A large class of high-speed visualization applications use image acquisition and 3D volume reconstruction techniques in cylindrical sampling grids; these include real-time 3D medical reconstruction, and reverse engineering. This paper presents the novel use of Chebyshev bases in such cylindrical grid- based volume applications, to allow efficient computation of cross-sectional planes of interest and partial volumes without the computationally expensive step of volume rendering, for subsequent transmission in constrained bitrate environments. This has important consequences for low-bitrate applications such as video-conferencing and internet-based visualization environments, where interaction and fusion between independently sampled heterogenous data streams (images, video and 3D volumes) from multiple sources is beginning to play an important part. Volumes often embody widely varying physical signals such as those acquired by X-rays, ultrasound sensors in addition to standard c.c.d. cameras. Several benefits of Chebyshev expansions such as fast convergence, bounded error, computational efficiency, and their optimality for cylindrical grids are taken into account. In addition, our method exploits knowledge about the sampling strategy (e.g. position and trajectory of the sensor) used to acquire the original ensemble of images, which in turn makes the overall approach very amenable to internet-based low-bitrate applications.