This paper describes a method of displaying the structure of branching three-dimensional anatomic structures (like blood vessels or airways). While such structures are normally displayed with a standard orthographic or perspective projection and some sort of shading, the authors are demonstrating the value of a display that, like curvilinear planar reformatting, is not restricted by the exact rigid 3D shape, but instead perserves only some aspects of shape.
In this sense, the design problem the authors solve can be described as “choose your confuser” (Section 6.1 of Algebraic Vis paper). The overall goal of the visualization is to remove confusers caused by 3D occlusion: the whole branching tree should be visible at once. The local 3D orientation of the branches is purposefully being lost, while the length of segments, and the angle of the branches, are explicitly included as quantities to be preserved in the optimization problem the authors solve. That is, the authors are designing their method to maximize visual-data correspondence for alphas that change lengths, angles, and branching order, while making confusers out of alphas that change branch orientation. On the other hand, Fig 8b shows how the orientation confuser can be removed with a colormap of surface normals.