Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <s> A general automatic approach is presented for accommodating local shape variation when mapping a two-dimensional (2-D) or three-dimensional (3-D) template image into alignment with a topologically similar target image. Local shape variability is accommodated by applying a vector-field transformation to the underlying material coordinate system of the template while constraining the transformation to be smooth (globally positive definite Jacobian). Smoothness is guaranteed without specifically penalizing large-magnitude deformations of small subvolumes by constraining the transformation on the basis of a Stokesian limit of the fluid-dynamical Navier-Stokes equations. This differs fundamentally from quadratic penalty methods, such as those based on linearized elasticity or thin-plate splines, in that stress restraining the motion relaxes over time allowing large-magnitude deformations. Kinematic nonlinearities are inherently necessary to maintain continuity of structures during large-magnitude deformations, and are included in all results. After initial global registration, final mappings are obtained by numerically solving a set of nonlinear partial differential equations associated with the constrained optimization problem. Automatic regridding is performed by propagating templates as the nonlinear transformations evaluated on a finite lattice become singular. Application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability. <s> BIB001 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <s> We present a topology preserving image matching method, based on a hierarchical parametric model of the deformation map. The transformation is parameterized at different scales, using a decomposition of the deformation vector field over a sequence of nested subspaces, generated from a single compactly supported scaling function. To preserve topology, the positivity of the Jacobian of the continuous deformation is enforced. We establish that, for the proposed hierarchical parametric model, the Jacobian J may be easily constrained over the whole deformation field, by only constraining its values at a limited number of control points, yielding a fast topology preserving matching algorithm. Experimental results are presented both on simulated and real-world medical MR images. <s> BIB002 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <s> In this paper we show first that a non-rigid registration algorithm used to register time-series MR images of the breast, can result in significant volume changes in the region of the enhanced lesion. Since this is physically implausible, given the short duration of the MR time series acquisition, the non-rigid registration algorithm was extended to allow the incorporation of rigid regions. In this way the registration is done in two stages. The enhanced lesions are first detected using the non-rigid registration algorithm in its original form. Secondly, the region of the enhanced lesion is set to be rigid and the new algorithm is applied to integrate this rigid region into the existing registration. By definition, volume and shape will be preserved in this rigid region. Preliminary results of applying this algorithm to 15 datasets are described. <s> BIB003 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <s> This paper presents a new method for image registration based on jointly estimating the forward and reverse transformations between two images while constraining these transforms to be inverses of one another. This approach produces a consistent set of transformations that have less pairwise registration error, i.e., better correspondence, than traditional methods that estimate the forward and reverse transformations independently. The transformations are estimated iteratively and are restricted to preserve topology by constraining them to obey the laws of continuum mechanics. The transformations are parameterized by a Fourier series to diagonalize the covariance structure imposed by the continuum mechanics constraints and to provide a computationally efficient numerical implementation. Results using a linear elastic material constraint are presented using both magnetic resonance and X-ray computed tomography image data. The results show that the joint estimation of a consistent set of forward and reverse transformations constrained by linear-elasticity give better registration results than using either constraint alone or none at all. <s> BIB004 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <s> In this paper, we extend a previously reported intensity-based nonrigid registration algorithm by using a novel regularization term to constrain the deformation. Global motion is modeled by a rigid transformation while local motion is described by a free-form deformation based on B-splines. An information theoretic measure, normalized mutual information, is used as an intensity-based image similarity measure. Registration is performed by searching for the deformation that minimizes a cost function consisting of a weighted combination of the image similarity measure and a regularization term. The novel regularization term is a local volume-preservation (incompressibility) constraint, which is motivated by the assumption that soft tissue is incompressible for small deformations and short time periods. The incompressibility constraint is implemented by penalizing deviations of the Jacobian determinant of the deformation from unity. We apply the nonrigid registration algorithm with and without the incompressibility constraint to precontrast and postcontrast magnetic resonance (MR) breast images from 17 patients. Without using a constraint, the volume of contrast-enhancing lesions decreases by 1%-78% (mean 26%). Image improvement (motion artifact reduction) obtained using the new constraint is compared with that obtained using a smoothness constraint based on the bending energy of the coordinate grid by blinded visual assessment of maximum intensity projections of subtraction images. For both constraints, volume preservation improves, and motion artifact correction worsens, as the weight of the constraint penalty term increases. For a given volume change of the contrast-enhancing lesions (2% of the original volume), the incompressibility constraint reduces motion artifacts better than or equal to the smoothness constraint in 13 out of 17 cases (better in 9, equal in 4, worse in 4). The preliminary results suggest that incorporation of the incompressibility regularization term improves intensity-based free-form nonrigid registration of contrast-enhanced MR breast images by greatly reducing the problem of shrinkage of contrast-enhancing structures while simultaneously allowing motion artifacts to be substantially reduced. <s> BIB005 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <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> BIB006 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <s> Voxel-based nonrigid image registration can be formulated as an optimisation problem whose goal is to minimise a cost function, consisting of a first term that characterises the similarity between both images and a second term that regularises the transformation and/or penalties improbable or impossible deformations. Within this paper, we extend previous works on nonrigid image registration by the introduction of a new penalty term that expresses the local rigidity of the deformation. A necessary and sufficient condition for the transformation to be locally rigid at a particular location is that its Jacobian matrix J T at this location is orthogonal, satisfying the orthogonality condition \(J_{\rm T} J_{\rm T}^{T} = {\bf 1}\). So we define the penalty term as the weighted integral of the Frobenius norm of \(J_{\rm T} J_{\rm T}^{T} - {\bf 1}\) integrated over the overlap of the images to be registered. We fit the implementation of the penalty term in a multidimensional, continuous and differentiable B-spline deformation framework and analytically determine the derivative of the similarity criterion and the penalty term with respect to the deformation parameters. We show results of the impact of the proposed rigidity constraint on artificial and clinical images demonstrating local shape preservation with the proposed constraint. <s> BIB007 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <s> This paper deals with topology preservation in three-dimensional (3-D) deformable image registration. This work is a nontrivial extension of, which addresses the case of two-dimensional (2-D) topology preserving mappings. In both cases, the deformation map is modeled as a hierarchical displacement field, decomposed on a multiresolution B-spline basis. Topology preservation is enforced by controlling the Jacobian of the transformation. Finding the optimal displacement parameters amounts to solving a constrained optimization problem: The residual energy between the target image and the deformed source image is minimized under constraints on the Jacobian. Unlike the 2-D case, in which simple linear constraints are derived, the 3-D B-spline-based deformable mapping yields a difficult (until now, unsolved) optimization problem. In this paper, we tackle the problem by resorting to interval analysis optimization techniques. Care is taken to keep the computational burden as low as possible. Results on multipatient 3-D MRI registration illustrate the ability of the method to preserve topology on the continuous image domain. <s> BIB008 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <s> In this paper we propose a diffeomorphic non-rigid registration algorithm based on free-form deformations (FFDs) which are modelled by B-splines. In contrast to existing non-rigid registration methods based on FFDs the proposed diffeomorphic non-rigid registration algorithm based on free-form deformations (FFDs) which are modelled by B-splines. To construct a diffeomorphic transformation we compose a sequence of free-form deformations while ensuring that individual FFDs are one-to-one transformations. We have evaluated the algorithm on 20 normal brain MR images which have been manually segmented into 67 anatomical structures. Using the agreement between manual segmentation and segmentation propagation as a measure of registration quality we have compared the algorithm to an existing FFD registration algorithm and a modified FFD registration algorithm which penalises non-diffeomorphic transformations. The results show that the proposed algorithm generates diffeomorphic transformations while providing similar levels of performance as the existing FFD registration algorithm in terms of registration accuracy. <s> BIB009 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <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> BIB010 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <s> Medical images that are to be registered for clinical application often contain both structures that deform and ones that remain rigid. Nonrigid registration algorithms that do not model properties of different tissue types may result in deformations of rigid structures. In this article a local rigidity penalty term is proposed which is included in the registration function in order to penalize the deformation of rigid objects. This term can be used for any representation of the deformation field capable of modelling locally rigid transformations. By using a B-spline representation of the deformation field, a fast algorithm can be devised. The proposed method is compared with an unconstrained nonrigid registration algorithm. It is evaluated on clinical three-dimensional CT follow-up data of the thorax and on two-dimensional DSA image sequences. The results show that nonrigid registration using the proposed rigidity penalty term is capable of nonrigidly aligning images, while keeping user-defined structures locally rigid. © 2007 American Association of Physicists in Medicine. <s> BIB011 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <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> BIB012 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <s> This paper presents a new nonrigid monomodality image registration algorithm based on B-splines. The deformation is described by a cubic B-spline field and found by minimizing the energy between a reference image and a deformed version of a floating image. To penalize noninvertible transformation, we propose two different constraints on the Jacobian of the transformation and its derivatives. The problem is modeled by an inequality constrained optimization problem which is efficiently solved by a combination of the multipliers method and the L-BFGS algorithm to handle the large number of variables and constraints of the registration of 3-D images. Numerical experiments are presented on magnetic resonance images using synthetic deformations and atlas based segmentation. <s> BIB013 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <s> B-splines are a convenient tool for nonrigid registration, but ensuring invertibility can be challenge. This paper describes a new penalty method that is devised to enforce a sufficient condition for local invertibility and smoothness of nth order B-spline based deformations. Traditional direct Jacobian penalty methods penalize negative Jacobian determinant values only at grid points. In contrast, our new penalty method enforces the sufficient condition for invertibility directly on the B-spline coefficients by using a modified quadratic penalty function so that it enforces invertibility globally over a 3D continuous domain. This approach also saves computation time and memory compared to using Jacobian determinant values. We apply this method to 3D CT images of a thorax at inhale and exhale. <s> BIB014 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <s> This paper presents a method for biventricular myocardial deformation recovery from cine MRI. The method is based on a deformable model that is nearly incompressible, a desirable property since the myocardium has been shown to be nearly incompressible. The model uses a matrix-valued radial basis function to represent divergence-free displacement fields, which is a first order approximation of incompressibility. This representation allows for deformation modeling of an arbitrary topologies with a relatively small number of parameters, which is suitable for representing the motion of the multi-chamber structure of the heart. The myocardium needs to be segmented in an initial frame after which the method automatically determines the tissue deformation everywhere in the myocardium throughout the cardiac cycle. Two studies were carried out to validate the method. In the first study the myocardial deformation was recovered from a 3D anatomical cine MRI sequence of a healthy volunteer and then validated against the manual segmentation of the biventricular wall and against the corresponding 3D tagged cine MRI sequence. The average volume agreement between the model and the manual segmentation had a false positive rate of 3.2%, false negative rate of 2.8% and true positive rate of 91.4%. The average distance between the model and manually determined intersections of perpendicular tag planes was 1.7mm (1.2 pixel). The same procedures was repeated on another set of 3D anatomical and tagged MRI scans of the same volunteer taken four months later. The recovered deformation was very similar to the one obtained from the first set of scans. In the second study the method was applied to 3D anatomical cine MRI scans of three patients with ventricular dyssynchrony and three age-matched healthy volunteers. The recovered strains of the normal subjects were clearly stronger than the recovered strains of the patients and they were similar to those reported by other researchers. The recovered deformation of all six subjects was validated against manual segmentation of the biventricular wall and against corresponding tagged MRI scans. The agreement was similar to that of the first study. <s> BIB015 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <s> A constrained non-rigid registration (CNRR) algorithm for use in prostate image-guided adaptive radiotherapy is presented in a coherent mathematical framework. The registration algorithm is based on a global rigid transformation combined with a series of local injective non-rigid multi-resolution cubic B-spline Free Form Deformation (FFD) transformations. The control points of the FFD are used to non-rigidly constrain the transformation to the prostate, rectum, and bladder. As well, the control points are used to rigidly constrain the transformation to the estimated position of the pelvis, left femur, and right femur. The algorithm was tested with both 3D conformal radiotherapy (3DCRT) and intensity-modulated radiotherapy (IMRT) dose plan data sets. The 3DCRT dose plan set consisted of 10 fan-beam CT (FBCT) treatment-day images acquired from four different patients. The IMRT dose plan set consisted of 32 cone-beam CT (CBCT) treatment-day images acquired from 4 different patients. The CNRR was tested with different combinations of anatomical constraints and each test significantly outperformed both rigid and non-rigid registration at aligning constrained bones and critical organs. The CNRR results were used to adapt the dose plans to account for patient positioning errors as well as inter-day bone motion and intrinsic organ deformation. Each adapted dose plan improved performance by lowering radiation distribution to the rectum and bladder while increasing or maintaining radiation distribution to the prostate. <s> BIB016 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <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> BIB017 </s> Deformable Medical Image Registration: A Survey <s> D. Task-Specific Constraints <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> BIB018
According to Hadamard's definition of well-posed problems  , unregularized optimization of similarity measures for high-dimensional deformable transformation models is, in general, an ill-posed problem. In order to cope with the difficulty associated with the ill-posedness of the problem, regularization is necessary. Moreover, regularization allows us to introduce any prior knowledge we may have regarding the physical properties of the underlying anatomical structure and helps optimization avoid local minima. There are two possible ways to regularize the problem: implicitly and explicitly. Implicit regularization may be achieved by parameterizing the deformation field with smooth functions. Explicit regularization may be achieved through the use of either hard constraints or soft constraints. Hard constraints are the constraints that the solution must satisfy in order for the registration to be successful. Soft constraints are introduced as additional terms in the energy function that penalize nonregular configurations. Soft constraints encode our preference regarding specific configurations, but deviations from the preferred configurations are allowed if driven by the other term(s) of the energy function. Physics-based deformation models are typical examples of explicit regularization. Moreover, explicit regularization may be used to achieve specific goals that are tailored to the problem at hand. Such goals include (see Fig. 1 ): 1) topology preservation, 2) volume preservation, and 3) rigidity constraints. Task-specific constraints can be, and often are, used in conjunction with physics-based models (Section II-A) and interpolation-based models (Section II-B). 1) Topology Preservation: One of the most important properties that a registration algorithm should exhibit is the preservation of topology. The preservation of topology is equivalent to the invertibility of the deformation field. The Jacobian of the deformation field is very informative regarding the local properties of the deformation field. In order to avoid singularities in the deformation field, Christensen et al. BIB001 proposed to track the values of the Jacobian. When its value dropped below a threshold, an intermediate deformed image was created and the registration process was reinitialized. Another way to enforce the preservation of topology is through the use of constraints, i.e., by including in the objective function an appropriate term that acts upon the Jacobian. Christensen and Johnson BIB004 added to the objective function a term that penalizes small and large Jacobian values for both the forward and backward transformation. Similarly, Rueckert et al. BIB009 introduced a term in the objective function that penalizes values of the Jacobian determinant that are close to zero. A different strategy is to formulate registration as a inequality constraint optimization problem. Musse et al. derived linear inequality constraints so that the topology is preserved BIB002 . The optimization was solved by employing a fast method that bears a resemblance to sequential linear programming. Noblet et al. extended the previous framework in the case BIB008 . The authors optimized the energy under the constraint that the Jacobian will stay between user specified bounds. Interval analysis techniques were used in order to solve the optimization problem. Haber and Modersitzki BIB010 also used inequality constraints. They used a variant of a log-barrier method to solve the optimization problem. Instead of solving the initial constrained problem, a sequence of unconstrained ones was employed. The weight for the barrier terms increased gradually for each unconstrained problem that was optimized by applying a variant of the Gauss-Newton's method. Sdika BIB013 also proposed a constrained optimization framework to ensure that the transformation is invertible. Two constraints were investigated for the case of a transformation model parametrized by cubic -splines. The first constrained the Jacobian of every pixel to be greater than a threshold. This constraint did not control the value of the Jacobian between the voxels. To account for that, the author proposed a second constraint that relates the Jacobian with its derivative. In that way, the Jacobian was restricted to be within a range of values. Moreover, its derivatives were constrained to be close to zero when approaching values close to the bounds. Chun and Fessler devised a simpler penalty for the case of -splines BIB014 . The penalty takes into account the difference between two adjacent nodes and is memory efficient. 2) Volume Preservation: In many applications, volume preservation is also important. Such a constraint is of particular interest when we know that the imaged anatomical structure is not compressible and that all changes are due to either motion or intensity changes provoked by the action of a contrast agent. A simple example is a rigid part of the body such as a bone structure. More complicated cases include deformable structures that preserve their volume such as breast, myocardium and liver. Tanner et al. BIB003 proposed a sequential approach for volume preserving deformable registration using an FFD model. First, a standard registration was performed. Based on its result, the areas whose volume should be preserved were identified. Then, the control points of the FFD that influenced these areas were grouped and restricted to move by a constant displacement that is equal to the mean value of their displacements during the initial registration step. Finally, the registration was solved again for the rest of the variables. Greene et al. also presented a sequential approach for image-guided adaptive radiotherapy using an FFD model BIB016 . First, the organs of interest and the bones were segmented and independently registered. Then, a constrained framework was used to estimate the FFD transformation that maps from one image to another. The displacements of the control points that influence the segmented objects were constrained to be close to the displacements that were calculated during the individual object registrations. Rohlfing et al. employed a volume preserving strategy to register contrast-enhanced MR breast images BIB005 . The objective function comprised an image matching term and a term that penalized volume changes. The penalty integrated the absolute logarithm of the Jacobian determinant and was zero only when local volume was preserved. Haber and Modersitzki BIB006 presented a constrained optimization approach for volume preservation. The proposed energy function, consisting of a matching and regularization term, was minimized under the constraint that the determinant of the transformation is equal to one . The myocardium is known to be a nearly incompressible material. Therefore, applications involving the deformation of the myocardium may profit from including an incompressibility constraint. Bistoquet et al. BIB015 approximated the previous constraint with . This constraint was enforced by the use of divergence-free radial basis functions as deformation model  . In addition, a hard constraint was introduced in the objective function to penalize deviations from incompressibility. Dauguet et al. BIB017 constrained the determinant of the Jacobian to be close to one in a predefined region by using Lagrange multipliers. Mansi et al. took a different approach in BIB018 . They constrained the velocity field to be divergence-free. This method was based on the fact that the integration over time of divergence-free velocities results in incompressible deformations. 3) Rigidity Constraints: The presence of rigid anatomical structures in medical images motivates the incorporation of rigidity constraints in image registration. Loeckx et al. BIB007 locally constrained a nonrigid FFD registration method by penalizing deviations of the Jacobian from orthogonality. Staring et al. BIB011 imposed rigidity by introducing three conditions. The first condition required the second derivatives of the transformation to be zero. The second condition forced the orthonormality of the rotation matrix, while the third condition required the determinant of the Jacobian to be equal to one. Modersitzki BIB012 has also investigated local rigidity in a variational setting. Modersitzki introduced a third term in an objective function comprising a matching and a regularization term. The additional term controlled the rigidity of the transformation by forcing its Jacobian to be linear, orthogonal and orientation preserving.