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Spike noise removal in the scanning laser microscopic image

of diamond abrasive grain using a wavelet transform

Kazuhiro Koshino, Noriyuki Saito, Shigehito Suzuki*, Jun?ichi Tamaki

Department of Computer Science, Kitami Institute of Technology, 165, Koen-cho, Kitami 090-8507, Japan

Received 17 January 2002; received in revised form 5 May 2002; accepted 6 August 2002

Abstract

To remove spike noise in the scanning laser microscopic image of diamond abrasive grain without blurring the sharp

edges, a new smoothing technique that combines a conventional averaging technique with wavelet transforms is pro-

posed. The diamond abrasive grain image is decomposed into high- and low-frequency subimages using wavelet filters,

and all subimages except the lowest frequency one are synthesized to obtain a high-frequency image, from whose pixel

values spike noise points are extracted. A conventional averaging technique is then applied to the same points in the

original image as the spike noise points in the high-frequency image. The smoothing technique successfully removes

both clustered and unclustered spike noise while preserving the sharp edges. Spike noise is removed without a loss in the

original grain shape. This smoothing technique will surely be effective for other applications.

? 2002 Elsevier Science B.V. All rights reserved.

Keywords: Spike noise removal; Scanning laser microscopic image; Diamond abrasive grain; A conventional averaging; Wavelet

transform; Edge-preserving smoothing

1. Introduction

Scanning laser microscopic (SLM) images of

diamond abrasive grains are degraded by spike

noise due to the weak detection of laser reflection

intensity caused by slanting surfaces, hollows, and

so on. Spike noise thus makes it difficult to obtain

accurate information about diamond grains. To

improve the quality of such degraded images, the

noise interference must be removed. In this study,

we propose a smoothing technique that combines

a conventional averaging technique with a 2-D

discrete wavelet transform in order to remove

spike noise in the SLM diamond abrasive image

without the blurring of sharp edges generally

caused by smoothing.

This new smoothing technique decomposes the

original image corrupted by noise into high- and

low-spatial frequency subimages using high- and

low-pass wavelet filters. All subimages except the

lowest frequency one are synthesized to obtain the

high-frequency image, from whose pixel values

spike noise points are extracted. A conventional

averaging technique is then applied to the same

1 October 2002

Optics Communications 211 (2002) 73–83

www.elsevier.com/locate/optcom

*Corresponding author. Fax: +81-157-26-9344.

E-mail address: suzuki@cs.kitami-it.ac.jp (S. Suzuki).

0030-4018/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.

PII: S0030-4018(02)01867-9

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points in the original image as the spike noise

points in the high-frequency image. Although our

proposed smoothing technique employs a con-

ventional averaging technique which has a draw-

back of causing spatial blurring, it produces

excellent smoothing results with sharp edges pre-

served because it takes advantage of the spike

noise point information for smoothing. To dem-

onstrate its effectiveness, we compare our tech-

nique with representative smoothing methods.

In our technique, the wavelet transform is only

used to extract noise information from the high-

frequency image, which is a difference from

wavelet-based de-noising techniques. Since many

wavelet de-noising techniques have been reported,

we briefly introduce the most recent ones [1–10].

Most of them [1–6] were threshold-based tech-

niques using hard or soft thresholding [11,12]

although they took various approaches to the

problem of noise removal. In principle, the tech-

niques either set to zero all wavelet coefficients

which have an absolute value lower than a

threshold (hard thresholding), or shrink coeffi-

cients larger than the threshold towards zero with

an amount equal to the threshold value (soft

thresholding). The de-noising image is recon-

structed from the modified wavelet coefficients.

Soft thresholding is applied to wavelet coefficients

obtained by wavelet decomposition [1], and to

those that are iteratively obtained using a regu-

larization method [2], and is included in the op-

timizationproblem

thresholding and soft thresholding are applied to

wavelet packet [4]. The two are used in the

technique [5] using both wavelet and Karhunen–

Loeve transforms for calculating the weighting

coefficients for wavelet coefficients and obtaining

the correlation matrix from the weighted wavelet

coefficients. Thresholding rules are obtained using

a Bayesian wavelet de-noising technique [6].

Several of the wavelet-based techniques [7,8]

modify coefficients according to statistical prop-

erties of wavelet coefficients such as kurtosis. The

wavelet coefficients that are definitely corrupted

by noise are set to zero [9] or are replaced by

interpolated values [10].

In the present work, wavelet coefficients of

spike noise had much larger absolute values than

approach[3].Hard

the signal components. This noise-interference

situation is greatly different from general situa-

tions where wavelet-coefficient levels of spike are

smaller than those of the signal components and

hence thresholding is effective for eliminating

noise. For this reason, we use the wavelet trans-

forms to extract spike noise information from their

wavelet reconstructed images by discriminating

spike noise from the signal component. For the

same points in the original image as the noise

points in the reconstructed image an averaging

technique is iteratively applied. The smoothing

method is therefore fundamentally different from

the above wavelet-based de-noising techniques.

2. Theory

The SLM image of diamond abrasive grain to

be smoothed is decomposed into subbands using

wavelet filters. Then, all subbands except the

lowest spatial frequency subband are synthesized

to obtain the high-frequency image. The high-fre-

quency image is used to distinguish between noise

and edge components and to extract information

about the magnitude and position of noise. On the

basis of the noise information, pixel points to be

processed are selected and smoothing filters for the

points are determined. In this way, the present

smoothing technique attempts to remove spike

noise from the original SLM diamond grain image

without a loss in edge sharpness.

A 2-D discrete wavelet transform can be im-

plemented using 1-D low- and high-pass wavelet

filters on the rows in the 2-D image array and then

on the columns. The 1-D discrete wavelet filtering

[13] is implemented as follows:

sðlþ1Þ

m

¼

X

k

hðk ? 2mÞsðlÞ

k;

ð1Þ

wðlþ1Þ

m

¼

X

k

gðk ? 2mÞsðlÞ

k;

ð2Þ

where fhg and fgg are the low- and high-pass fil-

ters, respectively, and sðlÞ

and high-frequency components in the level l,

respectively. Taking sð0Þ

components are decomposed into subbands using

mand wðlÞ

mare the mth low-

mas the original value, the

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K. Koshino et al. / Optics Communications 211 (2002) 73–83

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Eqs. (1) and (2). The inverse operation, namely,

reconstruction [4] is given by

sðl?1Þ

m

¼

X

k

hðk ? 2mÞsðlÞ

kþ

X

k

gðk ? 2mÞwðlÞ

k:

ð3Þ

Thus, the low-frequency component in level l ? 1

is obtained by synthesizing the low- and high-

frequency components in the next upper level. By

continuing this process down to l ¼ 1, the original

data are reconstructed.

Fig. 1 shows a three-level wavelet decomposi-

tion. Of the 10 subimages, the one obtained by

low-pass filtering of the rows and columns is re-

ferred to as the LL image, and the one obtained by

low-pass filtering of the rows and high-pass filter-

ing of columns is referred to as the LH image.

Similarly, the other images are called the HL and

HH images. The numbers attached to the sub-

images designate the level numbers. All decom-

posed subimages except the lowest-frequency one,

the LL image, are synthesized to yield the high-

frequency image, from which information about

the magnitude and position of noise is extracted.

Since dyadic wavelet transforms are used in the

present work, if the original image is M ? M in

size, in Fig. 1 every subimage at decomposition

level l is M=2l? M=2lin size, where l ¼ 1;2;...

Thus, the approximation to the original image

becomes coarser as the decomposition level is in-

creased; it is coarser according to 2l.

The reconstructed image rijprovides two kinds

of information, signal information that contains

edge, and noise one. The signal and noise are

discriminated based on the deviation of the pixel

value rijfrom the average pixel value over the re-

constructed image, since signal components were

experimentally found to take in general fairly

smaller deviations than spike noise, as will be

shown later. Hence, the pixel values rijthat satisfy

the following relation are determined to be spike

noise:

jrij? ljPnr:

Here l and r are the average pixel value over the

reconstructed image and the standard deviation,

respectively, and n is a fixed positive real number.

Since l was close to zero in this work and signal

components except edges also took values close to

zero, the signal and noise can be discriminated

based on the magnitude of the pixel value rij, jrijj.

A group of large noise is clearly distinguished

from the edges by Eq. (4). However, there is a

group of small noise that lies in the interval

jrij? lj < nr, and is hardly distinguishable from

the edges. To cope with this problem, smoothing is

performed in two steps as illustrated in Fig. 2. In

the first step, smoothing is iterated several times

for the noisy pixels fij in the original SLM dia-

mond abrasive image whose corresponding rij

values are definitely determined as noise by Eq.

(4). This repeated smoothing is performed because

a single smoothing was insufficient to remove the

large spike noise from the image. In the second

step, a single smoothing using filters different from

that used in the first step is applied for all pixels in

the original SLM image fij. The weighting coeffi-

cients of the averaging filters used in the first- and

second-step smoothing will be given below. In the

example in Fig. 2, smoothing is iterated a total of

K þ 1 times through the first- and second-steps.

A pixel value fijin the original SLM diamond

abrasive image is averaged over an N ? N local

area to yield the smoothed value gijas follows:

ð4Þ

gij¼

1

Wij

X

ðN?1Þ=2

k¼?ðN?1Þ=2

X

ðN?1Þ=2

l¼?ðN?1Þ=2

wklfði?kÞðj?lÞ;

ð5Þ

where wkldenote the weighting coefficients of the

averaging filter, and Wij is the sum of the coeffi-

cients wkl over the N ? N local area. In the first-

step smoothing shown in Fig. 2, all wklare set to be

Fig. 1. A three-level wavelet decomposition.

K. Koshino et al. / Optics Communications 211 (2002) 73–83

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the same because this averaging filter is the most

effective for removing large spike noise. Although

averaging tends to blur sharp edges, in this case we

avoid blurring because the averaging is applied

only to the pixels that were definitely determined

as noise by Eq. (5). In the second-step smoothing,

the weighting coefficients are determined accord-

ing to the magnitude of the pixel values in the

reconstructed image:

wkl¼

where jrmaxj is the largest magnitude in the recon-

struction and i ? ðN ? 1Þ=26k 6i þ ðN ? 1Þ=2,

j ? ðN ? 1Þ=26l6j þ ðN ? 1Þ=2. The weighting

coefficients are designed to remove noise while

preserving signal information, including edges. To

illustrate, we will consider a simple case in which

noise is absent at the pixel ði;jÞ and at the sur-

rounding pixels. In this case, as stated above, the

signal components take fairly small jrklj values,

and it hence gives wkl? 1 for the pixels except

ðk;lÞ ¼ ði;jÞ, which then yields gij? fij. The signal

component fijis thus almost completely preserved

in the smoothed value gij. On the other hand, in

the presence of noise at the point ði;jÞ and at some

surrounding pixels, the relation wkl? 1 does not

hold and a smoothing term dij is added to

fij: gij¼ fijþ dij. The noisy pixel fij is thus

smoothed with the weighting coefficients given by

Eq. (6).

1

ðk;lÞ ¼ ði;jÞ;

otherwise;

jrklj=jrmaxj

?

ð6Þ

3. Results

Figs. 3(a) and (b) show an SLM diamond

abrasive grain image and the smoothed image,

Fig. 3. (a) The original SLM image of diamond abrasive grain of size 256 ? 256 and (b) the smoothed image obtained at a decom-

position level of six using the Db12 filter. The horizontal lines are 96 lm in length.

Fig. 2. The two-step smoothing procedure of the present

technique.

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K. Koshino et al. / Optics Communications 211 (2002) 73–83

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respectively, both 256 ? 256 in size and 0:375 lm

in pixel width. The SLM diamond grain image was

observed at a sampling interval of 0:375 lm with

numerical aperture of the objective lens set to 0.95,

which gives a resolving power of 0:10 lm. The

width of the grain in Fig. 3 is about 43 lm along

the horizontal line. The smoothed image was ob-

tained using the Daubechies wavelet filter with 12

coefficients (Db12) [13] from the high-frequency

image that was reconstructed from all subband

images at a decomposition level of six except the

lowest frequency one (LL6). In Fig. 3(a) large

spike noise gathers in large clusters and forms dark

defect areas, as shown in the 46 ? 32 rectangular

region enclosed by black lines, and as also illus-

trated in large fluctuations in the profile curve. The

defects cause a large loss in the original diamond

grain shape. In Fig. 3(b) the clustered spike noise is

hardly observed. This smoothed image shows no

appreciable loss in sharp edges and a good resto-

ration of the grain shape, which suffered consid-

erable loss due to the clustered spike noise. The

profile curve also shows a nearly flat change over a

wide interval where interval large deviations were

caused in Fig. 3(a) by the clustered spike noise. To

extract spike noise points from the reconstructed

image rij, the pixel points ði;jÞ that satisfied

jrij? ljPnr, where n ¼ 2 in Eq. (4) were deter-

mined as the spike noise point. The same points

ði;jÞ in the original noisy image, Fig. 3(a), were

subjected to10iterations

smoothing with a 5 ? 5 averaging filter of the same

weighting coefficients, as illustrated in Fig. 2.

Then, the second-step smoothing was applied once

to all pixels, with the result shown in Fig. 3(b).

To show the smoothing effect on spike noise

over a certain image area, pixels in the 46 ? 32

region shown in Fig. 3 are three-dimensionally

of the first-step

plotted in Fig. 4(a) and (b) that correspond to

Figs. 3(a) and (b), respectively. A visual compari-

son between the two shows that the removal of

spike noise was excellent. To evaluate the spike

noise removal quantitatively, we introduce the

normalized largest difference defined by D ¼

ðmax?minÞ=ðmaxþminÞ, where max and min are

the largest and smallest pixel values in the 46 ? 32

(17:3 ? 12:0 lm2) region, respectively. For Figs.

4(a) and (b) D ¼ 0:354 and D ¼ 0:062, respec-

tively, were obtained; the latter small value corre-

sponds to the fact that the clustered spike noise in

Fig. 3(a) is hardly observed in Fig. 3(b), namely, in

Fig. 4(b).

Fig. 5 shows smoothed images for Fig. 3(a) at

different composition levels and for different

wavelet filters, where the images were obtained in

the same way as for the image in Fig. 3(b). Their

normalized largest differences D are shown in Ta-

ble 1. The values correspond well to the quality

evaluated visually for the smoothing results in Fig.

5. Fig. 5 shows that clustered spike noise is satis-

factorily removed at a decomposition level of six

for the Coiflet (Coif) [14], Daubechies, Symlet

(Sym) [14], and 9/7 [15] filters; all have 12 coeffi-

cients except the 9/7 filter. The Db12 also gave a

good result at a decomposition level of five. These

results gave similar small D values of 0.062–0.068,

as shown in Table 1. The Coif, Db, and Sym filters

with coefficients ranging from 6 to 20 could elim-

inate the clustered noise satisfactorily at a de-

composition level of six, although their smoothed

results are not shown here. On the other hand, the

clustered spike noise still remains at the other de-

composition levels, as shown in Fig. 5. Naturally,

their corresponding D values were larger than

those at a level of six, as shown in Table 1.

The optimum level depended on the smoothing

method as shown in Fig. 2, and on the value of n

used to discriminate between spike noise and the

signal component according to Eq. (4). Noise im-

ages jrijj, which satisfied jrij? ljPnr, where n ¼ 2

in Eq. (4), affected the quality of the smoothing

result. To illustrate this, noise images for Fig. 3(a)

at decomposition levels of four and six are pre-

sented in Figs. 6(a) and (b), both of which were

obtained using Db12. These noise images are a

binary representation of the bright and dark levels;

Fig. 4. The 3-D plots in (a) and (b) show the variations of pixel

values in a 46 ? 32 region of Fig. 3(a) and (b), respectively. The

region is enclosed with black lines.

K. Koshino et al. / Optics Communications 211 (2002) 73–83

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