14t0119t34857680322布*1平1 86L3≤97202992997215100467 0」3a色441S91010615401056 3k△641110304752t20099799 66B41天自眉女7855辛1314379554 60t【牙2501母711又99寸0899707 840非09707597331972015519ò 56【07ss1?S5(gz8143580909 43↓787￥1名55"(@554t0354(0 55】82551065030←75R0439+01 Figure 2:Examples of normalized digits from the testing set. is a common feature to all real-world databases,is that both the training set and the testing set contain numerous examples that are ambiguous,unclassifiable,or even misclassified. 3 PREPROCESSING Acquisition,binarization,location of the zipcode,and preliminary segmentation were performed by Postal Service contractors (Wang and Srihari,1988).Some of these steps constitute very hard tasks in themselves.The segmentation(separating each digit from its neighbors)would be a relatively simple task if we could assume that a character is contiguous and is disconnected from its neighbors,but neither of these assumptions holds in practice.Many ambiguous characters in the database are the result of mis-segmentation (especially broken 5's)as can be seen on figure 2. At this point,the size of a digit varies but is typically around 40 by 60 pixels.Since the input of a back-propagation network is fixed size,it is necessary to normalize the size of the characters.This was performed using a linear transformation to make the characters fit in a 16 by 16 pixel image.This transformation preserves the aspect ratio of the character,and is performed after extraneous marks in the image have been removed.Because of the linear transformation,the resulting image is not binary but has multiple gray levels,since a variable number of pixels in the original image can fall into a given pixel in the target image.The gray levels of each image are scaled and translated to fall within the range-1 to 1. 4 THE NETWORK The remainder of the recognition is entirely performed by a multi-layer network.All of the connections in the network are adaptive,although heavily constrained,and are trained using back-propagation.This is in contrast with earlier work (Denker et al.,1989)where the first few layers of connections were hand-chosen constants The input of the network is a 16 by 16 normalized image and the output is composed￾✂✁✄✝✆✞✡✠ ✁ ✌❨❯❤➟✍❳❩❧✖⑤✌❝❣✼●❈✜❆❲⑧➊❁✌❆❉❋❊❧✲❳❲❝❣❦❣➅●✼❍P☎P◗❦❣❷❉❦❣❥❙❈❇⑧➄❋❊❆❉❧✠❥❊❸✌✼✿❥❙✼❘❈❙❥❙❦❣❁✌❷❻❈❊✼●❥❍✸ ❦❣❈✵❳✧❜✐❆▼❧✖❧✖❆▼❁❵⑧➄✼❍❳❩❥❙❀✌❋❊✼❻❥❙❆✲❳❩❝❣❝❃❋❙✼✕❳❲❝➂⑩❶❱✜❆❉❋❊❝⑦P✰P✌❳❲❥✐❳❲❫✍❳❩❈❙✼●❈✕❂✽❦❣❈✩❥❙❸✍❳❩❥✑❫✻❆❉❥❙❸②❥❙❸✣✼✫❥❊❋❬❳❲❦❣❁✌❦❣❁✌❷☎❈❙✼●❥✿❳❩❁✍P ❥❙❸✣✼✖❥❙✼❘❈❙❥❙❦❣❁✌❷②❈❙✼❘❥✹❜✐❆▼❁◗❥❬❳❲❦❣❁✢❁◗❀✌❧☎✼●❋❙❆▼❀✌❈✩✼✐➟✍❳❲❧✖⑤✣❝⑥✼❘❈✿❥❙❸✍❳❩❥✫❳❲❋❊✼✲❳❩❧❻❫✌❦❣❷❉❀✌❆▼❀✌❈❍❂✻❀✌❁✍❜❹❝ ❳❩❈❙❈❊❦➫➁✽❳❲❫✌❝❣✼❉❂➊❆▼❋ ✼●➎▼✼●❁✰❧✖❦❣❈✐❜✐❝⑦❳❲❈❙❈❊❦➂➁✍✼❍P✽✸ ✂ ✗①✛✙✗①✈→ ③✙ ✉❅✉ ➓❛➔✜ ❞❜❘❽❉❀✣❦⑥❈❊❦❣❥❙❦❣❆❉❁❃❂❇❫✌❦❣❁✍❳❲❋❊❦❣➅❍❳❲❥❊❦⑥❆▼❁❃❂❴❝❣❆✌❜●❳❩❥❙❦❣❆❉❁t❆❲⑧✩❥❙❸✌✼ ➅●❦❣⑤➃❜✐❆✌P◗✼▼❂✑❳❲❁✍P✈⑤✌❋❊✼●❝❣❦❣❧✖❦❣❁✍❳❲❋❊❾④❈❊✼●❷❉❧☎✼●❁◗❥❬❳❩❥❙❦❣❆❉❁ ❱✜✼●❋❙✼❻⑤✻✼●❋♠⑧❆❉❋❙❧☎✼❍P ❫❛❾✰➑❤❆❉❈❊❥❬❳❩❝⑨❏◗✼●❋❊➎❛❦⑦❜✐✼✖❜❹❆❉❁◗❥❙❋✐❳❉❜✐❥❊❆❉❋❙❈➞➜➄❭✝❳❲❁✣❷✲❳❲❁✻P④❏◗❋❊❦❣❸✍❳❲❋❊❦❶❂➊➇✕➋❉➠❉➠▼➝●✸❵❏◗❆❉❧☎✼✿❆❲⑧ ❥❙❸✣✼●❈❙✼✩❈❙❥❊✼●⑤✌❈✑❜✐❆▼❁✌❈❙❥❊❦❣❥❙❀✌❥❊✼✵➎▼✼●❋❙❾❻❸✻❳❲❋❬P❵❥✐❳❲❈❊▲❛❈✜❦❣❁❻❥❙❸✌✼❘❧✖❈❙✼❘❝⑥➎▼✼●❈❍✸ ❡❸✣✼✑❈❙✼❘❷❉❧✖✼❘❁❛❥✐❳❲❥❊❦⑥❆▼❁✧➜➄❈❊✼●⑤✍❳❲❋✐❳❲❥❊❦⑥❁✣❷ ✼❍❳▼❜❙❸✰P◗❦❣❷▼❦⑥❥❇⑧➄❋❊❆❉❧✠❦❣❥❙❈☛❁✣✼●❦❣❷❉❸◗❫✍❆▼❋❙❈✐➝✜❱⑨❆▼❀✌❝⑦P❻❫✻✼✫❳✿❋❊✼●❝⑦❳❲❥❊❦⑥➎▼✼●❝❣❾❻❈❙❦❣❧✖⑤✣❝⑥✼❅❥❬❳❩❈❙▲✖❦➂⑧➊❱⑨✼✿❜✐❆▼❀✌❝⑦P✖❳❲❈❊❈❙❀✌❧☎✼ ❥❙❸✻❳❲❥✿❳✖❜❊❸✍❳❩❋❬❳❉❜❹❥❙✼●❋✿❦❣❈✩❜✐❆❉❁◗❥❙❦❣❷▼❀✌❆❉❀✌❈✿❳❩❁✍P✧❦❣❈✿P◗❦❣❈❬❜✐❆▼❁✌❁✌✼❍❜✐❥❊✼❍P ⑧➄❋❊❆❉❧ ❦❣❥❙❈❅❁✌✼●❦❣❷❉❸◗❫✻❆❉❋❙❈✕❂✽❫✌❀✌❥✩❁✌✼❘❦⑥❥❊❸✌✼●❋ ❆❲⑧✽❥❙❸✣✼●❈❙✼✿❳❲❈❊❈❙❀✌❧☎⑤✌❥❙❦❣❆▼❁✌❈➊❸✌❆❉❝⑦P◗❈❴❦❣❁❵⑤✌❋❬❳▼❜✐❥❙❦⑦❜✐✼▼✸❴❺✰❳❲❁◗❾❻❳❲❧❻❫✌❦❣❷❉❀✣❆❉❀✌❈➊❜❙❸✻❳❲❋❬❳▼❜✐❥❙✼❘❋❙❈☛❦❣❁✫❥❊❸✌✼✵P✣❳❲❥❬❳❩❫✍❳❲❈❊✼ ❳❲❋❊✼☛❥❙❸✌✼❅❋❙✼●❈❊❀✌❝❣❥❴❆❲⑧✽❧✖❦❣❈⑩❶❈❊✼●❷❉❧☎✼●❁◗❥❬❳❲❥❊❦❣❆❉❁❵➜➄✼●❈❊⑤✍✼✕❜✐❦⑦❳❲❝❣❝❣❾✫❫✌❋❊❆❉▲▼✼●❁✮✫✌➢ ❈❬➝❇❳❲❈⑨❜●❳❩❁❻❫✍✼☛❈❊✼●✼●❁☎❆❉❁✱➁✍❷❉❀✌❋❊✼ ✤✣✸ ❞❥❴❥❊❸✌❦❣❈❴⑤✍❆▼❦❣❁❛❥✕❂❉❥❊❸✌✼☛❈❙❦❣➅●✼❅❆❲⑧❃❳✿P◗❦❣❷❉❦❣❥➊➎❉❳❲❋❊❦❣✼●❈❴❫✌❀✌❥❇❦⑥❈❇❥♠❾❛⑤✣❦ ❜❘❳❲❝❣❝❣❾✫❳❲❋❊❆❉❀✌❁✍P ✪◗♦❅❫❛❾☎✄❉♦❅⑤✌❦➂➟✌✼●❝❣❈❍✸➊❏◗❦❣❁✍❜✐✼ ❥❙❸✣✼✫❦❣❁✌⑤✌❀✣❥✵❆❩⑧❨❳☎❫✍❳❉❜❊▲▼⑩♠⑤✌❋❙❆▼⑤✍❳❲❷◗❳❲❥❙❦❣❆▼❁✲❁✣✼●❥♠❱⑨❆▼❋❙▲✰❦❣❈❅➁✌➟✌✼❍P ❈❙❦❣➅●✼❉❂✏❦⑥❥✩❦❣❈✑❁✣✼❍❜✐✼●❈❊❈❬❳❲❋❊❾✢❥❙❆✧❁✌❆▼❋❙❧✲❳❩❝❣❦⑥➅❘✼ ❥❙❸✣✼✲❈❊❦❣➅●✼✰❆❲⑧✑❥❊❸✌✼④❜❊❸✍❳❩❋❬❳❉❜❹❥❙✼●❋❊❈❍✸ ❡❸✣❦⑥❈❵❱☛❳❲❈❵⑤✍✼●❋⑧➄❆❉❋❊❧✖✼❍P✈❀✌❈❊❦❣❁✌❷❿❳ ❝❣❦❣❁✌✼❍❳❲❋❵❥❙❋✐❳❲❁✌❈⑧➄❆❉❋❊❧✲❳❩❥❙❦❣❆❉❁✢❥❊❆ ❧✲❳❩▲❉✼✿❥❊❸✌✼✖❜❊❸✍❳❲❋✐❳❉❜✐❥❊✼●❋❙❈✱➁✍❥✿❦❣❁✢❳②➇✆✄☎❫◗❾ ➇✝✄✖⑤✣❦➫➟✌✼●❝❇❦⑥❧✧❳❲❷▼✼❉✸ ❡❸✌❦❣❈✩❥❙❋✐❳❲❁✌❈⑧➄❆❉❋❊❧✲❳❩❥❙❦❣❆❉❁✰⑤✣❋❙✼●❈❊✼●❋❙➎▼✼●❈ ❥❙❸✣✼✖❳❲❈❙⑤✻✼❍❜✐❥✱❋❬❳❲❥❊❦❣❆✲❆❩⑧❨❥❊❸✌✼✖❜❊❸✍❳❲❋✐❳❉❜✐❥❊✼●❋❍❂➊❳❲❁✻P✰❦❣❈✿⑤✻✼●❋♠⑧❆❉❋❙❧☎✼❍P②❳✕⑧➄❥❊✼●❋✩✼✐➟✌❥❙❋❬❳❩❁✌✼●❆▼❀✌❈✿❧✲❳❩❋❙▲◗❈✩❦❣❁✰❥❙❸✌✼ ❦❣❧✲❳❩❷❉✼➊❸✍❳✕➎❉✼✜❫✻✼●✼●❁❻❋❙✼❘❧✖❆❲➎❉✼✕P✽✸❃❄❇✼❍❜●❳❲❀✣❈❙✼☛❆❲⑧✽❥❙❸✣✼⑨❝❣❦❣❁✌✼❍❳❩❋❃❥❙❋✐❳❲❁✌❈⑧➄❆❉❋❊❧✲❳❩❥❙❦❣❆❉❁❃❂✕❥❊❸✌✼⑨❋❊✼●❈❙❀✣❝⑥❥❊❦❣❁✌❷✩❦⑥❧✧❳❲❷▼✼ ❦❣❈✑❁✌❆▼❥☛❫✌❦❣❁✍❳❲❋❊❾✧❫✌❀✌❥✑❸✻❳❲❈☛❧❻❀✌❝❣❥❙❦❣⑤✌❝❣✼❅❷❉❋❬❳✕❾☎❝❣✼●➎❉✼●❝❣❈✕❂✍❈❊❦❣❁✍❜✐✼✫❳➞➎❉❳❲❋❙❦⑦❳❩❫✌❝❣✼✑❁◗❀✌❧❻❫✻✼●❋☛❆❲⑧❴⑤✌❦➂➟✌✼●❝❣❈✑❦❣❁✧❥❙❸✌✼ ❆❉❋❊❦❣❷❉❦❣❁✍❳❲❝➊❦❣❧✲❳❩❷❉✼✫❜❘❳❲❁✢⑧♠❳❲❝❣❝➊❦⑥❁◗❥❊❆④❳✧❷❉❦❣➎▼✼●❁④⑤✣❦➫➟✌✼●❝✜❦❣❁✢❥❙❸✌✼☎❥❬❳❩❋❙❷❉✼❘❥✿❦❣❧✲❳❲❷▼✼❉✸ ❡❸✌✼☎❷❉❋✐❳✕❾✰❝❣✼●➎❉✼●❝❣❈❵❆❲⑧ ✼❍❳▼❜❙❸✧❦❣❧✲❳❩❷❉✼✑❳❩❋❙✼✿❈✐❜●❳❲❝❣✼❍P✧❳❲❁✻P✖❥❊❋❬❳❩❁✌❈❙❝⑦❳❲❥❊✼❍P☎❥❙❆❵⑧❶❳❩❝❣❝✍❱☛❦❣❥❙❸✣❦⑥❁☎❥❙❸✌✼✱❋❬❳❲❁✣❷❉✼✟✞✿➇✩❥❙❆✰➇❉✸ ✠ ✇☛✡✙ ➔✙✧✇✌☞→✝①✎✍ ❡❸✌✼✜❋❙✼●❧✧❳❲❦❣❁✍P◗✼❘❋❃❆❲⑧✻❥❙❸✌✼☛❋❊✼❍❜✐❆❉❷▼❁✌❦❣❥❙❦❣❆❉❁✱❦⑥❈➊✼●❁◗❥❊❦⑥❋❊✼●❝❣❾✿⑤✻✼●❋♠⑧➄❆▼❋❙❧☎✼❍P✩❫◗❾✿❳✑❧❻❀✣❝⑥❥❊❦➂⑩❶❝⑦❳❘❾❉✼●❋✽❁✌✼❘❥♠❱⑨❆▼❋❙▲✽✸ ❞❝❣❝ ❆❲⑧✜❥❙❸✌✼✫❜❹❆❉❁✌❁✌✼✕❜✐❥❙❦❣❆❉❁✣❈✿❦❣❁✲❥❊❸✌✼❵❁✌✼●❥♠❱✜❆❉❋❙▲②❳❲❋❙✼❵❳❉P✌❳❩⑤✌❥❙❦❣➎❉✼▼❂➃❳❩❝❣❥❙❸✌❆▼❀✌❷❉❸✧❸✌✼❍❳✕➎◗❦❣❝❣❾✲❜✐❆▼❁✌❈❙❥❊❋❬❳❩❦⑥❁✣✼❍P✽❂❃❳❩❁✍P ❳❲❋❊✼✫❥❊❋❬❳❩❦⑥❁✣✼❍P✰❀✣❈❙❦❣❁✌❷✖❫✻❳❉❜❊▲▼⑩❶⑤✌❋❊❆❉⑤✍❳❩❷❛❳❩❥❙❦❣❆❉❁❃✸ ❡❸✌❦❣❈✑❦❣❈✩❦❣❁④❜✐❆▼❁❛❥❊❋❬❳❩❈❙❥✩❱☛❦❣❥❙❸✰✼❍❳❩❋❙❝❣❦❣✼●❋✩❱⑨❆▼❋❙▲②➜❶❑❅✼●❁✣▲❉✼●❋ ✼●❥✿❳❲❝♠✸❣❂✽➇❍➋▼➠❉➋▼➝☛❱☛❸✌✼●❋❊✼✫❥❊❸✌✼✱➁✍❋❙❈❊❥✑⑧✼●❱➉❝⑦❳✕❾❉✼●❋❊❈✑❆❲⑧⑨❜✐❆▼❁✌❁✌✼❍❜❹❥❙❦❣❆❉❁✌❈✱❱⑨✼●❋❊✼✫❸✻❳❲❁✍P▼⑩❙❜❊❸✌❆❉❈❊✼●❁✢❜✐❆❉❁✌❈❊❥❬❳❩❁❛❥❊❈❍✸ ❡❸✌✼✜❦❣❁✌⑤✌❀✌❥❤❆❲⑧✍❥❊❸✌✼⑨❁✣✼●❥♠❱⑨❆▼❋❙▲✩❦❣❈✮❳❵➇✆✄✜❫◗❾❻➇✆✄❖❁✌❆❉❋❊❧✲❳❩❝⑥❦❣➅●✼✕P✑❦❣❧✧❳❲❷❉✼➊❳❲❁✍P✱❥❙❸✣✼⑨❆▼❀✌❥❙⑤✌❀✣❥❃❦❣❈✮❜✐❆▼❧✖⑤✻❆❉❈❊✼❍P