  In this paper, we present a novel data structure, called the Mixture Graph.
This data structure allows us to compress, render, and query segmentation
histograms. Such histograms arise when building a mipmap of a volume containing
segmentation IDs. Each voxel in the histogram mipmap contains a convex
combination (mixture) of segmentation IDs. Each mixture represents the
distribution of IDs in the respective voxel's children. Our method factorizes
these mixtures into a series of linear interpolations between exactly two
segmentation IDs. The result is represented as a directed acyclic graph (DAG)
whose nodes are topologically ordered. Pruning replicate nodes in the tree
followed by compression allows us to store the resulting data structure
efficiently. During rendering, transfer functions are propagated from sources
(leafs) through the DAG to allow for efficient, pre-filtered rendering at
interactive frame rates. Assembly of histogram contributions across the
footprint of a given volume allows us to efficiently query partial histograms,
achieving up to 178$\times$ speed-up over na$\mathrm{\"{i}}$ve parallelized
range queries. Additionally, we apply the Mixture Graph to compute correctly
pre-filtered volume lighting and to interactively explore segments based on
shape, geometry, and orientation using multi-dimensional transfer functions.
