EXAMPLE :Principle of Optimality

and Dynamic Programming

1.4

1.3

1.4

Dynamic Programming

Dynamic Programming

• Search for best path• Dynamic programming principle

– If B is on the shortest path from A to C, then the path from A to B that lies on the path from A to C is the shortest path from A to B

AB

C

… …

Dynamic Programming• If DP principle holds, then efficient search can be

implemented

• When at node B in search you can prune all paths to B that are not the shortest

• You should not expand these paths

AB

C

… …

Applications to Handwriting Recognition• Address Interpretation – USPS

• Word Recognition

– ZIP + Street No. Hypotheses

– “Small” Lexicons for Word Rec

1. Problem Statement and History

2. Picture of Parsed Address Block

3. Word Recognition Problem

4. Results for NIST OCR Tests with refs

Character Ambiguity in Handwriting Recognition

word image

segmentation

dynamic programming

matchinglexicon

primitivesprimitives

and unions of primitives

match scores between word

image and lexicon strings

character class membership assignment

3 modules for character class

membership assignment

Baseline module: MLP

New module: SOFM and MLP

New module: SOFM and FI

Handwritten Word Recognition System

fuzzy integral aggregation

DP Approach to handwriting recognition

Self-Organizing Feature Maps

Mixed Linear Programming

Fuzzy Inferfence

Segmentation-Based Handwritten Word Recognition

Search Problem – How do we put the pieces together?

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25

Inp u t Imag e

Primitiv es

Best Match

to "Richmo nd"

Best Match

to "Ed mun d"

Image Segmentatio n

R=5 3 i=2 7 c=5 2 h=6 1 m=7 0 o=4 3 n =61 d=8 8

E=1 2 d=7 9 m-85 u=25 n=6 1 d =88

DP Approach

• Let I be a word image

• L = {L1, L2, ..., LT} be a lexicon of strings representing all possible identities of I.

• Word recognition requires finding the string in L that best matches I.

• Search – DP, Heuristic

DP ApproachLet Lm = c1c2...cn. Let the primitives of I be S1, S2, ..., Sp , (ordered from left to right). Denote

the set of all legal unions of primitives by

UI = { Sjk : Sjk = k

jh

hS

, j k, and k

jh

hS

is legal }.

A segmentation of length n of I is a sequence of elements of UI of the form

S = Sk0+1,k1, Sk1+1,k2, ..., Skn-1+1, kn where k0 = 0 and k n = p.

Let Sn denote the set of all segmentations of I of length n. For each character class c, let xc(s)

denote the neural network output corresponding to class c given s UI as input. A match score

between Lm and a segmentation S Sn can be computed as

segmatch ( Lm , S) =

n

ikkc iii

Sxn 1

,1 )(11 .

DP ApproachFor i = 1, ..., w

For j = i, ..., p 1 IF i = 1, (that is we are matching against the first character) THEN

2 IF Legal(S1j) THEN

3 dp(1,j) = ( S1j, C1)

4 ELSE

5 dp(1,j) = -?

6 ENDIF

7 ELSEIF Legal(Sij) THEN

8 dp ( i , j ) maxk

dp ( i 1, k ) ( S k 1 , j , C i ) | i k j and Legal Sk j is TRUE

9 ELSE

10 dp(i,j) = -?

11 ENDIF

12 ENDIF

13 IF dp(i,j) > -? THEN

14 j2 arg maxk

dp (i 1, k ) (S k1, j , Ci ) | i k j and Legal Skj is TRUE

15 path(i,j) = (i - 1, j2)

16 ENDIF

ENDFOR

ENDFOR

DP Approach

Table 1. Dynamic Programming Array for Match of Richmond to "Richmond"

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

"R" 11 38 53 45 44 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

"i" 0 23 27 71 80 52 53 54 52 51 55 0 0 0 0 0 0 0 0 0 0 0 0 0 0

"c" 0 0 0 38 90 132 108 108 86 86 87 63 63 63 63 62 65 65 0 0 0 0 0 0 0

"h" 0 0 0 0 42 95 144 147 157 193 189 166 162 141 145 121 100 95 85 77 76 0 0 0 0

"m" 0 0 0 0 0 50 104 154 156 167 202 208 219 227 261 263 272 258 178 177 216 231 0 0 0

"o" 0 0 0 0 0 0 74 114 169 163 179 206 241 261 236 236 279 309 312 290 281 280 281 0 0

"n" 0 0 0 0 0 0 0 83 122 178 181 190 223 261 269 269 275 285 322 337 352 370 356 321 0

"d" 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 458

Table 2. Dynamic Programming Path Array for Match of Richmond to "Richmond"

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

"R" 0 0 0 0 0 0 0 1 8 8 6 9 11 11 12 12 13 16 15 15 19 20 20 19 20

"i" 0 1 1 3 3 5 5 5 5 5 5 5 7 7 14 10 16 16 16 15 15 19 16 21 19

"c" 0 0 2 3 3 5 5 5 5 5 5 7 7 7 8 11 11 11 12 14 14 16 21 19 21

"h" 0 0 0 3 4 5 6 6 6 6 6 6 6 8 8 10 11 11 15 15 15 16 16 17 21

"m" 0 0 0 0 4 5 6 7 8 9 10 10 10 11 11 10 10 11 12 13 15 15 16 21 19

"o" 0 0 0 0 0 5 6 7 8 9 10 11 12 13 14 14 16 16 16 15 17 17 17 19 21

"n" 0 0 0 0 0 0 6 7 8 9 9 9 11 13 14 14 13 17 18 18 17 18 18 19 23

"d" 0 0 0 0 0 0 0 7 8 9 10 10 11 12 13 13 16 17 17 17 20 20 20 22 22

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25

Inp u t Image

Primitives

Best Match

to "Rich mon d"

Best Match

to "Ed mu nd "

Image Seg mentatio n

R=53 i=2 7 c=5 2 h=61 m=70 o=43 n=6 1 d=88

E=1 2 d=7 9 m-8 5 u=25 n=61 d=88