# Overview

TubeTK provides algorithms for deformable registration of images depicting multiple organs in which the organs may have shifted, expanded, or compressed independently.

Traditional deformable registration imposes a uniform smoothness constraint on the deformation field. However, discontinuities in the deformation field are expected with sliding motion, and this constraint is not appropriate. This ultimately leads to registration inaccuracies.

TubeTK provides deformable image registration incorporating a deformation field regularization term that is based on anisotropic diffusion. A cost function ${\displaystyle C(u)}$ is a function of the current estimation of the deformation field ${\displaystyle u}$. ${\displaystyle C(u)}$ is iteratively optimized using finite differences and is the sum of of two terms:

• Intensity-based distance measure: captures intensity differences between the fixed image and the transformed moving image (sum of squared differences)
• Anisotropic diffusive regularization term: penalizes unrealistic deformation fields, while considering sliding motion

The anisotropic diffusive regularization is based on decomposing the deformation field into normal and tangential components, which are defined with respect to a given organ boundary along which sliding motion is expected to occur. These two components are handled differently:

• Motion normal to the organ boundary should be smooth both across organ boundaries and deep within organs. The motion normal to the organ boundary must be smooth in both the normal and tangential directions. The former condition enforces coupling between neighboring organs under the assumption that organs do not pull apart. The later forces smooth motion of individual organs.
• Motion tangential to the organ boundary should be smooth in the tangential direction within each individual organ. However, smoothness is not required across organ boundaries, therefore sliding transformations are not penalized.

These conditions are implemented by defining the anisotropic regularizer as:

${\displaystyle S_{\mathrm {a} }(u)={\frac {1}{2}}\sum _{l=x,y,z}\sum _{\mathbf {x} \in \Omega }\|P\nabla u_{l}(\mathbf {x} )\|^{2}+w\left(n^{T}\nabla u_{l}^{\perp }(\mathbf {x} )\right)^{2}}$

where

${\displaystyle P=I-wnn^{T}}$

and

• ${\displaystyle n}$ is the organ boundary in the vicinity of ${\displaystyle {\textbf {x}}}$,
• ${\displaystyle u(\mathbf {x} )}$ is the vector within the deformation field ${\displaystyle u}$ at location ${\displaystyle {\textbf {x}}}$
• ${\displaystyle \nabla u_{l}(\mathbf {x} )}$ is the gradient of the ${\displaystyle l}$-th component of ${\displaystyle u({\textbf {x}})}$
• ${\displaystyle u_{l}^{\perp }(\mathbf {x} )}$ is the component of ${\displaystyle u_{l}({\textbf {x}})}$ in the normal direction
• ${\displaystyle w}$ is a weighting term which decays exponentially from 1 to 0 as a function of distance to the organ boundary.

Close to organ boundaries, where ${\displaystyle w}$ is close to 1:

• ${\displaystyle \|P\nabla u_{l}(\mathbf {x} )\|^{2}}$ penalizes any discontinuities in the deformation field that are in the plane tangential to the organ boundary. This anisotropically smooths:
• Discontinuities in the deformation field's normal component that occur in the tangential plane
• Discontinuities in the deformation field's tangential component that occur in the tangential plane
• ${\displaystyle w\left(n^{T}\nabla u_{l}^{\perp }(\mathbf {x} )\right)^{2}}$ penalizes any discontinuities in the deformation field's normal component that occur in the normal direction
• Discontinuities in the deformation field's tangential component that occur in the normal direction are allowed: these are sliding motions!

The gradient is defined with respect to ${\displaystyle u}$ and is implemented in itkImageToImageDiffusiveDeformableRegistrationFilter and itkImageToImageDiffusiveDeformableRegistrationFunction using ITK's finite differences framework:

${\displaystyle c_{S_{\mathrm {a} }}\left(u(\mathbf {x} ,t)\right)=\sum _{l=x,y,z}{\textrm {div}}\left(P^{T}P\nabla u_{l}(\mathbf {x} )\right)(e_{l})+{\textrm {div}}\left(w\left(n^{T}\nabla u_{l}^{\perp }(\mathbf {x} )\right)n\right)n_{l}n}$

where ${\displaystyle e_{l}}$ is the ${\displaystyle l^{th}}$ canonical unit vector, i.e ${\displaystyle e_{x}=[1,0,0]^{T}}$

Further away from organ boundaries, ${\displaystyle w}$ approximates 0 and this tends to the diffusive regularization, which is equivalent of Gaussian smoothing. Therefore, uniformly smooth motion is required within each individual organ.

# Related Works

• Slipping objects in image registration: Improved motion field estimation with direction-dependent regularization
• Alexander Schmidt-Richberg, Jan Ehrhardt, Rene Werner and Heinz Handels
• MICCAI 2009, Lecture Notes in Computer Science, Volume 5761, pp.755-762, 2009
• Abstract:
• The computation of accurate motion fields is a crucial aspect in 4D medical imaging. It is usually done using a non-linear registration without further modeling of physiological motion properties. However, a globally homogeneous smoothing (regularization) of the motion field during the registration process can contradict the characteristics of motion dynamics. This is particularly the case when two organs slip along each other which leads to discontinuities in the motion field. In this paper, we present a diffusion-based model for incorporating physiological knowledge in image registration. By decoupling normal- and tangential-directed smoothing, we are able to estimate slipping motion at the organ borders while ensuring smooth motion fields in the inside and preventing gaps to arise in the field. We evaluate our model focusing on the estimation of respiratory lung motion. By accounting for the discontinuous motion of visceral and parietal pleurae, we are able to show a significant increase of registration accuracy with respect to the target registration error (TRE).
• An investigation of smoothness constraints for the estimation of displacement vector fields from image sequences
• Hans-Hellmut Nagel and Wilfried Enkelmann
• IEEE Transactions on Pattern Analysis and Machine Intelligence, 8(5), pp 565-593, 1986
• http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=4767833
• Abstract:
• A mapping between one frame from an image sequence and the preceding or following frame can be represented as a displacement vector field. In most situations, the mere gray value variations do not provide sufficient information in order to estimate such a displacement vector field. Supplementary constraints are necessary, for example the postulate that a displacement vector field varies smoothly as a function of the image position. Taken as a general requirement, this creates difficulties at gray value transitions which correspond to occluding contours. Nagel therefore introduced the oriented smoothness requirement which restricts variations of the displacement vector field only in directions with small or no variation of gray values. This contribution reports results of an investigation about how such an oriented smoothness constraint may be formulated and evaluated.
• A review of nonlinear diffusion filtering
• Joachim Weickert
• Scale-Space Theory in Computer Vision, Lecture Notes in Computer Science, Volume 1252, pp. 1-28, 1997