Polar Fitting and Hermite Interpolation for Freeform Droplet Geometry Measurement

Droplet-based microlens fabrication using Ultra Violet (UV) curable polymers demands the precise measurement of three-dimensional geometries, especially for non-axisymmetric shapes influenced by electric field deformation. In this work, we present a polar coordinate-based contour fitting method combined with Hermite interpolation to reconstruct 3D droplet geometries from two orthogonal shadowgraphy images. The image segmentation process integrates superpixel clustering with active contours to extract the droplet boundary, which is then approximated using a spline-based polar fitting approach. The two resulting contours are merged using a polar Hermite interpolation algorithm, enabling the reconstruction of freeform droplet shapes. We validate the method against both synthetic Computer-Aided Design (CAD) data and precision-machined reference objects, achieving volume deviations below 1% for axisymmetric shapes and approximately 3.5% for non-axisymmetric cases. The influence of focus, calibration, and alignment errors is quantitatively assessed through Monte Carlo simulations and empirical tests. Finally, the method is applied to real electrically deformed droplets, with volume deviations remaining within the experimental uncertainty range. This demonstrates the method’s robustness and suitability for metrology tasks involving complex droplet geometries.