## Numerical Methods for Image Registration (2004)

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Citations: | 170 - 26 self |

### BibTeX

@ARTICLE{Haber04numericalmethods,

author = {Eldad Haber and Jan Modersitzki},

title = {Numerical Methods for Image Registration},

journal = {},

year = {2004}

}

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### Abstract

Abstract. In this paper we introduce a new framework for image registration. Our formulation is based on consistent discretization of the optimization problem coupled with a multigrid solution of the linear system which evolves in a Gauss–Newton iteration. We show that our discretization is h-elliptic independent of parameter choice, and therefore a simple multigrid implementation can be used. To overcome potential large nonlinearities and to further speed up computation, we use a multilevel continuation technique. We demonstrate the efficiency of our method on a realistic highly nonlinear registration problem.

### Citations

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Citation Context ...r. This paper is organized as follows. In section 2, we discuss the discretization of the registration problem (1). Our optimization approach, which is based on a Gauss–Newton-type scheme (cf., e.g., =-=[20]-=-), is discussed in section 3. In section 4, we propose a multigrid method for a solution of the linearized problems. Using a local mode analysis we show that our discretization is particularly amenabl... |

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Citation Context ...ontinuous image model. For noisy images we use a smoothing B-spline to approximate the images where the smoothing parameter is chosen using the GCV [9, 22]. For data interpolation using B-splines see =-=[25]-=-. Since the grid is regular, we can quickly evaluate the spline coefficients using a cosine transform. The continuous smooth approximation is denoted by T spline . We are looking for a fast and effici... |

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Citation Context ...becomes similar to the reference image. Image registration has to be applied whenever images resulting from different times, devices, and/or perspectives need to be compared or integrated; see, e.g., =-=[6, 19]-=- and references therein. The computation of nonrigid image registration has two main building blocks. The first one is a distance measure D quantifying distance or similarity of images, and the second... |

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Citation Context ...h mixed derivatives is a delicate matter. Here, we use staggered grids (cf. Figure 1), which are very common for stable discretizations of fluid flow (see, e.g., [8]) and electromagnetics (see, e.g., =-=[11, 27]-=-), where operators such as the gradient, curl, and divergence are discretized. It is also well known that staggered grids are tightly connected to mixed finite element methods which are commonly used ... |

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Citation Context ...xample, [14, 24]. If the difference between the images is associated with random noise, then standard methods that are based on the statistics of the noise, such as generalized cross validation (GCV) =-=[9]-=- or χ 2 [21], can be used. However, in image registration the difference between two images (at least in most medical applications) is not associated with random noise. In fact, the difference may be ... |

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Citation Context ...data fitting term D is balanced with a regularization term using the regularization parameter α. Many authors have dealt with the optimal choice of α, using a variety of approaches; see, for example, =-=[14, 24]-=-. If the difference between the images is associated with random noise, then standard methods that are based on the statistics of the noise, such as generalized cross validation (GCV) [9] or χ 2 [21],... |

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Citation Context ...tational work and to deal with nonlinearities is to use a multilevel continuation. Multilevel continuation is well established for optimization problems and systems of nonlinear equations; see, e.g., =-=[1, 2]-=-. However, in image registration it has an additional advantage. Similar to [10, 4], we use the multilevel approach to efficiently identify the relevant range of α’s. Thus, our multilevel approach use... |

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Citation Context ..., 24]. If the difference between the images is associated with random noise, then standard methods that are based on the statistics of the noise, such as generalized cross validation (GCV) [9] or χ 2 =-=[21]-=-, can be used. However, in image registration the difference between two images (at least in most medical applications) is not associated with random noise. In fact, the difference may be highly struc... |

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Citation Context ... of partial differential equations with mixed derivatives is a delicate matter. Here, we use staggered grids (cf. Figure 1), which are very common for stable discretizations of fluid flow (see, e.g., =-=[8]-=-) and electromagnetics (see, e.g., [11, 27]), where operators such as the gradient, curl, and divergence are discretized. It is also well known that staggered grids are tightly connected to mixed fini... |

106 | Optimization of Mutual Information for Multiresolution Image Registration
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Citation Context ...ously differentiable objective function, we need a continuous image model. For noisy images we use a smoothing B-spline to approximate the images where the smoothing parameter is chosen using the GCV =-=[9, 22]-=-. For data interpolation using B-splines see [25]. Since the grid is regular, we can quickly evaluate the spline coefficients using a cosine transform. The continuous smooth approximation is denoted b... |

57 |
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Citation Context ...on in α. As discussed in the introduction, the optimal value of the regularization parameter α is in general unknown a priori. In order to estimate a reasonable α, we follow the strategy suggested in =-=[12]-=-. Starting with a large α0,we compute a sequence of solutions u αj , where αj+1 <αj. This sequence can be viewed as a continuation with respect to the regularization parameter α. For a large regulariz... |

47 | Mixed finite elements for elasticity
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Citation Context ...ors such as the gradient, curl, and divergence are discretized. It is also well known that staggered grids are tightly connected to mixed finite element methods which are commonly used for elasticity =-=[3]-=-. In this section we briefly summarize the discretization we use. Further discussion and details are given in [13]. 2.1. Discretizing u. We assume that our discrete images have m1 ×···×md pixels, wher... |

42 | A multigrid method for distributed parameter estimation problems
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Citation Context .... The convergence properties of the linear iteration was only mildly influenced by the decreasing regularization parameter. This observation is similar to other multigrid methods for inverse problems =-=[5]-=-. 6.1.3. Sensitivity of the algorithm to the ratio μ/λ. In the previous experiments, common to many other image registration algorithms [15], we used the ratio μ/λ = 1. However, as Theorem 1 shows, th... |

36 | Towards fast non–rigid registration
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Citation Context ...imize-discretize approach, one forms the objective function and then differentiates to obtain the continuous Euler–Lagrange equations, which are finally discretized and solved numerically; see, e.g., =-=[7, 15, 19]-=-. The second approach is the discretize-optimize approach. Here, one directly discretizes the problem and then solves a finite (but typically high) dimensional optimization problem; see, e.g., [13]. T... |

34 |
Grid refinement and scaling for distributed parameter estimation problems: Inverse Problems
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- 2001
(Show Context)
Citation Context ... Multilevel continuation is well established for optimization problems and systems of nonlinear equations; see, e.g., [1, 2]. However, in image registration it has an additional advantage. Similar to =-=[10, 4]-=-, we use the multilevel approach to efficiently identify the relevant range of α’s. Thus, our multilevel approach uses continuation in both the grid and the regularization parameter. Note that even fo... |

31 | Fast finite volume simulation of 3D electromagnetic problems with highly discontinuous coefficients
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- 2001
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Citation Context ...h mixed derivatives is a delicate matter. Here, we use staggered grids (cf. Figure 1), which are very common for stable discretizations of fluid flow (see, e.g., [8]) and electromagnetics (see, e.g., =-=[11, 27]-=-), where operators such as the gradient, curl, and divergence are discretized. It is also well known that staggered grids are tightly connected to mixed finite element methods which are commonly used ... |

30 | The orthogonal decomposition theorems for mimetic finite difference methods
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Citation Context ... A := B ⊤ B. A discretization of the product S h u ≈B ∗ B may not be directly linked to a discrete objective function of the form � (Bu) 2 dx. Thus our discretization is a mimetic discretization (see =-=[16]-=-).s1598 ELDAD HABER AND JAN MODERSITZKI The advantage of this approach is that the derivative S h u is automatically the derivative of the discrete function S h . Moreover, boundary conditions are nat... |

24 |
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Citation Context ...imize-discretize approach, one forms the objective function and then differentiates to obtain the continuous Euler–Lagrange equations, which are finally discretized and solved numerically; see, e.g., =-=[7, 15, 19]-=-. The second approach is the discretize-optimize approach. Here, one directly discretizes the problem and then solves a finite (but typically high) dimensional optimization problem; see, e.g., [13]. T... |

16 |
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Citation Context ...a system of partial differential equations (PDE’s) with mixed derivatives is a delicate matter. Here, we use staggered grids which are very common for stable discretizations of fluid flow (see, e.g., =-=[7]-=-) and electromagnetics (see, e.g., [23, 10]) where operators such as the gradient, curl, and divergence are discretized. It is also well known that staggered grids are tightly connected to mixed finit... |

15 | multilevel, level-set method for optimizing eigenvalues in shape design problems
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- 2004
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Citation Context ... Multilevel continuation is well established for optimization problems and systems of nonlinear equations; see, e.g., [1, 2]. However, in image registration it has an additional advantage. Similar to =-=[10, 4]-=-, we use the multilevel approach to efficiently identify the relevant range of α’s. Thus, our multilevel approach uses continuation in both the grid and the regularization parameter. Note that even fo... |

13 |
Rank Deficient and Ill-Posed Problems
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Citation Context ...he data fitting term D is balanced with a regularization term using the regularization parameter α. Many authors have dealt with the optimal choice of α using a variety of approaches; see for example =-=[13, 21]-=-. If the difference between the images is associated with random noise then methods which are based on the statistics of the noise such as GCV [8] or χ 2 [19] can be used. However, in image registrati... |

10 |
B-spline registration of 3D images with levenberg-marquardt optimization
- Kabus, Netsch, et al.
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Citation Context ...s provided by Thomas Netsch, Philips Medical Systems, Hamburg, Germany). The problem is known to be hard because large nonlinear deformations are needed in order to perform the registration; see also =-=[17]-=-. As is apparent from Figure 2, the reference shows the knee in an almost straight position, and the template shows the knee in a bent position. The deformation is thuss3D views 2D views MIP A MULTILE... |

8 |
The homotopy continuation method: numerically implementable topological procedures
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Citation Context ...tational work and to deal with nonlinearities is to use a multilevel continuation. Multilevel continuation is well established for optimization problems and systems of nonlinear equations; see, e.g., =-=[1, 2]-=-. However, in image registration it has an additional advantage. Similar to [10, 4], we use the multilevel approach to efficiently identify the relevant range of α’s. Thus, our multilevel approach use... |

7 |
U.: A variational multigrid for computing the optical flow. Vision, Modeling and Visualization 2003 (Berlin 2003) 577–584
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Citation Context ...own in [13], this discretization is also stable for pointwise volume preserving constraints. Our work relates to the work of Henn and Witsch [15], Clarenz, Droske, and Rumpf [7], and Kalmoun and Rüde =-=[18]-=-. In [15], an FAS based on the discretized optimality conditions for the continuous problem is presented. In [7], a diffusive rather than an elastic regularizer is used. A d-linear interpolation schem... |

4 |
Numerical solutions of volume preserving image registration
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- 2004
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Citation Context ...15, 19]. The second approach is the discretize-optimize approach. Here, one directly discretizes the problem and then solves a finite (but typically high) dimensional optimization problem; see, e.g., =-=[13]-=-. The advantage of the latter approach is that standard optimization methods can be used. We prefer the discretize-optimize approach; however, in order to take advantage of efficient optimization tech... |

4 |
Rank-deficient and ill-posed problems: Numerical aspects of linear inversion (SIAM
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Citation Context ...data fitting term D is balanced with a regularization term using the regularization parameter α. Many authors have dealt with the optimal choice of α, using a variety of approaches; see, for example, =-=[14, 24]-=-. If the difference between the images is associated with random noise, then standard methods that are based on the statistics of the noise, such as generalized cross validation (GCV) [9] or χ 2 [21],... |

4 |
Mixed finite elements for elasticity. Numerische Mathematik 2002
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Citation Context ...rs such as the gradient, curl, and divergence are discretized. It is also well known that staggered grids are tightly connected to mixed finite elements methods which are commonly used for elasticity =-=[3]-=-. In this section we shortly summarize the discretization we use. Further discussion and details are given in [12]. 2.1 Discretizing u We assume that our discrete images have m1 ×...×md pixels, where ... |