Abstract We present a new descriptor for activity recognition from videos acquired by a depth sensor. Previous descriptors mostly compute shape and motion features independently; thus, they often fail to capture the complex joint shapemotion cues at pixel-level. In contrast, we describe the depth sequence using a histogram capturing the distribution of the surface normal orientation in the 4D space of time, depth, and spatial coordinates. To build the histogram, we create 4D projectors, which quantize the 4D space and represent the possible directions for the 4D normal. We initial ize the projectors using the vertices of a regular polychoro Consequently, we re?ne the projectors using a discriminative density measure, such that additional projectors are induced in the directions where the 4D normals are more dense and discriminative. Through extensive experiments, we demonstrate that our descriptor better captures the joint shape-motion cues in the depth sequence, and thus outperforms the state-of-the-art on all relevant benchmarks.