Deformable Medical Image Registration: A Survey <s> 5) Locally Affine Models: <s> A new method for medical image registration is formulated as a minimization problem involving robust estimators. The authors propose an efficient hierarchical optimization framework which is both multiresolution and multigrid. An anatomical segmentation of the cortex is introduced in the adaptive partitioning of the volume on which the multigrid minimization is based. This allows to limit the estimation to the areas of interest, to accelerate the algorithm, and to refine the estimation in specified areas. At each stage of the hierarchical estimation, the authors refine current estimate by seeking a piecewise affine model for the incremental deformation field. The performance of this method is numerically evaluated on simulated data and its benefits and robustness are shown on a database of 18 magnetic resonance imaging scans of the head. <s> BIB001 </s> Deformable Medical Image Registration: A Survey <s> 5) Locally Affine Models: <s> In this paper we present a novel deformable registration algorithm for diffusion tensor (DT) MR images that enables explicit analytic optimization of tensor reorientation. The optimization seeks a piecewise affine transformation that divides the image domain into uniform regions and transforms each of them affinely. The objective function captures both the image similarity and the smoothness of the transformation across region boundaries. The image similarity enables explicit orientation optimization by incorporating tensor reorientation, which is necessary for warping DT images. The objective function is formulated in a way that allows explicit implementation of analytic derivatives to drive fast and accurate optimization using the conjugate gradient method. The optimal transformation is hierarchically refined in a subdivision framework. A comparison with affine registration for inter-subject normalization of 8 subjects shows that our algorithm improves the alignment of manually segmented white matter structures (corpus callosum and cortio-spinal tracts). <s> BIB002 </s> Deformable Medical Image Registration: A Survey <s> 5) Locally Affine Models: <s> Abstract This manuscript tackles the reconstruction of 3-D volumes via mono-modal registration of series of 2-D biological images (histological sections, autoradiographs, cryosections, etc.). The process of acquiring these images typically induces composite transformations that we model as a number of rigid or affine local transformations embedded in an elastic one. We propose a registration approach closely derived from this model. Given a pair of input images, we first compute a dense similarity field between them with a block matching algorithm. We use as a similarity measure an extension of the classical correlation coefficient that improves the consistency of the field. A hierarchical clustering algorithm then automatically partitions the field into a number of classes from which we extract independent pairs of sub-images. Our clustering algorithm relies on the Earth mover’s distribution metric and is additionally guided by robust least-square estimation of the transformations associated with each cluster. Finally, the pairs of sub-images are, independently, affinely registered and a hybrid affine/non-linear interpolation scheme is used to compose the output registered image. We investigate the behavior of our approach on several batches of histological data and discuss its sensitivity to parameters and noise. <s> BIB003 </s> Deformable Medical Image Registration: A Survey <s> 5) Locally Affine Models: <s> Many types of transformations are used to model deformations in medical image registration. While some focus on modeling local changes, some on continuity and invertibility, there is no closed-form nonlinear parametric approach that addresses all these properties. This paper presents a class of nonlinear transformations that are local, continuous and invertible under certain conditions. They are straightforward to implement, fast to compute and can be used particularly in cases where locally affine deformations need to be recovered. We use our new transformation model to demonstrate some results on synthetic images using a multi-scale approach to multi-modality mutual information based image registration. The original images were deformed using B-splines at three levels of scale. The results show that the proposed method can recover these deformations almost completely with very few iterations of a gradient based optimizer. <s> BIB004 </s> Deformable Medical Image Registration: A Survey <s> 5) Locally Affine Models: <s> Abstract We describe in this paper a novel kind of geometrical transformations, named polyrigid and polyaffine. These transformations efficiently code for locally rigid or affine deformations with a small number of intuitive parameters. They can describe compactly large rigid or affine movements, unlike most free-form deformation classes. Very flexible, this tool can be readily adapted to a large variety of situations, simply by tuning the number of rigid or affine components and the number of parameters describing their regions of influence. The displacement of each spatial position is defined by a continuous trajectory that follows a differential equation which averages the influence of each rigid or affine component. We show that the resulting transformations are diffeomorphisms, smooth with respect to their parameters. We devise a new and flexible numerical scheme to allow a trade-off between computational efficiency and closeness to the ideal diffeomorphism. Our algorithms are implemented within the Insight Toolkit, whose generic programming style offers rich facilities for prototyping. In this context, we derive an effective optimization strategy of the transformations which demonstrates that this new tool is highly suitable for inference. The whole framework is exemplified successfully with the registration of histological slices. This choice is challenging, because these data often present locally rigid deformations added during their acquisition, and can also present a loss of matter, which makes their registration even more difficult. Powerful and flexible, this new tool opens up large perspectives, in non-rigid 3D rigid registration as well as in shape statistics. <s> BIB005 </s> Deformable Medical Image Registration: A Survey <s> 5) Locally Affine Models: <s> In this article, we focus on the parameterization of non-rigid geometrical deformations with a small number of flexible degrees of freedom. In previous work, we proposed a general framework called polyaffine to parameterize deformations with a finite number of rigid or affine components, while guaranteeing the invertibility of global deformations. However, this framework lacks some important properties: the inverse of a polyaffine transformation is not polyaffine in general, and the polyaffine fusion of affine components is not invariant with respect to a change of coordinate system. We present here a novel general framework, called Log-Euclidean polyaffine, which overcomes these defects. ::: ::: We also detail a simple algorithm, the Fast Polyaffine Transform, which allows to compute very efficiently Log-Euclidean polyaffine transformations and their inverses on regular grids. The results presented here on real 3D locally affine registration suggest that our novel framework provides a general and efficient way of fusing local rigid or affine deformations into a global invertible transformation without introducing artifacts, independently of the way local deformations are first estimated. <s> BIB006 </s> Deformable Medical Image Registration: A Survey <s> 5) Locally Affine Models: <s> Groupwise image registration algorithms seek to establish dense correspondences between sets of images. Typically, they involve iteratively improving the registration between each image and an evolving mean. A variety of methods have been proposed, which differ in their choice of objective function, representation of deformation field, and optimization methods. Given the complexity of the task, the final accuracy is significantly affected by the choices made for each component. Here, we present a groupwise registration algorithm which can take advantage of the statistics of both the image intensities and the range of shapes across the group to achieve accurate matching. By testing on large sets of images (in both 2D and 3D), we explore the effects of using different image representations and different statistical shape constraints. We demonstrate that careful choice of such representations can lead to significant improvements in overall performance. <s> BIB007 </s> Deformable Medical Image Registration: A Survey <s> 5) Locally Affine Models: <s> Abstract Non-rigid image registration techniques are commonly used to estimate complex tissue deformations in medical imaging. A range of non-rigid registration algorithms have been proposed, but they typically have high computational complexity. To reduce this complexity, combinations of multiple less complex deformations have been proposed such as hierarchical techniques which successively split the non-rigid registration problem into multiple locally rigid or affine components. However, to date the splitting has been regular and the underlying image content has not been considered in the splitting process. This can lead to errors and artefacts in the resulting motion fields. In this paper, we propose three novel adaptive splitting techniques, an image-based, a similarity-based, and a motion-based technique within a hierarchical framework which attempt to process regions of similar motion and/or image structure in single registration components. We evaluate our technique on free-breathing whole-chest 3D MRI data from 10 volunteers and two publicly available CT datasets. We demonstrate a reduction in registration error of up to 49.1% over a non-adaptive technique and compare our results with a commonly used free-form registration algorithm. <s> BIB008
Locally affine models parametrize the transformation by locally linear deformations. One may discern two different cases: 1) piecewise affine models, and 2) poly-affine ones. In the first case, the image is mosaicked by a set of triangles or tetrahedra whose nodes parametrize the deformation. Inside each region, affine interpolation takes place. Efficiency and invertibility are the main strengths of this method, while lack of smoothness in the region boundaries is its main limitation. In the second case, fuzzy regions are used in order to tackle the aforementioned drawback and produce a smooth transformation. a) Piecewise Affine Models: Some of the most recent approaches using a piecewise affine model include, but are not limited to, the following. Hellier et al. BIB001 proposed a multi-resolution and multi-grid approach. The image was partitioned adaptively into cubes and an affine transformation was inferred for each one. A regularization energy term encouraged neighboring pairs to deform similarly. In a similar fashion, Zhang et al. BIB002 tackled diffusion tensor registration by taking into consideration tensor reorientation. The images were separated into contiguous blocks and an affine transformation was recovered for each one of them. Regularization on the interface of regions ensured the global smoothness of the transformation. Pitiot et al. BIB003 reconstructed volumes of histological images by employing a piecewise affine transformation model. The images were separated into independent components through hierarchical clustering. In a subsequent step, affine registration was performed for each pair of regions. The final transformation was estimated by calculating the affine transformation for each region and applying a nonlinear interpolation in between the regions. Commowick et al. presented similar approach was presented in  . The main difference between the two methods lies in the fact that a regularization step followed to improve the smoothness in the interpolated areas. The regularization was based on the Log-Euclidean framework using Euclidean differences between the logarithms of the affine transformations. Two more recent applications of piecewise affine models were presented in BIB007 , BIB008 . Cootes et al. BIB007 favored the use of piecewise affine transformations as they can be easily inverted. Buerger et al. BIB008 proposed a hierarchical framework to adaptively separate the images into regions. Splitting was formulated as an energy minimization problem and three criteria were used. The first criterion tried to group regions with rich structural information. The second criterion grouped regions with significant residual error in large blocks, while the last criterion encouraged regions with similar motion to be considered together. The second was found to perform best. Most approaches that employ piecewise linear strategies consider the affine transformations independently. As a result, singularities may occur and the transformation is not globally invertible. To account for this drawback, sophisticated methods have been introduced. Narayanan et al. proposed a transformation model that is affine at the center of a region and reduces to identity as the distance from the center increases BIB004 . This novel transformation model has a closed form and can be computed efficiently. Moreover, constraints were given in the form of bounds on the translation so that invertibility is ensured. b) Poly-Affine Models: Arsigny et al. BIB005 presented a poly-rigid/affine transformation model. the transformation is parametrized by a set of anchor points , a parameter that defines the importance of every point and a distance . Fuzzy regions are defined by calculating the influence of an anchor point at each position of the image as denotes a Gaussian function parametrized by a mean value and a standard deviation . Given the transformation of the set of anchor points, the global transformation at each point is given by a distance-weighted sum of infinitesimal velocities at the known points, integrated over time. No closed form exists and a computationally expensive integration of ODEs is necessary. Arsigny et al. BIB006 extended the poly-affine transformation so that its inverse is also poly-affine. Moreover, the fusion of affine transformations was rendered invariant to affine changes of the coordinate system.