# I. Introduction esolution has been always an important property in images and videos. High resolution (HR) image/video has a desired and strong demand in most imaging applications as it contains more details that can be crucial in these applications [1]. Resolution enhancement based on a single low-resolution (LR) image or multiple LR images has been used for different applications in various fields, such as satellite imaging [2]-[5], medical imaging [6], [7], and video enhancement [8]- [10]. Interpolation is one of the most commonly used techniques for increasing the resolution of a digital image [11]- [13]. There are four well-known interpolation methods, namely, nearest neighbor, bilinear, bicubic, and Lanczos. Nearest neighbor interpolation is the simplest method where the intensity of the new location point is assigned as that of the old location point which is the nearest neighbor to the new point. Although it is simple to implement, it produces undesirable artefacts, such as distortion of straight edges. In the bilinear interpolation, the value of a new pixel is interpolated linearly using the four nearest neighbour pixels by taking Resolution enhancement techniques in the wavelet domain have attracted more and more investigations to address the problems associated with conventional interpolation methods. Wavelet-Zero Padding (WZP) is relatively simple to implement and is capable of outperforming the conventional interpolation methods but it commonly introduces artefacts such as smoothing and ringing in the neighbourhood of edges in the reconstructed HR image. Addressing this problem, a Cycle-Spinning (CS) based WZP method was proposed [17]. Hidden Markov Tree (HMT) based resolution enhancement method is capable of modelling the statistical relationships between coefficients at different scales [18], but the main drawback is that the used Gaussian model does not take into account to keep track of the sign coefficients since the Gaussian is symmetrical around zero and the signs of these coefficients are randomly generated. To reduce this shortcoming, a refined HMT based method was proposed in [19], where the magnitude parameters are estimated using the HMT model, and the sign parameters are estimated based on a higher correlation among the parameters between a high-pass filtered version of the LR image and the high-frequency subbands. A Directional Cycle-Spinning (DCS) method was introduced in [20], where approximates of edge orientation information are derived from a wavelet decomposition of the LR image and used to affect the choice of CS parameters. It can refine better edge orientation and prevent staircase artefacts. More recently, a new dual-tree complex wavelet transform (DT-CWT) technique [4] # ( ) F a weighted average of these pixels [14]. Bicubic interpolation preserves fine details better and is more complex than bilinear interpolation where sixteen nearest neighbour pixels are used to estimate the value of the new pixel by taking a weighted average of these points. This method is more efficient and accurate and has become the most popular image interpolation method [15]. Lanczos interpolation increases the capability to detect linear features [16]. However, the main drawback of most interpolation-based methods is that the produced images suffer from blurring and staircase artefacts. improves Wasnaa Witwit ? , Yitian Zhao ? , Karl Jenkins ? & Yifan Zhao ? method, the high-frequency coefficients produced by CWT and the input image are interpolated using the Lanczos interpolation. A Demirel-Anbarjafari Super Resolution (DASR) method [21] was proposed based on Discrete Wavelet Transform (DWT), where three highfrequency components produced by DWT as well as the input image are interpolated using the bicubic interpolation. An updated DASR technique was proposed in [2] with its application in satellite images. Although DWT has been used to preserve the highfrequency details of the image, but downsampling in each of the DWT sub-bands and then the interpolation of the high-frequency sub-bands generate information loss in each of these sub-bands. More recently, a technique based on DWT and stationary wavelet transform (SWT) [22] was proposed to correct the estimated high-frequency sub-bands produced using DWT by adding the high-frequency sub-bands obtained by using SWT. that the assumptions they make are not always satisfied for real applications. For example, the detail of a physical object that an optical instrument can reproduce in an image has limits that are mandated by laws of physics, whether formulated by the diffraction equations in the wave theory of light or the Uncertainty Principle for photons in quantum mechanics. There is no such a wellaccepted model can fully describe the underlying mechanism. This mechanism can also be various case by case. In other words, the superior of one method than other methods claimed in the literatures is conditional. Although it has been reported that the performance of resolution enhancement methods can be affected by the methods to produce the lowresolution image, and other factors [16], there is very limited literatures investigating how to utilise these factors to assess and improve the resolution enhancement performance. Addressing this problem, this paper proposes an Optimal Factor Analysis method to increase the applicability and fidelity of the existing methods. Although the authors are aware that machinelearning-based super-resolution methods have attracted more and more interests recently [23]- [28], this paper focuses on wavelet-based methods and interpolation methods only. Section 2 initially identifies the important factors that affect the performance and analyses corresponding importance, and then proposes the new method as well as a new Figure of Merit to assist the selection of parameters. Section 3 presents the results of quantitative analysis using the proposed method and associated discussions. Conclusions are presented in the final section. # II. Method a) Important Factors Table 1 summarises the reviewed waveletbased image resolution enhancement techniques in terms of the way to evaluate their performance. The inconsistency of assumption of the considered factors for each individual technique has been observed. For example, the considered methods make the assumption that the observed LR image is produced by either applying a low-pass filtering and then downsampling, or achieving the low-frequency (LL) sub-band of DWT. For some methods, the description of these factors is either neglected or unclear. The performance of these methods is unknown when such an assumption is not satisfied. A method to compare the resolution enhancement methods in a more comprehensive # Global Journal of Computer Science and Technology Volume XVI Issue III Version I and equitable way is required. Such a method can also be used to further improve the overall performance of existing methods. Each potential factor that affects the performance has been studied one by one. In order to quantitatively evaluate the performance, the widely used Peak-signal-to-noise-ratio (PSNR) has been employed in this paper, and it can be calculated by (1) where ?? denotes the maximum fluctuation in the input image. Mean Square Error (MSE) measures the error between the super resolved image ?????? and the original HR image ??????ð??"ð??". It can be calculated by (2) where ??,?? denote the width and height of the HR image respectively. # i. The mechanism to produce low-resolution images It has been identified from the literature review that there are various ways to generate LR image including (a) downsampling of the original HR image through DWT, (b) bicubic interpolation, (c) bilinear interpolation, (d) nearest neighbour, and (e) low-pass filtering. Table 2 shows the resulting PSNR values for the Lena image using different resolution enhancement methods by considering different LR image generation methods. Inspection of Table 2 shows that WZP with the wavelet function db.9/7 has the best performance among the considered ???? = 10 log 10 ( ii. Wavelet Function There are several well-known wavelet families such as Daubechies (db) (db.1 is also referred as Haar), Symlets (sym), Biorthogonal (bior), Coiflets (coif) etc [29]. In this paper, the behaviour of the considered resolution enhancement techniques has been studied for a wide range of wavelet families as well as their various parameters, including db.1-20, sym.2-20, bior.1-6 and coif.1-5. Note that db.9/7 is equivalent to bior4.4 [30]. Table 3 shows the PSNR values for three test images (Lena, Baboon, and Elaine) using the WZP method with various wavelet functions, where only the parameters producing high PSNR values are shown to save space. The input LR image was produced by downsampling using DWT with db.9/7 wavelet function as suggested in [21]. The quantitative results show that considered methods decreases following the increase of scale factor, which indicates that the scale factor is an important factor to be considered for performance assessment. # iv. Interpolation Function Because of the obvious weakness, in this paper, the nearest method has been neglected, and bilinear, bicubic and Lanczos have been tested. The input LR image has been produced by downsampling using DWT with db.9/7 wavelet function. The PSNR results for Lena image are shown in Table 5, inspection of which indicates that there is no significant difference in performance for different interpolation methods. Moreover, the interpolation method producing the highest PSNR is not consistent for different methods. These observations indicate that the selection of interpolation function for wavelet-based techniques can affect the performance, but not significantly. # b) Optimal Factor Analysis The behaviour of resolution enhancement methods has been assessed above by varying one factor and fixing other factors, which aims to identify the important factors but it cannot reveal the best technique with the optimal parameter selection. Addressing this challenge, this paper proposes an Optimal Factors Analysis (OFA) approach in order to increase the performance of the existing methods, and also better assess their overall performance. OFA considers a resolution enhancement technique, ?, as a Multi-Input and Multi-Output (MIMO) model, which includes 5 inputs variables: the way to produce LR image ??????(??=1,2,?,??), the scale factor ???? ?? (??=1,2,?,??), the testing image ???? ?? (??=1,2,..,????), the (5) WZP+sym, WZP+coif, and WZP+bior). Three interpolation methods (??=3) were considered, namely bilinear, bicubic and Lanczos. Fig. 1 illustrates the performance of the WZP method before and after applying the proposed method, where the LR images were super-resolved from 128×128 to 512×512. The blue and red bars plot the Table 6 shows the results including the bestperformed method with its parameter selection, as well as the highest PSNR and RPSNR value for different factors. For the LR image obtained from DWT with db. 9/7 wavelet function, the optimal class corresponding with the optimal interpolation method is WZP using "sym" with bilinear interpolation for the Lena image with scale factor 2. However, for the Baboon and the Elaine images, the best class is WZP using "bior" with bilinear interpolation. For scale factor 4 and 8, the best class with the best interpolation method is WZP using "coif" with Lanczos interpolation for all three images. For This study considered Six methods to generate scale factors 2, 4, and 8 (??=3) and Three testing PSNR values before and after applying OFA respectively. It is clearly shown that the proposed method significantly improves the performance for all 7 ways to produce LR image and all three tested images. over the interpolation methods. This justifies that for almost all papers about wavelet-based techniques, the LR image was produced by either DWT with db. 9/7 wavelet function or low-pass filtering. In order to show the sensitivity for the selection of class of technique with different scale factors, input LR image producing methods and test images, the highest PSNR value for each However, for the LR images obtained by DWT with db. 9/7, the result of sensitivity analysis is different, as illustrated in Fig. 2 (d), (e) and (f). The values of std show that the selection of class of technique has significant effect on the results, and it is almost new technique is not important. Another observation is that the above conclusions are almost independent on independent on the scale factor. In other words, the selection of scale factor to demonstrate the superior of a test images due to the fact that Fig. 2 # III. Conclusions The wavelet-based image resolution enhancement techniques have been reviewed in this paper, especially the way to assess the performance. The inconsistency of assumptions has been observed, and for some methods, the description of these assumptions is either neglected or unclear. Due to the fact that the laws of physics to generate LR images are unclear and also various case by case, the current ways to assess performance assumptions may result in a biased conclusion. The importance of each factor has then been analysed by varying this factor and fixing other factors. It has been revealed that the way of producing LR image, the variation of wavelet family and its wavelet functions, and the scale factor can substantially affect the performance of techniques. ![?? (?,?)?? ??ð??" (?,?)Analysis Approach to Improve the Wavelet-based Image Resolution Enhancement Techniques](image-2.png "") 1![Figure 1: Performance improvement for the WZP technique by applying the proposed OFA method for the scale factor of 4](image-3.png "Figure 1 :") ![has been detected and the results are shown in Fig.2. The standard deviation (std) for each scale factor has been calculated to describe the performance variation of each class. Table7shows the std values for the three test images generated by lowpass filtering and DWT with db.9/7 respectively for scalefactors 2, 4, 8 and 16. A high std value indicates that the selection of class is important because the performance for different classes of techniques is significantly varied. A low std value indicates that the performance for each class of technique is relatively similar. Fig. 2 (a), (b) and (c) illustrate the sensitivity of the class](image-4.png "") 2![Figure 2: Highest PSNR values for each class of super resolution technique for different test images and low resolution image producing ways. (a) Lena + low pass filtering; (b) Baboon + low pass filtering; (c) Elaine + low pass filtering; (d) Lena + DWT with db. 9/7; (e) Baboon + DWT with db. 9/7; Elaine + DWT with db. 9/7 (d) (e) (f)](image-5.png "Figure 2 :") ![(a), (b) and (c) have similar patterns, as well as Fig.2 (d), (e) and (f).](image-6.png "") 1Year 201612A major limitation of most above methods is)(assessmentTechniquesInput LR ImageScale FactorInterpolation MethodWavelet FunctionTest ImageWZP-CS [17]LL sub-band of DWT2 & 4N/ADb.9/7Lena, Elaine, Baboon, and PeppersWZP-DCS [20]Low-pass filtering and downsampling2 & 4N/ADb.9/7Lena, Elaine, Baboon, and PeppersHMT [18]HR image Downsampling ofN/AN/AN/ALenaHMT [5]Downsampling of HR image2N/AN/ALenaHMT [19]low-pass filtering and downsampling2 & 4N/ADb.9/7Lena, Elaine, Baboon, and PeppersCWT [3]LL sub-band of DWT2 & 4BicubicN/A5 Satellite Images 2results for Lena image using different techniques for resolution enhancement from 128×128 to 512×512 for several LR image generation methods methods for the input image produced by DWT with the wavelet function db.9/7. For the LR images obtained by DWT (Haar), bicubic, bilinear interpolations and lowpass filtering methods, the bicubic interpolation method has the best performance, but for the LR images produced by the nearest neighbor, the bilinear interpolation technique has the best performance. These observations clearly indicate that the method to produce LR images has significant effect on the performance of different techniques. 3DT-CWT [4]Downsampling of HR image4LanczocN/A1 Satellite Image (Washington DC)DASR [21]LL sub-band of DWT4BicubicDb.9/7Lena, Elaine, Baboon, and PeppersDWT-Difference [2]4BicubicDb.9/75 Satellite ImagesDWT-SWT [22]4BicubicDb.9/7Lena, Elaine, Baboon, and PeppersDWT-SWT [6]4BicubicN/ALena, Elaine, Head, and Brain 4 Year 2016151Volume XVI Issue III Version I)F(Global Journal of Computer Science and Technology 59/7 for resolution enlargement factor from? (???? ?? ,???? ?? ,???? ?? ,???? ?? ,???? ?? ) (3)Depending on the value of ??,??,??,??, and ??, Eq. 6TestingScaleMethods to produce LR imageimagefactorDWT by DB.9/7DWT by HaarBicubicBilinearNearestLow-pass2WZP(sym20)InterpolationInterpolationInterpolationWZP(sym18)WZP(sym18)BilinearLanczosLanczosLanczosBilinearBilinear32.98(1.1762)32.63(1.0221)32.48(1.0237)30.85(1.0218)31.22(1.1321)31.18(1.1233)4WZP(coif2)InterpolationInterpolationInterpolationWZP(sym18)WZP(sym18)LenaLanczosLanczosLanczosLanczosBicubicLanczos26.88(1.1945)26.56(1.0097)26.58(1.0117)25.89(1.0126)25.40(1.0244)26.45(1.0305)8WZP(coif4)InterpolationInterpolationInterpolationWZP(sym8)WZP(sym9)LanczosLanczosLanczosLanczosBilinearBicubic23.14(1.2008)23.05(1.0058)23.09(1.0078)22.64(1.0096)21.76(1.0051)22.35(1.0035)2WZP(bior4.4)InterpolationInterpolationWZP(sym13)WZP(sym6)WZP(sym18)BilinearLanczosLanczosBilinearBilinearBilinear30.09(1.0733)29.65(1.0035)29.68(1.0088)28.98(1.0100)28.09(1.0375)29.21(1.0546)4WZP(coif2)InterpolationInterpolationInterpolationWZP(sym18)WZP(sym18)BaboonLanczosLanczosLanczosLanczosBilinearBicubic26.44(1.0903)26.34(1.0028)26.40(1.0052)26.04(1.0063)25.25(1.0257)26.22(1.0119)8WZP(coif4)InterpolationInterpolationWZP(bior5.5)WZP(bior3.1)WZP(sym6)LanczosLanczosLanczosLanczosBilinearBilinear24.22(1.1063)24.06(1.0030)24.10(1.0050)23.86(1.0066)22.76(1.0219)23.49(1.0038)2WZP(bior4.4)InterpolationInterpolationInterpolationWZP(sym6)WZP(sym18)BilinearLanczosLanczosLanczosBilinearLanczos34.96(1.0824)34.54 (1.0043)34.56(1.0073)33.56(1.0084)32.71(1.0402)33.73(1.0652)4WZP(coif2)InterpolationInterpolationInterpolationWZP(sym18)WZP(sym18)ElaineLanczosLanczosLanczosLanczosBicubicLanczos30.64(1.1785)30.42 (1.0090)30.49(1.0096)29.70(1.0111)29.35(1.0259)30.39(1.0323)8WZP(coif4)InterpolationInterpolationInterpolationWZP(sym17)WZP(sym17)LanczosLanczosLanczosLanczosBicubicBicubic26.58(1.2371)26.60 (1.0121)26.63(1.0124)25.89(1.0139)25.26(1.0069)26.08(1.0073)the LR image obtained from DWT with Haar, bicubic, and bilinear, the best technique with the highest PSNR value is Lanczos interpolation for most of the cases. For the LR image produced by nearest and low-pass filtering, the best class is WZP using "sym" for almost all cases. These observations conclude that, for the LR image obtained from DWT with db. 9/7 wavelet function, nearest and low-pass filtering, the wavelet-based techniques have the biggest potential to outperform the conventional interpolation methods, due the fact that they have relatively large RPSNR values. For the LR image obtained from Haar, bicubic, and bilinear, the wavelet-based methods have no significant advantages 7 performance moderately. 19 Year 2016LenaBaboonElaineScaleLow-DWT by db.DWT by db.Low-DWT by db.FactorLow-pass)pass9/79/7pass9/7( F2 techniques have no pronounced improvements over 2.50 2.16 4 0.88 1.92 conventional interpolation methods. All these observations conclude that in order to assess more1.27 0.430.83 0.991.71 0.971.05 2.018 comprehensivelyand0.16 equitablyfor1.70 resolution0.071.030.282.2716 enhancement techniques, variation of LR image 0.03 1.40 generation method, scale factor, and wavelet functions0.030.920.041.91must be considered, otherwise observed performancecould be limited and biased. © 2016 Global Journals Inc. (US) © 2016 Global Journals Inc. (US) 1 ## IV. Acknowledgements This work was supported by the Through-life Engineering Services Centre. * Super-resolution image reconstruction: a technical overview MinKyuSung Cheol Park Moon GiPark Kang IEEE Signal Process. Mag 20 3 May 2003 * Discrete Wavelet Transform-Based Satellite Image Resolution Enhancement HDemirel GAnbarjafari IEEE Trans. Geosci. Remote Sens 49 6 1997-2004, Jun. 2011 * Wavelet domain image resolution enhancement using cycle-spinning ATemizel TVlachos Electron. 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