*Jan Modersitzki*

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198528418
- eISBN:
- 9780191713583
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528418.003.0012
- Subject:
- Mathematics, Applied Mathematics

Elastic, fluid, and diffusion registrations are sensitive to affine linear displacements. As a consequence, an affine linear pre-registration is unavoidable. The second order derivatives-based ...
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Elastic, fluid, and diffusion registrations are sensitive to affine linear displacements. As a consequence, an affine linear pre-registration is unavoidable. The second order derivatives-based curvature regularizer is introduced to circumvent this additional pre-registration. The registration thus becomes less dependent on the initial position of reference and template images. For the practical implementation of the curvature registration, the continuous and discrete bi-harmonic equation are considered and numerical solution schemes are discussed. The natural relation to thin plate splines is shown. Various examples demonstrate the performance of curvature registration and the difference to elastic, fluid, and diffusion registration.Less

Elastic, fluid, and diffusion registrations are sensitive to affine linear displacements. As a consequence, an affine linear pre-registration is unavoidable. The second order derivatives-based curvature regularizer is introduced to circumvent this additional pre-registration. The registration thus becomes less dependent on the initial position of reference and template images. For the practical implementation of the curvature registration, the continuous and discrete bi-harmonic equation are considered and numerical solution schemes are discussed. The natural relation to thin plate splines is shown. Various examples demonstrate the performance of curvature registration and the difference to elastic, fluid, and diffusion registration.

*Jan Modersitzki*

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198528418
- eISBN:
- 9780191713583
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528418.003.0010
- Subject:
- Mathematics, Applied Mathematics

This chapter introduces the fluid registration scheme of Christensen and relates this to the elastic registration. Though the fluid approach is more related to a flow approach rather than to an ...
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This chapter introduces the fluid registration scheme of Christensen and relates this to the elastic registration. Though the fluid approach is more related to a flow approach rather than to an optimization approach, the analogy becomes visible using iterative Tychonov regularization. The chapter summarizes basic conservation laws like conservation of mass and conservation of linear momentum. Assuming a Stokes fluid, the underlying partial differential equation for the fluid registration is derived. It is shown that these equations can be solved numerically, and a fluid registration algorithm is presented together with various comments and remarks concerning implementation details. Various examples and a discussion of the fluid registration are given.Less

This chapter introduces the fluid registration scheme of Christensen and relates this to the elastic registration. Though the fluid approach is more related to a flow approach rather than to an optimization approach, the analogy becomes visible using iterative Tychonov regularization. The chapter summarizes basic conservation laws like conservation of mass and conservation of linear momentum. Assuming a Stokes fluid, the underlying partial differential equation for the fluid registration is derived. It is shown that these equations can be solved numerically, and a fluid registration algorithm is presented together with various comments and remarks concerning implementation details. Various examples and a discussion of the fluid registration are given.

*Jan Modersitzki*

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198528418
- eISBN:
- 9780191713583
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528418.003.0011
- Subject:
- Mathematics, Applied Mathematics

A gradient-based regularization for registration is introduced, and a fast and stable implementation is developed. In contrast to the physically motivated elastic and fluid registrations, the ...
More

A gradient-based regularization for registration is introduced, and a fast and stable implementation is developed. In contrast to the physically motivated elastic and fluid registrations, the diffusion regularizer is motivated by smoothing properties of the displacement. Another important motivation is that a registration step can be performed in linear complexity of the number of given data. The main tool is the so-called additive operator splitting scheme (AOS). The idea is to split the original problem into a number of simpler problems which allow for a fast numerical solution. A new proof for the accuracy of AOS is given, which is based purely on matrix analysis. Thus, the result also applies to more general situations. Thirion's demons registration is discussed.Less

A gradient-based regularization for registration is introduced, and a fast and stable implementation is developed. In contrast to the physically motivated elastic and fluid registrations, the diffusion regularizer is motivated by smoothing properties of the displacement. Another important motivation is that a registration step can be performed in linear complexity of the number of given data. The main tool is the so-called additive operator splitting scheme (AOS). The idea is to split the original problem into a number of simpler problems which allow for a fast numerical solution. A new proof for the accuracy of AOS is given, which is based purely on matrix analysis. Thus, the result also applies to more general situations. Thirion's demons registration is discussed.

*Jan Modersitzki*

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198528418
- eISBN:
- 9780191713583
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528418.003.0009
- Subject:
- Mathematics, Applied Mathematics

This chapter introduces elastic registration using the general framework introduced in Chapter 8. A physical motivation for this particular regularizer is given. The two images are viewed as two ...
More

This chapter introduces elastic registration using the general framework introduced in Chapter 8. A physical motivation for this particular regularizer is given. The two images are viewed as two different observations of an elastic body, one before and one after deformation. The displacement (or transformation) of the elastic body is derived using a linear elasticity model. In order to introduce different boundary conditions and for later reference, the eigenvalues and eigenfunctions of the Navier-Lame operator are computed. It is shown that the Navier-Lame equations can be derived from the general framework. The continuous Euler-Lagrange equations are discretized and the discretized partial differential operator can be diagonalized using Fast-Fourier type techniques. The Moore-Penrose pseudo inverse of the discrete Navier-Lamé operator is explicitly given. Registration results for academic as well as real life problems are presented.Less

This chapter introduces elastic registration using the general framework introduced in Chapter 8. A physical motivation for this particular regularizer is given. The two images are viewed as two different observations of an elastic body, one before and one after deformation. The displacement (or transformation) of the elastic body is derived using a linear elasticity model. In order to introduce different boundary conditions and for later reference, the eigenvalues and eigenfunctions of the Navier-Lame operator are computed. It is shown that the Navier-Lame equations can be derived from the general framework. The continuous Euler-Lagrange equations are discretized and the discretized partial differential operator can be diagonalized using Fast-Fourier type techniques. The Moore-Penrose pseudo inverse of the discrete Navier-Lamé operator is explicitly given. Registration results for academic as well as real life problems are presented.