Point-Based Global Illumination (PBGI) is a popular rendering method in special effects and motion picture productions. This algorithm provides a diffuse global illumination solution by caching radiance in a mesh-less hierarchical data structure during a pre-process, while solving for visibility over this cache, at rendering time and for each receiver, using microbuffers, which are localized depth and color buffers inspired from real time rendering environments. As a result, noise free ambient occlusion, indirect soft shadows and color bleeding effects are computed efficiently for high resolution image output and in a temporally coherent fashion. We propose an evolution of this method to address the case of non-diffuse inter-reflections and refractions. While the original PBGI algorithm models radiance using spherical harmonics, we propose to use wavelets parameterized on the direction space to better localize the radiance representation in the presence of highly directional reflectance. We also propose a new importance-driven adaptive microbuffer model to capture accurately incoming radiance at a point. Furthermore, we evaluate outgoing radiance using a fast wavelet radiance product and contain the induced larger memory footprint by encoding hierarchically the wavelets in the PBGI tree. As a result, our algorithm can handle non-lambertian BSDF in the light transport simulation, reproducing caustics and multiple reflections/refractions bounces with a similar quality to bidirectional path tracing in a large number of cases and for only a fraction of its computation time. Our approach is simple to implement and easy to integrate into any existing PBGI framework, with an intuitive control on the approximation error. We evaluate it on a collection of example scenes.