This paper introduces the generalised relaxed Radon transform (GR2T) as an extension to the generalised radon transform (GRT) [1]. This new modelling allows us to define a new framework for robust inference. The resulting objective functions are probability density functions that can be chosen differentiable and that can be optimised using gradient methods. One of this cost function is already widely used in the forms of the Hough transform and generalised projection based M-estimator, and it is interpreted as a conditional density function on the latent variables of interest. In addition the joint density function of the latent variables is also proposed as a cost function and it has the advantage of including a prior about the latent variable. Several applications, including lines detection in images and volume reconstruction from silhouettes captured from multiple views, are presented to underline the versatility of this framework.