Digital Halftoning

Digital Image Processing 

Ming-Sui (Amy) Lee 

Apr. 26, 2007

Announcement 

Class Information 

Course 

No class on May 10 

Homework #3 

Due on 5/17 

Other information 

資訊系專題研究說明會 

4/27 Fri. 12:40pm, Room103

Digital Halftoning

Digital Halftoning 

Goal 

Render the illusion of a continuous-tone 

image based on two-tone (half-tone) 

display 

Applications 

Computer hardcopies 

Laser printers/dot-matrix printers/color printers 

Fax machine 

Implementation 

Thresholding at 1/2 ?


Basic idea 

Spatial modulation 

Black/white gray-level image 

Darker area denser black points per area 

Whiter area sparser black points per area 

Three approaches 

Patterning 

Dithering 

Error Diffusion


Patterning 

p 

p 

1 Dot pattern 1 Gray-level pixel 

If p=4 

16 binary pixels 

256 gray levels 

Quantization 

17 levels (0~16) 

Rylander’s recursive 

patterning matrices


Patterning 

Four steps 

Read in the given grey-level image 

Quantization 

Design the patterning table 

Map each pixel to its corresponding pattern 

Notes 

Simplest way 

Generates image with higher spatial resolution 

than the source image 

Point-to-point ? OR region-to-region?


Dithering 

Create an image with the same number of 

dots as the number of pixels in the source 

image 

Idea 

F( 

j, 

k) 

N( 

j, 

k) 

H ( j, 

k) 

Threshold 

G( 

j, 

k)


Dithering 

Why adding noise? 

Under fixed thresholding taking MSB 

E.g. before and after adding noise 

To break the monotonicity of accumulated error 

in the area of constant (nearly constant) gray 

level 

White noise/pink noise/blue noise/green noise


Dithering 

Noise Type 

Power spectral density 

White noise Pink noise Blue noise 

f f f 

Robert Ulichney, “Digital Halftoning” 

http://www.hpl.hp.com/personal/Robert_Ulichney/


Dithering 

Another approach 

Adaptive thresholding 

Threshold the source image with a dither matrix 

Whenever the pixel value of the image is greater than 

the value in the matrix, a dot on the output image is 

filled 

Notes 

No randomness 

Region-to-region 

Recursive definition allowed


Dithering 

Dither matrix 

⎡1 

2⎤ 

⎡3 

1⎤ 

I2 ( i, 

j) 

= ⎢ ; 

3 0 

⎥ I 

2( 

i, 

j) 

= ⎢ ⎥ 

⎣ ⎦ 

⎣0 

2⎦ 

0 lowest threshold 

3 highest threshold


Dithering 

The general form of the dither matrix 

2x2 4x4 8x8 16x16… 

I 

2n 

( i, 

j) 

= 

⎡4I 

⎢ 

⎣4I 

I 

( i, 

( i, 

Eg. What is ? 

4 

n 

n 

( i, 

j) 

j) 

+ 1 

j) 

+ 

3 

4I 

4I 

n 

n 

( i, 

( i, 

j) 

+ 2⎤ 

j) 

+ 0 

⎥ 

⎦


Dithering 

Determine the threshold matrix 

T ( i, 

j) 

= 

255⋅ 

I( 

i, 

j) 

+ 

2 

N 

0. 

5 

Eg. 

⎡ 5 9 6 10⎤ 

⎢ 

13 1 14 2 

⎥ 

I 4( 

i, 

j) 

= ⎢ 

⎥, 

T4 

( i, 

j) 

= ? 

⎢ 7 11 4 8 ⎥ 

⎢ 

⎥ 

⎣15 

3 12 0 ⎦


Dithering 

128 128 128 128 

128 128 128 128 

128 128 128 128 

128 128 128 128


Error diffusion 

1975 Floyd & Steinberg 

A practical algorithm to implement blue noise 

dithering 

Framework 

F( 

j, 

k) 

~ 

F( 

j, 

k) 

Error 

diffusion 

Threshold 

G( 

j, 

k) 

error



1975 Floyd & Steinberg 

Normalize F(j,k) to lie between [0,1] 

Set threshold=0.5 

Output image: 0 or 1 

if 

~ 

F( 

j, 

k) 

≥ 

0. 

5 → G( 

j, 

k) 

= 1 

if 

~ 

F( 

j, 

k) 

< 0. 

5 → G( 

j, 

k) 

= 0 

Define 

~ 

E( 

j, 

k) 

= F( 

j, 

k) 

− G( 

j, 

k)



Error diffusion filter masks 

1975 Floyd Steinberg: 

1976 Jarvis et al: 

1 

48 

⎛0 

⎜ 

⎜0 

⎜0 

⎜ 

⎜3 

⎜ 

⎝1 

1 

16 

0 

0 

0 

5 

3 

⎛0 

⎜ 

⎜0 

⎜ 

⎝3 

0 

0 

0 

7 

5 

0 

0 

5 

0 

0 

7 

5 

3 

0⎞ 

⎟ 

7⎟ 

1⎟ 

⎠ 

0⎞ 

⎟ 

0⎟ 

5⎟ 

⎟ 

3⎟ 

1 

⎟ 

⎠



Error diffusion + serpentine scanning 

Right to Left 

Step 1 Left to Right 

1 

16 

⎛0 

⎜ 

⎜0 

⎜ 

⎝3 

0 

0 

5 

0⎞ 

⎟ 

7⎟ 

1⎟ 

⎠ 

1 

16 

⎛0 

⎜ 

⎜7 

⎜ 

⎝1 

0 

0 

5 

0⎞ 

⎟ 

0⎟ 

3⎟ 

⎠ 

Left to Right Right to Left 

Step 2 

//One complete cycle//


Multi-scale Error diffusion 

Several issues 

Region-to-region mapping 

Multi-resolution 

Time series/causal error diffusion process 

Easy to implement 

Causality appears to be artificial in images 

Is non-causal error diffusion possible? 

Quality metrics of halftoned images



“A multiscale error diffusion technique for digital halftoning” 

Ioannis Katsavounidis and C. –C. Jay Kuo 

Problem set-up 

X ( i, 

j) 

∈[ 

0, 

1] 

Input image 

Output image B( 

i, 

j) 

∈{ 

0, 

1} 

Error image E( i, 

j) 

= X ( i, 

j) 

− B( 

i, 

j) 

Intermediate stage 

X 

( i , j ), 0 ≤ k ≤ r, 

r = log N 

X 

k 

k 

( i 

k 

k 

, 

j 

k 

k 

) 

= 

1 

1 

∑∑ 

i= 0 j= 

0 

X 

k + 1 

( 2i 

k 

2 

+ i, 

2 j 

k 

+ 

j)



input output error 

X 

X 

M 

X 

0 

1 

r 

B 

B 

M 

0 

1 

Br 

Ek ( ik 

, jk 

) = X k ( ik 

, jk 

) − Bk 

( ik 

, jk 

), 0 ≤ k ≤ r 

Goal: minimize the error pyramid in a certain way! 

E 

E 

M 

0 

1 

Er



//Step 1// Initialization 

Set the entire output image pyramid to “0” 

//Step 2// Dot assignment 

1 parent node distributes its dots (integer numbers) 

to 4 children 

//Step 3// Error diffusion process 

1 

12 

⎛1 

2 1⎞ 

⎛0 

0 0⎞ 

⎛0 

0 0⎞ 

⎜ ⎟ 1 ⎜ ⎟ 1 

⎜ ⎟ 

⎜2 

−12 

2⎟ 

⎜2 

−8 

2⎟ 

⎜0 

− 5 2⎟ 

⎜ ⎟ 8 

⎝1 

2 1⎠ 

⎜ ⎟ 

5 ⎜ ⎟ 

⎝1 

2 1⎠ 

⎝0 

2 1⎠ 

center side corner



Quality management 

MSE vector 

MSEV 

Notes 

⎛ MSE 

⎜ 

⎜ MSE 

= ⎜ M 

⎜ 

⎝ 

0 

1 

MSEr 

⎞ 

⎟ 

⎟ 

⎟ 

⎟ 

⎠ 

MSE 

−12 

∑∑ 

i= 0 j= 

0 

Preserve contrast of the original image 

Does not over-smooth the image 

k 

= 

1 

N 

2 

2 

k k 

−1 

E 

2 

k 

( i, 

j)


Experimental results


Experimental results


Experimental results 

Visual cryptography 

“visual cryptography based on void-and-cluster halftoning 

technique” E. Myodo, S. Sakazawa and Y. Takishima 

+ =?

Texture Analysis

Texture Analysis 

Image analysis and its applications 

Noise removal 

Enhancement 

restoration 

Edge detection, 

Texture analysis 

Input image 

preprocessing 

Feature 

extraction 

segmentation 

Classification & 

Symbolic 

representation 

Object understanding & 

Scene interpretation 

Image analysis 

( Low-level of 

Computer vision tasks) 

Computer vision


What is texture? 

No mathematical definition 

Semi-regular structured patterns of object 

surfaces such as sand, grass, wool, cloth, leaves, 

etc. 

Why texture analysis? 

People started to be interested in late 50’s and 

early 60’s 

Analyze aerial images, e.g.


Example


History of texture analysis 

Fourier Spectra methods 

Edge Detection Methods 

Autocorrelation Methods 

Decorrelation Methods 

Dependency Matrix Method


Fourier Spectra methods 

No continuous work for a long while 

Edge Detection Methods 

Edge detection 

Use edge density and orientation as texture 

features


Autocorrelation Methods 

Treat the texture pattern as a 2D random process 

Statistical approach 

E{ F( 

x, 

y) 

F( 

x − 

Δx, 

y − Δy)}


Decorrelation Methods 

2D whitening filter 

2D 

whitening 

filter 

2 

σ


Dependency Matrix Method 

Joint probability 

P( 

a, 

b | 

j, 

k, 

Δj, 

Δk) 

= Pr ob{ 

F( 

j, 

k) 

= a, 

F( 

j − Δj, 

k − Δk) 

= b, 

0 ≤ a, 

b ≤ L −1} 

Not successful!!


Laws, 1979, a conference paper 

Micro structure method 

Multi-channel method 

Find local features 

F ( j, 

k) 

m i 

( j, 

k) 

For 3x3 mask 

i=1,2,3,…,9 

M i 

( j, 

k) 

Energy 

computation 

i T

Digital Halftoning

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