Visualizing Exact and Approximated 3D Empirical Attainment Functions

Table 1

Summarized properties of the presented visualization methods.

	Slicing	MIP	DVR
Exact EAF values	+ Enables detailed analysis of the approximation sets; − Visualizes only one slice at a time	− Not sensible to use since usually a large portion of the objective space has the maximum EAF value	− Not possible to use
Exact EAF differences	+ Enables detailed analysis of the approximation sets; + Capable of simultaneously visualizing positive and negative differences without overlapping; − Visualizes only one slice at a time	+ No parameters to set; + Shows all values in a single image; − No sense of depth; − Feasible only for a limited number of cuboids (impractical)	− Not possible to use
Approximated EAF values	− No need to use it on the approximation as it works well for exact EAFs	− Not sensible to use since usually a large portion of the objective space has the maximum EAF value	+ Nice visualizations; + Enables looking through the volume and preserves the sense of depth; − Requires definition of the transfer function
Approximated EAF differences	− No need to use it on the approximation as it works well for exact EAFs	+ No parameters to set; + Shows all values in a single image; − No sense of depth	+ Nice visualizations; + Enables looking through the volume and preserves the sense of depth; − Requires definition of the transfer function