Sensor shapes and weak modes of the ATLAS Inner Detector track-based alignment

The alignment of the ATLAS Inner Detector is performed with a track-based algorithm. The aim of the detector alignment is to provide an accurate description of the detector geometry such that track parameters are accurately determined and bias free. The detector alignment is validated and improved by studying resonant decays ($J/\psi$ and $Z$ to $\mu^+\mu^-$). The detailed study of these resonances (together with the properties of the tracks of their decay products) allows to detect and correct for alignment weak modes such as detector curls and radial deformations that may bias the momentum and/or the impact parameter measurements. Here, radial distortions were investigated. Furthermore, a new analysis with a detailed scrutiny of the track-to-hit residuals allowed to study the deformation shape of the Pixel and IBL modules. The sensor distortion can result in track-to-hit residual biases of up to $10\mu m$ within a given module. The shape of the IBL modules was parametrised with Bernstein-B\'ezier functions and used to correct the hit position in the track fitting procedure.