Keywords

1 Our Approach to Image Registration

This work connects the resulting registration deformations to the solution pool of VP in [1], which achieves a recent progression in describing non-folding grids in a diffeomorphism group. Hence, to restrict the image registration method built in [3] satisfying the constraint of VP, it is reformulated and proposed as follows: let \(I_{\pmb {m}}\) be a \(\pmb {moving}\) image is to be registered to a \(\pmb {fixed}\) image \(I_{\pmb {f}}\) on the fixed and bounded domain \((\pmb {\omega }=<x,y,z>\in )\mathrm {\Omega } \subset \mathbb {R}^{3}\), the energy function Loss is minimized over the form \(\pmb {\phi }=\pmb {id}+\pmb {u}\) on \(\mathrm {\Omega }\) with \(\pmb {u} = \pmb {0}\) on \(\partial \mathrm {\Omega }\),

$$\begin{aligned} Loss(\pmb {\phi }) = \dfrac{1}{2}\int _{\mathrm {\Omega }} [I_{\pmb {m}}(\pmb {\phi }) - I_{\pmb {f}}]^2 d\pmb {\omega } \quad \text { subjects to } \mathrm {\Delta } \pmb {\phi } = \pmb {F}(f, \pmb {g}) \text { in } \mathrm {\Omega }, \end{aligned}$$
(1)

where the scalar-valued f and the vector-valued \(\pmb {g}\) are the control functions in the sense of VP that mimic the prescribed JD and curl, respectively.

1.1 Gradient with Respect to Control \(\pmb {F}\)

The variational gradient of (1) with respect to \(\delta \mathrm {\Delta } \pmb {\phi }=\delta \mathrm {\Delta }\pmb {u}=\delta \pmb {F}\) is derived. For all \(\delta \pmb {F}\) vanishing on \(\partial \mathrm {\Omega }\) and by Green’s identities with fixed boundary condition,

$$\begin{aligned} \begin{aligned} \delta Loss(\pmb {\phi })&= \delta ( \dfrac{1}{2}\int _{\mathrm {\Omega }} [I_{\pmb {m}}(\pmb {\phi }) - I_{\pmb {f}}]^2 d\pmb {\omega }) = \int _{\mathrm {\Omega }} [(I_{\pmb {m}}(\pmb {\phi }) - I_{\pmb {f}}) \nabla I_{\pmb {m}}(\pmb {\phi }) \cdot \delta \pmb {\phi }]d\pmb {\omega }\\ =\int _{\mathrm {\Omega }} [&\mathrm {\Delta }\pmb {b} \cdot \delta \pmb {\phi } ]d\pmb {\omega } =\int _{\mathrm {\Omega }} [\pmb {b} \cdot \delta \mathrm {\Delta } \pmb {\phi } ]d\pmb {\omega }= \int _{\mathrm {\Omega }} [\pmb {b} \cdot \delta \pmb {F} ]d\pmb {\omega }\quad \Rightarrow \frac{\partial Loss}{\partial \pmb {F}}=\pmb {b},\\ \end{aligned} \end{aligned}$$
(2)

where \(\mathrm {\Delta }\pmb {b} = (I_{\pmb {m}}(\pmb {\phi }) - I_{\pmb {f}}) \nabla I_{\pmb {m}}(\pmb {\phi })\), so, a gradient-based algorithm can be formed.

1.2 Hessian Matrix with Respect to Control Function \(\pmb {F}\)

In case of a Newton optimizing scheme is applicable, from (2), one can derive the Hessian matrix \(\pmb {H}\) of (1) with respect to \(\pmb {F}\) as follows,

$$\begin{aligned} \delta ^2Loss(\pmb {\phi }):=\delta (\delta Loss(\pmb {\phi })) = \delta (\int _{\mathrm {\Omega }} [(I_{\pmb {m}}(\pmb {\phi }) - I_{\pmb {f}}) \nabla I_{\pmb {m}}(\pmb {\phi }) \cdot \delta \pmb {\phi }]d\pmb {\omega })=\int _{\mathrm {\Omega }} [\delta \pmb {\phi }^{\top }\pmb {K}\delta \pmb {\phi }]d\pmb {\omega }, \end{aligned}$$
$$\begin{aligned} \begin{aligned} \text {where }&\mathrm {\Delta }^{2}\pmb {H}=\pmb {K} = \nabla I_{\pmb {m}}(\pmb {\phi }) [\nabla I_{\pmb {m}}(\pmb {\phi })]^{\top } +(I_{\pmb {m}}(\pmb {\phi }) - I_{\pmb {f}}) \nabla ^{2} I_{\pmb {m}}(\pmb {\phi }),\\ \text { and }&\nabla ^{2} I_{\pmb {m}}(\pmb {\phi }) = \begin{pmatrix} I_{\pmb {m}}(\pmb {\phi })_{xx} \quad I_{\pmb {m}}(\pmb {\phi })_{xy} \quad I_{\pmb {m}}(\pmb {\phi })_{xz} \\ I_{\pmb {m}}(\pmb {\phi })_{yx} \quad I_{\pmb {m}}(\pmb {\phi })_{yy} \quad I_{\pmb {m}}(\pmb {\phi })_{yz} \\ I_{\pmb {m}}(\pmb {\phi })_{zx} \quad I_{\pmb {m}}(\pmb {\phi })_{zy} \quad I_{\pmb {m}}(\pmb {\phi })_{zz} \\ \end{pmatrix}, \end{aligned} \end{aligned}$$
$$\begin{aligned} \text {so, } \delta ^2Loss(\pmb {\phi }) = \int _{\mathrm {\Omega }} [\delta \pmb {\phi }^{\top }\mathrm {\Delta }^{2}\pmb {H}\delta \pmb {\phi }]d\pmb {\omega }=\int _{\mathrm {\Omega }} [\delta \mathrm {\Delta } \pmb {\phi }^{\top }\pmb {H}\delta \mathrm {\Delta } \pmb {\phi }]d\pmb {\omega } \Rightarrow \frac{\partial ^2Loss}{(\partial \pmb {F}) ^{2}}=\pmb {H}. \end{aligned}$$
(3)

A necessary condition that ensures a Newton scheme works is to show such Hessian \(\pmb {H}\) must be of Semi-Positive Definite matrix. This is left for future study.

1.3 Partial Gradients with Respect to Control Functions \(\hat{f}\) and \(\pmb {g}\)

To ensure (1) producing diffeomorphic solutions that is controlled by \(J_{min}\in (0,1)\), instead of optimizing along \(\pmb {F}\) by (2), it can be set that \(f:= J_{min}+\hat{f}^{2}\) in (1). Since it is known \( \delta \mathrm {\Delta } \pmb {u} = \delta \pmb {F} = \delta (\nabla f-\nabla \times \pmb {g})\), then, it carries to,

$$\begin{aligned} \begin{aligned} \delta Loss(\pmb {\phi })&=\int _{\mathrm {\Omega }} [\pmb {b} \cdot \delta \mathrm {\Delta } \pmb {\phi } ]d\pmb {\omega }= \int _{\mathrm {\Omega }} [\pmb {b} \cdot \delta \pmb {F} ]d\pmb {\omega } =\int _{\mathrm {\Omega }}[ \pmb {b} \cdot \delta ( \nabla f -\nabla \times \pmb {g})]d\pmb {\omega }\\&=\int _{\mathrm {\Omega }} [\pmb {b} \cdot ( \nabla \delta (J_{min}+\hat{f}^{2})] d\pmb {\omega }+\int _{\mathrm {\Omega }}[- \pmb {b} \cdot \nabla \times \delta \pmb {g}]d\pmb {\omega } \end{aligned} \end{aligned}$$
$$\begin{aligned} =\int _{\mathrm {\Omega }} [\pmb {b} \cdot (2\hat{f} \nabla \delta \hat{f} )]d\pmb {\omega }+\int _{\mathrm {\Omega }}[- \pmb {b}\cdot \nabla \times \delta \pmb {g}]d\pmb {\omega }=\int _{\mathrm {\Omega }}[-2\hat{f} \nabla \cdot \pmb {b} \delta \hat{f}] d\pmb {\omega }+\int _{\mathrm {\Omega }}[-\nabla \times \pmb {b}\cdot \delta \pmb {g}]d\pmb {\omega } \end{aligned}$$
$$\begin{aligned} \Rightarrow \quad \frac{\partial Loss}{\partial \hat{f}}=-2\hat{f}\nabla \cdot \pmb {b} \quad \text { and } \quad \frac{\partial Loss}{\partial \pmb {g}}=-\nabla \times \pmb {b}. \end{aligned}$$
(4)

2 Numerical Examples

In our algorithms, \(J_{min}=0.5\) is artificially set. It is desirable to design a mechanism that yields optimal values of \(J_{min}\). The gradient-based algorithms can be structured with (1) the coarse-to-fine multiresolution technique, which fits better in large deformation problems over binary images, as it did in [2]; and (2) the function composition regriding technique, which divides the problem difficulty and prevent non-diffeomorphic solutions on medical image registrations. These observations are demonstrated by the next example.

2.1 A Large Deformation Test and a MRI Registration Test

The J-to-V part of this example is done with multiresolution and the Brain Morph part is done with regriding. In Fig. 1(c, j), \(\pmb {\phi }\) is the diffeomorphic solution found by the proposed method; Fig. 1(d, k), \(I_{\pmb {m}}(\pmb {\phi })\) is the registered image that is close to \(I_{\pmb {f}}\), Fig. 1(b, i). Next, \(\pmb {\phi }_{vp}^{-1}\) is the inverse of \(\pmb {\phi }\) that constructed by VP. In Fig. 1(f,m), \(\pmb {\phi }\) is composed by \(\pmb {\phi }^{-1}\), in Red grid, and superposed on Black grid \(\pmb {id}\) but the Black grid barely shows. This shows the composition \(\pmb {T}=\pmb {\phi }_{vp}^{-1}\circ \pmb {\phi }\) is very close to \(\pmb {id}\). Therefore, \(\pmb {\phi }_{vp}^{-1}\) can be treated as the inverse to \(\pmb {\phi }\) and they are of the same diffeomorphism group which VP focuses (Fig. 1).

Table 1. Evaluation of the proposed image registration
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Fig. 1.
figure 1

Resulting registration deformations and their inverses by VP

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The question is whether \(\pmb {\phi }_{vp}^{-1}\) is also a valid inverse registration deformation that moves \(I_{\pmb {f}}\) back to \(I_{\pmb {m}}\). The answer is YES, at least in our tested examples. \(I_{\pmb {f}}(\pmb {\phi }_{vp}^{-1})\) is indeed close to \(I_{\pmb {m}}\). That means \(\pmb {\phi }_{vp}^{-1}\) can be treated as a valid registration deformation from \(I_{\pmb {f}}\) to \(I_{\pmb {m}}\), as it is confirmed by the Table 2 records.

Table 2. Evaluation of \(\pmb {\phi }_{vp}^{-1}\) by VP in the sense of Image Registration
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3 Discussion

This note provides the analytic description with simple demonstration of the proposed method. A full paper with extensive experiments will be available soon.