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General first-order target registration error model considering a coordinate reference frame in an image-guided surgical system

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Abstract

Point-based rigid registration (PBRR) techniques are widely used in many aspects of image-guided surgery (IGS). Accurately estimating target registration error (TRE) statistics is of essential value for medical applications such as optically surgical tool-tip tracking and image registration. For example, knowing the TRE distribution statistics of surgical tool tip can help the surgeon make right decisions during surgery. In the meantime, the pose of a surgical tool is usually reported relative to a second rigid body whose local frame is called coordinate reference frame (CRF). In an n-ocular tracking system, fiducial localization error (FLE) should be considered inhomogeneous, that means FLE is different between fiducials, and anisotropic that indicates FLE is different in all directions. In this paper, we extend the TRE estimation algorithm relative to a CRF from homogeneous and anisotropic to heterogeneous FLE cases. Arbitrary weightings can be assumed in solving the registration problems in the proposed TRE estimation algorithm. Monte Carlo simulation results demonstrate the proposed algorithm’s effectiveness for both homogeneous and inhomogeneous FLE distributions. The results are further compared with those using the other two algorithms. When FLE distribution is anisotropic and homogeneous, the proposed TRE estimation algorithm’s performance is comparable with that of the first one. When FLE distribution is heterogeneous, proposed TRE estimation algorithm outperforms the other two classical algorithms in all test cases when ideal weighting scheme is adopted in solving two registrations. Possible clinical applications include the online estimation of surgical tool-tip tracking error with respect to a CRF in IGS.

This paper provides the target registration error model considering a coordinate reference frame in surgical navigation.

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Funding

This project is partially supported by the Hong Kong RGC GRF grant no. 14210117, RGC NSFC/RGC Joint Research Scheme no. N_CUHK448/17, ITC ITF grant no. ITS/236/15, and Shenzhen Science and Technology Innovation projects JCYJ20170413161616163 awarded to Prof. Max Q.-H. Meng.

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  1. Department of Electronic Engineering, The Chinese University of Hong Kong, Hong Kong SAR, N.T., Hong Kong

    Zhe Min & Max Q.-H. Meng

  2. Department of Electronic and Electrical Engineering, Southern University of Science and Technology, Shenzhen, China

    Max Q.-H. Meng

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  1. Zhe Min
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Correspondence to Max Q.-H. Meng.

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Appendix

Appendix

In this section, we verify that the TRE vector calculated using the proposed algorithm is consistent with that using Wiles’ algorithm. In Wiles’ algorithm, the estimated TRE expression in a single registration is in the principal axes’ frame of a rigid body. In the last row of Eq. 2, the term \({{~}_{ots}^{crf} }{\mathbf {R}^{\star }}\cdot ({~}^{ots}{\mathbf {t}\mathbf {r}\mathbf {e}_{to}({{~}^{ots}}{\mathbf {p}})})\) becomes \({{~}_{trf}^{crf} }{\mathbf {R}^{\star }}\cdot (^{trf}{\mathbf {t}\mathbf {r}\mathbf {e}_{to}({{~}^{trf}}{\mathbf {p}})})\) when Wiles’ algorithm is used, where \(^{trf}{\mathbf {t}\mathbf {r}\mathbf {e}_{to}({{~}^{trf}}{\mathbf {p}})}\) is defined as,

$$ \begin{array}{ll} ^{trf}{\mathbf{t}\mathbf{r}\mathbf{e}_{to}({{~}^{trf}}{\mathbf{p}})}&=\mathbf{T}_{e,p}\cdot^{trf}{\mathbf{p}}^{\star}-^{trf}{\mathbf{p}}^{\star}\\ &=\mathbf{R}_{e,p}\cdot^{trf}{\mathbf{p}}^{\star}+\mathbf{t}_{e,p}-^{trf}{\mathbf{p}}^{\star} \end{array} $$
(12)

where the term Te,p represents the erroneous transformation matrix relating OTS frame and TRF. We can rewrite the Te,p explicitly:

$$ \mathbf{T}_{e,p}=({{~}_{trf}^{ots}}{\mathbf{T}}^{\star})^{-1}\cdot{{~}_{trf}^{ots}}{\mathbf{T}}={{~}_{ots}^{trf}}{\mathbf{T}}^{\star}\cdot{{~}_{trf}^{ots}}{\mathbf{T}} $$
(13)

where we have \({{~}_{trf}^{ots}}{\mathbf {T}}^{\star }=[{{~}_{trf}^{ots}}{\mathbf {R}}^{\star },{{~}_{trf}^{ots}}{\mathbf {t}}^{\star }]\), we then have the inverse:

$$ \begin{array}{ll} ({{~}_{trf}^{ots}}{\mathbf{T}}^{\star})^{-1}&=[({{~}_{trf}^{ots}}{\mathbf{R}}^{\star})^{\mathsf{T}},-({{~}_{trf}^{ots}}{\mathbf{R}}^{\star})^{\mathsf{T}}\cdot{{~}_{trf}^{ots}}{\mathbf{t}}^{\star}]\\&=[{{~}_{ots}^{trf}}{\mathbf{R}}^{\star},-{{~}_{ots}^{trf}}{\mathbf{R}}^{\star}\cdot{{~}_{trf}^{ots}}{\mathbf{t}}^{\star}] \end{array} $$
(14)

With \({{~}_{trf}^{ots}}{\mathbf {T}}=[{{~}_{trf}^{ots}}{\mathbf {R}},{{~}_{trf}^{ots}}{\mathbf {t}}]\), we can have the explicit expression of Te,p: \(\mathbf {T}_{e,p}=[\mathbf {R}_{e,p},\mathbf {t}_{e,p}]=[{{~}_{ots}^{trf}}{\mathbf {R}}^{\star }\cdot {{~}_{trf}^{ots}}{\mathbf {R}},{{~}_{ots}^{trf}}{\mathbf {R}}^{\star }\cdot {{~}_{trf}^{ots}}{\mathbf {t}}-{{~}_{ots}^{trf}}{\mathbf {R}}^{\star }\cdot {{~}_{trf}^{ots}}{\mathbf {t}}^{\star }]\). Substituting the expression of Te,p in Eq. 13 into the first row of Eq. 12, we can have another expression of \(^{trf}{\mathbf {t}\mathbf {r}\mathbf {e}_{to}({{~}^{trf}}{\mathbf {p}})}\):

$$ \begin{array}{ll} ^{trf}{\mathbf{t}\mathbf{r}\mathbf{e}_{to}({{~}^{trf}}{\mathbf{p}})}&={{~}_{ots}^{trf}}{\mathbf{T}}^{\star}\cdot{{~}_{trf}^{ots}}{\mathbf{T}}\cdot^{trf}{\mathbf{p}}^{\star}-^{trf}{\mathbf{p}}^{\star}\\ &={{~}_{ots}^{trf}}{\mathbf{T}}^{\star}\cdot({{~}_{trf}^{ots}}{\mathbf{T}}\cdot^{trf}{\mathbf{p}}^{\star}-{{~}_{trf}^{ots}}{\mathbf{T}}^{\star}\cdot^{trf}{\mathbf{p}}^{\star}) \end{array} $$
(15)

Noticing that \(({{~}_{trf}^{ots}}{\mathbf {T}}\cdot ^{trf}{\mathbf {p}}^{\star }-{{~}_{trf}^{ots}}{\mathbf {T}}^{\star }\cdot ^{trf}{\mathbf {p}}^{\star })\) in Eq. 15 equals the value of \(^{ots}{\mathbf {t}\mathbf {r}\mathbf {e}_{to}({{~}^{ots}}{\mathbf {p}})}\) (referring to Eq. 4), together with Eq. 15, we can have:

$$ ^{trf}{\mathbf{t}\mathbf{r}\mathbf{e}_{to}({{~}^{trf}}{\mathbf{p}})}={{~}_{ots}^{trf}}{\mathbf{R}}^{\star}\cdot^{ots}{\mathbf{t}\mathbf{r}\mathbf{e}_{to}({{~}^{ots}}{\mathbf{p}})} $$
(16)

Finally, multiplying both sides of Eq. 16 with \({{~}_{trf}^{crf} }{\mathbf {R}^{\star }}\), we can have the following:

$$ \begin{array}{ll} {{~}_{trf}^{crf} }{\mathbf{R}^{\star}}\cdot (^{trf}{\mathbf{t}\mathbf{r}\mathbf{e}_{to}({{~}^{trf}}{\mathbf{p}})})&={{~}_{trf}^{crf} }{\mathbf{R}^{\star}}\cdot{{~}_{ots}^{trf}}{\mathbf{R}}^{\star}\cdot^{ots}{\mathbf{t}\mathbf{r}\mathbf{e}_{to}({{~}^{ots}}{\mathbf{p}})}\\ &={{~}_{ots}^{crf} }{\mathbf{R}^{\star}}\cdot (^{ots}{\mathbf{t}\mathbf{r}\mathbf{e}_{to}({{~}^{ots}}{\mathbf{p}})}) \end{array} $$
(17)

Until now, we have shown that the expression of the composed TRE vector in CRF space derived in this article is equal to the classical one. We can further derive the mathematical relationship between the covariance matrices of random variables \(^{trf}{\mathbf {t}\mathbf {r}\mathbf {e}_{to}({{~}^{trf}}{\mathbf {p}})}\) and \(^{ots}{\mathbf {t}\mathbf {r}\mathbf {e}_{to}({{~}^{ots}}{\mathbf {p}})}\) on both sides of Eq. 16 as the following:

$$ cov[^{trf}{\mathbf{t}\mathbf{r}\mathbf{e}_{to}({{~}^{trf}}{\mathbf{p}})}]={{~}_{ots}^{trf}}{\mathbf{R}}^{\star}\cdot cov[^{ots}{\mathbf{t}\mathbf{r}\mathbf{e}_{to}({{~}^{ots}}{\mathbf{p}})}]\cdot{{{~}_{ots}^{trf}}{\mathbf{R}}^{\star}}^{\mathsf{T}} $$
(18)

For the sake of clarity, the covariance matrix of composed TRE vector \(cov[{{~}_{comb}}{\mathbf {t}\mathbf {r}\mathbf {e}}{({{~}^{crf}}{\mathbf {p}})}]\) in Wiles’ algorithm is as the following:

$$ \begin{array}{ll} &{{~}^{crf}}{{\varSigma}}_{tre,comb}({^{crf}{\mathbf{p}}})\\ &={{~}^{crf}}{{\varSigma}}_{tre,oc}({^{crf}{\mathbf{p}}})+{{~}_{trf}^{crf} }{\mathbf{R}^{\star}}\cdot cov[^{trf}{\mathbf{t}\mathbf{r}\mathbf{e}_{to}({{~}^{trf}}{\mathbf{p}})}]({{~}_{trf}^{crf} }{\mathbf{R}^{\star}})^{\mathsf{T}} \end{array} $$
(19)

where the value of \({{~}^{crf}}{{\varSigma }}_{tre,oc}({^{crf}{\mathbf {p}}})\) and \(cov[^{trf}{\mathbf {t}\mathbf {r}\mathbf {e}_{to}({{~}^{trf}}{\mathbf {p}})}]\) can be acquired directly from Wiles’ algorithm in a single registration as the two terms are expressed in their respective principal axes’ frames . Substituting Eqs. 8 and 18 into 19, we can conclude expression in Eq. 19 is in fact equivalent to that in Eq. 8.

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Min, Z., Meng, M.QH. General first-order target registration error model considering a coordinate reference frame in an image-guided surgical system. Med Biol Eng Comput 58, 2989–3002 (2020). https://doi.org/10.1007/s11517-020-02265-y

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