In the diffusion MRI domain, the HARDI methods were proposed to better characterize the complex biological tissues such as the white matter. In fact, they allow to overcome the problem of crossing fibers detection in the case of the Diffusion Tensor Imaging (DTI). However, the HARDI technique requires a very large number of Diffusion Weighted (DW) magnetic resonance images and thus it takes a long acquisition time, restricting its use in the clinical practice. We propose, in this paper, to develop a novel method for accelerating the reconstruction of the HARDI signal from a few number of DW images of the brain. The approach is a triangulation-based geometrical interpolation of existing signals. It consists on recovering non-acquired data according to their neighborhood from a reduced set of diffusion orientations on the sphere of the q-space. The accuracy of the proposed method was performed on two different phantom datasets for a qualitative and quantitative evaluation against known ground truths. We test also the robustness of the proposed approach under the noise. The obtained results demonstrate that the novel method can approximately halve the scan time and simultaneously obtain a proper fiber orientation estimation. In fact, the estimated FOD (e.g. FOD, Fiber Orientation Distribution) function based on only 34 directions is nearly identical to the one calculated with the full HARDI acquisitions (65 directions). Comparative evaluation on standard phantoms show that the proposed approach outperforms a large selection of methods of state-of-the-art according to the success rate and the angular error criteria.