Depth-Dependent Point Spread Function Calibration using Spline Interpolation in 3D Single Molecule Localization Microscopy

Abstract

In three-dimensional single-molecule localization microscopy (SMLM), emitter positions are estimated by fitting a model of the microscope’s Point Spread Function (PSF) to measured data. In practice, PSF models are typically calibrated using bead data acquired near the coverslip, and are assumed to remain valid representations at larger imaging depths. However, refractive index mismatch between the immersion medium, coverslip, and sample introduces depth-dependent spherical aberrations, causing the PSF shape to vary with imaging depth. As a result, a PSF model calibrated at the coverslip leads to degraded lateral localization precision and substantial axial bias when applied several micrometers deep into the sample. In this work, we introduce a depth-dependent PSF calibration approach that interpolates between calibration datasets acquired at multiple imaging depths. Calibration stacks are reconstructed at arbitrary depths using Catmull–Rom spline interpolation and used to calibrate cubic spline (cspline) models for localization. Simulations show that a conventional coverslip-calibrated model results in mean absolute axial biases exceeding 294 nm at an imaging depth of 5 μm. In contrast, the proposed approach reduces the axial bias up to 99%, consistently achieving axial bias below 5 nm. In addition, the lateral localization precision improves by 62% and 61% in x and y, respectively. Validation on experimentally acquired bead data demonstrates an axial bias reduction of 80% compared to coverslip calibration. These results show that interpolation of calibration data across depth compensates for depth-dependent PSF mismatch, enabling accurate and precise 3D localization over extended imaging depths without requiring additional optical hardware.