Signal Processing in

Digital Image Processing

P. NagabhushanUniversity of Mysore

pnagabhushan@hotmail.com

Entropy Analysis

of

Line Scan Signals

for Establishing the

Equivalence of Document

Images

11

Entropy Computation

s

Histogramming

Regression

Line Scan Signals

Equivalence Test

12

Generation of Line Scan Signals

Transforming a text document image’s line scan signals into ENTROPY

Row Transitions

Column Transitions

The structure of the text components are the foreground overlay on the back ground of the document image resulting in a 0/1 transition line scan signals.

‘0 - 1’ Text to Background transition (+ve)

‘1 – 0’ Background to Text transition (-ve)

Entropy – Measure of change in energy caused due to transitions. 13

Entropy is defined as :

E = p Log(1/p) + (1-p) Log(1/(1-p))

‘p’ Probable occurrence of transition.

‘(1-p)’ Non probable occurrence of transition.

Veronique Eglin and Stephane Bres, Document Page Similarity Based on Layout Visual Saliency: Application to Query By Example And Document Classification, International Conference on Document Analysis and Recognition, 2003.

For recognizing the unique structure of the components and to comprehend the content of the components at three hierarchical levels, in this research work, two new models namely,

Conventional Entropy Quantifier (CEQ)

Spatial Entropy Quantifier (SEQ)

have been developed originating with the entropy formulation

14

Conventional Entropy Quantifier (CEQ)

This is based on the number of times the transition occurs in each row and column of the image matrix.

))1/(1log()1()/1log()( pppptE ‘t’ is the transition from 0-1 and 1-0.

‘p’ is calculated based on the number of +ve and -ve transitions in each row or column divided by the total number of columns or rows respectively.

‘1-p’ is the probable non-occurrences of transition.

E(t) is the total entropy.

15

Development of the Computational Model

Depending upon the transition (1/0 or 0/1) as described above E(t) could be E-(t) or E+(t).

Let R represent the set of sequence of ‘m’ rows in a component and C represent the set of sequence of ‘n’ columns in a component.

R= {r / every r is a horizontal row of consecutively placed n pixels one after the other} and 1 m.

C={r / every r is a vertical column of consecutively placed m pixels one below the other} and 1 n.Total entropy along an th row is defined as

E() = E+() + E-()

where E+() = E+(t) at all packets of transitions scanning over ;E-() = E-(t) 1 n.

Hence the net horizontal entropy for a component is defined as EH = R E()

16

Similarly total entropy along a th column is

E() = E+() + E-()

Where E+() = E+(t) at all packets of transitions scanning over ; and E-() = E-(t) 1 m.

Hence the net vertical entropy for a component is EV = C E()

Finally the overall entropy of the component is

EO=EH +EVentropy feature is represented as a 4-dimentional feature set F1, F2, F3 and F4. where,

F1 stands for the total row entropy for transition 1-0 (E-( ))

F2 stands for the total row entropy for transition 0-1(E+( ))

F3 stands for the total column entropy for transition 1-0 (E-())

F4 stands for the total column entropy for transition 0-1(E+())

1-dimensional feature F5 where, F5=F1+F2+F3+F4 is the overall entropy EO

17

Component F1 F2 F3 F4 F5 difference

Hydrobia 5.23 5.23 22.58 22.58 55.63 0.52

Hydrobic 5.22 5.22 22.33 22.33 55.11  

             

Impunity 5.21 5.21 14.58 14.58 39.59 1.11

Impurity 5.22 5.22 14.02 14.02 38.48  

             

Metc 4.77 4.77 46.37 46.37 102.29 2.27

Meta 4.88 4.88 47.97 47.97 105.69  

Component F1 F2 F3 F4 F5 difference

Hydrobia 5.08 5.12 26.59 25.60 62.40 1.46

Hydrobic 5.07 5.10 26.00 25.02 61.20  

             

Impunity 5.05 5.08 16.45 16.18 42.76 1.62

Impurity 5.06 5.09 15.78 15.52 41.45  

             

Metc 4.63 4.63 53.38 52.98 115.61 4.52

Meta 4.74 4.73 55.17 54.73 119.38  

Distance between the features extracted with tight segmentation

Distance between the features extracted without tight segmentation

18

Experimental Analysis with Highly Resembling Finer Components.

Set Character F1 F2 F3 F4 F5

(0,o)0 5.33 5.33 2.17 2.17 14.99

O 4.28 4.28 3.72 3.72 16.00

(u,v)u 5.08 5.08 4.88 4.88 18.32

v 4.88 5.08 4.88 5.08 18.32

(b,p,d,q)

b 5.02 5.32 3.09 2.64 16.07

p 5.23 4.85 3.09 2.98 16.15

d 5.00 4.28 3.39 3.39 16.05

q 4.67 5.42 3.02 3.33 16.43

(6,9)6 5.14 5.06 2.73 2.90 15.83

9 5.51 5.14 2.79 2.84 16.28

Set of highly resembling finest components

19

Experimental Analysis with Variation in Font Size and Font Style

Font size

F1 F2 F3 F4 F5

8 3.6902 3.8274 7.3345 7.3216 22.1738

10 4.0664 4.337 8.4822 8.4495 25.3352

12 4.7016 4.8436 8.821 8.7091 27.0754

14 5.0523 5.1278 9.1954 9.1356 28.5111

16 5.3422 5.5107 10.1921 10.1921 31.237

18 5.6999 5.7931 10.9236 10.8178 33.2343

Feature vector for a component (word) in different font sizes.

Font style F1 F2 F3 F4 F5

TNR 5.1054 4.9964 6.9746 6.4661 23.5424

Arial 4.9007 5.7858 6.596 5.7773 23.0598

courier 4.0503 4.2283 9.5296 9.3543 27.1625

courierN 3.8731 4.0407 10.5304 9.8292 28.2734

Feature vector for a component (word) in different font styles.

20

Experimental Analysis with Medium (Line) Components

Line is a group of words. Experiments are conducted on lines with different combinations of fine level components.

Line 1: Radha went to school in the afternoon.Line 2: in the afternoon Radha went to school.Line 3: went to school Radha in the afternoon.Line 4: went to school in the afternoon Radha.Line 5: in the afternoon went to school Radha.

sample F1 F2 F3 F4 F5

Line1 5.28 5.28 77.16 77.16 164.86

Line2 5.28 5.28 77.16 77.16 164.86

Line3 5.2753 5.2753 77.1597 77.1597 164.86

Line4 5.2753 5.2753 77.1597 77.1597 164.86

Line5 5.2753 5.2753 77.1597 77.1597 164.86

Feature vector of medium components with tight segmentation

21

Experimental Analysis with Coarse (Paragraph) Components.

paragraph 1 Paragraph 2 Paragraph 3

sample F1 F2 F3 F4 F5

Para1 20.8874 20.9908 38.4377 37.719 118.0349

Para2 36.1343 36.4856 23.7066 23.2941 119.6205

Para3 23.9655 24.0126 33.4495 32.828 114.2556

Feature vector for coarse components with tight segmentation

22

Spatial Entropy Quantifier (SEQ)

The formulation of SEQ essentially emanates from CEQ, which is morphed to take an improved form.

position of transition is represented by pos.

Where pos is the row number in column transition and column number in row transition.

23

Development of the Computational Model

Consider a component of size m x n, obtained from any hierarchical level.

Let R represent the set of sequence of ‘m’ rows and C represents the set of sequence of ‘n’ columns.

R= {r / every r is a horizontal row of consecutively placed n pixels one after the other} and 1 m.

C= {c / every c is a vertical column of consecutively placed m pixels one below the other} and 1 n.

If the transition occurs in a row rα then it contributes to row Entropy at position pos and it is formulated as:

If the transition occurs in a column cβ then it contributes to column entropy at position pos and it is given by

)))))((log())((())log())(((()( posnmnnposmposnnposmrrE

)))))((log())((())log())(((()( posnmmmposnposmmposnccE

24

Total entropy for each row E(r) is the summation of entropy at +ve (E+( r)) and –ve (E-(r)) transitions in a particular row which is formulated as,

nn

rErErE )()()(

Similarly for each column,

mm

cEcEcE )()()(

Then the total horizontal entropy Eh(R) is given by,

nh rERE )()(

Similarly for vertical entropy Ev(C) is given by,

mv cECE )()(

Total entropy of the component is given by

)()()( CEREtE vh 25

Spatial-entropy quantifier ,

Position of occurrence of transition.

Based on the position of the component the entropy also changes.

A Hypothetical Experiment

Component (EMPTY) Position -1

Position -2 Position –3

26

Position-1

 

Position-3 Position-2

Features Features Features

F1 F2 F3 F4 F1 F2 F3 F4 F1 F2 F3 F4

-34.34 -34.37 -23.17 -23.28 -137.85 -137.97 -11.31 -11.35 -258.00 -258.23 -16.58 -16.63

-34.13 -34.16 -23.04 -23.15 -136.94 -137.04 -11.26 -11.30 -256.38 -256.57 -16.52 -16.57

-38.61 -38.67 -26.10 -26.16 -142.03 -142.28 -14.15 -14.18 -262.08 -262.53 -19.36 -19.39

-38.40 -38.46 -25.98 -26.01 -141.20 -141.44 -14.10 -14.11 -260.65 -261.04 -19.30 -19.31

-42.86 -42.96 -28.96 -29.03 -146.20 -146.60 -16.97 -17.00 -266.16 -266.84 -22.10 -22.14

-42.73 -42.76 -28.90 -28.93 -145.72 -145.84 -16.94 -16.95 -265.32 -265.52 -22.07 -22.09

-42.67 -42.70 -31.82 -31.90 -145.50 -145.61 -19.78 -19.81 -264.92 -265.12 -24.85 -24.89

-47.22 -47.26 -34.72 -34.76 -150.77 -150.91 -22.60 -22.62 -270.90 -271.14 -27.61 -27.63

-47.11 -47.18 -37.57 -37.70 -150.37 -150.63 -25.41 -25.47 -270.23 -270.67 -30.35 -30.42

-47.04 -47.07 -40.60 -40.64 -150.13 -150.25 -28.30 -28.33 -269.81 -270.02 -33.18 -33.21

-51.31 -51.43 -40.41 -40.50 -154.42 -154.80 -28.20 -28.25 -274.09 -274.74 -33.08 -33.13

-55.81 -55.85 -43.55 -43.60 -159.38 -159.53 -31.16 -31.19 -279.50 -279.75 -35.98 -36.00

-55.72 -55.76 -43.25 -43.35 -159.10 -159.24 -31.00 -31.05 -279.04 -279.27 -35.81 -35.86

-55.55 -55.63 -46.40 -46.56 -158.58 -158.84 -33.96 -34.06 -278.16 -278.59 -38.71 -38.80

-55.47 -55.51 -46.09 -46.29 -158.34 -158.46 -33.78 -33.90 -277.74 -277.95 -38.54 -38.65

-60.05 -60.15     -163.54 -163.84     -283.56 -284.05    

-59.73 -59.87     -162.62 -163.00     -282.02 -282.65    

-64.39 -64.45     -168.00 -168.15     -288.10 -288.35    

-64.00 -64.09     -166.90 -167.15     -286.29 -286.72    

-68.69 -68.74     -172.31 -172.47     -292.40 -292.66    

-68.27 -68.32     -171.17 -171.31     -290.56 -290.78    27

Experimental Analysis with Highly Resembling Components

CEQ F1 F2 F3 F4 F5 Distance(F5)

U 4.8765 4.8765 2.4371 3.69 15.88010

V 4.8765 4.8765 2.4371 3.69 15.8801

SEQ

U 278.18 275.13 66.66 169.14 789.1000.99

V 277.59 275.10 68.54 168.66 789.88

CEQ F1 F2 F3 F4 F5 Distance(F5)

intended 4.69 4.72 21.91 19.2 50.720

indented 4.69 4.72 21.91 19.2 50.72

SEQ

intended 1657.4 1671.3 7707 5546.9 16582.60.2

indented 1656.5 1670.4 7793.4 5462.1 16582.4

Entropy values extracted using CEQ and SEQ

28

Sample TNR cap

O

Arial cap

O

TNR small

O

Arial small

o

TNR digit

0

Arial digit

0

TNR cap - O0 167.4 958.7 645.32 449.2 192

Arial cap - O167.4 0 1126.1 812.72 616.6 359.4

TNR small l – o958.7 1126.1 0 313.38 509.5 768.7

Arial small -o645.32 812.72 313.38 0 196.12 453.32

TNR digit - 0449.2 616.6 509.5 196.12 0 257.2

Arial digit -0192 359.4 768.7 453.32 257.2 0

Distance using feature F5 extracted using SEQ

Experimental Analysis with Variation in Font Size and Font Style.

29

Experimental Analysis with Fine (word) Components

Sample F1 F2 F3 F4 F5Distance using F5

but 680.93 647.51 709.80 472.91 2,511.20

8.24tub 664.00 617.77 661.07 485.95 2,428.80

hydrobia 2970.1 2972.9 17955 18224 42122

2761hydrobic 2899.5 2890.1 16651 16920 39361

impunity 2093.5 2121.1 7263.3 6871.7 18350

1489impurity 1991.1 2018.8 6614.6 6236.2 16861

bombay 1830.9 1819.3 7262.9 7031.6 17945

549dombay 1802 1849.5 6977.7 6767.1 17396

intended 1657.4 1671.3 7707 5546.9 16583

1indented 1656.5 1670.4 7793.4 5462.1 16582

korna 1089.4 1091.6 3411 1958.1 7550

185korne 1026.1 1002 3275.2 2061.8 7365

Distance between words using SEQ

30

Features extracted through CEQ Features extracted through SEQ.

Distance using F5 extracted through CEQ Distance using F5 extracted through SEQ

31

Experimental Analysis with Medium (line) Components

Line 1: Radha went to school in the afternoon.Line 2: in the afternoon Radha went to school.Line 3: went to school Radha in the afternoon.Line 4: went to school in the afternoon Radha.Line 5: in the afternoon went to school Radha.

Feature vectors for 5 line samples using CEQ Feature vector for 5 line samples using SEQ

Distance using F5 extracted through CEQ Distance using F5 extracted through SEQ

32

Wavesim Transform:

A Novel Perspective Of

Wavelet Transform For Signals

The CWT is essentially a collection of inner products of a signal

f(t) and the translated and dilated wavelet for all a and b

where the values W(a,b) represents the correlation coefficients:

CWT has a cross correlation interpretation as well.

−∞

The WaveSim(WS) transform of f(t) with

respect to a wavelet Ψa,b(t) is defined as

where Sim(.,.) is the similarity measure

computed as illustrated in figure with a

Haar Wavelet. The amplitude of Ψ is

max(f(t)).

Wavesim Transform

A1 = Area Covered by the positive half of the Haar WaveletA2 = Area Covered by the negative half of the Haar WaveletA3 = Positive area spanned by f(t) within the positive half of the wavelet.A4 = Negative area spanned by f(t) within the negative half of the waveletA5 = Negative area spanned by f(t) within the postive scale range of the waveletA6 = Positive area spanned by f(t) within the negative scale range of the wavelet

(A3/A1) is the similarity measure between the positive half of the wavelet and f(t)

in the positive scale range, (A5/A1) is the dissimilarity measure between the

positive half of the wavelet and f(t) in the positive scale range.

Likewise (A4/A2) is the similarity measure between the negative half of the

wavelet and f(t) in the negative scale range, (A6/A2) is the dissimilarity measure

between the negative half of the wavelet and f(t) in the negative scale range.

The WaveSim coefficient is computed as follows

Sim(f(t), Ψ(t)) = ½((A3 – A5)/A1 + (A4 – A6)/A2))

½ helps in normalizing the coefficient values to lie in the interval [-

1,1].

Case 1: Input signal is a set of 200 samples with equal values (f(t) is a horizontal straight line)

Case 2: Input signal is a set of 200 samples with the first 100 samples holding a value 20 and the second 100 samples holding a value 80.

Case 3: Input signal is a set of 200 samples linearly increasing from 1 to 200. (f(t) is a straight line)

Case 4: Input signal is a set of 200 samples with a curved structure.

Adaptive WaveSim

Thank You