2007年美国数学建模MCMA一等奖.doc

For office use only T1 ________________ T2 ________________ T3 ________________ T4 ________________ Team Control Number307 ? Problem ChosenA For office use only F1 ________________ F2 ________________ F3 ________________ F4 ________________ 2004 Mathematical Contest in Modeling (MCM) Summary Sheet  Unique or not ? Fingerprint says “Yes!” Summery It is hard to believe that everyone’s name is unique to himself. But “are your fingerprints unique to yourself?” many people will answer “yes”. In order to prove this problem, we create models to explain it. At first we create a simple and comprehensible model to quickly and coarsely estimate this problem with fingerprints patterns. It’s easy to see that our fingerprints vary in general patterns. There are commonly 9 patterns in our fingerprints. Assume that we can distinguish c kinds of fingerprints in a certain pattern for sampled population A. With the variety of parameter c and A, we get an interesting conclusion that even we can distinguish a huge amount of fingerprints in a certain pattern (c=1000 billion) for a local city’s population (A=30 million), the probability of two or more people have the same fingerprints, or similar exactly, is nearly 100%. To obtain an available model, we create an advanced model more strictly and precisely. The ridges and furrows in fingerprint have some similar characteristics. We select 3 kinds of minutiae points including their numbers, positions and directions and develop a statistical algorithm to calculate the uniqueness of fingerprints. The answer is that fingerprints are unique, as expected! Those two models are under a primary assumption that all characteristics of a given fingerprint can be well recognized. But, if the fingerprints at hand are mostly incomplete, polluted or distorted, odds of misidentification will occur. Based on our second model, we analyze how the incompleteness and distortion of minutiae points influence precision of fingerprints identification, and obta