , BIB002 , BIB009 . 8) Constrained Optimization Methods: All the previous approaches aim to solve an unconstrained optimization problem. As discussed in Section II-D, constrained optimization problems arise when trying to impose task-specific conditions on the deformation field. The solution of such optimization problems is more challenging. The optimization strategies that are usually employed transform the constrained to an unconstrained one that can be solved efficiently. For example, a log-barrier method was used in BIB004 . Another way to solve the problem is by augmenting the dimensionality of the problem using the method of Lagrange multipliers BIB003 , BIB010 .