Deformable Medical Image Registration: A Survey <s> 4) Basis Functions From Signal Representation: <s> We describe a comprehensive framework for performing rapid and automatic nonlabel-based nonlinear spatial normalizations. The approach adopted minimizes the residual squared difference between an image and a template of the same modality. In order to reduce the number of parameters to be fitted, the nonlinear warps are described by a linear combination of low spatial frequency basis functions. The objective is to determine the optimum coefficients fur each of the bases by minimizing the sum of squared differences between the image and template, while simultaneously maximizing the smoothness of the transformation using a maximum a posteriori (MAP) approach. Most MAT approaches assume that the variance associated with each voxel is already known and that there is no covariance between neighboring voxels. The approach described here attempts to estimate this variance from the data, and also corrects fur the correlations between neighboring voxels. This makes the same approach suitable for the spatial normalization of both high-quality magnetic resonance images, and low-resolution noisy positron emission tomography images. A fast algorithm has been developed that utilizes Taylor's theorem and the separable nature of the basis functions, meaning that most of the nonlinear spatial variability between images can be automatically corrected within a few minutes. Hum. Brain Mapping 7:254-266, 1999. (C) 1999 Wiley-Liss, Inc. <s> BIB001 </s> Deformable Medical Image Registration: A Survey <s> 4) Basis Functions From Signal Representation: <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> 4) Basis Functions From Signal Representation: <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> BIB003 </s> Deformable Medical Image Registration: A Survey <s> 4) Basis Functions From Signal Representation: <s> A three-dimensional wavelet-based algorithm for nonlinear registration of an elastic body model of the brain is developed. Surfaces of external and internal anatomic brain structures are used to guide alignment. The deformation field is represented with a multiresolution wavelet expansion and is modeled by the partial differential equations of linear elasticity. A progressive estimation of the registration parameters and the usage of an adaptive distance map reduce algorithm complexity, thereby providing computational flexibility that allows mapping of large, high resolution datasets. The performance of the algorithm was evaluated on rat brains. The wavelet-based registration method yielded a twofold improvement over affine registration. <s> BIB004 </s> Deformable Medical Image Registration: A Survey <s> 4) Basis Functions From Signal Representation: <s> An image registration algorithm is developed to estimate dense motion vectors between two images using the coarse-to-fine wavelet-based motion model. This motion model is described by a linear combination of hierarchical basis functions proposed by Cai and Wang (SIAM Numer. Anal., 33(3):937–970, 1996). The coarser-scale basis function has larger support while the finer-scale basis function has smaller support. With these variable supports in full resolution, the basis functions serve as large-to-small windows so that the global and local information can be incorporated concurrently for image matching, especially for recovering motion vectors containing large displacements. To evaluate the accuracy of the wavelet-based method, two sets of test images were experimented using both the wavelet-based method and a leading pyramid spline-based method by Szeliski et al. (International Journal of Computer Vision, 22(3):199–218, 1996). One set of test images, taken from Barron et al. (International Journal of Computer Vision, 12:43–77, 1994), contains small displacements. The other set exhibits low texture or spatial aliasing after image blurring and contains large displacements. The experimental results showed that our wavelet-based method produced better motion estimates with error distributions having a smaller mean and smaller standard deviation. <s> BIB005 </s> Deformable Medical Image Registration: A Survey <s> 4) Basis Functions From Signal Representation: <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> BIB006 </s> Deformable Medical Image Registration: A Survey <s> 4) Basis Functions From Signal Representation: <s> Abstract This paper proposes a comprehensive evaluation of a monomodal B-spline-based non-rigid registration algorithm allowing topology preservation in 3-D. This article is to be considered as the companion of [Noblet, V., Heinrich, C., Heitz, F., Armspach, J.-P., 2005. 3-D deformable image registration: a topology preservation scheme based on hierarchical deformation models and interval analysis optimization. IEEE Transactions on Image Processing, 14 (5), 553–566] where this algorithm, based on the minimization of an objective function, was introduced and detailed. Overall assessment is based on the estimation of synthetic deformation fields, on average brain construction, on atlas-based segmentation and on landmark mapping. The influence of the model parameters is characterized. Comparison between several objective functions is carried out and impact of their symmetrization is pointed out. An original intensity normalization scheme is also introduced, leading to significant improvements of the registration quality. The comparison benchmark is the popular demons algorithm [Thirion, J.-P., 1998. Image matching as a diffusion process: an analogy with Maxwell’s demons. Medical Image Analysis, 2 (3), 243–260], that exhibited best results in a recent comparison between several non-rigid 3-D registration methods [Hellier, P., Barillot, C., Corouge, I., Gibaud, B., Le Goualher, G., Collins, D.L., Evans, A., Malandain, G., Ayache, N., Christensen, G.E., Johnson, H.J., 2003. Retrospective evaluation of intersubject brain registration. IEEE Transactions on Medical Imaging, 22 (9), 1120–1130]. The topology preserving B-spline-based method proved to outperform the commonly available ITK implementation of the demons algorithms on many points. Some limits of intensity-based registration methods are also highlighted through this work. <s> BIB007 </s> Deformable Medical Image Registration: A Survey <s> 4) Basis Functions From Signal Representation: <s> With the growing acceptance of nonrigid registration as a useful tool to perform clinical research, and in particular group studies, the storage space needed to hold the resulting transforms is deemed to become a concern for vector field based approaches, on top of the traditional computation time issue. In a recent study we lead, which involved the registration of more than 22,000 pairs of T1 MR volumes, this constrain appeared critical indeed. In this paper, we propose to decompose the vector field on a wavelet basis, and let the registration algorithm minimize the number of non-zero coefficients by introducing an L1 penalty. This enables a sparse representation of the vector field which, unlike parametric representations, does not confine the estimated transform into a small parametric space with a fixed uniform smoothness : nonzero wavelet coefficients are optimally distributed depending on the data. Furthermore, we show that the iconic feature registration framework allows to embed the non-differentiable L1 penalty into a C1 energy that can be efficiently minimized by standard optimization techniques. <s> BIB008
Inspired by the mathematical tools that are available to represent and analyze signals, many researchers have used Fourier and Wavelet analysis to model transformations. An important reason to use them is the fact that they can naturally provide a multi-resolution decomposition of the displacement field. This is a useful property for the coarse-to-fine schemes that are commonly applied in medical image registration to ease the computations and handle large deformations. Christensen and Johnson employed a Fourier-based transformation scheme in their consistent registration framework BIB003 . The Fourier series representation of the transformation simplifies the linear elasticity constraint, thus allowing an efficient numerical implementation. Ashburner and Friston BIB001 tackled nonlinear registration by employing a transformation model that was parametrized as a linear combination of discrete cosine turansform basis functions. The separable nature of the basis functions was exploited by the authors to accelerate calculations. Fourier basis functions are well localized in the frequency domain. On the contrary, they are not localized at all in the spatial domain. Wavelet basis functions, being localized in both domains, can model local deformations more efficiently than Fourier basis  . Amit  presented two variational approaches for image matching. In the first case, the transformation was parametrized with respect to the Fourier basis, while in the second case, it was parametrized with respect to a wavelet basis. The reported experimental results indicated that the second method was able to capture local deformations with more accuracy than the Fourier method. Wu et al. BIB005 used a wavelet-based deformation model. The Cai-Wang wavelet was employed to generate a multi-resolution description in Sobolev space yielding intrinsically smooth deformations. Based on this model, the authors were able to treat global and local information simultaneously in a coarse-to-fine approach. Gefen et al. BIB004 modeled the deformation field with a finite-supported, semi-orthogonal wavelet toward tackling the problem of aligning rat brain histological images. In order to ease the optimization burden, the authors exploited the natural multi-resolution and multi-band decomposition of the wavelet coefficients. The transformation parameters were first inferred for low resolution levels, separately for each subband, before proceeding to finer resolution levels. Musse et al. BIB002 presented a topology-preserving multi-resolution approach for images. The authors used nonorthogonal Riesz basis of polynomial splines due to their compactness. The topology was preserved by controlling the Jacobian through hard linear constraints. Noblet et al. extended this approach to the domain in BIB006 and further validated it in BIB007 . In the case, the same multi-resolution framework was used, though the topology could not be preserved by satisfying linear constraints. This was made possible by solving a constrained optimization problem where the Jacobian was enclosed between two user specified bounds. Cathier BIB008 used the same wavelet basis as in BIB005 to decompose the transformation in a multi-resolution fashion. An penalty on the wavelet coefficients was used to regularize the registration problem. This regularization led to sparse transformations with respect to the wavelet basis and thus facilitated their storage in memory.