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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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the bright level expresses the pixel points ði;jÞ that

satisfied the relation jrij? ljPnr, and the dark

level represents the other pixel points. For the

same bright points ði;jÞ in the original noisy im-

age, Fig. 3(a), 10 rounds of the first-step smooth-

ing were applied, as already stated. Then, the

second-step smoothing was applied once to all

pixels, with the results shown in Figs. 6(c) and (d)

which were obtained from Figs. 6(a) and (b),

respectively. In Fig. 6(c) the clustered spike noise

still remains, whereas it is not observed in Fig.

6(d). Corresponding to this result, the D value for

Fig. 6(c), 0.137, was two times as large as that

for Fig. 6(d). The result implies that the noise

image in Fig. 6(b) was much more suitable for the

clustered noise removal than that in Fig. 6(a). The

bright area showing spike noise is much larger in

Fig. 6(a) than in Fig. 6(b). The difference in noise

area produced a markedly different quality in the

smoothed result. The noise area depended on the

value of n that was used to discriminate spike noise

from the signal component according to Eq. (4).

For the noise area, the K times of the first-step

smoothing shown in Fig. 2 were applied. The op-

timum level thus depended on both the value of n

and the smoothing method.

Fig. 5. Smoothed images for Fig. 3(a) obtained at different composition levels and for different wavelet filters.

Table 1

The normalized largest difference D for the smoothed results in

Fig. 5

Wavelet filter

Decomposition level

4567

Coif12

Db12

Sym12

9/7

0.138

0.137

0.127

0.121

0.075

0.068

0.071

0.087

0.065

0.062

0.065

0.063

0.087

0.104

0.080

0.221

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We compared our smoothing technique with

three others: a technique using a 5 ? 5 averaging

filter that was adopted in the first-step smoothing,

a technique using a weighted 5 ? 5 averaging filter

with different weighting coefficients (Fig. 7), and a

5 ? 5 median filtering method. The three tech-

niques were repeatedly applied to the image in Fig.

3(a), and their respective smoothing results after

10 iterations are shown in Figs. 8(a)–(c). The

smoothing results are compared with Fig. 6(d),

which was obtained by 10 iterations of the first-

step smoothing, and one iteration of the second-

step smoothing. The clustered spike noise in Fig.

3(a) still remains in Fig. 8(a)–(c), but it is hardly

observed in Fig. 6(d). The results indicate that only

the present technique was sufficiently effective for

removing the clustered spike noise. It should be

Fig. 7. Weighting coefficients of a 5 ? 5 averaging filter.

Fig. 6. (a) and (b) The noise images for Fig. 3(a) at decomposition levels of four and six; (c) and (d) the smoothed results obtained

from (a) and (b), respectively. Here, the Db12 filter was used.

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Fig. 8. Smoothed results for Fig. 3(a) obtained using three different smoothing filters: (a) a 5 ? 5 averaging filter with the same

weighting coefficients; (b) a 5 ? 5 averaging filter with the weighting coefficients shown in Fig. 7; and (c) a 5 ? 5 median filter.

Fig. 9. (a) Another original SLM image of diamond abrasive grain; (b), (c), and (d) the smoothing results obtained at a composition

level of five using the Coif12, Db12, and Sym12 filters, respectively. The horizontal lines are 96 lm in length.

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noted that our smoothing technique also employed

a 5 ? 5 averaging filter that was also used the

technique using the 5 ? 5 averaging filter, but ours

gave an excellent result whereas the technique us-

ing the 5 ? 5 averaging filter did not.

Another SLM image of a diamond abrasive

grain is presented in Fig. 9(a), where its width is

about 30:4 lm along the horizontal line. The large

variations in the profile curve in this figure are

caused by large spike noise, but spike noise does

not gather in large clusters as seen in Fig. 3(a).

Figs. 9(b)–(d) show the smoothed results at a

composition level of five obtained using Coif12,

Db12, and Sym12, respectively; all were obtained

by five iterations of the first-step smoothing and

one iteration of the second-step smoothing. These

figures show an excellent spike noise removal,

which is evident from their profile curves. Further,

no appreciable blurring is seen at the edge of the

diamond grain, no appreciable dependence of the

wavelet filters on smoothing results was found, as

shown in Figs. 9(b)–(d), and no distinct effect of

filter size on smoothing results was observed.

These results were similar to those for the clustered

spike noise in Fig. 3(a).

4. Discussion

The distinct difference seen in Figs. 6(d) and 8

between the present smoothing technique and the

other ones is attributable to the method of

smoothing. The present technique first applies it-

erative smoothing not for all pixels in the image,

but for definitely determined noisy pixels. As a re-

sult of the iterative smoothing for large noise

components, the pixel values, which were mostly

quite low (see the profile curve in Fig. 3(a)), became

progressively higher, because there were still some

signals having high-pixel values in a 5 ? 5 neigh-

borhood of the noisy points. The values were fur-

ther increased with the successive iterations of the

first-step smoothing and approached signal com-

ponent values, which could account for the absence

of spike noise in Figs. 3(b) and 6(d). To demon-

strate this by example of 1-D data simulating

clustered spike noise, the variation of spike noise

value with iteration of the first-step smoothing is

illustrated in Fig. 10(a). This figure shows that

spike noise values progressively return to the signal

value with increasing iteration number.

In contrast, the remaining methods smoothed

all the pixels simultaneously. Because of the low-

pixel value of spike noise, the two 5 ? 5 averaging

methods caused a decrease in the signal compo-

nent value in the vicinity of spike noise, together

with an increase in the spike noise value. The in-

crease and decrease were repeated throughout the

iterative smoothing processes. Here, let DI and DD

be the increase at a noisy point and the decrease at

a surrounding pixel, respectively. If DI < DD, the

noisy pixel cannot increase in value to a signal

component level. This case will occur if spike noise

takes a very small pixel value, and it can explain

the presence of spike noise in Figs. 8(a) and (b), as

Fig. 10. The variation of spike noise values with iteration of

smoothing: (a) the present smoothing technique; (b) a 5 ? 5

averaging filter with the same weighting coefficients. The sym-

bols " and # show an increase in spike noise value and a de-

crease in the signal component, respectively.

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

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also

smoothing technique gives DD ¼ 0 in the first-step

smoothing, which would mean an increase in spike

noise values, as stated above. The spike noise re-

maining in Fig. 8(c) is due to the fact that a 5 ? 5

median filter was unable to yield signal compo-

nents because more than half of the 5 ? 5 points

were occupied by spike noise.

Optimum decomposition levels were five or six

for the clustered spike noise in Fig. 3(a), and five

for the unclustered one in Fig. 9(a). Thus, the

most suitable decomposition level for noise re-

moval was dependent on the noise-interference

situation in the image to be smoothed. As stated

above, the optimum level depended on the value

of n in Eq. (4) used to discriminate between spike

noiseand the signal

smoothing method shown in Fig. 2. The quality

of the smoothing result was determined by the

noise images at different decomposition levels.

The noise image at a level of six shown in Fig.

6(b) yielded the best smoothing result for the

clustered noise in Fig. 3(a), whereas the noise

image at a level of four had a larger noise area

than that at level six, and could not eliminate the

spike noise. The noise image at level seven also

took a much larger noise area, as shown in Fig.

11(a), and it too could not eliminate the spike

noise satisfactorily, as shown in Fig. 11(b). The

reconstructed image at a lower level than level six

is richer in high-frequency components than that

illustratedin Fig. 10(b). The present

component,and the

at level six, whereas the reconstructed image at a

higher level than six is richer in low-frequency

components. Generally, the former case can result

in larger uncertainties in the spike noise position

than the case for level six, and the latter case will

also provide uncertain noise information because

the fraction of low-frequency components is re-

duced. A well-balanced compromise between the

low- and high-frequency components to remove

spike noise most effectively was attained at a

decomposition level of six; as a result, the best

smoothing results were produced. Of the wavelet

filters considered here, the Daubechies wavelet

filters showed the smallest dependence on de-

composition level, as shown in Fig. 5 and Table

1. Although it would be difficult to determine the

optimum level theoretically, but it can be deter-

mined empirically from experiments.

There were problems in the present smoothing

technique. Firstly, since spike noise points were

repeatedly smoothed, as illustrated in Fig. 2,

points at which spike noise was already removed

sufficiently were further smoothed. This will cause

a loss in edge sharpness. Secondly, if wavelet co-

efficients of diamond-grain edges are of similar

magnitude to those of spike noise, the edge and

spike are not distinguishable. To solve the first

problem, it is necessary to distinguish spike noise

points at which spike noise has already been re-

moved and those where it has not yet been re-

moved. To solve the second problem, it is

Fig. 11. (a) The noise images for Fig. 3(a) at a decomposition level of seven, and (b) the smoothed result obtained from (a), where

Db12 filter was used.

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

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necessary to discern the edge and spike noise using

geometrical structures of the edges such as edge

lines. These are subjects for a further study.

5. Conclusion

To remove spike noise in the SLM image of

diamond abrasive grain without blurring sharp

edges, a new smoothing technique that combines a

conventional averaging technique with a 2-D dis-

crete wavelet transform was proposed. This tech-

nique proved effective for removing both clustered

and unclustered spike noise while preserving sharp

edges. Because the spike noise was eliminated

without loss of the original shape, good restora-

tion of the original diamond shape could be

achieved. This smoothing technique will surely be

effective for other applications.

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