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This paper describes an extension to the technical framework of Constructive Volume Geometry (CVG) in order to accommodate point clouds in volume scene graphs. It introduces the notion of point-based volume object (PBVO) that is characterized by the opacity, rather than the geometry, of a point cloud. It examines and compares several radial basis functions (RBFs), including the one proposed in this paper, for constructing scalar fields from point clouds. It applies basic CVG operators to PBVOs and demonstrates the inter-operability of PBVOs with conventional volume objects including those procedurally defined and those constructed from volume datasets. It presents an octree-based algorithm for reducing the complexity in rendering a PBVO with a large number of points, and a set of testing results showing a significant speedup when an octree is deployed for rendering PBVOs.