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Current technology makes it easy for humans to take an image and convert it to digital content, but sometimes there is additional noise in the image so it looks damaged. The damage that often occurs, like blurring and excessive noise in digital images, can certainly affect the meaning and quality of the image. Image restoration is a process used to restore the image to its original state before the image damage occurs. In this research, we proposed an image restoration method by combining Wavelet transformation and Akamatsu transformation. Based on previous research Akamatsu’s transformation only works well on blurred images. In order not to focus solely on blurry images, Akamatsu’s transformation will be applied based on Wavelet transformations on high- low (HL), low-high (LH), and high-high (HH) subunits. The result of the proposed method will be comparable with the previous methods. PSNR is used as a measure of image quality restoration. Based on the results the proposed method can improve the quality of the restoration on image noise, such as Gaussian, salt and pepper, and also works well on blurred images. The average increase is around 2 dB based on the PSNR calculation. Keywords Akamatsu transform, Blurred image, Discrete wavelet transform, Image Denoising, Image restoration Paper type Original Article 1. Introduction An image is a discrete representation of data that has layout information and the color intensity of an object [1]. Nowadays, an image can be easily converted into digital content, but sometimes there is additional noise in the image during camera capture so that the image appears damaged. This operation of repairing damage is called image restoration. Damage can occur when shooting can be caused by the quality of the digitizer, poor camera focus, and signal noise from sensors, so that noise or image appears blurry [2]. Image restoration is the process to obtain a clean original image of the damaged image or that exposed to noise [3,4]. © Jihad Maulana Akbar and De Rosal Ignatius Moses Setiadi. Published in Applied Computing and Informatics. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http:// creativecommons.org/licences/by/4.0/legalcode Declaration of Competing Interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Publishers note: The publisher wishes to inform readers that the article “Joint method using Akamatsu and discrete wavelet transform for image restoration” was originally published by the previous publisher of Applied Computing and Informatics and the pagination of this article has been subsequently changed. There has been no change to the content of the article. This change was Applied Computing and necessary for the journal to transition from the previous publisher to the new one. The publisher Informatics Vol. 19 No. 3/4, 2023 sincerely apologises for any inconvenience caused. To access and cite this article, please use Akbar, J. M., pp. 226-238 De Rosal., I. M. S. (2019), “Joint method using Akamatsu and discrete wavelet transform for image Emerald Publishing Limited e-ISSN: 2210-8327 restoration”, Applied Computing and Informatics. Vol. ahead-of-print No. ahead-of-print. https://10.1016/ p-ISSN: 2634-1964 DOI 10.1016/j.aci.2019.10.002 j.aci.2019.10.002. The original publication date for this paper was 12/10/2019. The image restoration has been done with various methods, such as Akamatsu transform [5], Joint method Wavelet Transform [6–8], Anscombe transforms [9], Stochastic Resonance [7], Singular Value for image Decomposition(SVD) [6,10], Adaptive Histogram Equalization [6]. restoration Karungaru et al. [5] used Akamatsu transformation to restore the image. Akamatsu transform is a transformation technique developed by Norio Akamatsu. Akamatsu transform produces a differential value showing the characteristics of the image and by scaling between the integral and differential values can perform image restoration. From this integral and differential values, there is a scale adjustment between the integral and the differential values to obtain the image of the restoration result. The result of restoration with this method is very good only on the blurred image because it can increase the energy in the image, but produce side effects as other noises appear sharper. Megha et al. [6] proposed Adaptive Histogram Equalization and DWT techniques to restore the image. The first step of image restoration is done by decomposing using Discrete Wavelet Transform to get four low-low (LL), low–high (LH), high-low (HL), and high-high (HH) sub-domains. Sub-band LL is selected then copied to LL2 to be applied to Adaptive Histogram Equalization. Sub-band LL and LL2 subsequently decomposed using SVD to produce U1, S1, V1 and U2, S2, V2. Then the calculation of both SVD decomposition to determine the improvement factor. The results of this restoration method are good for removing noise but making the energy in the image is reduced. SVD is a transformation that has been widely used in digital image processing [10–12]. SVD is a technique for obtaining geometric features of an image. Basically, SVD is used to handle matrices that do not have an inverse. The SVD has several characteristics that the singular value change does not significantly alter the overall image and the singular value contains illumination information, while the singular vector retains the information from the image. Wavelet Transform is a technique for stable signals and not stationary. Because of its efficiency in analyzing local discontinuities of a signal, the Wavelet Transform has been widely used in various disciplines [13–16]. DWT is one form of Wavelets Transform in which the image signal is passed through a pile of analysis filters followed by a subtraction operation. DWT decomposes images into 4 sub-domains: low-low (LL), low–high (LH), high- low (HL), and high-high (HH). LL is the low-frequency signal which is the approximation of the original image. LH and HL are intermediate frequency signals. And HH is a high- frequency signal. LH, HL, and HH represent changes in information in the image [13,17]. So DWT is very good at representing a signal with singularity. However, DWT is not suitable for representing fine image signals and colored noise so it needs to be combined with other methods to improve image restoration results [18]. Akamatsu transform proved excellent for restoring the image blur due to produce a different value which is characteristic of the image. This value will be strengthened and can reduce the blur in the image. While the use of low sub-band on wavelet transforms can reduce noise but make significant energy changes in the image. Based on this background, this research proposes to combine DWT and Akamatsu to get more optimal results of image restoration. Sub-bands LH, HL, and HH were chosen to apply Akamatsu transformation. To test the quality of restoration method will be given various image noise such as Gaussian noise, salt & pepper, and blur. The result of the restoration will be calculated by the PSNR and compared with the method contained in the research Karungaru et al. [5]and Megha P et al. [6]. 2. Akamatsu and wavelet transform 2.1 Akamatsu transform Akamatsu transformation is a transformation that produces the integral and differential values of an input signal [5,19]. Akamatsu transformation requires a target signal that is defined as Pðx; yÞ Where the target signal is in the transformation. With the presence of the ACI target signal, the result of the transformation will be divided into two parts, namely the right 19,3/4 of the target signal and the target signal left. The signals to the right of Pðx; yÞ are defined as VRSðxÞ and left as VLSðxÞ. The target signal is used to determine the result of Akamatsu transformation. To see more clearly the description of the target signal on Akamatsu transform can see Figure 1. To achieve the target signal required the calculation of the integral value and the differential value. To get an Integral value can use Eq. (1). ½AvrS ðxÞþ AvrS ðxÞ R L AðxÞ¼ (1) with AvrSRðxÞ and AvrSLðxÞ obtained from Eqs. (2) and (3). S ðxÞ AvrS ðxÞ¼ (2) S ðxÞ AvrS ðxÞ¼ (3) While the first order of the integral value of the Akamatsu transform is defined by Eq. (4). 1st ½AðxÞ ¼ AðxÞ (4) Thus, the member values of VRSðxÞ and VLSðxÞ can be written by Eqs. (5) and (6). VRSðxÞ¼ Pðx þ 1; yÞ; Pðx þ 2; yÞ; ::; Pðx þ N ; yÞ (5) VLSðxÞ¼ Pðx 1; yÞ; Pðx 2; yÞ; ::; Pðx N ; yÞ (6) N is an optional constant value that can be changed in every order. While the sum of VRSðxÞ, defined as S can be calculated by Eq. (7). S ¼ Pðx þ kÞ (7) k¼1 and the sum of VLSðxÞ, defined as S which can be calculated by Eq. (8). S ¼ Pðx kÞ (8) k¼1 Having obtained the value of Integral it can be obtained deferential value. The differential value ðDÞ is obtained by calculating the difference of the original pixel value with the integral value found in Eq. (9). Figure 1. Target signal in Akamatsu Transform. 1st 1st Joint method ½DðxÞ ¼ PðxÞ ½AðxÞ (9) for image In previous research, the Akamatsu transformation was calculated in the x-direction. But in restoration this research Akamatsu is also applied in the y-direction. Akamatsu transforms applied in y-direction also divides the signal into two parts corresponding to the target signal Pðx; yÞ. The section located above the Pðx; yÞ is called VTSðyÞ, which can be calculated by Eq. (10) and the part that is under the Pðx; yÞ is called VBSðyÞ, which can be calculated by Eq. (11). VTSðyÞ¼ Pðx; y þ 1Þ; Pðx; y þ 2Þ;: :; Pðx; y þ NÞ (10) VBSðyÞ¼ Pðx; y 1Þ; Pðx; y 2Þ;: :; Pðx; y NÞ (11) Similar to the x-direction Akamatsu transform, the integral value of AðyÞ is obtained by Eq. (12). ½AvrS ðyÞþ AvrS ðyÞ T B AðyÞ¼ (12) With AvrS ðyÞ and AvrS ðyÞ derived from Eqs (13) and (14). T B S ðyÞ AvrS ðyÞ¼ (13) S ðyÞ AvrS ðyÞ¼ (14) With the first known order from Akamatsu transform then it can be continued to the next order with the above equation as many as n order, so that is obtained Eq. (15) to calculate Akamatsu transform. 1st ½AðxÞ ¼ AðxÞ 1st 1st ½DðxÞ ¼ PðxÞ½AðxÞ (15) nth ½AðxÞ ¼ AðxÞ nth nth ½DðxÞ ¼ PðxÞ ½AðxÞ After obtained the integral and differential values, the new pixel value for x-direction is obtained by Eq. (16). P ðx; yÞ¼ up enh 3 DðxÞþ down enh 3 AðxÞ (16) new As for y-direction, a new pixel value is obtained by Eq. (17). P ðx; yÞ¼ up enh 3 DðyÞþ down enh 3 AðyÞ (17) new where up enh is an increasing value to increase the value of the differential waveform to obtain a clearer image. While down enhis increasing value to decrease the integral waveform value so that the new pixel value does not exceed the 255 limit value. 2.2 Discrete wavelet transform The use of wavelet transforms has been widely used in digital image processing and pattern recognition [20]. This technology also takes an important role in digital image processing. Decomposition using wavelet transform produces two-dimensional functions of time and space. Wavelet decomposition is also helpful in recognizing singularity details by analyzing the time frequencies of non-stationary signals [21–23]. Furthermore, outside there have been many forms of developed wavelet transforms and each has different characteristics. The ACI discrete wavelet transform is one form of the wavelet transform, such as complex wavelet 19,3/4 transform (CWT), Slantlet transforms (SLT) [23,24]. DWT has many good features such as multi-resolution capabilities and space-frequency localization property that makes DWT can be applied to the entire image and can be reconstructed in several image size [25]. In DWT, images are divided into 4 sub-bands: Low-Low (LL), Low-High (LH), High-Low (HL), High- High (HH). The subband distribution process is carried out with two kinds of filters namely low pass and high pass filter. There are various kinds of filters that can be used such as Daubechies and Haar which are the most popular filters [26–28]. Haar filters have the advantage of being simple in structure so that they can reduce memory usage and are preferred for analyzing compact discrete signals [27]. Each sub-band holds important information about the image. LL is representative of the image and is the sub-band most similar to the original image before it is decomposed. LH, HL, and HH are wavelets of variations vertically, horizontally, and diagonally [6,13,21,25]. Decomposition on DWT can be done using Eqs. (18) and (19)[29]. j j j Φ ðx; yÞ¼ 2 w 2 m; 2 m; 2 y n (18) j;m;n i j j j Ψ ðx; yÞ¼ 2 2 m; 2 m; 2 y n ; i ¼ fH ; V ; Dg (19) j;m;n where: i 5 wavelet {HL,LH,HH}, m; n 5 size of image Figure 2 is a description of sub-bands decomposition proccess using in DWT using filter Haar.where 2↓1: downsample columns and 1↓2: downsample rows. Figure 2 shows the decomposition process at the first level. In its development, decomposition is also done on several levels to get the most appropriate results. But in this research decomposition is only done at one level. Where later each subband will be carried out further different processes, according to the characteristics of each subband. While reconstruction is done by inverse DWT using Eq. (20). XX f ðx; yÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ w ðj ; m; nÞw ðx; yÞ w 0 j ;m;n MN m n X X XX i i þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 w ðj ; m; nÞΨ ðx; yÞ (20) ψ j ;m;n MN i¼H ;V Dj¼jo m n 3. Proposed restoration scheme The proposed restoration scheme in this study is to combine DWT and Akamatsu on the input image. The input image is an image that has been given manipulations such as Figure 2. Sub-bands decomposition in DWT using filter Haar. Gaussian noise, salt and pepper, and blurring. To be able to see more clearly the stages of the Joint method proposed scheme could see Figure 3. for image Here is a detailed stage of the proposed restoration process in Figure 3: restoration 1. Decompose the input image using DWT to get 4 subband LL, HL, LH, and HH. 2. Select subband HL, LH, and HH to apply Akamatsu transformation. 3. Apply y-direction Akamatsu transform on subband LH to get a new LH subband. 4. Apply x-direction Akamatsu transform on subband HL to get a new HL subband. 5. Apply two Akamatsu transforms on the HH subband. First, apply Akamatsu transform x-direction and the second Akamatsu transform y-direction. To obtain a new HH subband get the average value of both Akamatsu transforms with Eq. (21). ðHH þ HH Þ x−direction y−direction HH ¼ (21) new 6. Perform a DWT inverse to get the image of the restoration. 4. Implementation and results In this study, the proposed method will be simulated using five grayscale images with 512x512 size. This image is a standard image that can be downloaded on the internet. After the image is downloaded, the image is immediately used without preprocessing. In this study, the whole trial process uses Matlab R2015a software. The image used is shown in Figure 4. To test the proposed method, three kinds of manipulations are a blur, Gaussian noise, and salt and pepper using Matlab function. Figure 5 shows a sample that has been given manipulation. Peak to Signal Noise Ratio (PSNR) is used to determine the degree of damage to the noised image. The value of PSNR is obtained by comparing the original image and the noised image. Equation (22) is used to calculate PSNR. Figure 3. The flow of Proposed Restoration Scheme. ACI 19,3/4 Figure 4. Original Image Used {(a) Babbon; (b) Barbara; (c) Cameraman; (d) Goldhill; (e) Lena}. Figure 5. Noise Cameraman Image {(a) Blur; (b) Gaussian Noise 0.01; (c) Salt and Pepper 0.01}. PSNR ¼ 10 log (22) P P H−1 G−1 O ðh; gÞ N ðh; gÞ i i h¼1 g¼1 where: h; g is the size of the image, O is an original image, and N is a noised image. i i Table 1 shows the PSNR value of the original image after given three kinds of noise models. Furthermore, the restoration of the noised image with the proposed method. The proposed method combines two methods, namely wavelet transform with Haar filter and then transformed again with Akamatsu transform. Wavelet transform is performed to get four subbands, where three of the four subbands are further processed using Akamatsu transform. The processing of the Akamatsu transformation in each DWT subband is adjusted to the characteristics of each subband to optimized restoration results. Perform the Joint method Image Noise Type PSNR for image Lena Gaussian 20.0750 restoration Salt & Pepper 25.3945 Blur 27.4355 Barbara Gaussian 20.0847 Salt & Pepper 25.4396 Blur 23.5422 Baboon Gaussian 20.0327 Salt & Pepper 25.6330 Blur 20.6008 Cameraman Gaussian 20.3898 Salt & Pepper 25.0744 Blur 26.1126 Goldhill Gaussian 20.1134 Table 1. Salt & Pepper 25.2362 PSNR of the Blur 26.7294 noise image. Akamatsu y-direction transform on the LH subband, the x-direction on the HL subband and the xy-direction on the HH subband. This is done because the LH subband has a middle frequency horizontally, HL has a vertically middle frequency and HH has a diagonal high frequency [6,13,21,25]. Table 2 shows the results and PSNR values of the reconstructed images from the proposed restoration method. 5. Comparative and analysis To evaluate the performance of the proposed method, the restoration of the proposed method were compared with the results of method restoration on the research of Karungaru et al. [5] and Megha et al. [6]. In this research, both methods have been replicated to compare the results of image restoration. This is done because the data set of images used are different, so it needs to be replicated so that it can be compared with the same image dataset. Comparative measurements were made with PSNR shown in Table 3. As can be seen in Table 3, the PSNR value of the proposed method of restoration appears better than the two previous methods. Visually the difference of image result of restoration can be seen in Figures 6–8. Based on the PSNR values contained in Table 3, proves that the results of image restoration by the proposed method appear superior to the previous method. Visually, Figures 6–8 also show that the proposed method appears to be superior. Only, in the blur, the results of restoration by the method [5] appear more clear and sharp when compared with the proposed method. However, the result of restoration on the method [6] creates new noise in the form of lines on the edge of the image, as shown in Figure 6(a). 6. Conclusions Based on the results of the analysis and comparison, it can be concluded that the proposed method has better restoration compared to the method [5] and method [6] in all types of manipulations. The restoration of Gaussian noise and salt and pepper manipulations also reduce and smooth the noise generated. While the restoration using the method [5] even causes a blur effect. This happens because the Akamatsu transformation has characteristics that can change the image information by changing the differential value with the down enh value. While the proposed method applies changes to this information on sub-band HL, LH, ACI 19,3/4 Table 2. Reconstructed results with PSNR values using the proposed method. Joint method Image Noise type Proposed method The method in [5] The method in [6] for image Lena Gaussian 24.9357 23.4659 17.0212 restoration Salt & Pepper 28.9584 25.9581 19.0746 Blur 27.4395 26.3382 19.2786 Barbara Gaussian 23.2848 23.0617 18.0083 Salt & Pepper 25.5292 25.1769 20.3796 Blur 23.5623 23.0914 19.6397 Baboon Gaussian 22.2379 22.1477 17.1517 Salt & Pepper 24.0013 23.8401 19.2273 Blur 20.5861 20.4113 17.5664 Cameraman Gaussian 25.3617 22.8571 17.2854 Table 3. Salt & Pepper 29.2486 24.7714 18.6267 Comparison of PSNR Blur 26.1307 25.3826 18.8358 value proposed Goldhill Gaussian 24.7563 23.2829 17.1602 method, the method Salt & Pepper 28.4402 25.4686 18.7861 in [5] and the method Blur 26.7336 25.6285 18.6463 in [6]. Figure 6. Reconstructed Cameraman Image from Blur manipulations {(a) Method in [5]; (b) Method in [6]; (c) Proposed Method}. Figure 7. Reconstructed Cameraman Image from Gaussian Noise manipulations {(a) Method in [5]; (b) Method in [6]; (c) Proposed Method}. and HH. So after the inverse DWT, the original image characteristics did not change significantly but can eliminate the noise that occurs in the image. Method [6] has a different character, although noise is reduced, the energy in the image of the restoration in the image is also reduced so that the image appears darker and the resulting PSNR value is less good. So it can be concluded that the proposed method can work better on various manipulations and can maintain the value of image approximation. In future studies, several wavelet filters can ACI 19,3/4 Figure 8. Reconstructed Cameraman Image from Salt and Peppers manipulations {(a) Method in [5]; (b) Method in [6]; (c) Proposed Method}. be used, as well as improved wavelet transforms and wavelet transform levels to analyze the results so that more optimal restoration is obtained. 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Corresponding author De Rosal Ignatius Moses Setiadi can be contacted at: moses@dsn.dinus.ac.id For instructions on how to order reprints of this article, please visit our website: www.emeraldgrouppublishing.com/licensing/reprints.htm Or contact us for further details: permissions@emeraldinsight.com
Applied Computing and Informatics – Emerald Publishing
Published: Jun 9, 2023
Keywords: Akamatsu transform; Blurred image; Discrete wavelet transform; Image Denoising; Image restoration
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