Modeling the Histogram of Modeling the Histogram of the Halftone Image to the Halftone Image to Determine the Area Determine the Area

Fraction of InkFraction of Ink

Yat-Ming WongMay 8,1998

Advisor: Dr. Jonathan Arney

BackgroundBackground

Drawing useful information from an image is important in various fields that depend upon them

Tools used to interpret an image need to be good enough to give meaningful data

HistogramHistogram

The histogram is a tool that gives a graphical interpretation of an image

It give us an idea of the make up of the image, such as the amount of ink in its composition

HistogramHistogram

The image is read pixel by pixel for their reflectance values

R1,9 = 0.1

R7,10 = 0.9

HistogramHistogram

0

2000

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10000

12000

0 0.2 0.4 0.6 0.8 1

Reflectance

Freq

uenc

y

Histogram of halftone dotsHistogram of halftone dots

0

2000

4000

6000

8000

10000

12000

0 0.2 0.4 0.6 0.8 1

Reflectance

Frequency

-

0

2000

4000

6000

8000

10000

12000

0 0.2 0.4 0.6 0.8 1

Reflectance

Frequency

-

Ink Population

PaperPopulation

HistogramHistogram

Segmentation of the histogram has so far been done by visual approximation

Visual approximation is a highly inaccurate method of measurement in cases where data needs to be in significant figures

ThresholdThreshold

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12000

0 0.2 0.4 0.6 0.8 1

Reflectance

Freq

uenc

y Threshold, RT (?)

SolutionSolution

Models to segment histogram computationally:

Gaussian Model

Straight-Edge Model

Gaussian ModelGaussian Model

Reflectance

G1 G2

G1+G2

Gaussian ModelGaussian Model

G1( ) [ /( )]*exp[ ( ) / ]R R R 1 2 21 1

212

G2( ) [ /( )]*exp[ ( ) / ]R R R 1 2 22 2

222

+

Gaussian ModelGaussian Model

f(i) = F*G1(R) + (1-F)*G2(R)

R1 R2

1 2

F 1-F

REFLECTANCE

G1+G2

Sum of two gaussians vs. offset lithographic print dataSum of two gaussians vs. offset lithographic print data

PROBLEM

REFLECTANCE

G1+G2

Data

Sum of two gaussians vs. inkjet “stochastic halftone” dataSum of two gaussians vs. inkjet “stochastic halftone” data

REFLECTANCE

G1+G2

Data

PROBLEM

Straight Edge ModelStraight Edge Model

Halftone dots are a collection of edges

Straight Edge ModelStraight Edge Model

Model of the Halftone Reflection Distribution as a Single “Equivalent Edge”

H

R

Model the Halftone “Equivalent EdgeModel the Halftone “Equivalent Edge

Vary F

H

R

Model the Halftone “Equivalent Edge”Model the Halftone “Equivalent Edge”

H

Change Rmin or Rmax

R

Model the Halftone “Equivalent Edge”Model the Halftone “Equivalent Edge”

x scan

R

x

1

10

0

RR R

a x FR

max minminexp{ ( )}1

where:

R

x

1

10

0

RR R

a x FR

max minminexp{ ( )}1

The Model

H RdR

dx( )

1H

R0 1

The Noise Model

-0.1 0.1R

S(R)

Add A Noise Metric Assume A Reflectance Variation

S RR

( ) exp

1

2 2

2

2

S RR

( ) exp

1

2 2

2

2 h RdR

dx( )

1

H

R0 1

*S(R)

The Noise Model

R

Straight Edge ModelStraight Edge Model

Rmin Rmax

F

1-F

a

Straight edge model vs. offset lithographic print dataStraight edge model vs. offset lithographic print data

H(R)

R0 0.2 0.4 0.60

0.02

0.04

0.06

0.08

Straight edge model vs. inkjet “stochastic halftone” dataStraight edge model vs. inkjet “stochastic halftone” data

0.1 0.2 0.3 0.4 0.5 0.60

0.01

0.02

0.03

H(R)

R

Comparison of models in matching offset lithographic print dataComparison of models in matching offset lithographic print data

Sum of two gaussians Straight Edge

vs.

Comparison of models in matching inkjet “stochastic halftone” dataComparison of models in matching inkjet “stochastic halftone” data

Sum of two gaussians Straight Edge

vs.

Automated computationAutomated computation

Program written in Visual Basic

Opens up a data file and automatically find the best computational match by looking for the set of variables that yields the lowest RMS deviation value.

Problems with the straight edge modelProblems with the straight edge model

H(R)

R

H(R)

R0 10

0.1

Expand

Problems with the straight edge modelProblems with the straight edge model

H(R)

R

H(R)

R

Expand

ConclusionConclusion

Model fits well for Ri and Rp close to each other

For Ri and Rp widely spaced, a single noise metric is inadequate.

The EndThe End