We present an asymptotic treatment of errors involved in point-based image registration where control point (CP) localization is subject to heteroscedastic noise; a suitable model for image registration in fluorescence microscopy. Assuming an affine transform, CPs are used to solve a multivariate regression problem. With measurement errors existing for both sets of CPs this is an errors-in-variable problem and linear least squares is inappropriate; the correct method being generalized least squares. To allow for point dependent errors the equivalence of a generalized maximum likelihood and heteroscedastic generalized least squares model is achieved allowing previously published asymptotic results to be extended to image registration. For a particularly useful model of heteroscedastic noise where covariance matrices are scalar multiples of a known matrix (including the case where covariance matrices are multiples of the identity) we provide closed form solutions to estimators and derive their distribution. We consider the target registration error (TRE) and define a new measure called the localization registration error (LRE) believed to be useful, especially in microscopy registration experiments. Assuming Gaussianity of the CP localization errors, it is shown that the asymptotic distribution for the TRE and LRE are themselves Gaussian and the parameterized distributions are derived. Results are successfully applied to registration in single molecule microscopy to derive the key dependence of the TRE and LRE variance on the number of CPs and their associated photon counts. Simulations show asymptotic results are robust for low CP numbers and non-Gaussianity. The method presented here is shown to outperform GLS on real imaging data.

中文翻译：

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基于点的图像配准误差分析在单分子显微镜中的应用

我们对基于点的图像配准中涉及的错误进行渐近处理，其中控制点 (CP) 定位受异方差噪声的影响；一种适用于荧光显微镜图像配准的模型。假设仿射变换，CP 用于解决多元回归问题。由于两组 CP 都存在测量误差，这是一个可变误差问题，线性最小二乘法是不合适的；正确的方法是广义最小二乘法。为了允许点相关误差，实现了广义最大似然和异方差广义最小二乘模型的等效性，从而允许将先前发布的渐近结果扩展到图像配准。对于一个特别有用的异方差噪声模型，其中协方差矩阵是已知矩阵的标量倍数（包括协方差矩阵是身份倍数的情况），我们为估计量提供了封闭形式的解决方案并推导出它们的分布。我们考虑了目标配准误差 (TRE) 并定义了一种称为定位配准误差 (LRE) 的新度量，该指标被认为是有用的，尤其是在显微镜配准实验中。假设 CP 定位误差的高斯性，表明 TRE 和 LRE 的渐近分布本身是高斯分布，并且导出了参数化分布。结果已成功应用于单分子显微镜中的配准，以推导出 TRE 和 LRE 方差对 CP 数量及其相关光子计数的关键依赖性。模拟显示渐近结果对于低 CP 数和非高斯性是稳健的。此处介绍的方法在真实成像数据上的表现优于 GLS。