International Journal of Advance Engineering and Research Development. Comparative analysis of Wavelet and Curvelet based Image Compression

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1 Scientific Journal of Impact Factor (SJIF): 4.14 International Journal of Advance Engineering and Research Development Volume 3, Issue 3, March e-issn (O): p-issn (P): Comparative analysis of Wavelet and Curvelet based Image Compression NIKUNJ SHINGALA #1, TEJAL TANDEL *2 DEVEN TRIVEDI *3 SUNAYANA DOMADIA *4 #1,#2,#3,#4 Assistant Professor, Electronic and Communication Department (EC), Madhuben And Bhanubhai Patel Institute Of Engineering, For Studies And Research In Computer And Communication Technology., New V.V. Nagar, India. Abstract: The paper includes comparative analysis of wavelet and curvelet based image compression. Brief introduction about curvelet and Wavelet transform is also included. Wavelet is proven most powerful tool for image compression as compared to DCT (Discrete Cosine Transform) but it has got certain limitation when it is handling line and curve singularities in the image. The solution is Curvelet Transformwhich is able to handle image with line and curves. Comparasion is done on the basis of Compression Ratio and PSNR. Keywords Image compression, Curvelet transform, Wavelet Transform 1. INTRODUCTION Image compression is the application of data compression on digital image. In this era, because of internet and multimedia sizes of images and video is increasing rapidly. For example, a standard 35-mm photograph digitized at 12μm per pixel requires about 18Mbytes of storage and one second of NTSC-quality color video requires 23 Mbytes of storage [1]. Compression of the data is worthy because the data whichis uncompressed such as audio, video and graphics requireslarge transmission bandwidth and storage capacity. Imagecompression is also a type of compression technique whichis used to reduce the size of graphics in terms of bytes toinhibit an acceptable quality of an image [2].So for efficient transmission bandwidth it needs to be compressed. Many compression standards have been developed in last two decades which is used for image compression. But all of them have few limitations. So day by day new standard has been developed. The use of image is becoming every day more popular in our society; remote sensing centers have currently more and more available images of the earth surface, the medical community is already used to diagnosis with images, in our everyday lives digital cameras with resolution are becoming usual. It is well known that sizes of images are becoming large. So images are needed to be compressed before it is transmitted via channel for efficient bandwidth transmission so that the storage space and transmission time can be reduced. Digital image is compressed by removing redundancy present in the image. Different type of compression technique like Scalar/Vector quantization, Differential Coding, transform coding are used [3]. Transform coding provides higher compression with quality reconstruction. One of the transformation is wavelet transform. The introduction of wavelets gave a different dimension to the compression. It is proven better than discrete cosine transform DCT. But there are some limitations of wavelets while handling the line and curve singularities in the image. There are some new version beyond wavelets namely Curvelet and Ridgelet Transforms. Compression are basically of two types 1.Lossy compression 2. lossless compression (1)In lossless compression, during compression every bit of information is preserved. The image recovered from compressed image is exact replica of original image. The compression ratio achieved with this scheme is low. It is used in applications where no loss of image data can be compromised.(2)in lossy compression, some information is not recovered. a perfect reconstruction of the image is sacrificed by the elimination of some amount of redundancies in the image to achieve higher compression ratio. loss of information is not perceived under normal viewing conditions. In this paper, lossy image compression using two transforms- Curvelet and Wavelet is discussed. Another application of image compression is in the medical imaging. Medical images such as MRI, CT scan images are large in size. This kind of image has dark background so it can be compressed with higher compression ratio There are some parameter that is associated with the the image compression is PSNE and MSE. A general lossy image encoder system consists of several operations as shown in Figure All rights Reserved 215

2 Figure-1 Image Encoder The transform operation aims to reduce the entropy of the image. This is reversible operation and does not cause any loss of information to the image. An examples of such transform are Discrete cosine transform, wavelet transform and multiscale curvelet transform. Form the figure-1 it can be seen that first operation is quantization it is a lossy operation. it maps a large set of input image data to a smaller set of output image data, attempting to remove redundancies in the image. The quantization process is irreversible and it introduces distortion because of rounding off the sample values. There are two types of quantization scalar and Vector quantization. The next step is entropy coding operation which further compresses image without any information loss. In entropy coding, average number of bits required to represent a symbol is reduced by assigning a longer codeword to a less probable symbol and a shorter codeword to a most likely symbol. Example of entropy coding are Huffman coding, Run Length coding and Arithmetic coding.finally after entropy coding compresed image is achieved which is used for transmission in raw data format. At the receiver reverse procedure is followed. To reconstruct the image, we will reverse the three stages as shown in Figure 2. At each stage, an inverse operation will be carried out. The image decoder is shown below. Figure-2 Image Decoder 2. WAVELET TRANSFORM (WT) Many transforms have been developed day by day but they all have certain disadvantages. Fourier Transform has sinusoid as its basis function. This function does not have limited time duration. When a small change occurs in the signal in the time domain, it will affect all the components in the frequency domain. Therefore, it is not suitable for analyzing the non-stationary signal using Fourier Transform [4]. Discrete cosine transform is similar to the discrete Fourier transform, it transforms a signal or image in the spatial domain to the frequency domain and obtains DCT coefficients which can be used in various image processing purpose. It is used in JPEG. It produces blocking effect in image and less compression ratio. To overcome the limitation of the FT, a window-version of Fourier Transform known as Short Time Fourier Transform (STFT) was developed [1]. In STFT, we can divide the non-stationary signal into small segments where each segment of the signal is assumed to be stationary. But we will have a resolution problem here. Once the size of the STFT s window is chosen, the time-frequency resolution is fixed for the entire time-frequency plane. This inequality means that we have to trade off time resolution for frequency resolution and vice-versa. That is, if we want a good frequency resolution, we have to settle for poor time resolution. Likewise, if we want a good time resolution, we have to settle for poor frequency resolution. Wavelet transform has been used in recent years as a promising tool for image compression and noise reduction. It has a set basis functions that can be used to analyze signals in both time and frequency domains simultaneously. This analysis is accomplished by the use of a scalable window to cover the time-frequency plane, providing a convenient means for the analyzing of non-stationary signal that is often found in most applications. Wavelet has infinite functions that can be used for many applications. Discrete wavelet transform (DWT) is one of the most promising multi resolution approaches used in wavelet based image compression [6]. Wavelet transform cuts up the data or function into different frequency components to study each component with resolution matched to its scale due to better frequency and time localization. Wavelets have become main tool for All rights Reserved 216

3 processing as process of creating edge sub-images at resolutions are analogous to a process performed by mammalian vision system including human visual system(hvs). DWT has a number of advantages over other transforms (DFT or DCT) e.g. progressive and low bit rate transmission, quality scalability and region of interest (ROI) coding demand more efficient and versatile image. The decomposition of an image using DWT involves a pair of waveform, one for high frequencies corresponding to detail part of image (wavelet function ( (t)) and another for low frequencies or smoother part (scaling function (ϕ(t)). wavelet function ( (t)) is high pass filter and allows high frequencies components of signal and HVS is less sensitive to it. Scaling function(ϕ (t)) is a low pass filter that allows low frequencies and sensitive to HVS[5]. Wavelet decomposition of image is shown in figure-3. It decomposes image in several bands. Figure-3 wavelet decomposition of image Discrete wavelet transform(dwt) is applied on the original image. Then it does high pass filtering which yields three detail images, describing the local changes in horizontal, vertical and diagonal direction of the original image. Then DWT does low pass filtering which provides an approximated image[6]. This image is again filtered in the same manner to generate high and low frequency sub bands at the next lower resolution level. This process is continued until the whole image is processed or a level is determined as the lowest to stop decomposition. This continuing decomposition process is known as down sampling and shown in Fig 3. The whole decomposition process provides us with an array of DWT coefficients obtained from each subbands at each scale. These coefficients can then be used to analyse the texture patterns of an image. Figure -4 shows first level decomposition of Lena image. After first decomposition we get four sub band LL, LH, HL,HH. The first letter in the subband name identifies the filter that has been applied horizontally, the second letter identifies the vertical filtering, and the number identifies the decomposition level. Figure-4 First level decomposition of Lena image. LL subband is futher decomposed to achieve level-2 decomposition. Level-1 and Level-2 decomposition is shown in All rights Reserved 217

4 Figure-5 Level-1 and level-2 decomposition of Lena image obtained using MATLAB. 3. CURVELET TRANSFORM (CT) Wavelet transform has some limitation when it is handling an image with line and curve singularities. To overcome the problems associated with wavelets, Candes,E. and Donoho,D.(1999) introduced new multiscale systems like curvelets, which is very different from Wavelet-like systems[7 8]. Curvelets take the form of basis elements which exhibit very high directional sensitivity and is highly anisotropic. The curvelet transform is a new family of multi-scale representation containing the information about the scale and location parameters. Unlike wavelets, it also contains the directional parameters. Though wavelet transform has been explored widely in various branches of image processing, it fails to represent objects containing randomly oriented edges and curves as it is not good at representing line singularities. The orientation selectivity behavior and anisotropic nature of the curvelet transform helps to represent suitably the objects with curves and handles other two-dimensional singularities better than wavelets, which makes it a more proficient transformation for image compression application[8]. So, Curvelet transform overcomes the limitations of wavelet. As we have seen Wavelet provides three directional parameters like Horizontal, Verticla, Diagonal. But curvelet provides multi-directional parameters which is shown in figure 6 [9]. Figure-6 5-Level curvelet decomposition Curvelet Decomposition The procedural definition of Curvelet transform for an object X (image in our case) is actually a combination of four ideas, which are briefly reviewed as under [10]: Subband Decomposition: Divide image X into several resolution layers and each layer contains details of different frequencies X ( P0X, Δ 1 X, Δ 2 X, ) Here P0 and Δs (where s 0) are low pass and high pass filters respectively. P 0 f is the smooth low-pass layer that efficiently represented by using wavelet base and ΔsX are high-pass layers effected by discontinuity curves. At the end, each ΔsX layer contains objects near high frequencies with fine details Smooth Partitioning:To represent high-pass layers ΔsX efficiently, dissect layers into small partitions by definingsmooth windows WQ(y1, y2) localized around dyadic squares. This windowing is a nonnegative smooth function and creates ridges of width = 2-2s and length =2 s so that width leangth 2. Multiplication of ΔsX with WQ produces a smooth dissection (hq) of function into squares. The mathematical form is as follows: hq = W Q. ΔsX 4.7 Renormalization:each square from previous stage is renormalized into the unit square. Ridgelet Analysis: each normalized square analysed with discrete ridgelet transform that in turn produce curvelet coefficients. Block of image decomposition using curvelet transformsis shown All rights Reserved 218

5 P 0 C 0 I 256x256 Δ1 Tilling 32 Ridgelet Transform On till C 1 Δ 2 Tilling 32 Ridgelet C 2 Transform on till Figure-7: Block of image decomposition using Curvelet transforms 4. WAVELET BASED COMPRESSION ALGORITHM Step-1: Different Types of images are taken as input in MATLAB. Medical images are also taken as input. Step-2: Discrete Wavelet Transform (DWT) is applied to inputimage. Step-3: Decomposition Level is selected Step-4: Thresholding technique is applied. Here Different threshold value is selected. The coefficient of subbands above threshold are retained and other are set to zero Step-5: Finally Entropy coding is done. Step-6: From encoded bit stream measure the Compression Ratio. Image is reconstructed using reverse procedure. Then from the Reconstructed image PSNR is measured. 5. CURVELET BASED COMPRESSION ALGORITHM Step-1: Different class of image are taken as input. Step-2: Unequally spaced fast Fourier transform based Fast discrete Curvelet transform (fdct_usfft.m) is applied to image. 4-level image decomposition is done. Step-3: Soft Thresholding technique is applied to the decomposed image. Threshold function is applied to only detail and fine level cell to maintain the quality of image. It is not applied to coarse scale coefficient. Step-4: Finally Entropy coding is done and Compression Ratio is measure from encoded bit Stream Step-5: The image is reconstructed by doing reverse procedure Step-6: PSNR is measured from the reconstructed image. 6. SIMULATION AND EXPERIMENTAL RESULT Algorithms are implemented using MATLAB. For wavelet, wavelet toolbox is used and for curvelet, CurveLab tool box is used[11]. CurveLab is a collection of Matlab and C++ programs for the Fast Discrete Curvelet Transform in two and three dimensions. For the 2D curvelet transform, the software package includes two distinct implementations: the wrapping-based transform and the transform using unequally-spaced fast Fourier transform (USFFT). Curve Lab tar file is downloaded from website[11]. It is then expanded in one of the local drive. Different types of images are taken for experimental purpose like lena.bmp, NPH_MRI_276.bmp. Each image is of size 512x512. Comparative analysis is done on the basis of Compression Ratio, the Root Mean Square Error (RMSE) and Peak Signalto-noise Ratio (PSNR). At same compression ratio Value of PSNR and MSE is measured. The PSNR is the most commonly used as a measure of quality of reconstruction in image compression.. Mean Square Error (MSE) which requires two m x n grey-scale images I and K where one of the images is considered as approximation of the other is defined as:. m 1,n 1 MSE = 1 I i, j K(i, j) 2 mn i,j =0 The PSNR for both original and reconstructed images were identified using the following formulae: PSNR = 20 log MAXi 10 MSE Here, MAXI is the maximum pixel value of the image. The PSNR of the reconstructed image is compared for wavelet and Curvelet transform for same compression ratio. Compression Ratio is defined as All rights Reserved 219

6 M N 8 CR = C Where M and N are the size of image. C is length of compressed bit stream. Compressed image is represented as bit stream. Original lena image and its decomposition using wavelet and curvelet is shown in figure-8. (a) (b) (c) Figure-8 (a) original Lena image (b) Level-4 Decomposition using Wavelet (c) Curvelet Decomposition. Compression results of lena.bmp image for wavelet and curvelet transform are shown in figure 9. a. Wavelet(6.72:1) b. All rights Reserved 220

7 c. Wavelet(7.52:1) d. Curvelet(7.52:1) Figure-9 Lena image at Different Compression Ratio using Wavelet and Curvelet Compression results of NPH_MRI_276.bmp image for wavelet and curvelet transform are shown in figure 10. a. Original image b. Wavelet(7.69:1) c. Curvelet(7.69:1) Figure-10 NPH_MRI_276.BMP image at different compression ratio using Wavelet and Curvelet transform. PSNR Results for same compression using wavelet and Curvelet for Lena and NPH_MRI_276.BMP is shown in Table-1. Table-1 PSNR Results for same compression using wavelet and Curvelet for Lena.bmpand NPH_MRI_276.bmp Name of Compression PSNR(dB) image Ratio Wavelet Curvelet Lena.bmp NPH_MRI_ 276.bmp CONCLUSION Wavelet is proven promising tool for image compression and image denoising. From figure-9 we can see at compression ratio 6.72 the quality of reconstructed image for both curvelet and wavelet is good but at compression ratio 7.52 for lena image, visual inspection of reconstructed image show that Curvelet transform is superior over wavelet transform. From figure-10 we can see at compression ratio 7.69 for medical image the quality of reconstructed image using curvelet All rights Reserved 221

8 good as compared to the wavelet. For medical image, at compression ratio 7.69 the PSNR for wavelet is and curvelet is Wavelet is not able to handle the image with line and curve singularities. For NPH_MRI_276.bmp image curvelet provides good compression ratio with good PSNR value. So from above discussion it is proven that Curvelet transform is better as compared to wavelet transform for certain class of image. 8. REFERENCES [1] M L Hilton, B D Jawerth and Ayan Sengupta, Compressing still and moving images with wavelets published in the journal Multimedia Systems vol 2 no-3,1994. [2] K. S. Solanki and Nitu Senani A Survey on Compression of an Image Using Wavelet Transform International Journal of Computer Science and Information Technologies, Vol. 6 (4), 2015, ,ISSN [3] Choo Li Tan Still Image Compression Using Wavelet Transform Thesis submitted to university of queensland [4] Robi Polikar; Wavelet Tutorials. [5] Gurparkash Singh Kang Blind Digital Image Watermarking Using Adpative Casting Energy In Different Resolutions Of Wavelet Transform, Int l Conf. on Computer & Communication Technology[ICCCT-10] [6] Amir Averbuch, Danny Lazar, and Moshe Israeli,"Image Compression Using Wavelet Transform and Multiresolution Decomposition"IEEE Trans. on Image Processing, Vol. 5, No-1, JANUARY [7] E.J. Candes, Laurent Demanet, D.L. Donoho, Lexing Ying, Fast Discrete Curvelet Transforms, Stanford University Press Notes, July [8] E. J. Cand`es and D. L. Donoho. Curvelets a surprisingly effective nonadaptive representation for objects with edges. In C. Rabut A. Cohen and L. L. Schumaker, editors, Curves and Surfaces, pages , Vanderbilt University Press, Nashville, TN. [9]Isharat Sumana Image Retrieval Using Discrete Curvelet Transform Thesis submitted to Monash University, Australia November, 2008 [10] Muhammad Azhar Iqbal, Dr Muhammad Younus Javed, Usman Qayyum Curvelet-based Image Compression with SPIHT 2007 International Conference on Convergence Information Technology [11] All rights Reserved 222