Purpose Non-invasive axon diameter distribution (ADD) mapping using diffusion MRI is an ill-posed problem. Current ADD mapping methods require knowledge of axon orientation before performing the acquisition. Instead, ActiveAx uses a 3D sampling scheme to estimate the orientation from the signal, providing orientationally invariant estimates. The mean diameter is estimated instead of the distribution for the solution to be tractable. Here, we propose an extension (ActiveAx(ADD)) that provides non-parametric and orientationally invariant estimates of the whole distribution. Theory The accelerated microstructure imaging with convex optimization (AMICO) framework accelerates mean diameter estimation using a linear formulation combined with Tikhonov regularization to stabilize the solution. Here, we implement a new formulation (ActiveAx(ADD)) that uses Laplacian regularization to provide robust estimates of the whole ADD. Methods The performance of ActiveAx(ADD) was evaluated using Monte Carlo simulations on synthetic white matter samples mimicking axon distributions reported in histological studies. Results ActiveAx(ADD) provided robust ADD reconstructions when considering the isolated intra-axonal signal. However, our formulation inherited some common microstructure imaging limitations. When accounting for the extra axonal compartment, estimated ADDs showed spurious peaks and increased variability because of the difficulty of disentangling intra and extra axonal contributions. Conclusion Laplacian regularization solves the ill-posedness regarding the intra axonal compartment. ActiveAx(ADD) can potentially provide non-parametric and orientationally invariant ADDs from isolated intra-axonal signals. However, further work is required before ActiveAx(ADD) can be applied to real data containing extra-axonal contributions, as disentangling the 2 compartment appears to be an overlooked challenge that affects microstructure imaging methods in general.