Copyright Ⓒ 2013 JSPE㪬㪉㪊Development of Areal Wavelet Transform applying to the 2D image
Graduate School, Ibaraki University, Taiju SUZUKI, Ibaraki University, Hirotaka OJIMA,Ibaraki University, Libo ZHOU, Jun SHIMIZU, Teppei ONUKI
Today, there is an increased need for quality control in the manufacturing sectors. Particularly, the automated detectionof the product surface defect is essential to fulfill the requirement of quality assurance.
In this research, the algorithm of surface defect detection by use of wavelet transform has been developed. This method
is established by one-dimensional wavelet filter. However, the image processing of 2D image for defect detection should
be applied by two-dimensional filter. Therefore, in this report, a new wavelet transform is developed with 2D filter andthe method is called for areal wavelet transform. This areal wavelet transform is proposed and is confirmed the efficiency
of denoising 2D image.
Key words: wavelet transform, Haar wavelet, areal wavelet transform

1. Introduction
Most recently, wavelet analysis stands out to be a powerful tool for
signal processing including data compression, data transmission anddenoising.

Similar to moving average, mathematical calculation ofwavelet transform (WT) falls in a form of convolution.

In thehorizontal direction and perpendicular both to a two-dimensional
object image.

However, it is rational that they are tworeconstruction

processing is enabled by processing a one-dimensional filter to a

decomposition

present wavelet image processing, two-dimensional filter

Input signal 䢳s䢲anddiscretization 䢯䢳䢲 䢳 䢴 䢵 䢶 䢷 䢸 䢹level 䢳s-1
䢲
-1

-3

dimensions as for the filter which processes a two-dimensional
object image.

-2

Therefore, in this research, the two-dimensional

filter is designed and proposed and it is called areal wavelet䢯䢳䢲 䢳 䢴 䢵 䢶 䢷 䢸 䢹䢳s-2䢲䢯䢳䢲 䢳 䢴 䢵 䢶 䢷 䢸 䢹䢳s-3䢲䢯䢳䢲 䢳 䢴 䢵 䢶 䢷 䢸 䢹䢳d-1䢲䢯䢳䢲 䢳 䢴 䢵 䢶 䢷 䢸 䢹
䢳
d-2
䢲
䢯䢳
䢲 䢳 䢴 䢵 䢶 䢷 䢸 䢹䢳d-3䢲䢯䢳䢲 䢳 䢴 䢵 䢶 䢷 䢸 䢹

Low frequency component

High frequency component

Fig. 1 Concept of wavelet transform

transform.

2. Conventional wavelet transformIn this research, wavelet transform has been investigated fordenoising and defect detection.

For denoising and defectdetection, the wavelet transformation is a powerful tool because
it employs time-frequency windows to decompose the original
signal into a sequence of subsets in scalar space as shown in Fig.
1 1).

As the wavelet functions are compactly supported in bothFig. 2

Process of conventional 2D-Haar wavelet transformspatial and frequency domains, the different components in thescalar space can be decomposed and reconstructed perfectly.
The nature of wavelet transformation also reserves both
frequency and spatial properties.

Wavelet transform is thus able

to give not only the amplitude of a specific spatial frequency(vertical level -1, -2, -3… in Fig. 1), but also the location ofcorresponding spatial frequency (horizontal axis in Fig. 1).

Fig. 3In this paper, conventional Daubechies wavelet is used as thescaling and wavelet functions as following equations (1), (2).

(t )

(t )

 2 / 2䚷 0  t  1 / 2
 2 / 2䚷䚷䚷1 / 2  t  1
 0otherwise 2 /2 0  t 1䚷otherwise 0

Process of areal Haar wavelet transform

As shown in Fig. 2, in conventional wavelet transform algorithmfor 2D image, the decomposition and reconstruction process are(1)applied in row direction first and in column direction by turns.
Then, the results components are four component (LL, HL, LH,HH).

Next level components is obtained by processing LL

component similarly.
(2)Thus, this algorithm is computationallyexpensive.

ISUPEN 2013
International Symposium on Ultraprecision Engineering and Nanotechnology, Tokyo, Japan, March 13, 2013.

43

Copyright Ⓒ 2013 JSPE㪬㪉㪊
3. Areal wavelet transformIn this research, the areal wavelet transform is introduced for
2D image wavelet transform.

In Fig. 3, the concept of areal

wavelet transform is shown.

In this figure, 2D areal filter is100100formed by the tensor product of two one-dimensional mother200200wavelets

300

300

400400in

orthogonal

directions.

Incomparison

toconventional wavelet transform, this algorithm is only performedby convolution between input original image and 2D areal filter.
In this paper, 2D areal filter is formed by the tensor product of500

500
100200

300

400

500

100

200

300

400500

Daubechies mother wavelet.
In order to confirm the validity of this 2D filters, difference(a) Original image(b) Noise addition image

between the input image and reconstructed image is investigated.

Fig. 4 Original image and noise addition image

Figure 4 shows the original image and the image with additional
noise.

In Fig. 5, the histogram of difference between original

image and reconstructed image is shown.

From this figure, it isfound that the differences are almost zero.

3x 105

Therefore, the2.5

proposed 2D filters are valid for wavelet transform filter.

24. Denoising results for 2D image

1.5

In this section, the denoising capability of areal wavelet
transform is confirmed.

1The areal wavelet transform algorithm

is compared with conventional wavelet transform algorithm.

As0.5

the input signal is decomposed into scalar domain, WT furtherprovides the capability of removing such a component that has

0-20-1001020

amplitude less (or larger) than a specified threshold. For theFig. 5

purpose of reducing small-amplitude noises, the filtering
algorithm performs decomposition and reconstruction by setting

Difference between original image andreconstructed image

the absolute value of scalar coefficients smaller than theunwanted amplitude to zero.

In Fig. 6, the algorithm ofdenoising 2D image by wavelet transform is shown.

In thispaper, the HH component is only filtered by threshold value
AL1~AL2 in Fig, 6.

In Fig. 7, the results of denoising by

conventional wavelet transform and areal wavelet transform areshown.

In comparison to input image shown in Fig. ??, thedenoising result by conventional wavelet transform is reduced
noise.

However, whole color is paled out.

Therefore, thisresult is filtered not only noise but also original image.

On theother hand, the denoising result by areal wavelet transform isFig. 6also reduced noise, and whole color of this result is near theoriginal image.

A method of denoising

As a result, the proposed areal wavelet

transform is better than conventional wavelet transform.
100

100

200

200

300

300

4004005. Conclusion
In this paper, in order to introduce the 2D filter for denoising
2D image the areal wavelet transform is proposed.

The

proposed areal wavelet transform filter is able to decompose andreconstruct the image completely.

Compared to the filters withconventional wavelet transform, the new 2D filter can separate

500500100

the noise from 2D image much more efficiently.

200300400(a) Conventional WT500100200300400500(b) Areal WTReferencesFig. 71) Jiang, X., 2004, Third generation of morphological features from micro andnano scalar surfaces, pp.1235-1240.

conventional Haar wavelet transform and areal Haar wavelet

44Denoising results for including noise image withtransform

ISUPEN 2013
International Symposium on Ultraprecision Engineering and Nanotechnology, Tokyo, Japan, March 13, 2013.