Tone Dependent Color Error Diffusion Halftoning

Multi-Dimensional DSP Project

Vishal Monga,

April 30, 2003

3/16

7/16

5/16 1/16

+ _

_+

e(m)

b(m)x(m)

difference threshold

compute error

shape error

u(m)

)(mh

Grayscale Error Diffusion

• 2- D sigma delta modulation [Anastassiou, 1989]

• Shape quantization noise into high frequencies• Linear Gain Model [Kite, Evans, Bovik, 1997]

– Replace quantizer by scalar gain Ks and additive noise image

current pixel

weights z

z

HK

XK

s

s

11

)(

)()(1 zz NH

Transfer functions

Direct Binary Search [Analoui, Allebach 1992]

• Computationally too expensive for real-time applns. viz. printing

• Used in screen design

• Serves as a practical upper bound for achievable halftone quality

Tone Dependent Error Diffusion

• Train error diffusionweights and thresholdmodulation[Li & Allebach, 2002]

b(m)+ _

_+

e(m)

x(m)

Tone dependent error filter

mxh

mxQ

Tone dependent threshold modulation

Graylevel patch x

HVS

mm xx Qh ,Halftone pattern

for graylevel x FFT

FFT

DBS pattern

for graylevel x

mm xx Qh ,

Halftone pattern

for graylevel x FFT

FFT

Midtone regions

Highlights and shadows

Tone Dependent Color Error Diffusion

• Color TDED, Goal– Obtain optimal (in visual quality) error filters with filter

weights dependent on input RGB triplet (or 3-tuple)

• Extension to color is non-trivial– Applying grayscale TDED independently to the 3 color

channels ignores the correlation amongst them

• Choice of error filter – Separable error filters for each color channel– Matrix valued filters [Damera-Venkata, Evans 2001]

• Design of error filter key to quality– Take human visual system (HVS) response into

account

Tone Dependent Color Error Diffusion

• Problem(s):– Criterion for error filter design ?

– (256)3 possible input RGB tuples

• Solution– Train error filters to minimize the visually weighted

squared error between the magnitude spectra of a “constant” RGB image and its halftone pattern

– Design error filters along the diagonal line of the color cube i.e. (R,G,B) = {(0,0,0) ; (1,1,1) …(255,255,255)}

– Color screens are designed in this manner

– 256 error filters for each of the 3 color planes

Color Transformation

sRGB Yy Cx Cz

(Linearized CIELab)

FFT

FFT

Input RGB Patch

Halftone Pattern

Perceptual Error Metric

HVS ChrominanceFrequency Response

HVS Luminance Frequency Response

HVS ChrominanceFrequency Response

Total Squared Error (TSE)

Yy

Cx

Cz

Perceptual Error Metric

• Find optimal error filters that minimize TSE subject to diffusion and non-negativity constraints, m = r,g,b; a (0,255)

1ak;hk

m k S 0ak;h m (Floyd-Steinberg)

I)3/1(),,(),,( baLzxy CCY

Linear CIELab Color Space Transformation

• Linearize CIELab space about D65 white point [Flohr, Kolpatzik, R.Balasubramanian, Carrara, Bouman, Allebach, 1993]

Yy = 116 Y/Yn – 116 L = 116 f (Y/Yn) – 116

Cx = 200[X/Xn – Y/Yn] a = 200[ f(X/Xn ) – f(Y/Yn ) ]

Cz = 500 [Y/Yn – Z/Zn] b = 500 [ f(Y/Yn ) – f(Z/Zn ) ]wheref(x) = 7.787x + 16/116 0 ≤ x < 0.008856f(x) = x1/3 0.008856 ≤ x ≤ 1

• Decouples incremental changes in Yy, Cx, Cz at white point on (L,a,b) values

• Transformation is sRGB CIEXYZ YyCx Cz

HVS Filtering

• Filter chrominance channels more aggressively– Luminance frequency response [Näsänen and Sullivan, 1984]

L average luminance of display

weighted radial spatial frequency

– Chrominance frequency response [Kolpatzik and Bouman, 1992]

– Chrominance response allows more low frequency chromatic error not to be perceived vs. luminance response

~)(

)( )()~( LY eLKW

y

),( )( AeW

zx CC

~

Set p(0) to be the optimal value from the last designed “3 tuple”

(First choice: p(0) Floyd-Steinberg)

hw = 1/16 , i = 0

while (p(i) p(i-1)) {

find p(i+1) Nhw (p(i)) that minimizes the total squared error (TSE)

i i + 1 }

while (hw 1/256) {

hw hw/2

find p(i+1) Nhw (p(i)) that minimizes TSE

i i + 1 }

Search Algorithm[Li, Allebach 2002]

a))(k;h a),(k;h , a)(k;h( bgrp Let p be the vector of “filter weights”

define the neighborhood of a))(k;h a),(k;h , a)(k;h( )()()()( ib

ig

ir

ip

bgrmhpN wi

mmbgri

hw ,,,a)(k;h - a)(k;h :)a)(k;h , a)(k;h , a)(k;h()( )()(

a (0,255), k = (k1, k2), k S

a) Original b) FS Halftone c) TDED Serpentine

Results

a)

b)

c)

d) TDED Raster e) TDED 2-row serp f) Detail of FS (left) and TDED

d)

e)

f)

Original

House Image

Floyd Steinberg Halftone

TDED Halftone

Conclusion

• Color TDED– Worms and other directional artifacts removed

– False textures eliminated

– Visibility of “halftone-pattern” minimized (HVS model)

– More accurate color rendering at extreme levels

• Scan path choice– Serpentine scan gives best results (not parallelizable)

– 2-row serpentine gives comparable quality

• Future Work – Design “optimum” matrix valued filters ?

– Look for better HVS models/transformations

Back Up Slides

HVS model details, Monochrome images Yy, Cx planes of color

halftones

Floyd Steinberg Yy

component

Floyd Steinberg Cx

component

TDED Yy

component

TDED Cx

component

HVS Filtering contd….

2

1)4cos(

2

1)(

wws

where p = (u2+v2)1/2 and )arctan(u

v

)(~

s

pp

w – symmetry parameter

)(s reduces contrast sensitivity at odd multiples of 45 degrees

• Role of frequency weighting– weighting by a function of angular spatial frequency [Sullivan, Ray, Miller 1991]

equivalent to dumping the luminance error across the diagonals

where the eye is least sensitive.