As modern high-resolution imaging devices allow to acquire increasingly large and complex volume data sets, their effective and compact representation for visualization becomes a challenging task. The Tucker decomposition has already confirmed higher- order tensor approximation (TA) as a viable technique for compressed volume representation; however, alternative decomposition approaches exist. In this work, we review the main TA models proposed in the literature on multiway data analysis and study their application in a visualization context, where reconstruction performance is emphasized along with reduced data representation costs. Progressive and selective detail reconstruction is a main goal for such representations and can efficiently be achieved by truncating an existing decomposition. To this end, we explore alternative incremental variations of the CANDECOMP/PARAFAC and Tucker models. We give theoretical time and space complexity estimates for every discussed approach and variant. Addi- tionally, their empirical decomposition and reconstruction times and approximation quality are tested in both C++ and MATLAB implementations. Several scanned real-life exemplar volumes are used varying data sizes, initialization methods, degree of com- pression and truncation. As a result of this, we demonstrate the superiority of the Tucker model for most visualization purposes, while canonical-based models offer benefits only in limited situations.