Suppression of moire´ patterns in scanned halftone images

* Corresponding author. Tel.: 035 715 900; fax: 035 721 490.

 This work was supported partially by National Science Council, Republic of China under grant NSC85-2221-E009-011.

 Also with Tungnan Junior College of Technology.

Signal Processing 70 (1998) 23—42

Suppression of moire´ patterns in scanned halftone images 

James Chingyu Yang , Wen-Hsiang Tsai*

Department of Computer and Information Science, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan 300, People+s Republic of China

Received 18 January 1996; received in revised form 6 July 1998

Abstract

Moire´ patterns often appear in the image obtained from scanning an image printed on a magazine or a newspaper. The patterns do not exist in the original printing but come from alias sampling of screened halftone pictures. A strategy to design a moire´ suppression scanning procedure is proposed. Fourier analysis is presented for both the screening and the scanning processes, from which a formula is derived to calculate a special scanning resolution, called moire´ controlling scanning resolution. With the moire´ scanning resolution, an input halftone image is scanned and the moire´ signals can be confined in programmed frequency areas. After filtering out the signals in those areas, the original can be restored. Some useful spatial filters for this purpose are designed. Experimental results are shown to demonstrate the feasibility of the proposed approach.  1998 Elsevier Science B.V. All rights reserved.

Zusammenfassung

Beim Scannen eines Zeitungs- oder Zeitschriftenbildes treten oft Moire´-Muster auf. Diese Muster existieren nicht im Original, sondern entstehen durch u¨berlappende Abtastung von Halbtonbildern. Vorgeschlagen wird eine Strategie zum Entwurf eines Abtastverfahrens mit Moire´-Unterdru¨ckung. Eine Fourieranalyse sowohl fu¨r die Rasterungs- als auch fu¨r die Abtastungsprozesse wird vorgestellt, aus der eine Formel zur Berechnung einer speziellen Abtastauflo¨sung hergeleitet wird, die Moire´-kontrollierende Abtastauflo¨sung genannt wird. Mit dieser Auflo¨sung wird ein Halbtonbild gescannt und die Moire´-Signale ko¨nnen auf einen programmierten Frequenzbereich beschra¨nkt werden. Nach dem Herausfiltern dieser Signale aus eben diesen Frequenzbereichen kann das Original wiederhergestellt werden. Zu diesem Zweck werden einige nu¨tzliche ra¨umliche Filter entworfen. Experimentelle Ergebnisse werden gezeigt, um die Durchfu¨hrbarkeit des vorgeschlagenen Vorgehensweise zu demonstrieren.  1998 Elsevier Science B.V. All rights reserved.

Re´sume´

Des motifs de moire´ apparaissent souvent dans des images obtenues par nume´risation d’images imprime´es dans un magazine ou un journal. Ces motifs n’existent pas dans l’impression originale mais proviennent du repli de spectre des images en demi-teintes affiche´es. Nous pre´sentons ici une strate´gie de conception d’une proce´dure de nume´risation supprimant ce moire´. Une analyse de Fourier est pre´sente´e a` la fois pour l’affichage et pour la nume´risation, dans laquelle

0165-1684/98/$ — see front matter  1998 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 5 - 1 6 8 4 ( 9 8 ) 0 0 1 1 1 - X

Fig. 1. A halftone image creates an illusion of a continuous tone image. (a) The original continuous tone image. (b) Part of the generated halftone image.

une formule est de´rive´e afin de calculer une re´solution de nume´risation spe´ciale, appele´e ‘‘re´solution de nume´risation pour le controˆle du moire´’’. Avec une telle re´solution, une image en demi-teintes est nume´rise´e et les signaux de moire´

peuvent eˆtre confine´s dans une plage de fre´quences programme´e. Apre´s filtrage des signaux dans cette plage, l’image originale peut eˆtre restaure´e. Des filtres spatiaux ont e´te´ conc7 us dans ce but. Des re´sultats expe´rimentaux sont pre´sente´s, pour de´montrer la faisabilite´ de cette approche.  1998 Elsevier Science B.V. All rights reserved.

Keywords: Moire´ controlling scanning resolution; Moire´ pattern; Halftone image; Printing; Scanning; Fourier analysis;

Spatial filter; Image restoration; Moire´ suppression

1. Introduction

Most printed images are produced as bi-level images [11]. And halftone screening is employed to convert a continuous-tone image into a halftoned one which consists of numerous tiny screening dots.

The size of each dot varies according to the various tones of the original image. Halftoning systems rely on human vision that integrates numerous small features to achieve an illusion of the original con- tinuous-tone image. The dots in the halftone image are spread periodically. They are also arranged orthogonally to comfort human eyes. A raster im- age processor (RIP) is required to generate the halftone images for digital reproduction. A sample continuous-tone image and a halftone image gener- ated from it are shown in Fig. 1.

When we scan an image with periodic structures or superpose multiple color screens, aliasing is un- avoidable and additional moire´ patterns will usu- ally appear in the scanning or superposition result [3,6,8]. Because screen dots are repeated period- ically in a halftone image, scanning a halftone im- age will generate additional moire´ patterns. Fourier analysis can be employed to describe this phenom- enon. Some researches [1,5] described the aliasing phenomenon for the screening process. In this paper, Fourier analyses of both the modern digital screen- ing and the scanning processes will be presented.

According to the functionality of scanner hard- ware, the image scanning process can be divided into three major stages. The first stage is pre-filter- ing which is an optical process related to the char- acteristics of the scanner lens. Using a large drum

Fig. 2. The screening and scanning process.

scanner, a user can adjust the focus and the aper- tures of the lens to make the scanning result sharper or smoother. The next stage is sampling that is realized by a motor moving or a CCD arrange- ment. A user can adjust the scanning resolution to change the spacing of the sampling grid. The brightness values of the scanned picture are quan- tized into digital values. The final stage is post- filtering which is a software process employed to correct digital image values by gamma correction, look up tables, or other filters.

Fig. 2 shows a flowchart of the screening and the scanning processes. For gray-scale images, the moire´ phenomenon mainly comes from sampling the screen of printing result in the scanning step.

Some works proposed to suppress moire´ patterns are reviewed in the following.

Smoothing is a natural way to remove high- frequency noise. A scanner operator usually adjusts the focus or the aperture in the pre-filtering stage to smooth the scanning result in order to suppress the moire´ patterns. As is well known, smoothing is a low-pass filter in the frequency domain [8], which reduces the high-frequency halftone screen signals.

It removes most periodic structures of the image.

There are however problems that are related to smoothing. First, the image will be blurred. Next, it is difficult to perform good aperture or focus ad- justment. Only high-end drum scanners support these kinds of adjustments in the pre-filtering stage.

Using inverse halftoning as a post-filter to re- move moire´ patterns has also been tried [2,4].

First, the details of the halftone image are scanned with very high resolution. By analysis of each screen dot, it is possible to derive a corresponding gray-scale image. However, much processing time

may be required, and high-resolution scanning takes longer time and larger memory space.

In Shu and Yeh [10], possible factors that cause moire´ patterns were analyzed. A rule was figured out to formulate the relationship between scanning resolution, screening resolution, their angles, and moire´ visibility. A suggestion to select a scanning resolution was proposed for scanning to yield min- imum visibility of moire´ patterns.

Russ [9] removed periodic information of im- ages in the frequency domain by manual masking operations. However, it is in general difficult to determine in the frequency domain the spots that cause moire´ patterns in the original image.

Roetling [7] proposed a halftoning method with moire´ suppression which can be used for the repro- duction of halftone images. Using an adaptive thre- sholding process, the average value of each halftone cell on the reproduced halftone result can be kept identical to the average value of the source halftone image, and the moire´ patterns on the high resolu- tion halftone output can then be suppressed. This moire´ suppression method keeps the screen struc- tures of the source halftone image, however, this is not necessary for the ordinary gray scale image representation.

A new strategy is proposed in the present paper to design a moire´ suppression procedure for halftone image scanning. After the screening angle and frequency are measured by a digital screen tester. We derive a formula for a moire´ controlling scanning resolution (MCSR). Using the derived res- olution, the scanning confines the positions of the moire´ signals in some frequency areas. Signals in those areas are then suppressed by some spatial filters designed in this study.

J.C. Yang, W.-H. Tsai / Signal Processing 70 (1998) 23—42 25

Fig. 3. The dynamic thresholding operation of the screening process. (a) The source intensity function (solid) and the screen- ing function (dashed). (b) The output halftone intensity function.

In the remainder of this paper, we formulate the screening and scanning processes in Section 2. Us- ing Fourier analysis, we point out how moire´ pat- terns are generated by halftone image scanning. In Section 3, we describe the proposed moire´ sup- pression strategy and the proposed digital screen tester. Some experimental results are shown in Sec- tion 4, followed by conclusions in Section 5.

2. Fourier analyses of moire´ phenomenon 2.1. Fourier analysis of screening

The halftoning process is accomplished by a dy- namic thresholding operation based on a thre- sholding function. The thresholding process can be modeled by the following formula:

h(r)"g(r)Ns(r)"



0, if g(r)(s(r), (1) where N denotes the thresholding operator, the vector r specifies a position in the spatial domain, the function g(r) defines the intensity of the source continuous-tone image with range [0,1], s(r) is the thresholding function, and the function h(r) defines the signals on the output halftone image. Here, we use intensity value 0 to represent black and 1 to represent white. Since a halftone image is bi-level, the value of the function for each pixel is either 0 or 1. Fig. 3 illustrates the conversion of a source con- tinuous-tone image into a halftone image.

The thresholding function s(r), called screening function, can be programmed to generate different halftone patterns. As discussed before, the patterns are created to generate a simulated illusion of the original continuous-tone image. Normally, an ar- ray of varying-sized black screen dots is generated.

Ideally, the screen dots are very small and can hardly be detected by human eyes. They are distrib- uted uniformly and orthogonally to create a stable illusion. We can use the following convolution for- mula to model the screening function s(r):

s(r)"sB(r) * mQ(r), (2)

where sB(r) specifies the screening dot function and mQ(r) specifies the screening grid. The screening dot

function defines the shape of the dot. Normally, it is a symmetric and homogeneous decreasing func- tion. Some frequently used screening dot functions have round, elliptical, diamond or square shapes.

Occasionally, people use stochastic screening func- tions for low-resolution printers; in the present con- tribution, however, we only deal with periodic screening functions. The screening grid can be es- tablished by two orthogonal bases, rQ and rQ, defined below:

mQ(r)"  K\



L\d(r!mrQ!nrQ). (3) Convolution of the screening grid with the screening dot function distributes the screening dot function uniformly and orthogonally on the screen- ing space. Obviously, the screening dot function is confined in a square or parallelogram range framed by each screening grid. The screen dot function is to be designed so as to avoid any overlapping caused by convolution. Fig. 4 shows the effective area of the screening dot function for the central dot.

In the following analysis, we formulate the digital screening process, in order to show some properties of the screened halftone images.

In a source digital gray-scale image, each pixel has a certain quantized gray value. Several

Fig. 4. The screening grids and the effective area for the screen- ing dot function.

Fig. 5. The screened halftone image is idential to the summation of all the screened halftone images of the component image plates of the source image. (a) The source gray-scaled image and the corresponding screened halftone image. (b) Four component image plates (left) obtained from the source image and their corresponding screened halftone images. (c) Adding all the screened halftone image of each component image plates in (b) yields the same screened halftone image as the right image in (a).

non-overlapping image plates can be generated from the source digital gray-scale image according to the gray values. All the pixels on an identical image plate have equivalent gray values. For example, on image plate gK(r), pixels with gray value m retain their values and the remaining pixels are cleared to 0, i.e.,

gK(r)"



0, elsewhere. (4)

For example, a four-colored digital multi-inten- sity image, as shown in Fig. 5(a), can be separated into four individual image plates g(r), g(r), g(r) and g(r) as shown in Fig. 5(b). If we perform a screening process on each image plate gK(r), it is easy to see that the resulting halftone image gK(r) N s(r), called component halftone image plate, will keep the non-overlapping prop- erty. Adding these component halftone image plates, a screened halftone image is generated which is equivalent to the one that is generated directly from the source image by the thresholding process with the same screening function s(r). Ac- cordingly, we can define the output halftone image as follows:

h(r)"g(r)Ns(r)",\

K[gK(r)Ns(r)]. (5)

Because each pixel value of gK(r) is either 0 or m, the thresholding process of gK(r)Ns(r) only alters those pixels with gray value m. All the pixels having gray values different to m must be 0, and are not altered by the thresholding operation. In other words, we can regard the image plate as an image mask which clears those pixels on a constant screen plate to 0 wherever the corresponding pixels on the mask have gray value 0. Here, the constant screen

J.C. Yang, W.-H. Tsai / Signal Processing 70 (1998) 23—42 27

Fig. 6. An example of the constant screen plate (left) and its corresponding Fourier spectrum (right).

plate has uniform screen dots and is defined as follows:

sK(r)"mNs(r). (6)

Normally people use the logical AND operation to describe the mask operation. Because images (1/m)gK(r) and mNs(r) are binary with intensity values either 0 or 1, we can also use multiplication to model the masking operation mathematically.

That is, the thresholding operation of the image plate gK(r) can also be written as

gK(r)Ns(r)"





h(r)",\

K

mgK(r)sK(r).1 (8)

By comparing Eqs. (1) and (8), we see that the thresholding operator does not appear explicitly. It is now possible to analyze the Fourier transform of the output halftone image. The Fourier transform of Eq. (8) is

H(w)",\

K



where GK(w) is the Fourier transform of image plate gK(r), and SK(w) is the Fourier transform of constant screen plate sK(r).

The constant screen plates sK(r) in Eq. (6) are an output of a thresholding process, i.e., a binary im- ages that contains pixels with gray values either 0 or 1. As described before, the thresholding func- tion s(r) in Eq. (2) is established by duplicating the screening dot function sB(r) on the screening grid.

The effective areas of the screening dot functions do not overlap. When thresholding a constant, an identical screen dot pattern will be generated in each effective area. In other words, the result of the thresholding is an array of screen dots, which is spread on the screening grid. Hence, the constant screen plates can be formulated as a convolution of the screening grid and the screened dot that is generated by the thresholding operation mNsB(r).

Therefore, the constant screen plates can be for- mulated as

sK(r)"[mNsB(r]*mQ(r),

wheremQ(r) is described by Eq. (3). And the Fourier transform of the above equation is

sK(w)"I[mNsB(r),NQ(w), (10)

whereNQ(w) is the reciprocal screening grid in the frequency domain defined as

NQ(w)"C  K\



L\d(w!mwQ!nwQ), (11) where C is a constant; and wQ and wQ are the reciprocals of the screening bases, rQ and rQ, re- spectively. Obviously, non-zero terms only exist at the nodes of the screening grid in the frequency domain. Fig. 6 shows an example of the constant screen plate in the spatial and the frequency domains.

Because the screening structure is used to create the illusion of a gray-scale image, the screen dots must be small enough to make the detail of the source image distinguishable, i.e., the screening sig- nal should have much higher frequency than that of the image plate. The signals in the image plate GK(w) should vanish outside the rectangle with the boundary of $wQ and $wQ. Convolution of GK(w) and the Fourier transform of the constant screen plate SK(w) in Eq. (9) creates significant sig- nal components at the nodes of the reciprocal screening grid in the frequency domain. Fig. 7 shows the result of such a convolution for the one-dimensional case, in which the signals of the

Fig. 7. The result of the convolution of GK(w) and SK(w). (a) The constant screen plate SK(w) in the frequency domain. (b) The signal of component image plate GK(w) in the frequency domain. (c) The result of the convolution of (a) and (b).

Fig. 8. The screen signal components.

image plate are shown as a dashed line and the result of the convolution is shown as a solid line.

Here, we call each significant signal component as a screen signal component (SSC). The screen signal components can be classified by frequency. We denote by SSC? the set of screen signal components placed within a circle with radiusa in the frequency domain. Fig. 8 shows the result of the convolution and the screen signal components of various radii in the frequency domain.

2.2. Fourier analysis of scanning

As described before, there are three stages in the scanning process. We use the following equation to model the first two stages of scanning, namely, pre-filtering and sampling:

g(r)"[h(r)*a(r)];mL(r), (12)

J.C. Yang, W.-H. Tsai / Signal Processing 70 (1998) 23—42 29

Fig. 9. H(w);A(w) in the frequency domain.

Fig. 10. Moire´ signals are generated by any screen signal com- ponents shifted into the low frequency area (one of the cases is shown).

where a(r) is the aperture function which defines the aperture transmittance of the scanner lens; h(r) is the source halftone image produced by the RIP and printed on paper; g(r) is the gray-scale image re- sulting from scanning; andmL(r) denotes the scann- ing grid defined as

mL(r)"  K\



L\d(r!ma!na2), (13)

wherea and a are the basis vectors of the scann- ing grid; and m and n are integers.

The first part in the right hand side of Eq. (12), the convolution h(r) * a(r), models the pre-filtering step of the scanning process, in which light is reflec- ted from the printed halftone image and collected by the optics structure of the scanner. After that, light is sampled at positions ma#na in the sampling step. The aperture function is a gaussian function. The larger the distance from the sampling point, the less the light can be transmitted. The Fourier transform of Eq. (12) is

G(w)"[H(w);A(w)]*NL(w), (14)

where H(w) is the Fourier transform of h(r), A(w) is the Fourier transform of a(r), andNL(w) denotes the reciprocal scanning grid defined as

NL(w)"C  I\



J\d(w!ku!lu), (15) where u and u are the reciprocal basis derived from a and a; k and l are integers; and C is a constant.

The first part of the right hand side of Eq. (14), i.e., the product H(w);A(w) of the aperture function and the original halftone image in the

frequency domain, is shown in Fig. 9 for the one-dimensional case. According to the previous discussion, the halftone image H(w) has signal components at frequencies mw#nw. The prod- uct H(w);A(w) should also have corresponding signal components at these positions.

Then, the product is convolved with the grid spread by basis vectors u and u. By the convolu- tion, the signal components of H(w);A(w) centered at mw#nw are reproduced at each node of the scanning grid. An illustration of the result of such convolution for the 2-dimensional case is shown in Fig. 10 in which some screen signal components are shifted into the low-frequency area (the shaded square area). These low-frequencies screen signal components introduce additional moire´ patterns.

Let us define moire& signal set MSS?@ as the set of the shifted screen signal components in the low frequency area which originally are in SSC? and are

Fig. 11. Moire´ signals sets. (a) MSS₍

⁽. (b) MSS.

shifted due to convolution with the impulses on the reciprocal scanning gridNL(w) with "w""b. Fig. 11 shows two examples of moire´ signal sets MSS₍

⁽ and MSS.

The scanning resolution, the screen frequency and the screen angle are parameters that may alter the positions of the moire´ signal sets in the fre- quency domain. For commercial reproduction, the screening resolution is normally 100 to 200 LPI and the screening angle is often fixed to 0° (90°), 15°

(105°), 45° (135°) and 75° (165°). For black and white printing, the 45° screening angle is the most comfortable screening angle for human eyes. The scanning resolution varies for different applica- tions. Resolutions between 100 DPI to 600 DPI are normally selected. Before scanning, the screen fre- quency and the screen angle of the source halftone image can be measured by a screen tester. The angle of the scanning grid is fixed by scanner hardware. If we want to alter the moire´ patterns that appear on the scanned image, changing the scanning resolu- tion becomes the only way.

3. Proposed moire´ suppression method

Moire´ patterns can be altered by adjusting the scanning resolution. In this section, we will show that a specific scanning resolution, called ‘‘moire´

controlling scanning resolution’’ (MCSR), can be derived from certain given screen angles and screen frequencies, and by using the MCSR, the moire´

signals of a halftone image can be confined in programmed areas. We will also show that band- pass filters can be employed to suppress the signals in these confining areas and a better image can be obtained. Each band-pass filtering operation is ac- complished by the use of a combination of some spatial filters. Some useful spatial filters are de- signed in this study and shown in this section.

Fourier transforms are not performed and the fil- tering speed is fast. Screen parameters, including the screen angle and the screen frequency, are re- quired for the calculation of the desired MCSR. We also propose a digital screen tester to measure the screen parameters precisely.

3.1. Moire´ controlling scanning resolution

The MCSR is a value that can be calculated by two parameters, the screen frequency and the screen angle, of the source halftone image. In order to describe the MCSR, three cases, which cover all possible conditions of source halftone images, are checked.

The first case occurs when the screen angle is 0°

(or 90°). This situation is shown in Fig. 12(a). If we select the scanning resolution defined in Eq. (16)

J.C. Yang, W.-H. Tsai / Signal Processing 70 (1998) 23—42 31

Fig. 12. Screen signal components (MCSR) of 0° screen. (a) The SSC of 0° halftone. (b) The SSC of 0° halftone on the image scanned by MCSR(0°) using n"1.

below to scan such an image, all the moire´ signal sets will perfectly overlap at the nodes of the screen- ing grid so that the screen signal components are enhanced and no additional moire´ patterns are generated:

MCSR(0°)"n;f, (16)

where MCSR(0°) denotes the MCSR for 0° (or 90°);

f is the screen frequency; and n is an integer number. This situation is shown in Fig. 12(b).

The next case occurs when the screen angle is 45°. As we know, the angle of 45° is the most frequently used halftone angle for black and white image reproduction. By using the MCSR for 45°

calculated by Eq. (17) below, the screen signal com- ponents also perfectly overlap on the screening grid:

MCSR(45°)"n;f; 1

(2, (17)

where MCSR(45°) denotes the MCSR for 45°;

f is the screen frequency; and n is an integer number. This situation is shown in Fig. 13. For example, for a 45° screened halftone image with the screen frequency of 150 LPI, the MCSR is, by Eq. (17) using n"3, calculated to be 318 DPI.

The third case occurs when the screen angle is not 0°, 90° or 45°. The positions of the screen signal components are altered if the screen angle is

changed. For this case, we generalize the MCSR calculated by Eq. (16) or Eq. (17) to be Eq. (18) as follows:

MCSR(h)"n;f;cosh, (18) where MCSR(h) denotes the MCSR of any angle h;

f is the screen frequency; and n is an integer number.

Tracking the occurrences of the SSC of varying screen angles, we found that the SSC are located on lines. All signal components in SSC?Z⁺^2, are kept on horizontal and vertical lines, or move along certain directions. This situation is shown in Fig. 14. Corresponding to the fact that the halftone angles range from 0° to 90°, the directions of the SSC movements ranges from !45° to 45°. Because negative halftone angles create symmetric moving paths, only 0°—45° angle changes are illustrated in Fig. 15. As we know, the SSC of higher frequencies are weaker, and generally, the elements in SSC?

are too weak to generate significant moire´ patterns.

Therefore it is acceptable to analyze SSC?W only and to find an effective way for moire´ suppression.

Fig. 15(a) illustrates the moving paths of the moire´ signal sets for different screen angles, 0° to 45°, using the MCSR calculated by Eq. (18) with n"3. Two cases are discussed here. One is for the angles in 0°—22.5° and the other is for the angles in 22.5°—45°. The SSC moving paths for these two cases are illustrated in Fig. 15(b) and Fig. 15(c),

Fig. 13. Screen signal components (MCSR) of 45° screen. (a) The SSC of 45° halftone. (b) The SSC of 45° halftone on the image scanned by MCSR(45°).

Fig. 14. Locations of SSC for any angled screen scans using the moire´ controlling scanning resolution. (a) SSC and SSC⁽of 0°—45°

screens. (b) SSC, SSC⁽², SSC, SSC⁽⁵and SSC₂₍₂of 0°—45° screens.

respectively. The overlapping SSC?W signals are illustrated in Fig. 15(d) and Fig. 15(e). The overlap- ping SSC?W signals are confined in the shaded areas. This phenomenon is concluded in Fig. 16.

When the halftone angles are in !22.5°&22.5°, the MSS will be confined in the areas shown in Fig. 16(a). For the halftone angles out of

!22.5°—22.5°, the MSS is confined in the areas shown in Fig. 16(b). Obviously, all the moire´ sig-

nals are placed on band tracks of line segments.

And this property is useful for moire´ suppression as discussed in the following section.

3.2. Filters to suppress moire´ signals

From the previous analysis, we know that after a halftone image is scanned by the MCSR, the

J.C. Yang, W.-H. Tsai / Signal Processing 70 (1998) 23—42 33

Fig. 15. The result of 0° to 45° screen scans using the MCSR with n"3. (a) The SSC moving of 0°—45° halftone scanning using MCSR.

(b) The SSC moving of 0°—22.5° halftone scanning using MCSR. (c) The SSC moving of 22.5°—45° halftone scanning using MCSR. (d) The result of 0°—22.5° screened halftone scanning using MCSR. (e) The result of 22.5°—45° screened halftone scanning using MCSR.

Fig. 16. The effect areas of the moire´ signal components for n"3. (a)"h")22.5°. (b) 22.5°("h")45°.

Fig. 17. Two averaging filters in the frequency domain. (a) a"2/2. (b) a"3/2.

Fig. 18. The Fourier spectra of the digital averaging filters of Fig. 17 in the frequency domain. (a) 2;2. (b) 3;3.

moire´ signal sets are placed in known areas. By filtering out these signals, a better moire´ suppressed image can be obtained. Certainly, we may erase these signals in the frequency domain by the FFT and the inverse FFT. However, due to processing time consideration, we prefer spatial operations to frequency operations. Some useful spatial filters for moire´ suppression are proposed in the following.

The first useful spatial filter to suppress the moire´

signal is the averaging filter. The averaging filter is defined as a convolution of the image intensity function and a rectangle function. The rectangle function is defined as

rect(x, y)"



where a is a positive value. The Fourier transform of the rectangle function is a two-dimensional sinc function:

I+rect(u, v),"sin(2pau) pu

sin(2pav) pv .

The convolution of two functions in the spatial domain is equivalent to the multiplication in the frequency domain. When the value of the rectangle function in the frequency domain is zero, the signals of that frequency can be erased by convolution.

Therefore, the averaging filter can be employed to clear or weaken signals in certain high-frequency areas. Fig. 17 shows two rectangle functions in the frequency domain. For a digital image, the equiva- lent averaging filter in the spatial domain is a con- stant pixel matrix. Fig. 18 shows the 2;2 and the

3;3 digital averaging filters. The averaging filter clears four bands of signals in the frequency do- main. According to Fig. 16, the moire´ signal sets are connected as line segments. Since the connected line segments match the cleared bands, the aver- aging filter can be employed to suppress the moire´

signals. For example, when the MCSR is calculated with n"3, we can use the 3;3 averaging filter to suppress the moire´ signals. Certainly, one may se- lect bigger n values to calculate a higher MCSR for scanning. Practically, n"3 is acceptable for most cases.

According to the discussion in the previous sec- tion, moire´ signals are placed on the screening grid if the screen angle of the halftone is 0°, 45° or 90°.

And the signals are easy to erase by the averaging filters. For the other screen angles, the SSC is placed on the horizontal or vertical lines. The sig- nals on the lines can also be easily erased by the averaging filter. After that, one or more filters are required to erase the remaining moire´ signals that are generated by higher-ordered SSC. The aver- aging filters as well as a set of conceivable spatial filters that might be used to suppress moire´ signals are listed in Fig. 19. Areas where moire´ signals possibly exist for these cases can be determined by tracking the SSC. Then, a suitable filter to suppress the remaining moire´ signals can be selected. And in this way a moire´ suppression scanning procedure can be derived.

For example, for halftone images other than 45°, the MCSR can be calculated by (18). The n;n averaging filter then may be employed to suppress the major moire´ patterns generated by SSC. After that, a filter listed in Fig. 19 is selected to suppress the remaining moire´ signals. More specifically, we may first perform 3;3 averaging filtering, using the filter shown in Fig. 19(e), on an image that is scan- ned using the MCSR of n"3. If the screen angle is

J.C. Yang, W.-H. Tsai / Signal Processing 70 (1998) 23—42 35

Fig. 19. Some useful spatial filters and their Fourier spectra.

between 22.5° and 67.5°, the moire´ signals can be estimated by Fig. 16(a) and the filter shown in Fig. 19(k) can be used to suppress the remaining moire´ patterns. Otherwise, the filter shown in Fig. 19(j) can be used.

3.3. Image normalization

For low-end scanners, the scanning resolution cannot be freely adjusted. Therefore, a normaliz- ation of the scanned image is necessary. For this case, moire´ signals may not concentrate on lines.

We propose here a procedure that may be used to adjust the scanning result to have a desired MCSR.

Because the moire´ signals are already generated by the first scan, only the major low-ordered moire´

signal is considered to be suppressed.

First, scan the image using a resolution as high as possible in order to keep the generated moire´ sig- nals as weak as possible. Secondly, calculate a scal- ing ratio by the following formula:

scaling ratio" desired MCSR

current resolution. (19) Then, use the scaling ratio to re-sample the scanned image to generate a new image with the MCSR.

Finally, follow the procedure proposed in the pre- vious section to select a spatial filter to erase the moire´ signals.

3.4. Digital screen tester

Accurate screen parameters, including the screen angle and the screen frequency, are required for the calculation of the MCSR. A digital screen tester is proposed here to measure the screen parameters by analyzing the Fourier spectrum of screened halftone images. Digital screen testing using the tester is a separate process from the moire´ sup- pression process. One may perform the testing and keep the records of the resulting screen parameters for different types of document. The records may be used for future scanning to achieve moire´ suppression.

According to the previous analysis of the screen signals, we know that, in the frequency domain, the

screen signal components are placed on the reci- procal screening grid and can be classified by fre- quency. The most significant signals on the screening grid are the largest screen signal compo- nents, SSC. By checking the location of the spec- trum signal peaks, the screen parameters can be determined. According to the symmetry property of the screen signals, the detection of the screen para- meters may be simplified by expressing the spec- trum in polar coordinates to yield a function S(r,h), where S is the spectrum function, and r andh are the variables in this coordinate system. Because the screen signal components are symmetric signal peaks, four significant signal peaks with 90° inter- vals can be found in the function S(r,h). To detect these peaks, a function as follows is defined:

N(r,h)"min+S(r, h), S(r,h#90°), S(r, h#180°), S(r,h#270°),,

whereh is an angle in the range of 0°—90°. Let rL and hK be the values such that

N(rL , hK)"max PF

N(r,h),

then rL and hK are taken to be the desired screen frequency and screen angle, respectively.

Practically, the detection of the screen para- meters is designed in the following way. Firstly, a range of the possible screen frequency r is pre- determined, normally from 50 LPI to 200 LPI.

Then a pre-selected block of the screened halftone image is scanned by a high resolution (higher than a pre-determined largest frequency), and the scann- ing result is Fourier-transformed to yield a Fourier spectrum. Finally, in the pre-determined range of screen frequencies 50 LPI)r)200 LPI, the de- sired screen parameters, rL and hK, can be determined.

An example of the detection result is shown in Fig. 20.

3.5. Summary of proposed moire´ suppression scanning procedure

A summary of the proposed moire´ suppression scanning procedure described previously is given here. At the beginning, we measure the screen angle

J.C. Yang, W.-H. Tsai / Signal Processing 70 (1998) 23—42 37

Fig. 20. Detection of the screen parameters. (a) Halftone image (600 DPI). (b) Fourier spectrum of (a). (c) S(r,h). (d) Detection result. (e) N(r,h).

and frequency of a given image by a screen tester.

After that, depending on the scanner characters and the screen angles, three cases are treated.

Case 1. When the screen angle is 0°, 90° or 45° and the scanning resolution is freely adjustable, calcu- late the MCSR by Eq. (18) using any integer num- ber n. Then, scan the halftone image using the MCSR. Finally, apply an n;n averaging filter to the scanning result and a moire´-suppressed image can be obtained.

Case 2. When the screen angle is different from 0°, 90° and 45° and the scanning resolution is freely

adjustable, calculate the MCSR by Eq. (18) using any integer number n. Then, scan the halftone im- age using the MCSR. The moire´ signals are now confined in certain areas in the frequency domain.

Finally, use spatial filters in Fig. 19 to erase the moire´ signals and yield a moire´-suppressed image.

Here, n"3 is recommended to calculate the MCSR. Then, use the averaging filter in Fig. 19(e), to suppress the major moire´ signals, and apply the filter in Fig. 19(i) to erase the remaining moire´

signals if the screen angle is in the range of !22.5°

to 22.5° For other screen angles, apply the filter in Fig. 19(k) to remove the remaining moire´ signals.

Case 3. When the scanner resolution is fixed to some values that are different to the MCSR, a nor- malization process is necessary. First, scan the im- age using a resolution as high as possible. Then, resample the scanning result by the scaling ratio defined in Eq. (19). After this normalization, the procedure described in Case 1 and Case 2 can be employed to suppress the moire´ signals.

4. Experimental results

Two types of experiment were conducted. In the first type we scanned one halftone image by a high- end drum scanner whose scanning resolution is freely adjustable. And in the second, another halftone image was scanned by a low-end handy scanner so that the process of normalization can be tested.

The first original halftone printing was printed using a 100 LPI and 45° screen. Firstly, we cal- culated the MCSR that is 3;cos 45°; screen- ing resolution"3;cos 45°;100"212 DPI. After scanning with the MCSR, the image shown in Fig. 21 was obtained. By checking the Fourier spectrum, it is seen obvious that the moire´ signals are placed on lines. Then, we performed a 3;3 averaging filtering on the image. The result is shown in Fig. 22. Clearly, the major moire´ signals are suppressed. Finally, a sharpening filtering was performed to enhance the high-frequency signals.

The resulting image is shown in Fig. 23. By check- ing the Fourier spectrum, most of the moire´ signals are erased. The image is now ready to scale to any

Fig. 21. A halftone image scanned by the MCSR.

Fig. 22. Result of the 2;2 averaging filtering.

resolution for various usages, without producing significant moire´ patterns. To check this, we scale the image down to 70 DPI, a resolution we know to cause obvious moire´ patterns in usual cases. An additional scanning of the original printing using 70 DPI was performed for comparison. The two results are shown in Fig. 24. Obviously, the image produced by the proposed approach is much better than the one obtained from direct scanning which has a lot of moire´ patterns.

The second experiment is more complicated.

A handy scanner was used. The resolution of the scanner is fixed to 100, 150, 200, 300 and 400 DPI.

The second printing scanned has 120 LPI screen frequency and 105° (15°) screen angled halftone.

Initially, we use the highest available scanning res- olution, 400 DPI, to scan the printing. Then, the image is scaled to 3;120;cos 15°"348 DPI. Fi- nally, a 3;3 averaging filter and the spatial filter in Fig. 16(k) were applied to the image. Fig. 25 shows the intermediate result for each stage. For compari- son, we also scaled the result to 100 DPI and per- formed an additional scan using the same scanner.

The two images are shown in Fig. 26. Obviously, the image produced by our approach is superior to that scanned by the traditional way.

J.C. Yang, W.-H. Tsai / Signal Processing 70 (1998) 23—42 39

Fig. 23. Result of the sharpening.

Fig. 24. A comparision between the results of proposed and the traditional scanning. (a) The result from the proposed method. (b) The result of traditional scanning.

Fig. 25. Normalization and moire´ suppression results for a halftone image scanned by a handy scanner. (a) Result of initial scanning (400 DPI). (b) Result of normalization (scale to 348 DPI). (c) Result of 3;3 averaging filtering. (d) Result of spatial filtering 16(k).

Fig. 26. A comparison between the results of traditional scanning. (a) The result from the proposed method. (b) The result of traditional scanning.

J.C. Yang, W.-H. Tsai / Signal Processing 70 (1998) 23—42 41

5. Conclusions

When scanning a halftone printing, additional moire´ patterns will appear in the scanning result.

By Fourier analysis, we have shown that these are aliasing patterns coming from the sampling of the screened halftone image by a scanner. A strategy has been proposed to design a scanning procedure that can suppress the additional moire´ patterns.

Firstly, we calculate an MCSR to keep the moire´

signals in certain frequency domain areas. Then, we select specially-designed spatial filters to suppress the moire´ signals. Because the redundant high- frequency moire´ signals are suppressed, the result- ing image is good for reproduction, image compres- sion, and rescaling.

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