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## Non-linear Registration Using Block-Matching and B-Splines (2004)

### Citations

2223 |
Iterative methods for sparse linear systems
- Saad
- 2003
(Show Context)
Citation Context ...ompose (what will always work because is symmetric and positive definite). Working with the equation equation (4.6) that was given in section 4.2 we form where will be symmetric and positive definite =-=[28]-=-. The new - preconditioned - problem to be solved is now represented by the following equation with (6.4) To compute the control point values, we commence by determining using the conjugate gradient a... |

472 | An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
- Shewchuk
- 1994
(Show Context)
Citation Context ...ror causes the residual to gradually loose accuracy. However, this does not pose a problem as in general the results become satisfyingly precise before the procedure has actually undergone iterations =-=[26]-=-. In the conjugate gradient approach, we look for the solution ’gradient equals zero’ by analyzing a -orthogonal set of search directions. Let and (see equation (4.6)). The algorithm that is directly ... |

380 |
Fundamentals of Computer Aided Geometric Design, A K Peters
- Hoschek, Lasser
- 1993
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Citation Context ...tion 3.1 B-Spline basis function: for for - for (3.1) B-Spline basis functions possess the characteristic traits depicted in the theorem below: Theorem 3.1 Properties of B-Spline basis functions (see =-=[14]-=-, chapter 4):sB-Splines 23 1 1 0.7 β 0 (x) 0.8 0.6 0.4 0.2 0 −1 −0.5 0 0.5 1 x β 1 (x) 0.8 0.6 0.4 0.2 0 −1 −0.5 0 0.5 1 x β 3 (x) 0.6 0.5 0.4 0.3 0.2 0.1 0 −1 −0.5 0 0.5 1 x (a) B-Spline basis functi... |

102 | Fast parametric elastic image registration - Kybic, Unser - 2003 |

55 |
D.J.: Nonrigid registration using free-form deformations: Application to breast MR images
- Rueckert, Sonoda, et al.
- 1999
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Citation Context ... parameter defines the tradeoff between the alignment of the two images and the smoothness of the transformation. corresponds to the least squares estimation as seen in equation (4.6). As proposed in =-=[18]-=-, we use a biharmonic model as depicted in equation (4.11) to describe the4.5 The Multiresolution Strategy 37 penalty term. The physical interpretation of the biharmonic model is the idea of approxim... |

11 |
Nonrigid registration: concepts, algorithms, and applications
- Rueckert
- 2001
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Citation Context ... the transformation estimation: (4.10) Here, characterizes the similarity between the source and the target image, and represents the degree of discontinuity of the associated transformation function =-=[17]-=-. The parameter defines the tradeoff between the alignment of the two images and the smoothness of the transformation. corresponds to the least squares estimation as seen in equation (4.6). As propose... |

10 | Elastic Image Registration Using Parametric Deformation Models
- Kybic
- 2001
(Show Context)
Citation Context ...ne basis function in a B-Spline curve is equipped with its respective control point value as coefficient. Thus, all B-Spline curves are unambiguously characterized by their sequence of control points =-=[15, 16]-=-. Due to the fact that the basis functions have local support, each control point influences only for . To ensure that all basis functions are present in each interval considered, we have to choose an... |

9 |
A Perfect Fit for Signal and
- Unser, “Splines
- 1999
(Show Context)
Citation Context ...ne basis function in a B-Spline curve is equipped with its respective control point value as coefficient. Thus, all B-Spline curves are unambiguously characterized by their sequence of control points =-=[15, 16]-=-. Due to the fact that the basis functions have local support, each control point influences only for . To ensure that all basis functions are present in each interval considered, we have to choose an... |

7 |
Étude de la dynamique cardiaque par analyse d’images tridimensionnelles
- Declerck
- 1997
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Citation Context ...s 64 control points in its neighbourhood, each row of the matrix contains 64 entries that are non-zero. The matrix can be arranged to have a diagonal appearance like depicted in figure (4.1) (compare =-=[20]-=-). Here, each square stands for a non-zero entry in one column and in several rows where the number of rows with the same structure depends on the ’sampling point - control point’ ratio. is symmetrica... |

7 |
Numerik linearer Gleichungssysteme - Eine Einführung in moderne Verfahren, Friedrich Vieweg Verlagsgesellschaft
- Meister
- 1999
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Citation Context ...d. As is positive definite the function is shaped like a parabolic bowl. Hence, searching the minimum as demanded in equation (6.1) we have to determine the location where the gradient of equals zero =-=[25]-=-. The most popular approach to solve a non-symmetric linear system in the way described above employs the conjugate gradient procedure. It is efficient and will converge theoretically after a certain ... |

4 |
B-Spline Approximation with Energy Constraints
- Dietz
- 1996
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Citation Context ....2 3D B-Spline Least Squares Estimation 31 given by the set of matches at hand. We choose a least squares estimation as the criterion to be minimized (see equation (4.3)) which gives a linear problem =-=[19]-=-: Definition 4.3 Least squares estimation: Minimize the criterion (4.3) By decomposing the norm into three sums along its x-, y-, and z-direction we obtain As we deal with a tensor product, , and rely... |

3 |
Sumanaweera T, Hemler P, Napel S, Adler J: Grey value correlation techniques used for automatic matching
- P, Pol
- 1994
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Citation Context ...ce image and the voxels they are built out of. For images of different modalities, the registration methods are often based on statistical relationships between voxel values (for more information see =-=[21]-=-). There are various transformation classes used in the registration algorithms described above. They are defined by the space of research where the transformation is calculated, therefore, the transf... |

3 |
Wolberg G, Shin SY: Scattered data interpolation with multilevel B-splines
- Lee
- 1997
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Citation Context ...rget images of level . Now, we use these correspondences as a set of matches and apply a B-Spline approximation on basis of a least squares estimator to compute a transformation representing on level =-=[23]-=-. This approach is very convenient concerning the implementation because it resorts to the B-Spline algorithm already implemented (see section 4.1).sC H A P T E R 6 Implementation This chapter deals w... |

2 |
Subsol G, Pennec X, Sattonnet C. Automatic alignment of histological sections for 3D reconstruction and analysis
- Ourselin, Roche
- 1998
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Citation Context ...he corresponding target voxel is determined. To obtain the optimal result, there are 4 scheme parameters that can be modified so the algorithm works ideally according to the registration task at hand =-=[2, 11, 12]-=-:2.1 The Block-Matching Algorithm 11 I S I T Figure 2.1: The block-matching algorithm. For the grey shaded block in image the position that features the greatest similarity in image is searched. The ... |

1 |
Hung Y.P.; Fuh C.S.: Fast Block Matching Algorithm Based on the Winner-Update Strategy
- Chen
- 2001
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Citation Context ... where a number of point pairs that indicate the correspondences of voxels of the source image with voxels of the target image is stored. This information is used to estimate the image transformation =-=[9, 10]-=-. 2.1.1 Matching the Blocks To obtain the sparse displacement field, we cut into an ensemble of possibly overlapping subimages that we call blocks. Next, each block is moved around in the volume in or... |

1 |
Baladin: mode d’emploi
- Malandain
- 2003
(Show Context)
Citation Context ...he corresponding target voxel is determined. To obtain the optimal result, there are 4 scheme parameters that can be modified so the algorithm works ideally according to the registration task at hand =-=[2, 11, 12]-=-:2.1 The Block-Matching Algorithm 11 I S I T Figure 2.1: The block-matching algorithm. For the grey shaded block in image the position that features the greatest similarity in image is searched. The ... |

1 |
J.-J.: M thodes math matiques pour la
- Risler
- 1991
(Show Context)
Citation Context .... Here, determining the value of a cubic B-Spline basis function given an input takes 6 mathematic operations where each one consists of multiplications, 1 division and 1 substraction (see table 6.1) =-=[29]-=-. To evaluate the value of the B-Spline transformation as presented in equation 4.2 we need to evaluate 3 times a B-Spline basis function, we need to effactuate 2 multiplications for the tensor produc... |