Fig. 4

Validation using artificially generated data I. a A test of how the estimation performance is affected by spatially heterogeneous noise anisotropy resulting from metric anisotropy. We Inferred the deformation map of a surface whose shape before and after deformation can be represented as graphs. b Since the answer of the map was given, the performance of the proposed method was evaluated by its estimation error (see Supplementary Note 4–1). The labels, “True”, “Approx.”, and “No adjust.” indicate cases in which the noise anisotropy was calculated using the answer of the map given a priori (Eq. (2)), was approximated using the data point (Eq. (2)), or was not taken into consideration (i.e., 2D isotropic noise was assumed), respectively. Including the noise anisotropy in the likelihood improved the mean estimation error by 5–10%, but more importantly, c it prevented spatially biased error; otherwise the error strongly depends on position (more precisely, the error is highly correlated with the size of the induced metric that is measured by \(\det [\tilde{g}]\)), which was clearly observed when the magnitude of noise was not negligible. d Positional dependence of the mean estimation error over 10 estimation trials. The error bar indicates the standard deviation. Incorporating metric-dependent noise anisotropy into the statistical model improves the error bias. In particular, the error in the region with a steep gradient (e.g., at position 5) was clearly improved