Machine learning K-means algorithm applied to wavefront sensing in Bi-Ronchi/Hartmann tests with SLM

We apply new machine learning (ML) technologies to optimize the Hartmann test (HT) and Bi-Ronchi test (BRT) regarding the recognition, identification, and localization of the centroids in experimental Hartmanngrams and Bi-Ronchigrams. We replace the conventional rigid Hartmann screen (Hartmann mask, HM) with structured apertures implemented via a spatial light modulator (SLM), which enables the generation of multiple patterns with different aperture geometries. Based on the classical HM with circular apertures, we build square apertures for the Bi-Ronchi mask (BRM). We designed an experimental setup based on an SLM with a laser illumination system and implemented an unsupervised Centroid Clustering Algorithm (uCCA), based on the ML algorithm K-means, to identify the geometries of the centroids, followed by their segmentation and localization by clustering. We compare the experimental and theoretical Bi-Ronchigrams (or Hartmanngrams) to obtain a point cloud of transverse aberrations (𝑃𝐶_𝑇𝐴). We apply the point cloud method (PCM) to obtain an integrable surface from the points in 𝑃𝐶_𝑇𝐴. Finally, we replace the numerical integration of 𝑃𝐶_𝑇𝐴 with transverse aberrations (𝑇𝐴) and a directional derivative approach based on the Eikonal equation, solved using Gaussian quadrature to obtain the wavefront. We compare our results with the Zernike aberration polynomials for sensing optical elements from the aberrations of the system by means of the aberrations of its wavefront W(𝜌, 𝜃).