The predominant effect of the atmosphere on the incoming wavefront of an astronomical object is the introduction of phase distortion, resulting in an aberrated image from ground-based telescopes. Since wavefront perturbations cannot be directly measured from an image, a wavefront sensor can use intensity variations from a point source to measure specific wavefront aberrations. However, processing of measured aberration data from these sensors can be computationally intensive and this is a challenge for real-time image restoration. To accurately represent such wavefront aberrations with improved processing time, we analyse how the ridgelet transform can be used with the slope-based wavefront sensor i.e., geometric wavefront sensor, in an open-loop configuration. Ridgelet analysis is performed in the Radon domain, where each Radon line integral is computed over N angles, and is represented by a wavelet. Contrasting the behaviour of the ridgelet transform to generate Zernike polynomials with the geometric wavefront sensor, which uses the properties of geometric optics, is the main aim of this paper. We first decompose the image into a Radon domain, and then analyse each line integeral of a Radon transform by a wavelet transform. We show that multi-resolution geometric analysis with ridgelets results in lower wavefront errors, particularly for low photon counts, and computational efficiency of the geometric wavefront sensor is improved by almost a factor of 2.