What are the three main categories of halftones?

Existing halftoning techniques can be broadly classified as ordered dither, error diffusion, and optimization-based halftoning techniques.

Where did he receive his MSE and PhD degrees?

After receiving his BSc degree, he received his MSE and PhD degrees in systems engineering from the Moore School of Electrical Engineering at the University of Pennsylvania, Philadelphia.

What is the computational complexity of the QR-RLS algorithm?

Computational complexity of the QR-RLS algorithm is O(K2N2) for the composite multichannel error diffusion and it is O(KN2) for the channelby-channel implementation.

Where did she receive her BSc degree?

Gozde Bozkurt Unal received her BSc degree in electrical engineering from the Middle East Technical University, Ankara, Turkey, in 1996 and her MSc degree in electrical engineering from the Bilkent University, Ankara, Turkey, in 1998.

what is the optimum error diffusion filter coefficient?

G ,with the vector g̃(s) and the variable f̃ k(s) updating the present optimum error diffusion filter ashk !~s !5hk !~s21 !2 g̃~s !

What is the power spectrum of the quantization error diffusive method?

As the authors can observe from these plots, the power spectrum of the quantization error diffused by the QR-RLS type adaptive method has not only the lowest energy but also the flattest response, whereas the error diffusion with Floyd-Steinberg has the highest energy.

(Open Access) QR-RLS algorithm for error diffusion of color images (2000) | Gozde Unal

Q: how can i reduce the error diffusion filter coefficients matrix?

The minimization of the cost function Js(H) with respect to the error diffusion filter coefficients matrix H(s) can be carried out by determining the rows hkT of the optimum matrix H(s) by minimizing the individual cost functions Jk ,s(hk) with respect to hk , for k51,2,...,K .

Q: What is the simplest way to reduce the error diffusion coefficient?

By constraining the form of the error diffusion coefficient matrix H(s), as in Eq. ~4!, further savings in computational load can be achieved with tolerable degradation in the overall performance of the diffusion process.

Q: What is the coefficient of the adaptive error diffusion filter?

The coefficients of the adaptive error diffusion filter in both algorithms were scaled by 0.9, i.e., the authors allowed diffusion of not all but a fraction of the error made in quantization to neighboring pixels, and this resulted in a slight improvement in terms of color impulses.

QR-RLS algorithm for error diffusion

of color images

Gozde Bozkurt Unal

North Carolina State University

Electrical and Computer Engineering

Department

Raleigh, North Carolina 27695

Yasemin Yardimci

Middle East Technical University

Informatics Institute

Ankara 06531, Turkey

Orhan Arikan

A. Enis C¸ etin

Bilkent University

Electrical Engineering Department

Ankara 06533, Turkey

Abstract. Printing color images on color printers and displaying them on

computer monitors requires a signiﬁcant reduction of physically distinct

colors, which causes degradation in image quality. An efﬁcient method to

improve the display quality of a quantized image is error diffusion, which

works by distributing the previous quantization errors to neighboring pix-

els, exploiting the eye’s averaging of colors in the neighborhood of the

point of interest. This creates the illusion of more colors. A new error

diffusion method is presented in which the adaptive recursive least-

squares (RLS) algorithm is used. This algorithm provides local optimiza-

tion of the error diffusion ﬁlter along with smoothing of the ﬁlter coefﬁ-

cients in a neighborhood. To improve the performance, a diagonal scan

is used in processing the image.

2000 Society of Photo-Optical Instrumentation

Engineers.

[S0091-3286(00)00611-5]

Subject terms: halftones; color dithering; displays; printing; quantization; QR-RLS

algorithm.

Paper 990163 received Apr. 14, 1999; revised manuscript received June 2, 2000;

accepted for publication June 7, 2000.

1 Introduction

Color output devices such as halftone color printers and

palette-based displays are capable of producing only a lim-

ited number of colors, whereas the human eye can distin-

guish around 10 million colors under optimal viewing

conditions.

The eye perceives only a local spatial average

of the color spots produced by a printing device and is

relatively insensitive to errors made in high frequencies in

an image.

Halftoning algorithms therefore aim to preserve

these local averages while forcing the errors between the

continuous tone image and the halftone image to high-

frequency regions. Existing halftoning techniques can be

broadly classiﬁed as ordered dither, error diffusion, and

optimization-based halftoning techniques. A comparative

study of earlier image reproduction techniques can be

found in Stoffel and Moreland.

Ordered dither techniques are mainly based on thresh-

olding each pixel value after adding a pseudonoise se-

quence. These techniques are attractive in the sense that

they are simple to implement and computationally inexpen-

sive because they require pixelwise operations. However,

ordered dithering results in regular and periodic error pat-

terns, which lowers the quality of the output image.

Another group of halftoning methods are error diffusion

techniques ﬁrst introduced by Floyd and Steinberg.

They

proposed an algorithm that is predicated on distributing the

quantization error of the current pixel to neighboring pix-

els. Typically, at each pixel, the weighted sum of previous

quantization errors is added to the current pixel value, and

then the pixel is quantized to produce the output pixel

value. These weights form an error diffusion ﬁlter. Error

diffusion aims to preserve the local average value of the

image, therefore a unity gain low-pass ﬁnite impulse re-

sponse 共FIR兲 ﬁlter is used for distributing the error.

Error diffusion was ﬁrst developed for gray-scale im-

ages. For color images, error diffusion can be applied to

each color component independently or a color pixel can be

error diffused in a vectorized manner.

Some directional artifacts seen in error diffusion are due

largely to the traditional raster of processing.

Previous ap-

proaches for improving error diffusion employed various

choices of space ﬁlling curves to deﬁne the order of pro-

cessing, such as serpentine curves,

Peano curves,

and ran-

dom space-ﬁlling curves.

In contrast to deterministic error ﬁlter kernels, some re-

cent research employed dynamically adjusting the error ﬁl-

ter kernel using adaptive signal processing techniques.

Akarun et al.

used a vectorized error diffusion approach,

and updated the error diffusion ﬁlter coefﬁcients adap-

tively. Wong

minimizes a local frequency-weighted error

criterion to adjust the error diffusion kernel dynamically

using the well-known least mean-squares 共LMS兲

algorithm.

Kollias and Anastassiou

used neural networks

to minimize a frequency-weighted mean-squared-error cri-

terion.

In optimization-based halftoning techniques, the prob-

lem of halftoning is formulated as an optimization problem

that minimizes an error metric between the continuous tone

original image and its halftone version. Disadvantages of

optimization-based methods for halftoning are that there

are multiple optima, the methods are iterative, and they

require substantially high computational power. For color

images, processing requirements further increase.

Some hybrid schemes that combine different aspects of

halftoning methods are proposed in the literature such as

the blue-noise halftoning,

green-noise halftoning,

ran-

domized error propagation,

and using a nonlinear Laplac-

ian operator.

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2 Diagonal Error Diffusion

A block diagram of the standard error diffusion technique

is given in Fig. 1. Usually, the image is processed in a

raster scan fashion, and each input color pixel x(s)isa3

⫻ 1 vector, where the index s⫽ s

M⫹ s

and M is the num-

ber of horizontal pixels in the image. The current pixel x(s)

together with the diffused error is quantized. The resultant

image y(s) is the dithered image.

Here, Q is the quantizer, and H is the error diffusion

matrix. Some well-known error diffusion ﬁlter masks

3,15

are shown in Fig. 2, where the solid dots denote the pixel

located at s. These masks determine the support of the error

diffusion ﬁlter. A common characteristic of these ﬁlters is

that they are casual, i.e., their region of support is wedge to

ensure that these ﬁlters can be applied in a sequential

manner.

The ﬁlter coefﬁcients are deterministic, low pass

in nature, and add up to 1 so that errors are neither ampli-

ﬁed nor reduced.

The pixel at location s has the value

共

兲

⫽ x

共

兲

⫹ H

共

兲

共

兲

, 共1兲

before quantization. Here, the error diffusion ﬁlter H has

size K⫻L, where K is the number of channels, N is the size

of support of the error diffusion ﬁlter on each channel, and

L⫽KN. For an RGB image that uses an error diffusion

ﬁlter support of size 4 as the Floyd-Steinberg ﬁlter, K⫽3

and N⫽ 4. The composite error vector e

(s) consists of the

past quantization errors that must be diffused on the pixel at

location s

共

兲

⫽

关

共

兲

共

兲

...e

共

兲

兴

, 共2兲

where the subscript p indicates that the quantization errors

are made in the past. The past quantization error vector for

the k’th channel e

(s) has the open form:

共

兲

⫽

关

共

兲

共

兲

...e

共

兲

兴

, 共3兲

where e

(s) is the quantization error in the k’th channel

for the n’th neighbor of the pixel s. There are two cases of

interest:

1. Matrix H is a full matrix so that errors in different

channels may be diffused on each other. We refer to

this case as composite-multichannel error diffusion

and its block diagram is depicted in Fig. 3. It may be

also called vectorized error diffusion.

2. Matrix H has the following block diagonal structure:

H⫽

冋

0 0 ... 0

0 h

0 ... 0

⯗⯗⯗ 0

0 0 0 ... h

册

, 共4兲

with each h

designating the error diffusion ﬁlter for

the k’th channel as follows:

⫽

关

...h

兴

In this structure, the errors in one channel are only diffused

in the same channel. We call this strategy channel-by-

channel error diffusion and its block diagram is given in

Fig. 4. It is also called scalar error diffusion.

After error is diffused on the pixel s,(K⫻1) quantiza-

tion error vector e(s) for this pixel is formed as

共

兲

⫽ u

共

兲

⫺ y

共

兲

, 共5兲

共

兲

⫽ Q

关

共

兲

兴

. 共6兲

Fig. 1 Block diagram of the error diffusion method.

Fig. 2 Error diffusion ﬁlter masks.

Fig. 3 One channel of the composite-multichannel error diffusion;

where

is the error diffusion ﬁlter size and

is the channel index,

which takes a value from the set

⫽ 1,

⫽ 2, and

⫽ 3.

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The error between the original input pixel and the output

pixel is deﬁned as the output error,

共

兲

⫽ x

共

兲

⫺ y

共

兲

, 共7兲

which can be expressed in terms of the past and the present

quantization errors as

共

兲

⫽ e

共

兲

⫺ H

共

兲

共

兲

, 共8兲

by substituting Eqs. 共1兲 and 共5兲 in Eq. 共7兲.

The normal raster used in error diffusion causes vertical

or horizontal artifacts, and regular patterns that arise espe-

cially in uniform intensity regions. It is well known that the

human visual system is less sensitive to diagonal errors

compared to the vertical or horizontal errors. To take ad-

vantage of this fact we scanned the image diagonally, hence

the error is diagonally diffused. Causal prediction windows

shown in Fig. 5 are used in the error diffusion algorithm.

Here, we aim to break up the horizontal and vertical direc-

tionality of the possible error patterns, and force the accu-

mulation of the error to be in diagonal orientation to which

the human eye is less sensitive.

3 New Adaptive Error Diffusion

The error diffusion ﬁlter plays an important role in shaping

the output error spectrum. In contrast to deterministic error

diffusion ﬁlters, recent algorithms use the optimum ﬁlter

coefﬁcients for a given image, or update the coefﬁcients

adaptively using LMS type adaptive algorithms.

7,8

We would like to minimize the energy of the output

error z(s)

关

储

共

兲

储

兴

⫽ E

关

储

共

兲

⫺ y

共

兲

储

兴

⫽ E

关

储

共

兲

⫺ H

共

兲

共

兲

储

兴

共9兲

with respect to the ﬁlter coefﬁcients H(s). Since typical

image characteristics are locally nonstationary, an adaptive

algorithm is used for the minimization of the output error

sequence energy. However, to reduce the effects of noise

and provide further averaging of the ﬁlter coefﬁcients

around s we included the past quantization errors around s

in our cost function. We propose to use a recursive least-

squares 共RLS兲 type adaptation algorithm minimizing a cost

function of the form

共

兲

⫽

兺

j⫽ 1

␭

s⫺ j

储

共

兲

储

, 共10兲

where ␭苸

关

0,1

兴

is the forgetting factor. The main advan-

tages of this algorithm are the following:

1. fast adaptation to local features with an appropriately

selected forgetting factor ␭

2. numerical stability due to its lack of sensitivity to the

condition matrix of the quantization error autocorre-

lation matrix

3. one-step convergence to the optimum H(s) matrix

RLS algorithms that employ QR decomposition of the

quantization error autocorrelation matrix, also called QR-

RLS algorithms, have the added beneﬁt of having the abil-

ity to work in low-bit arithmetic, therefore making them

feasible to be implemented using very large scale integra-

tion 共VLSI兲.

3.1

Composite Multichannel Error Diffusion

Substituting the expression for the output error in Eq. 共10兲

we obtain

共

兲

⫽

兺

j⫽ 1

␭

s⫺ j

储

共

兲

⫺ H

共

兲

共

兲

储

⫽

兺

k⫽ 1

兺

j⫽ 1

␭

s⫺ j

关

共

兲

⫺ h

共

兲

共

兲

兴

⫽

兺

k⫽ 1

k,s

共

兲

, 共11兲

where the rows of H(s) are denoted by h

. The minimiza-

tion of the cost function J

(H) with respect to the error

diffusion ﬁlter coefﬁcients matrix H(s) can be carried out

by determining the rows h

of the optimum matrix H(s)by

minimizing the individual cost functions J

k,s

) with re-

spect to h

, for k⫽1,2,...,K. The optimum solution has the

form

Fig. 4 One channel of the channel-by-channel error diffusion:

the error diffusion ﬁlter size and

is the channel index which takes a

value from the set

⫽ 1,

⫽ 2, and

⫽ 3.

Fig. 5 Diagonal scanning: dots correspond to the current pixel, and

the L-shaped window contains the previous pixels.

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⫽ C

⫺ 1

共

兲

⌬

共

兲

, 共12兲

where C(s) and ⌬

(s) are the correlation estimates deﬁned

共

兲

⫽

兺

j⫽ 1

␭

s⫺ j

共

兲

共

兲

, 共13兲

⌬

共

兲

⫽

兺

j⫽ 1

␭

s⫺ j

共

兲

共

兲

, 共14兲

and the latter correlation has to be computed for each chan-

nel k⫽ 1,2,...,K.

In QR-RLS the optimal solution of Eq. 共12兲 is computed

by employing a recursive procedure operating on the

square-rooted covariance matrices providing numerical ro-

bustness even when very few bits are used for arithmetic

implementations. Introducing the unique upper triangular

Cholesky factor with positive diagonal entries R(s), and

vectors ⌫

(s):

共

兲

⫽ R

共

兲

共

兲

, 共15兲

⌫

共

兲

⫽ R

⫺ T

共

兲

⌬

共

兲

, 共16兲

the optimal error diffusion ﬁlter coefﬁcients can be ob-

tained as a function of the square-rooted covariance

쐓

共

兲

⫽ R

⫺ 1

共

兲

⌫

共

兲

, k⫽1,2,...K. 共17兲

The QR-RLS algorithm is effective because it can be

implemented in a recursive manner. Assuming the matrices

R(s⫺ 1) and R

(s⫺ 1) and the vectors ⌫

(s⫺ 1) are al-

ready computed for the previous pixel (s⫺1), they can be

updated so that

共

兲

冋

冑

␭R

共

s⫺1

兲

冑

␭⌫

共

s⫺1

兲

⫺ T

共

s⫺1

兲

冑

␭

共

兲

共

兲

册

⫽

冋

共

兲

⌫

共

兲

⫺ T

共

兲

共

兲

共

兲

共

兲

册

with the vector g

(s) and the variable f

(s) updating the

present optimum error diffusion ﬁlter as

쐓

共

兲

⫽ h

쐓

共

s⫺1

兲

⫺ g

共

兲

共

兲

k⫽1,2,...K. 共18兲

The orthonormal matrix Q(s) is of size (L⫹1)⫻ (L⫹ 1)

and consists of L Givens rotations

共

兲

⫽ Q

共

兲

L⫺ 1

共

兲

...Q

共

兲

each with the form

共

兲

⫽

冋

i⫺ 1

]]

¯ cos

␪

共

兲

¯ sin

␪

共

兲

] I

L⫺ i

]

¯ ⫺ sin

␪

共

兲

¯ cos

␪

共

兲

册

. 共19兲

The Givens rotation matrices Q

(s) differ from an identity

matrix of size (L⫹ 1)⫻(L⫹1) only at its four entries, as

shown in Eq. 共19兲. The rotation angle

␪

(s) is selected so

that the (L⫹1,i) element of the matrix to which Q

(s)is

applied is annihilated while R(s) remains upper triangular.

The Q

(s) matrix constructed in this manner simulta-

neously updates the matrix R

(s⫺ 1) and the vector ⌫

(s)

as indicated by the last two columns of Eq. 共18兲. The itera-

tions can be started with a diagonal matrix R(0)⫽

冑

␦

Yang and Bo

hme have described this and other rotation-

based RLS algorithms in a uniﬁed framework.

3.2

Channel-by-Channel Error Diffusion

In the previous subsection, error diffusion by using previ-

ously made errors in all of the channels is discussed. By

constraining the form of the error diffusion coefﬁcient ma-

trix H(s), as in Eq. 共4兲, further savings in computational

load can be achieved with tolerable degradation in the over-

all performance of the diffusion process. In this case, the

cost function of Eq. 共11兲 simpliﬁes to

共

兲

⫽

兺

j⫽ 1

␭

s⫺ j

储

共

兲

⫺ H

共

兲

共

兲

储

共20兲

⫽

兺

k⫽ 1

兺

j⫽ 1

␭

s⫺ j

关

共

兲

⫺ h

共

兲

共

兲

兴

共21兲

⫽

兺

k⫽ 1

k,s

共

兲

, 共22兲

where the previous quantization errors for the k’th channel

are given by

共

兲

⫽

关

共

兲

共

兲

...e

共

兲

兴

k⫽1,2,...,K. 共23兲

We deliberately skipped the subscript p on the right-hand

side to simplify the notation.

3.3

Computational Complexity

The computational complexity of one iteration step of the

standard LMS algorithm is known to be the length of the

input vector for every channel. Therefore, the computa-

tional load of composite multichannel error diffusion is

N and the load reduces to L⫽ KN for the channel-by-

channel error diffusion. Computational complexity of the

QR-RLS algorithm is O(K

) for the composite multi-

channel error diffusion and it is O(KN

) for the channel-

by-channel implementation. With typical selections of K

⫽ 3 and N⫽ 4, the load QR-RLS algorithm is not very high

compared to the LMS algorithm. Furthermore, systolic ar-

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rays can be utilized in the QR-RLS implementation and the

proposed approaches can be realized efﬁciently by VLSI

technology.

The diagonal and the traditional raster have the same

computational load but the memory requirements may vary

depending on the application. If the image is ‘‘scanned’’

diagonally the memory requirements only increase by a

factor of

冑

(2), butwhen the image is already scanned hori-

zontally almost the whole image has to be buffered.

4 Simulation Results

To demonstrate the performance of our error diffusion al-

gorithm, we carried out simulations with several images

among which a representative one is presented here.

First

the images were quantized to 16 levels using the median

cut algorithm.

We compared the new method with Floyd-

Steinberg’s method, and the adaptive error diffusion with

LMS algorithm both with raster scan and diagonal scan of

the image. Various values for the forgetting factor ␭ for the

QR-RLS algorithm were tried in the interval 关0.9, 0.99兴 and

the algorithm showed robust performance with these selec-

tions. We kept the forgetting factor as 0.95 in our simula-

tions for both types of scanning methods. The parameter

used in the initialization of the QR-RLS algorithm,

冑

␦

was chosen as 0.1. The coefﬁcients of the adaptive error

diffusion ﬁlter in both algorithms were scaled by 0.9, i.e.,

we allowed diffusion of not all but a fraction of the error

made in quantization to neighboring pixels, and this re-

sulted in a slight improvement in terms of color impulses.

The results for the ‘‘Peppers’’ image is shown

in Fig.

6. The image error diffused by Floyd-Steinberg’s method in

Fig. 6共b兲 contains color impulses, and the edges are

smeared to each other. These artifacts, color impulses and

false edges, are reduced in Figs. 6共c兲 and 6共d兲, which were

obtained by the LMS-based error diffusion method. The

image in Fig. 6共d兲 was obtained by diagonal processing

which shows some improvement when compared to Fig.

The color image outputs can be viewed in Ref. 20.

Fig. 6 (a) Original image, (b) standard error diffusion, (c) error diffusion with LMS (raster scan), (d)

error diffusion with LMS (diagonal scan), (e) error diffusion with QR-RLS (raster scan), and (f) error

diffusion with QR-RLS (diagonal scan).

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QR-RLS algorithm for error diffusion of color images

Figures

Citations

Colorant-based direct binary search halftoning

References

The effects of a visual fidelity criterion of the encoding of images

Color image quantization for frame buffer display

A survey of techniques for the display of continuous tone pictures on bilevel displays

Digital color imaging

Color image quantization for frame buffer display

Related Papers (5)

Optimized error diffusion for image display

Color diffusion: error diffusion for color halftones

Variance‐based color image quantization for frame buffer display

Joint quantization and error diffusion of color images using competitive learning

Improved quantization methods in color error diffusion

Frequently Asked Questions (11)

Q1. What are the contributions mentioned in the paper "Qr-rls algorithm for error diffusion of color images" ?

Q2. What are the three main categories of halftones?

Q3. how can i reduce the error diffusion filter coefficients matrix?

Q4. Where did he receive his MSE and PhD degrees?

Q5. What is the simplest way to reduce the error diffusion coefficient?

Q6. What are the disadvantages of optimization-based methods for halftoning?

Q7. What is the computational complexity of the QR-RLS algorithm?

Q8. What is the coefficient of the adaptive error diffusion filter?

Q9. Where did she receive her BSc degree?

Q10. what is the optimum error diffusion filter coefficient?

Q11. What is the power spectrum of the quantization error diffusive method?