Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> We study the use of the BFGS and DFP algorithms with step-lengths of one for minimizing quadratic functions of only two variables. The updating formulae in this case imply nonlinear three term recurrence relations between the eigenvalues of consecutive second derivative approximations, which are analysed in order to explain some gross inefficiencies that can occur. Specifically, the BFGS algorithm may require more than 10 iterations to achieve the first decimal place of accuracy, while the performance of the DFP method is far worse. The results help to explain why the DFP method is often less suitable than the BFGS algorithm for general unconstrained optimization calculations, and they show that quadratic functions provide much information about efficiency when the current vector of variables is too far from the solution for an asymptotic convergence analysis. <s> BIB001 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> A new approach to the problem of multimodality medical image registration is proposed, using a basic concept from information theory, mutual information (MI), or relative entropy, as a new matching criterion. The method presented in this paper applies MI to measure the statistical dependence or information redundancy between the image intensities of corresponding voxels in both images, which is assumed to be maximal if the images are geometrically aligned. Maximization of MI is a very general and powerful criterion, because no assumptions are made regarding the nature of this dependence and no limiting constraints are imposed on the image content of the modalities involved. The accuracy of the MI criterion is validated for rigid body registration of computed tomography (CT), magnetic resonance (MR), and photon emission tomography (PET) images by comparison with the stereotactic registration solution, while robustness is evaluated with respect to implementation issues, such as interpolation and optimization, and image content, including partial overlap and image degradation. Our results demonstrate that subvoxel accuracy with respect to the stereotactic reference solution can be achieved completely automatically and without any prior segmentation, feature extraction, or other preprocessing steps which makes this method very well suited for clinical applications. <s> BIB002 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> Mutual information has developed into an accurate measure for rigid and affine monomodality and multimodality image registration. The robustness of the measure is questionable, however. A possible reason for this is the absence of spatial information in the measure. The present paper proposes to include spatial information by combining mutual information with a term based on the image gradient of the images to be registered. The gradient term not only seeks to align locations of high gradient magnitude, but also aims for a similar orientation of the gradients at these locations. Results of combining both standard mutual information as well as a normalized measure are presented for rigid registration of three-dimensional clinical images [magnetic resonance (MR), computed tomography (CT), and positron emission tomography (PET)]. The results indicate that the combined measures yield a better registration function does mutual information or normalized mutual information per se. The registration functions are less sensitive to low sampling resolution, do not contain incorrect global maxima that are sometimes found in the mutual information function, and interpolation-induced local minima can be reduced. These characteristics yield the promise of more robust registration measures. The accuracy of the combined measures is similar to that of mutual information-based methods. <s> BIB003 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> A measure for registration of medical images that currently draws much attention is mutual information. The measure originates from information theory, but has been shown to be successful for image registration as well. Information theory, however, offers many more measures that may be suitable for image registration. These all measure the divergence of the joint distribution of the images' grey values from the joint distribution that would have been found had the images been completely independent. This paper compares the performance of mutual information as a registration measure with that of other f-information measures. The measures are applied to rigid registration of positron emission tomography(PET)/magnetic resonance (MR) and MR/computed tomography (CT) images, for 35 and 41 image pairs, respectively. An accurate gold standard transformation is available for the images, based on implanted markers. The registration performance, robustness and accuracy of the measures are studied. Some of the measures are shown to perform poorly on all aspects. The majority of measures produces results similar to those of mutual information. An important finding, however, is that several measures, although slightly more difficult to optimize, can potentially yield significantly more accurate results than mutual information. <s> BIB004 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> As image registration becomes more and more central to many biomedical imaging applications, the efficiency of the algorithms becomes a key issue. Image registration is classically performed by optimizing a similarity criterion over a given spatial transformation space. Even if this problem is considered as almost solved for linear registration, we show in this paper that some tools that have recently been developed in the field of vision-based robot control can outperform classical solutions. The adequacy of these tools for linear image registration leads us to revisit non-linear registration and allows us to provide interesting theoretical roots to the different variants of Thirion's demons algorithm. This analysis predicts a theoretical advantage to the symmetric forces variant of the demons algorithm. We show that, on controlled experiments, this advantage is confirmed, and yields a faster convergence. <s> BIB005 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> In this paper, several vision-based robot control methods are classified following an analogy with well known minimization methods. Comparing the rate of convergence between minimization algorithms helps us to understand the difference of performance of the control schemes. In particular, it is shown that standard vision-based control methods have in general low rates of convergence. Thus, the performance of vision-based control could be improved using schemes which perform like the Newton minimization algorithm that has a high convergence rate. Unfortunately, the Newton minimization method needs the computation of second derivatives that can be ill-conditioned causing convergence problems. In order to solve these problems, this paper proposes two new control schemes based on efficient second-order minimization techniques. <s> BIB006 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> Estimating a meaningful average or mean shape from a set of shapes represented by unlabeled point-sets is a challenging problem since, usually this involves solving for point correspondence under a non-rigid motion setting. In this paper, we propose a novel and robust algorithm that is capable of simultaneously computing the mean shape from multiple unlabeled point-sets (represented by finite mixtures) and registering them nonrigidly to this emerging mean shape. This algorithm avoids the correspondence problem by minimizing the Jensen-Shannon (JS) divergence between the point sets represented as finite mixtures. We derive the analytic gradient of the cost function namely, the JS-divergence, in order to efficiently achieve the optimal solution. The cost function is fully symmetric with no bias toward any of the given shapes to be registered and whose mean is being sought. Our algorithm can be especially useful for creating atlases of various shapes present in images as well as for simultaneously (rigidly or non-rigidly) registering 3D range data sets without having to establish any correspondence. We present experimental results on non-rigidly registering 2D as well as 3D real data (point sets). <s> BIB007 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <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> BIB008 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <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> BIB009 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> We propose conditional mutual information (cMI) as a new similarity measure for nonrigid image registration. We start from a 3D joint histogram incorporating, besides the reference and floating intensity dimensions, also a spatial dimension expressing the location of the joint intensity pair in the reference image. cMI is calculated as the expectation value of the conditional mutual information between the reference and floating intensities given the spatial distribution. ::: ::: Validation experiments were performed comparing cMI and global MI on artificial CT/MR registrations and registrations complicated with a strong bias field; both a Parzen window and generalised partial volume kernel were used for histogram construction. In both experiments, cMI significantly outperforms global MI. Moreover, cMI is compared to global MI for the registration of three patient CT/MR datasets, using overlap and centroid distance as validation measure. The best results are obtained using cMI. <s> BIB010 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> Registration is a technique nowadays commonly used in medical imaging. A drawback of most of the current registration schemes is that all tissue is being considered as non-rigid (Staring et al., Proceedings of the SPIE 2006, vol. 6144, pp. 1---10, 2006). Therefore, rigid objects in an image, such as bony structures or surgical instruments, may be transformed non-rigidly. In this paper, we integrate the concept of local rigidity to the FLexible Image Registration Toolbox (FLIRT) (Haber and Modersitzki, in SIAM J. Sci. Comput. 27(5):1594---1607, 2006; Modersitzki, Numerical Methods for Image Registration, 2004). The idea is to add a penalty for local non-rigidity to the cost function and thus to penalize non-rigid transformations of rigid objects. As our examples show, the new approach allows the maintenance of local rigidity in the desired fashion. For example, the new scheme can keep bony structures rigid during registration. ::: ::: We show, how the concept of local rigidity can be integrated in the FLIRT approach and present the variational backbone, a proper discretization, and a multilevel optimization scheme. We compare the FLIRT approach to the B-spline approach. As expected from the more general setting of the FLIRT approach, our examples demonstrate that the FLIRT results are superior: much smoother, smaller deformations, visually much more pleasing. <s> BIB011 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> We propose a non-parametric diffeomorphic image registration algorithm based on Thirion's demons algorithm. The demons algorithm can be seen as an optimization procedure on the entire space of displacement fields. The main idea of our algorithm is to adapt this procedure to a space of diffeomorphic transformations. In contrast to many diffeomorphic registration algorithms, our solution is computationally efficient since in practice it only replaces an addition of free form deformations by a few compositions. Our experiments show that in addition to being diffeomorphic, our algorithm provides results that are similar to the ones from the demons algorithm but with transformations that are much smoother and closer to the true ones in terms of Jacobians. <s> BIB012 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> To make up for the lack of concern on the spatial information in the conventional mutual information based image registration framework, this paper designs a novel spatial feature field, namely the maximum distance-gradient (MDG) vector field, for registration tasks. It encodes both the local edge information and globally defined spatial information related to the intensity difference, the distance, and the direction of a voxel to a MDG source point. A novel similarity measure is proposed as the combination of the multi-dimensional mutual information and an angle measure on the MDG vector field. This measure integrates both the magnitude and orientation information of the MDG vector field into the image registration process. Experimental results on clinical 3D CT and T1-weighted MR image volumes show that, as compared with the conventional mutual information based method and two of its adaptations incorporating spatial information, the proposed method can give longer capture ranges at different image resolutions. This leads to more robust registrations. Around 2000 randomized rigid registration experiments demonstrate that our method consistently gives much higher success rates than the aforementioned three related methods. Moreover, it is shown that the registration accuracy of our method is high. <s> BIB013 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> We propose an efficient non-parametric diffeomorphic image registration algorithm based on Thirion's demons algorithm. In the first part of this paper, we show that Thirion's demons algorithm can be seen as an optimization procedure on the entire space of displacement fields. We provide strong theoretical roots to the different variants of Thirion's demons algorithm. This analysis predicts a theoretical advantage for the symmetric forces variant of the demons algorithm. We show on controlled experiments that this advantage is confirmed in practice and yields a faster convergence. In the second part of this paper, we adapt the optimization procedure underlying the demons algorithm to a space of diffeomorphic transformations. In contrast to many diffeomorphic registration algorithms, our solution is computationally efficient since in practice it only replaces an addition of displacement fields by a few compositions. Our experiments show that in addition to being diffeomorphic, our algorithm provides results that are similar to the ones from the demons algorithm but with transformations that are much smoother and closer to the gold standard, available in controlled experiments, in terms of Jacobians. <s> BIB014 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> Modern morphometric studies use non-linear image registration to compare anatomies and perform group analysis. Recently, log-Euclidean approaches have contributed to promote the use of such computational anatomy tools by permitting simple computations of statistics on a rather large class of invertible spatial transformations. In this work, we propose a non-linear registration algorithm perfectly fit for log-Euclidean statistics on diffeomorphisms. Our algorithm works completely in the log-domain, i.e. it uses a stationary velocity field. This implies that we guarantee the invertibility of the deformation and have access to the true inverse transformation. This also means that our output can be directly used for log-Euclidean statistics without relying on the heavy computation of the log of the spatial transformation. As it is often desirable, our algorithm is symmetric with respect to the order of the input images. Furthermore, we use an alternate optimization approach related to Thirion's demons algorithm to provide a fast non-linear registration algorithm. First results show that our algorithm outperforms both the demons algorithm and the recently proposed diffeomorphic demons algorithm in terms of accuracy of the transformation while remaining computationally efficient. <s> BIB015 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> This paper proposes a new registration method for the in vivo quantification of cardiac deformation from a sequence of possibly noisy images. Our algorithm has been applied to 3D ultrasound (3D-US) images, which currently give a reasonable spatial and time resolution, but suffer from significant acquisition noise. Therefore, this modality requires the design of a robust strategy to quantify motion and deformation with the clinical aim of better quantifying cardiac function e.g. in heart failure. In the proposed method, referred to as Large Diffeomorphic Free Form Deformation (LDFFD), the displacement field at each time step is computed from a smooth non-stationary velocity field, thus imposing a coupling between the transformations at successive time steps. Our contribution is to extend this framework to the estimation of motion and deformation in an image sequence. Similarity is captured for the entire image sequence using an extension of the pairwise mutual information metric. The LDFFD algorithm is applied here to recover longitudinal strain curves from healthy and Left-Bundle Branch Block (LBBB) subjects. Strain curves for the healthy subjects are in accordance with the literature. For the LBBB patient, strain quantified before and after Cardiac Resynchronization Therapy show a clear improvement of cardiac function in this subject, in accordance with clinical observations. <s> BIB016 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> In this paper, we propose the DT-REFinD algorithm for the diffeomorphic nonlinear registration of diffusion tensor images. Unlike scalar images, deforming tensor images requires choosing both a reorientation strategy and an interpolation scheme. Current diffusion tensor registration algorithms that use full tensor information face difficulties in computing the differential of the tensor reorientation strategy and consequently, these methods often approximate the gradient of the objective function. In the case of the finite-strain (FS) reorientation strategy, we borrow results from the pose estimation literature in computer vision to derive an analytical gradient of the registration objective function. By utilizing the closed-form gradient and the velocity field representation of one parameter subgroups of diffeomorphisms, the resulting registration algorithm is diffeomorphic and fast. We contrast the algorithm with a traditional FS alternative that ignores the reorientation in the gradient computation. We show that the exact gradient leads to significantly better registration at the cost of computation time. Independently of the choice of Euclidean or Log-Euclidean interpolation and sum of squared differences dissimilarity measure, the exact gradient achieves better alignment over an entire spectrum of deformation penalties. Alignment quality is assessed with a battery of metrics including tensor overlap, fractional anisotropy, inverse consistency and closeness to synthetic warps. The improvements persist even when a different reorientation scheme, preservation of principal directions, is used to apply the final deformations. <s> BIB017 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> We present the Spherical Demons algorithm for registering two spherical images. By exploiting spherical vector spline interpolation theory, we show that a large class of regularizors for the modified Demons objective function can be efficiently approximated on the sphere using iterative smoothing. Based on one parameter subgroups of diffeomorphisms, the resulting registration is diffeomorphic and fast. The Spherical Demons algorithm can also be modified to register a given spherical image to a probabilistic atlas. We demonstrate two variants of the algorithm corresponding to warping the atlas or warping the subject. Registration of a cortical surface mesh to an atlas mesh, both with more than 160 k nodes requires less than 5 min when warping the atlas and less than 3 min when warping the subject on a Xeon 3.2 GHz single processor machine. This is comparable to the fastest nondiffeomorphic landmark-free surface registration algorithms. Furthermore, the accuracy of our method compares favorably to the popular FreeSurfer registration algorithm. We validate the technique in two different applications that use registration to transfer segmentation labels onto a new image 1) parcellation of in vivo cortical surfaces and 2) Brodmann area localization in ex vivo cortical surfaces. <s> BIB018 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> Tracking soft tissues in medical images using non-linear image registration algorithms requires methods that are fast and provide spatial transformations consistent with the biological characteristics of the tissues. LogDemons algorithm is a fast non-linear registration method that computes diffeomorphic transformations parameterised by stationary velocity fields. Although computationally efficient, its use for tissue tracking has been limited because of its ad-hoc Gaussian regularisation, which hampers the implementation of more biologically motivated regularisations. In this work, we improve the logDemons by integrating elasticity and incompressibility for soft-tissue tracking. To that end, a mathematical justification of demons Gaussian regularisation is proposed. Building on this result, we replace the Gaussian smoothing by an efficient elastic-like regulariser based on isotropic differential quadratic forms of vector fields. The registration energy functional is finally minimised under the divergence-free constraint to get incompressible deformations. As the elastic regulariser and the constraint are linear, the method remains computationally tractable and easy to implement. Tests on synthetic incompressible deformations showed that our approach outperforms the original logDemons in terms of elastic incompressible deformation recovery without reducing the image matching accuracy. As an application, we applied the proposed algorithm to estimate 3D myocardium strain on clinical cine MRI of two adult patients. Results showed that incompressibility constraint improves the cardiac motion recovery when compared to the ground truth provided by 3D tagged MRI. <s> BIB019 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> Purpose: Four-dimensional computed tomography (4D CT) can provide patient-specific motion information for radiotherapy planning and delivery. Motion estimation in 4D CT is challenging due to the reduced image quality and the presence of artifacts. We aim to improve the robustness of deformable registration applied to respiratory-correlated imaging of the lungs, by using a global problem formulation and pursuing a restrictive parametrization for the spatiotemporal deformation model. <s> BIB020 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> This paper presents a nonlinear image registration algorithm based on the setting of Large Deformation Diffeomorphic Metric Mapping (LDDMM), but with a more efficient optimisation scheme — both in terms of memory required and the number of iterations required to reach convergence. Rather than perform a variational optimisation on a series of velocity fields, the algorithm is formulated to use a geodesic shooting procedure, so that only an initial velocity is estimated. A Gauss–Newton optimisation strategy is used to achieve faster convergence. The algorithm was evaluated using freely available manually labelled datasets, and found to compare favourably with other inter-subject registration algorithms evaluated using the same data. <s> BIB021 </s> Deformable Medical Image Registration: A Survey <s> 3) Powell's Conjugate Directions Method: <s> We present a preconditioning scheme for improving the efficiency of optimization of arbitrary difference measures in deformable registration problems. This is of particular interest for high-dimensional registration problems with statistical difference measures such as MI, and the demons method, since in these cases the range of applicable optimization methods is limited. The proposed scheme is simple and computationally efficient: It performs an approximate normalization of the point-wise vectors of the difference gradient to unit length. The major contribution of this work is a theoretical analysis which demonstrates the improvement of the condition by our approach, which is furthermore shown to be an approximation to the optimal case for the analyzed model. Our scheme improves the convergence speed while adding only negligible computational cost, thus resulting in shorter effective runtimes. The theoretical findings are confirmed by experiments on 3D brain data. <s> BIB022
Powell's optimization approach or the Direction Set method  is another method that has been used in image registration. Powell's method aims to minimize the objective function by following conjugate directions. Contrary to the CG methods, the conjugate directions are calculated without the use of gradient information. The basic procedure that Powell proposed sets the initial direction to the basis vectors ; optimizes along each parameter axis independently from the rest; performs the replacement while adding and iterates until convergence. Powell's method is gradient free and has been applied in low degrees of freedom registration tasks e.g., BIB002 , BIB004 ,  , BIB003 , BIB013 . A drawback of Powell's method is that it tends to find search directions that are linearly dependent  . As a consequence, the optimization fails even for moderate scale problems. 4) Quasi-Newton Methods: Another class of optimization methods that has been tested in registration applications is that of Quasi-Newton (QN) methods  . This class of methods aims to accumulate information from the previous iterations and take advantage of it in order to achieve better convergence. More specifically, these methods aim to estimate the inverse Hessian matrix and use it to define the search direction. Thus, the search direction is defined as , where the denotes that an approximation is used (the true Hessian is used in the case of Newton's or the Newton-Raphson method). Two main algorithms exist in this category, the Davidon-Fletcher-Powell (DFP) and the Broyden-Fletcher-Goldfarb-Shanno (BFGS). BFGS is considered to be more efficient than DFP BIB001 ,  . A version of BFGS that uses less memory (L-BFGS) was tested in BIB009 . Other efforts where researchers have investigated the use of Quasi-Newton methods in image registration can be found in BIB016 , BIB020 , BIB007 , BIB010 . 5) Gauss-Newton Method: An optimization method of the same family is the Gauss-Newton (GN) algorithm. It is devised to solve optimization problems involving sum of squared function values. This is of particular interest for image registration as this type of objective function is common when aligning images of the same modality. This algorithm does not require the computation of second derivatives. Instead, the Hessian is approximated by ignoring derivatives higher than first order with where denotes the Jacobian. The search direction is now given as . The Gauss-Newton optimizer has been used in BIB021 , BIB008 , BIB011 . The Gauss-Newton algorithm is frequently used in the Demons registration framework to optimize the similarity measure when tackling mono-modal registration BIB005 , BIB012 , BIB017 , BIB018 , BIB014 . In the demons registration setting, an extension of Gauss-Newton by Malis BIB006 was employed to derive the symmetric demons forces BIB015 , BIB019 . This algorithm exploits more knowledge with respect to the problem at hand. More specifically, it takes advantage of the fact that when the images are aligned, the gradient of the source can be approximated by the gradient of the target. Recently, Zikic et al. BIB022 proposed a preconditioning scheme that improves the convergence speed of registration algorithms. The scheme is based on normalizing the length of the point force vectors.