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Lines give information, sometimes partial or approximate, about object boundaries, and hence about 3-D shape. They also inform about light intensity and texture. Through curvature, thickness, and other properties, they suggest emotion, mood, and character. Some of this may be because of their resemblance to poses and bodyparts, as well as through cross-sensory associations such as the bouba/kiki effect.
All this is well known to artists. Mathematicians and computer scientists need to understand it too, if writing programs that process or generate drawings. What mathematical techniques might we use to represent the semantics of line drawings?
One possibility that occurs to me is scale-space filtering for separating out lines used for cross-hatching and other shading. To represent "suprasegmental" information about line character, use sheaves. Let the basis sets be all open sets of the lines in the drawing, and define functors to a strength of association with poses and bodyparts, as well as to attributes such as those that Paul Klee writes about in in his Pedagogical Sketchbook.
A similar scheme could be used to represent the likelihood that incomplete or ambiguous figures, such as the duck's wing/hands in the clip art, and the people at the right-hand end of the Phiz crowd drawing, actually do denote a particular object.
Any semantics needs to work on all the drawings I've shown in this talk, including the caricatures and the abstract designs on the previous slide.