PSNR values in dB for 20:1 compression
Abstract
In this chapter, we propose a new lossy image compression technique that uses singular value decomposition (SVD) and wavelet difference reduction (WDR) technique followed by resolution enhancement using discrete wavelet transform (DWT) and stationary wavelet transform (SWT). The input image is decomposed into four different frequency subbands by using DWT. The lowfrequency subband is the being compressed by using DWR and in parallel the highfrequency subbands are being compressed by using SVD which reduces the rank by ignoring small singular values. The compression ratio is obtained by dividing the total number of bits required to represent the input image over the total bit numbers obtain by WDR and SVD. Reconstruction is carried out by using inverse of WDR to obtained lowfrequency subband and reconstructing the highfrequency subbands by using matrix multiplications. The highfrequency subbands are being enhanced by incorporating the highfrequency subbands obtained by applying SWT on the reconstructed lowfrequency subband. The reconstructed lowfrequency subband and enhanced highfrequency subbands are being used to generate the reconstructed image by using inverse DWT. The visual and quantitative experimental results of the proposed image compression technique are shown and also compared with those of the WDR with arithmetic coding technique and JPEG2000. From the results of the comparison, the proposed image compression technique outperforms the WDRAC and JPEG2000 techniques.
Keywords
 Lossy Image compression
 Singular Value Decomposition
 Wavelet Difference Reduction
 Stationary Wavelet Transform
 Discrete Wavelet Transform
 Image Super Resolution
1. Introduction
With the growing demand for multimedia applications, especially highdefinition images, efficient storage and transmission of images have been issues of great concern [1, 2, 3, 4]. Image processing deals with the reduction of the amount of bits used to represent an image. Not only that but also resolution of an image plays an important role in many image processing applications, such as video resolution enhancement [5], feature extraction [6], and satellite image resolution enhancement [7]. In general, there are two types of super resolution approaches, multiimage super resolution and single image. Multipleimage superresolution algorithms, like [8], [9], [10] to name a few, receive a couple of lowresolution images of the same scene as input and usually employ a registration algorithm to find the transformation between them. This transformation information is then used along with the estimated blurring parameters of the input lowresolution images, to combine them into a higherscale framework to produce a superresolved output image. For multipleimage superresolution algorithms to work properly there should be subpixels displacements between input lowresolution images. Furthermore, these subpixels displacements should be estimated properly by the registration algorithm, which is usually a challenging task, especially when complicated motion of nonrigid objects, like human body, needs to be modeled. These algorithms are guaranteed to produce proper higherresolution details; however, their improvement factors are usually limited by factors close to 2 [11].
Singleimage superresolution algorithms, like [12, 13, 14], to name a few, do not have the possibility of utilizing subpixel displacements, because they only have a single input. Instead, they employ a kind of training algorithm to learn the relationship between a set of highresolution images and their lowresolution counterparts. This learned relationship is then used to predict the missing highresolution details of the input lowresolution images. Depending on the relationship between the training low and highresolution images, these algorithms can produce highresolution images that are far better than their inputs, by improvement factors that are much larger than 2 [15]. Hence, compression of an image and yet reconstructing the image with good resolution is important. Information theory is playing an important role in image compression. Information theory can be used in order to reduce the dimensionality of data such as histogram [16, 17]
There are two categories of image compression techniques, namely lossless and lossy image compression techniques [18, 19]. In lossless image compression, the original image can be perfectly recovered from the compressed image while in lossy compression the original image cannot be perfectively recovered from the compressed image because some information is lost as a result of compression. Lossless compression is used in applications with high requirements such as medical imaging. Lossy compression techniques are very popular because they offer higher compression ratio. The objective of image compression is to achieve as much compression as possible with little loss of information [20, 21].
Wavelets are also playing significant role in many image processing applications [12, 22, 23, 24]. The twodimensional wavelet decomposition of an image is performed by applying the onedimensional DWT along the rows of the image first, and then the results are decomposed along the columns. This operation results in four decomposed subband images referred to LowLow (LL), LowHigh (LH), HighLow (HL), and HighHigh (HH). The frequency components of those subbands cover the full frequency spectrum of the original image. Figure 1 shows different subband images of Lena image where the topleft image is the LL subband and the bottomright image is the HH subband.
In this research work, a new lossy compression technique which employs singular value decomposition (SVD) and wavelet difference reduction (WDR) is presented. SVD is a lossy image compression technique which can be regarded as a quantization process where it reduces the physcovisual redundancies of the image [25, 26]. In order to enhance the resolution of the decompressed image, stationary wavelet transform (SWT) is used. WDR is one of the stateoftheart techniques in image compression which uses wavelet transform. It is a lossy image compression technique which achieves compression by first taking the wavelet transform of the input image and then applying the difference reduction method on the transform values [27, 28, 29, 30].
Wavelet transform based techniques also play a significant role in many image processing applications, in particular in resolution enhancement, and recently, many novel resolution enhancement by using wavelet transforms have been proposed. Demirel and Anbarjafari [31] proposed an image resolution enhancement technique based on the input image and interpolation of the highfrequency subband images obtained by DWT. In their technique, an SWT technique is used in order to enhance the edges. Then, at the same time input image is decomposed into four frequency subbands image by using DWT. After that the input image, as well as the highfrequency subbands are interpolated. The highfrequency subbands of SWT are used to modify the estimated highfrequency subbands. Finally, inverse DWT (IDWT) is applied to combine all frequency subbands in order to generate a highresolution image. Figure 2 shows the block diagram of the proposed method in [31].
The authors in [32] proposed a learningbased superresolution algorithm. In their proposed algorithm, a multiresolution wavelet approach was adopted to perform the synthesis of local highfrequency features. Two frequency subbands, LH and HL, were estimated based on wavelet frame in order to get a highresolution image. The LH and HL frequency subbands were used to prepare their training sets. Then, they used the training set in order to estimate wavelet coefficients for both LH and HL frequency subbands. Finally, the IDWT was used in order to reconstruct a highresolution image.
In [33], the authors used a complex waveletdomain image resolution enhancement algorithm based on the estimation of wavelet coefficients. Their method uses a dualtree complex wavelet transform (DTCWT) in order to generate a highresolution image. First, they estimate a set of wavelet coefficients from the DTCWT decomposition of the rough estimation of the highresolution image. Then, the inverse DTCWT is used to combine the wavelet coefficients and the lowresolution input image in order to reconstruct a highresolution image. Figure 3 shows the block diagram of the proposed method in [33].
Patel and Joshi [34] proposed a new learningbased approach for super resolution using DWT. The novelty of their method lies in designing applicationspecific wavelet basis (filter coefficients). First the filter coefficients and learning the highfrequency details in the wavelet domain is used to initial estimate of superresolution image. Then, they used a sparsely based regularization framework, in which image there was degradation. Finally, the superresolution image is estimated by the initial superresolution estimate and the estimated wavelet filter coefficients. Their algorithm has some advantages such as avoiding the use of registered images while learning the initial estimate, use of sparsity prior to preserving neighborhood dependencies in superresolution image and use of estimated wavelet filter coefficients to represent an optimal point spread function to model image acquisition process. Figure 4 illustrates the block diagram of the proposed method in [34].
In [35], similar to the proposed method in [30], the authors used wavelet domain in order to generate superresolution image from a single lowresolution image. They proposed an intermediate stage with the aim of estimating highfrequency subbands. The intermediate stage consists of an edge preservation procedure and mutual interpolation between the input lowresolution image and the HF subband images. Sparse mixing weights are calculated over blocks of coefficients in an image, which provides a sparse signal representation in the lowresolution image. Finally, they used IDWT to combine all frequency subbands in order to reconstruct a highresolution image. The block diagram of their proposed method is shown in Fig. 5.
In [36], they proposed a learningbased approach for superresolving an image captured at low spatial resolution. They used a low resolution and a database of high and lowresolution images as inputs to the proposed method. First, they used DWT in order to obtain highfrequency details of database images. Then, an initial highresolution image was decimated by using the highfrequency details. In their observation model, they modelled a lowresolution image as an aliased and noisy version of the corresponding highresolution image and then the initial highresolution and test image estimated the aliasing matrix entries. After that, the prior model for the superresolved image was chosen as an Inhomogeneous Gaussian Markov random field (IGMRF) and the model parameters were estimated using the same initial highresolution estimate. They used a maximum a posteriori (MAP) estimation in order to arrive at the cost function minimized using a simple gradient descent approach. Figure 6 shows the block diagram of the proposed method in [36].
In the proposed compression technique, the input image is firstly decomposed into its different frequency subbands by using 1 level DWT. The LL subband is then being compressed by using DWR and the highfrequency subbands, i.e., LH, HL, and HH, are being compressed by using SVD. The proposed technique has been tested on several wellknown images such as, Lena, Peppers, Boat, and Airfield. The results of this technique have been compared with those of JPEG2000 and WDR with arithmetic coding techniques. The quantitative experimental results based on PSNR show that the proposed technique overcomes the aforementioned techniques. The SVD and WDR image compression techniques are discussed in the next section.
2. Review of singular value decomposition and wavelet difference reduction
2.1. Singular value decomposition
From a mathematical point of view, an image can be represented by a matrix, which consists of one or three layers in the case the image is grayscale or RGB, respectively. The results of the implementation of SVD on a grayscale image, which is represented by the singlelayer image
Eqn. (2) shows how a matrix
Some columns of
The compressed image is then obtained as shown in Eqn. (4):
Because the singular matrix has sorted singular values (in descending order), by using the physcovisual concept, ignoring low singular value will not significantly reduce the visual quality of the image. Figure 7 shows Lena’s picture being reconstructed by using different amount of singular values. This characteristic that an image can be reconstructed by fewer amounts of singular values makes SVD suitable for compression. Because after reconstruction of the image the ignored singular values cannot be recovered, the compression by SVD is lossy [33].
2.2. Wavelet difference reduction
The WDR is a compression technique, which is based on the difference reduction method. The wavelet transform of the input image is first made; bit plane encoding is then applied to the transform values. The bit plane encoding procedure starts with the initialization stage, where a threshold
3. The proposed lossy image compression technique
The proposed image compression technique is a lossy compression technique. Firstly, the image is decomposed into its frequency subbands by using DWT. Among these subbands, LL subband is being compressed by using WDR. The highfrequency subband images are being compressed by using SVD. The number of singular values that are being used in order to reconstruct the highfrequency subbands can be reduced into 1, i.e., the highest singular value is enough to reconstruct the highfrequency subbands. If only one singular value is being used in order to reconstruct a matrix, this means that only one column of U and V matrices are being used. The qualitative loss is not psychovisually noticeable up to some point. In order to obtain the compression ratio of the proposed technique, the total number of bits required to represent the original image is divided by the total of number of bits which is obtained by adding the number of bit streams of WDR for LL and that of the SVD compression for LH, HL, and HH.
Decompression is carried out by taking the inverse WDR (IWDR) of the bit streams in order to reconstruct the LL subband and in parallel the matrix multiplications are conducted in order to reconstruct LH, HL, and HH subbands. Due to the losses by ignoring lowvalued singular values, highfrequency subbands need to be enhanced. For this purpose, stationary wavelet transform (SWT) is applied to the LL subband image which results in new low and highfrequency subbands. These highfrequency subbands will have the same direction as the ones obtained by DWT (e.g., horizontal, vertical, and diagonal), so they will be added to the respective ones reconstructed by matrix multiplications. Now, the LL subband image obtained by IWDR and the enhanced LH, HL, and HH subbands are combined by using inverse DWT (IDWT) in order to reconstruct the decompressed image. The enhancement of highfrequency subbands by using SWT results in more sharpened decompressed image. The block diagram of the proposed lossy image compression technique is shown in Fig. 8. The experimental qualitative and quantitative results are represented and discussed in the next section.
4. Experimental results and discussion
As it was mentioned in the Introduction, for comparison purposes, the proposed lossy image compression was tested on many benchmark images, namely, Lena, Pepper, Boats, Airfield, and Goldhill. All the input images were of resolutions
The foregoing tables illustrate the superiority of the proposed method in terms of its capability in leading to significantly higher PSNR values compared to the other techniques proposed, previously, in the literature. It is worth noticing that the improvement in the PSNR values brought about by considering the proposed method might better show its impact while keeping in mind the fact that they are calculated in dB, meaning that a logarithmic function determines them, which clarifies how considerable the difference between the actual values has been. To be more clear, if one calculates the difference between the PSNR values obtained using WDR and JPEG2000, and subsequently, that of JPEG2000 and the proposed method, it can be seen that the latter is much higher than the former, although JPEG2000 has always been deemed of significantly better performance than WDR. Thus, it can be concluded that the proposed method makes an enormous enhancement to the PSNR values compared to the ones obtained upon employing WDR or JPEG2000.







Lena  35.72  35.99 

Pepper  34.21  35.07 

Boats  32.42  33.18 

Airfield  27.02  27.32 

Goldhill  31.76  32.18 








Lena  32.44  32.75 

Pepper  31.67  32.40 

Boats  29.32  29.76 

Airfield  24.72  24.88 

Goldhill  29.43  29.72 








Lena  29.71  29.62 

Pepper  28.93  29.54 

Boats  26.96  26.76 

Airfield  22.71  22.64 

Goldhill  27.72  27.69 

In order to ensure the quality of the output of the proposed technique, and for visual illustration, the images resulting from the implementation of the foregoing approach were obtained, along with that of JPEG2000 and WDR. Figure 9 shows a part of the magnified Lena image having been compressed using the foregoing approaches, separately, with compression ratio 40:1. As sought from the outset, the proposed method is competent enough to maintain the quality of the image while compressing it, and at the same time, result in better PSNR, which shows its capability in correctly deciding on a reasonable tradeoff between the amount of data needed to be transferred, or kept, and the visibility and authenticity of the details in the image blocks, which is, probably, the most tricky criterion in devising image compression algorithms. As Fig. 9 illustrates, the overall quality of the Lena image being compressed by the proposed method is satisfactory despite possessing much higher PSNR value compared to the JPEG2000 and WDR techniques, and the details are clear and visible, even better than the output of the WDR.
5. Conclusion
In this research work, a new lossy image compression technique which uses singular value decomposition and wavelet difference reduction techniques, followed by resolution enhancement, using discrete wavelet transform and stationary wavelet transform was proposed.
As the first step in the proposed image compression technique, the input image was decomposed into four different frequency subbands using discrete wavelet transform. The lowfrequency subband was compressed using wavelet difference reduction, and in parallel, the highfrequency subbands were compressed using singular value decomposition. The compression ratio was obtained by dividing the total number of bits required to represent the input image over the total bit numbers obtained by wavelet difference reduction and singular value decomposition.
Reconstruction was carried out using inverse wavelet difference reduction to obtain lowfrequency subband and reconstructing the highfrequency subbands using matrix multiplications. The highfrequency subbands were enhanced using high frequency obtained by stationary wavelet transform. The reconstructed lowfrequency subband and enhanced highfrequency subbands were used to generate the reconstructed image using inverse discrete wavelet transform.
The visual and quantitative experimental results of the proposed image compression technique showed that the proposed image compression technique outperformed the wavelet difference reduction and JPEG2000 techniques.
Acknowledgments
The research was supported by the ERDF program, “Estonian higher education information and communications technology and research and development activities state program 20112015 (ICT program)” and Estonian Research Council grant (PUT638).
References
 1.
T. Fujii, D. Shirai, Y. Tonomura, M. Kitamura, T. Nakachi, T. Sawabe, M. Ogawara, T. Yamaguchi, M. Nomura and K. Shirakawa, "Digital cinema and superhighdefinition content distribution on optical highspeed networks," in Proceedings of the IEEE , pp. 140153, 2013.  2.
A. Qian, C. Rongguan, N. a. G. S. Ning, Z. Xiaochen, Z. Jie and K. Deyu, "HighDefinition Image Processing Algorithm and Digital Platform Design," in Computer and Information Technology (CIT), 2012 IEEE 12th International Conference on , 2012.  3.
K. Ishimaru, T. Fujii, T. Sawabe, J. Suzuki and S. Ono, "Transmission Characteristics of MPEG2 encoded super high definition images," in Global Telecommunications Conference, 1996. GLOBECOM'96.'Communications: The Key to Global Prosperity , pp. 289293 1996.  4.
T.J. Chen, H. Chiueh, C.C. Hsieh, C. Yin, W.H. Chang, H.H. Tsai and C.F. Chiu, "High Definition Image PreProcessing System for MultiStripe Satellites' Image Sensors," Sensors Journal, IEEE, vol. 12, no. 9, pp. 28592865, 2012.  5.
G. Anbarjafari, S. Izadpanahi and H. Demirel, "Video resolution enhancement by using discrete and stationary wavelet transforms with illumination compensation," Signal, Image and Video Processing, vol. 9, no. 1, pp. 8792, 2015.  6.
T. Celik, C. Direkoglu, H. Ozkaramanli, H. Demirel and M. Uyguroglu, "Regionbased superresolution aided facial feature extraction from lowresolution video sequences," in Acoustics, Speech, and Signal Processing, 2005. Proceedings.(ICASSP'05). IEEE International Conference on , pp. 785789 2005.  7.
H. Demirel and G. Anbarjafari, "Discrete wavelet transform based satellite image resolution enhancement," Geoscience and Remote Sensing, IEEE Transactions on, vol. 49, no. 6, pp. 19972004, 2011.  8.
C. Liu and D. Sun, "On Bayesian Adaptive Video Super Resolution," Pattern Analysis and Machine Intelligence, IEEE Transactions on, vol. 36, no. 2, pp. 346360, 2014.  9.
G. Polatkan, M. Zhou, L. Carin, D. Blei and I. Daubechies, "A Bayesian Nonparametric Approach to Image SuperResolution," Pattern Analysis and Machine Intelligence, IEEE Transactions on, vol. 37, no. 2, pp. 346358, 2015.  10.
P. Rasti, H. Demirel and G. Anbarjafari, "Image resolution enhancement by using interpolation followed by iterative back projection," in Signal Processing and Communications Applications Conference (SIU), 2013 21st , pp. 14, 2013.  11.
D. Glasner, S. Bagon and M. Irani, "Superresolution from a single image," in Computer Vision, 2009 IEEE 12th International Conference on , pp. 349356, 2009.  12.
G. Anbarjafari and H. Demirel, "Image Super Resolution Based on Interpolation of Wavelet Domain High Frequency Subbands and the Spatial Domain Input Image," ETRI journal, vol. 32, no. 3, pp. 390394, 2010.  13.
C. Dong, C. Loy, K. He and X. Tang, "Learning a Deep Convolutional Network for Image SuperResolution," in European Conference on Computer Vision (ECCV), 2014 Springer Conference on , pp. 184199, 2014.  14.
Y. Zhu, Y. Zhang and A. Yuille, "Single Image Superresolution Using Deformable Patches," in Computer Vision and Pattern Recognition (CVPR), 2014 IEEE Conference on , pp. 29172924 2014.  15.
K. Nasrollahi and T. Moeslund, "Superresolution: a comprehensive survey," Machine Vision and Applications, vol. 25, no. 6, pp. 14231468, 2014.  16.
H. Demirel and G. Anbarjafari, "Data fusion boosted face recognition based on probability distribution functions in different colour channels," EURASIP Journal on Advances in Signal Processing, vol. 2009, p. 25, 2009.  17.
H. Demirel and G. Anbarjafari, "Iris recognition system using combined colour statistics.," in Signal Processing and Information Technology, pp. 175179, 2008.  18.
X. Zhang, "Lossy compression and iterative reconstruction for encrypted image," Information Forensics and Security, IEEE Transactions on, vol. 16, no. 1, pp. 5358, 2011.  19.
F. Hussain and J. Jeong, "Efficient Deep Neural Network for Digital Image Compression Employing Rectified Linear Neurons," Journal of Sensors, 2015.  20.
M. Boliek, "Beyond compression: a survey of functionality derived from still image coding," in Signals, Systems and Computers, 2004. Conference Record of the ThirtySeventh Asilomar Conference on , pp. 19711974, 2003.  21.
R. C. Gonzalez and R. E. Woods, Digital Image Processing, Prentice Hall Upper Saddle River, NJ, 2002.  22.
A. Karami, M. Yazdi and G. Mercier, "Compression of Hyperspectral Images Using Discerete Wavelet Transform and Tucker Decomposition," Selected Topics in Applied Earth Observations and Remote Sensing, IEEE Journal of, vol. 5, no. 2, pp. 444450, 2012.  23.
H. a. A. G. Demirel, C. Ozcinar and S. Izadpanahi, "Video Resolution Enhancement By Using Complex Wavelet Transform," in Image Processing (ICIP), 2011 18th IEEE International Conference on , pp. 20932096, 2011.  24.
M. Ghazel, G. H. Freeman and E. R. Vrscay, "Fractalwavelet image denoising revisited," {Image Processing, IEEE Transactions on, vol. 15, no. 9, pp. 26692675, 2006.  25.
J.F. Yang and C.L. Lu, "Combined Techniques of Singular Value Decomposition and Vector Quantization for Image Coding," IEEE Transactions on Image Processing, vol. 4, no. 8, pp. 11411146, 1995.  26.
M. D. Greenberg, Differential equations & Linear algebra, Prentice Hall, 2001.  27.
J. Tian and R. O. Wells Jr, "Embedded image coding using wavelet difference reduction," in Wavelet image and video compression , pp. 289301, 2002.  28.
J. Tian and R. O. Wells Jr, "A lossy image codec based on index coding," in IEEE Data Compression Conference ,p. 456, 1996.  29.
J. Tian and R. O. Wells Jr, "Image data processing in the compressed wavelet domain," in Signal Processing, 1996., 3rd International Conference on , pp. 978981 1996.  30.
S. Raja and A. Suruliandi, "Image Compression Using WDR and ASWDR Techniques With Different Wavelet Codecs," ACEEE Int. J. Inform. Technol. v01, pp. 2326, 2011.  31.
H. Demirel and G. Anbarjafari, "Image resolution enhancement by using discrete and stationary wavelet decomposition.," Image Processing, IEEE Transactions on, vol. 20, no. 5, pp. 14581460, 2011.  32.
C. Kim, K. Choi, K. Hwang and J. B. Ra, "Learningbased superresolution using a multiresolution wavelet approach," in Iternational workshop on Advance Image Technology (IWAIT) , 2009.  33.
H. Demirel and G. Anbarjafari, "Satellite image resolution enhancement using complex wavelet transform.," Geoscience and Remote Sensing Letters, IEEE, vol. 7, no. 1, pp. 123126, 2010.  34.
R. C. Patel and M. V. Joshi, "SuperResolution of Hyperspectral Images: Use of Optimum Wavelet Filter Coefficients and Sparsity Regularization," Geoscience and Remote Sensing, IEEE Transactions on, vol. 53, no. 4, pp. 1728  1736, 2015.  35.
H. ChavezRoman and V. Ponomaryov, "Super resolution image generation using wavelet domain interpolation with edge extraction via a sparse representation," Geoscience and Remote Sensing Letters, IEEE, vol. 11, no. 10, pp. 17771781, 2014.  36.
P. P. Gajjar and M. V. Joshi, "New learning based superresolution: use of DWT and IGMRF prior," Image Processing, IEEE Transactions on, vol. 19, no. 5, pp. 12011213, 2010.  37.
D. Kaiman, "A Singularly Valuable Decomposition," College Mathematics Journal, vol. 27, no. 1, pp. 223, 1996.  38.
L. Knockaert, B. De Backer and D. De Zutter, "SVD compression, unitary transforms, and computational complexity," Signal Processing, IEEE Transactions on, vol. 47, no. 10, pp. 27242729, 1999.  39.
S. J. Nivedita, "Performance Analysis of SVD and SPIHT Algorithm for Image Compression Application," International Journal of Advanced Research in Computer Science and Software Engineering, vol. 2, no. 2, 2012.  40.
J. S. Walker, T. Q. Nguyen and Y.J. Chen, "A lowpower, lowmemory system for waveletbased image compression," Optical Engineering, Research Signposts, vol. 5, pp. 111125, 2003.  41.
J. S. Walker and T. Q. Nguyen, "Waveletbased image compression," in Transforms and Data Compression, CRC Press, 2001.