On the Complexity of Cognition

S. JAEGER Intera tive Systems Laboratories University of Karlsruhe Computer S ien e Department, 76128 Karlsruhe, Germany email: stefan.jaeger ira.uka.de This paper presents an investigation of a ognitive problem in terms of omplexity theory. Two global optimization approa hes are presented for re overing traje tories from stati , handwritten word images. Both take graph-theoreti al representations of symbols as input. The rst, polynomial approa h minimizes the length of the re overed traje tories. This approa h annot re over traje tories traversing parts of the word more than twi e. The se ond approa h, whi h minimizes osts at distinguished nodes of the traje tory, is more powerful in this respe t and is proved to be NP-hard. An eÆ ient divide-andonquer method is proposed that splits handwritten words into independent subparts and re overs traje tories for every subpart, whi h turn out to be very small in pra ti e. The splitting te hnique exploits morphologi al features from the stati word image. 1 Introdu tion This paper des ribes a theoreti al investigation towards re overing traje tories from stati word images. The solution to this problem would be a major step towards a uni ed view on handwriting re ognition that ombines online as well as o -line aspe ts [3, 9, 12℄. There is no doubt that re overing traje tories is a diÆ ult problem that tou hes aspe ts of many elds su h as per eption, ognition, and many more. This paper introdu es an approa h trying to measure this diÆ ulty in terms of omplexity theory. Figure 1 shows a s anned binary image of the letter \B" and its skeleton. The skeleton, whi h is shown on the right-hand side of Figure 1, is the graphtheoreti al abstra tion of the s anned word image. A skeleton is a graph omprising nodes and edges, where nodes o ur at rossing points or ends of strokes. It oin ides in most areas with the enterline of the writing tra e and provides us with the ne essary formalism for theoreti al investigations. In Figure 1, the original traje tory orresponds to the following sequen e of edges: A;B;C;B;D;E; F;G;H , where the edges C,B, and E must be retra ed. Se tions 2 and 3 present two graph-theoreti al methods for re overing the original traje tories from skeletons. While the rst approa h presumes that the original traje tory is a path with the minimum length, the se ond approa h tries to minimize the deviation from the straight writing line. Both methods are global optimization te hniques based on the assumption that the per eption 291