Geometry constrained sparse embedding for multi-dimensional transfer function design in direct volume rendering

Direct volume rendering (DVR) is commonly employed for the medical visualization. Multi-dimensional transfer functions are used in DVR to emphasize the region of interest in details. However, it is impractical to interact directly with the functions in more than three dimension. This paper proposes a novel framework called geometry constrained sparse embedding (GCSE) for dimensionality reduction (DR). GCSE allows the conventional DR methods to be applied to a dictionary with much smaller atoms instead. The mapping derived from the dictionary feeds to the original features to obtain the ones in the reduced dimension. To obtain a good dictionary, the intrinsic structure of features is encoded in the sparse embedding based on a geometry distance. In addition, stochastic gradient descent algorithm is employed to speed up the dictionary learning. Various experiments have been conducted using both synthetic and real CT data sets. Compared with conventional methods, GCSE not only produces the comparable results, but also performs well with the capability to handle the large data set more powerfully. The rendering results using the real CT data has demonstrated the effectiveness of GCSE.