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Table 1 Mathematical formulations of similarity metrics

From: Impact of deformable image registration on dose accumulation applied electrocardiograph-gated 4DCT in the heart and left ventricular myocardium during esophageal cancer radiotherapy

  Category Equation Description
Geometry or Feature-based Metrics Point matching \( \mathrm{R}=\sum {\left({p}_{A^{\prime }}-{p}_B\right)}^2/\mathrm{N} \) The registration metric R is defined as the sum of the squared distances between corresponding points \( {p}_{A^{\prime }}\ \mathrm{and}\ {p}_B \), where N is the total number of points [16].
Surface matching \( \mathrm{R}=\sum \mathrm{dist}{\left({p_{A^{\prime}}}^{'},{S}_B\right)}^2/\mathrm{N} \) The dist (\( {p}_{A^{\prime }} \), SB) computes the (minimum) distance between point \( {p}_{A^{\prime }} \) and the surfaces SB, N is the number of points in Study A [16].
Intensity-based Metrics Sum of Squared Differences (SSD) \( \mathrm{SSD}=\sum {\left({I}_{A^{\prime }}-{I}_B\right)}^2/\mathrm{N} \) The SSD metric is defined as the average squared intensity (\( {I}_{A^{\prime }} \) and IB) difference between Study A and Study B, where N is the number of evaluated voxels [16].
Correlation Coefficient (CC) CC=\( \frac{\sum_{\overrightarrow{x}}\left(A\left(\overline{x}\right)-\overline{A}\right)\left(T\left(B\left(\overrightarrow{x}\right)\right)-\overline{B}\right)}{\sqrt{\sum_{\overrightarrow{x}}{\left(A\left(\overline{x}\right)-\overline{A}\right)}^2{\sum}_{\overrightarrow{x}}{\left(T\left(B\left(\overrightarrow{x}\right)\right)-\overline{B}\right)}^2}} \) CC measures the similarity in the image signal, which assumes a linear relationship between voxel intensities in two images [16].
Mutual Information (MI) \( \mathrm{MI}\ \left({I}_{A^{\prime }},{I}_B\right)=\sum \limits_{\mathrm{B}}\sum \limits_Ap\left({I}_{A^{\prime }},{I}_B\right){\mathit{\log}}_2\left[p\left({I}_{A^{\prime }},{I}_B\right)/p\left({I}_{A^{\prime }}\right)p\left({I}_B\right)\right] \) MI has proven very effective for registering image data from different modalities, where p (IA’) and p (IB) are the probability distribution functions of the intensities IA’ and IB, respectively, and \( p\left({I}_{A^{\prime }},{I}_B\right) \) is the joint probability distribution function [16].