Image Compression And Denoising Algorithm Based On Multi .
110Int'l Conf. IP, Comp. Vision, and Pattern Recognition IPCV'16 Image Compression and Denoising Algorithm based onMulti-resolution Discrete Cosine TransformYanjun Zhao1, Saeid Belkasim2Computer Science Department, Troy University, Troy, AL, USA2Computer Science Department, Georgia State University, Atlanta, GA, USA1Abstract - Discrete cosine transform (DCT) and wavelettransform coding system are the most popular imagecompression methods. Although DCT has outstanding energycompaction properties, blocking artifacts impact itsperformance. Wavelet avoids blocking artifacts; it is also themost popular approach to doing image compression anddenoising simultaneously. However wavelet has highercomputational complexity. Exploring an image in differentresolutions reveals its dominant information in comparison toredundant one. We propose a novel multi-resolution DCT;based on our multi-resolution DCT, we propose a novelalgorithm to do image compression and denoisingsimultaneously. Our algorithm achieves multi-resolutionanalysis, avoids blocking artifacts, has excellent energycompaction property and is ideal for parallel computing.Compared to wavelet, our algorithm has good computationaccuracy and efficiency.Keywords: Multi-resolution Discrete Cosine Transform(Multi-resolution DCT); Multi-resolution Analysis; BlockingArtifacts; Compression and Denoising.1 IntroductionThe goal of image compression is to reduce redundancyof image data to efficiently store or transmit data whilepreserving quality required for a given application [1]. Manycompression schemes based on lossless or lossy criteria areproposed [1-8]. Although lossy compression is irreversible; itmaintains visually lossless data [9]. To achieve highercompression ratios, lossy compression is applied. Variouscoding schemes are proposed for lossy compression, includingpredictive coding [10], subband coding [11], transform coding[12], vector quantization [13, 14]. The most popular one istransform coding; discrete cosine transform (DCT) [15, 16]and wavelet transform [3, 16-18], are the most populartransform coding methods.DCT has excellent energy compaction property and highcomputation efficiency; however the performance of itstraditional coder generally degrades at high compression ratiosmainly due to the underlying block-based strategy [19]. Eachblock of DCT coefficients only represents the localinformation of an image. Separately compressing each blockbreaks correlation between the pixels at the borders of blocksand causes blocking artifacts [19].Wavelet avoids blocking artifacts and maintains highimage quality at high compression ratios; it is more robustunder transmission and decoding errors [16, 18]. Howeverwavelet has high computation complexity [16].Image compression removes redundant information; if theremoved information is noise, image compression anddenoising can be done simultaneously. Currently the mostpopular method for doing image compression and denoisingsimultaneously is also wavelet.Exploring an image in different resolutions reveals itsdominant information in comparison to redundant one. Wepropose a novel multi-resolution discrete cosine transform(multi-resolution DCT); based on our multi-resolution DCT,we propose a novel algorithm to do image compressing anddenoising simultaneously. Our algorithm achieves multiresolution analysis, avoids blocking artifacts, maintainsoutstanding energy compaction property of traditional DCT,and is ideal for parallel processing. Compared with wavelet,our algorithm has good computation accuracy and efficiency.2 Multi-resolution DCT2.1 An odd-even image treeA 2 ' 3 image can be represented as a 2 ' 3matrix J 2 3 K where 2 F 2 Dand 3 F 3 D . An odd-even image tree is constructedthrough dyadically dividing an image into four sets: a set ofodd-odd pixels that contains pixels from odd rows and oddcolumns; a set of odd-even pixels that contains pixels from oddrows and even columns, a set of even-odd pixels that containspixels from even rows and odd columns, and a set of even-evenpixels that contains pixels from even rows and even columns.The multi-resolution image at node of tree level ! whereF ; and ! F is represented as:: :2 :;; J 2 ; 3 ; K where; F J 2 K; D88:3 ; F J 3 K; D ; J 2 K; F B? and J 3 K; F B@ .33Each node :; represents the original image in a global view.ISBN: 1-60132-442-1, CSREA Press
Int'l Conf. IP, Comp. Vision, and Pattern Recognition IPCV'16 111Fig. 1 A DCT tree (Multi-resolution DCT)A: image; C: DCT coefficients set; The size of the original image F 2 E 3 .2.2 Multi-resolution DCTThrough applying traditional DCT to the odd-even imagetree, we generate our multi-resolution DCT:E9? AB 71J8@ KB 62:;9@ AB 71 :; J2 :; ; J 2 K; J 3 K; F3 :; KC E ;: E(E "EJ 2 K;E 3 :E%;: E(; CE " J K EJ 3 K;2 ;: FJ8? KB:Where &2 ; F;3 ;: F J 2 K; DE 22&3 J%;: K FJ8@ KBJ8? KB 62J8@ KB 62 B 71 B 71E ; J ; %; K E "2J8? KBWhere &2 ; F %;: F3 %;: F J 3 K; D2; ;: F J 2 K; D ; %;: F J 3 K; D .The multi-resolution DCT can be represented by a tree ofDCT sets, shown in Fig. 1. The DCT sets from different treelevels represent the original image in different resolutions andthe resolution is inversely proportional to the height of the tree.The highest resolution occurs at root whereas the lowest onesoccur at leaves; each other node has four children of lowerresolution in terms of odd-odd, odd-even, even-odd and eveneven DCT set and a parent node of higher resolution. At treelevel ! , each of ; DCT sets independently represents the2original image with B of its original size in a global view; all4 ; DCT sets together lossless represent the original image.At level !, by computing mean of ; DCT sets, we use aDCT set ; to represent all ; DCT sets together:&3 J%; K F&2 ; E&3 J%; KEJ 2 K; C E ; E(EJ 2 K;E%; E( J KEJ 3 K; CEJ 3 K;E "J8? KBJ8@ KB3 ;: ;: %;: J K ;:72Where ; F J 2 K; D ; %; F J 3 K; D .The original image can be reconstructed: ;: ;: %;: F &2 ;: E&3 J%;: KJ8? KB 624B ; ; %; FJ8? KBJ8@ KB3J8@ KB ; F ; F J 2 K; D; %; F %; F J 3 K; D;J 2 K; F J 2 K; D ; J 3 K; F J 3 K; D .2The size of reconstructed image ; J 2 K; J 3 K; is B of4the original image size.In traditional DCT coding system, each DCT set onlyrepresents local information of the original image, which leadsto blocking artifacts. In our multi-resolution DCT, each DCTset individually represents the global information of theoriginal image, which avoids blocking artifacts. By computingthe mean of the DCT sets at the same resolution level, wefurther enhance the correlations among pixels.Image noise is random variation of intensity informationin an image. At the same resolution level, each DCT set,ISBN: 1-60132-442-1, CSREA Press
112Int'l Conf. IP, Comp. Vision, and Pattern Recognition IPCV'16 globally representing the noised image, can be considered as asample of the noised image; computing mean of the DCT setsat the same resolution level is a good way to smooth this imagewhich in turn reduces its embedded noises.1.4 Threshold ; to ;I based on a threshold .Output ;I as the compressed and denoised image.Part Two: Decoding3 Compression & denoising algorithmOur algorithm has two parts: encoding and decoding.J8? KB 62 J8@ KB 62&2 B 71 B 713EJ9? KB 52 E B E A. Encoding"Mapper: map an image into a DCT tree to represent thisimage in multi-resolution levels.Quantizer: at tree level !, compute the mean of ; DCTsets; Since DCT has strong energy compaction property; basedon a threshold , threshold mean ; to ;I for furthercompression. ;I globally represents the original image; itscompression ratio ; H ; .Output ;I as compressed and denoised image.B. DecodingInverse Mapper: compute reconstructed image ;I byapplying invert DCT to ;I ; resize ;I to I; with the same sizeof the original image, by applying bicubic interpolation to ;I .Bicubic interpolation is good for image smoothing which inturn further reduces noises.Output I; as decompressed and denoised image.The odd-even image tree and DCT tree are also ideal forparallel computing.Our Image Compression and Denoising AlgorithmInput: Image Output: Compressed and denoised image ;IDecompressed and denoised image I;1.1 Transform the image into an odd-even image:::tree :; J 2 ; 3 ; K where 2 ; F J 2 K; D8@3B:;F J 3 K; D; J 2 K; F8?3B; J 3 K; F.; F ; ;! F .1.2 Through applying traditional DCT to the odd-even imagetree, generate DCT tree ;: ;: %;: F &2 ;: EJ8 KB 62J8@ KB 62 :: J 2 :&3 J%;: KE 9 ? A; 3 ; KE719 A 71 ;? B"@ BA3E 9? AB 52 E B E E3EJ8? KB"A3E 9@ AB 52 E B E 3EJ8@ KB1.3 Represent all of ; DCT sets at tree level ! , bycomputingtheirmean ; ; %; F24B4B::72 ; ;: %;: 3EJ8? KB ; E&3 %; E ;I ; %; EE "3EJ9@ KB 52 E B E 3EJ8@ KB 2.2 Generate decompressed and denoised image I; with thesame size of the original image , by applying bicubicinterpolation to image ;I .Output I; as the decompressed and denoised image.4 SimulationSince wavelet is the most widely used method for imagecompression and denoising, this simulation is to compare ouralgorithm with wavelet in terms of computation accuracy andefficiency.4.1 Simulation dataWe use one standard image “Lena”, and two imagedatabases “MITForest” and “PCA” as simulation data.“MITForest” has 328 images of different forests. It isdownloaded from http://www-cvr.ai.uiuc.edu/ponce grp/data/(under “Fifteen Scene Categories” of the webpage). PCA has91 images of different backgrounds. It is downloaded fromhttp://pics.psych.stir.ac.uk/Other image types.htm.We use Matlab function im2double to convert theintensity image to double precision; then we respectively addGaussian noise with mean 0 & variance 0.005 and “Salt&Pepper” noise with noise density 0.02 into image.4.2 Simulation processPart One: Encoding; 32.1 Through applying invert DCT to ;I , generateIreconstructedimage ; J 2 K; J 3 K; FWe respectively apply our algorithm and wavelets to theimages corrupted by noise for compression and denoising.We construct our multi-resolution DCT ;: where F ; ! F . Using threshold F , we getcompressed & denoised image 2I with compression ratio 2 H and compressed & denoised image 3I with compressionratio 3 H . Based on 2I and 3I , we computedecompressed & denoised image 2I and I3 respectively.Haar and Biorthogonal 1.5 [19] wavelet are widely usedin image compression and denoising; we respectively applythese wavelets in two levels.Applying wavelet transform to image , we getapproximation coefficient 2 with details 2 2 2 .WeISBN: 1-60132-442-1, CSREA Press
Int'l Conf. IP, Comp. Vision, and Pattern Recognition IPCV'16 considered 2 with threshold details # 2 , # 2 ,# 2 as compressed & denoised image in level one; itscompression ratio 2 G . Applying wavelet transform to 2 ,we get approximation coefficient 3 with details 3 3 3 .We considered 3 with threshold details# 3 , # 3 , # 3 , # 2 , # 2 , # 2 ascompressed & denoised image in level two; its compressionratio 3 G . Denoising threshold is generated by Matlabfunction ddencmp.Applying invert wavelet transform to 2 with # 2 ,# 2 , # 2 , we get decompressed & denoised image forlevel one. Applying invert wavelet transform to 3 with# 3 , # 3 , # 3 , we get 2 I ; applying invertwavelet transform to 2 I with # 2 , # 2 , # 2 , weget decompressed & denoised image for level two.We use two popular measurements to measurecomputation accuracy: (1) Mean Square Errors (MSE) betweenimage without noise and decompressed & denoised image.The lower MSE is, the higher quality of decompressed &denoised image is. (2) Peak Signal-to-Noise Ratio (PSNR)between image without noise and decompressed & denoisedimage. The higher PSNR is, the higher quality ofdecompressed & denoised image is. We use execute time ofeach method to measure computation time.4.3 Simulation resultsFigure 2-5 and Table I-II show simulation results. 2 and 3 represent decompressed & denoised image based on Haar113wavelet in level one and two respectively; 2 and 3 representdecompressed & denoised image based on Biorthogonal 1.5wavelet in level one and two respectively; 2 and 3represent decompressed & denoised image based on ouralgorithm in level one and two respectively.Figure 2 - 5 show simulation results about “Lena” imageFigure 2 and 4 show whole images. To highlight differences,we zoom out part of the images (right corner of Lena’sforehead with part of her hair and hat) and get Figure 3 and 5.Figure 3 and 5 show that both wavelets methods generateartefacts resulting into pixelated-like images; by contrast, ouralgorithm generates much smoother images. These figuresdemonstrate that our algorithm produces higher quality ofdecompressed & denoised images than wavelets.Table I shows simulation results about “Lena” image in aquantitative way. In the same level, our algorithm has theminimum MSE, maximum PSNR and minimum computationtime. Furthermore, MSE of our algorithm in level two is lowerthan MSE of wavelets in level one; PSNR of our algorithm inlevel two is higher than PSNR of wavelets in level one;computation time of our algorithm in level two is less thancomputation time of wavelets in level one.To show generalizability of our algorithm, we summed upsimulation results about “MITForest” and “PCA” imagedatabase into table II. For each database, in the same level, ouralgorithm always has the minimum average MSE, maximumaverage PSNR and minimum average computation time.Table I: MSE, PSNR and Computation Time for “Lena” ImageImageLenaAlgorithm *) *) .CompressRatio 2 G 3 G 2 G 3 G ,- H /,. H .001181180.00151805MSESalt & 329.276828.1871PSNRSalt & mputation TimeGaussianSalt & 8010.0458064Table II: Average MSE, Average PSNR and Average Computation Time for “MITForest” and “PCA” Image DatabaseImageDatabaseMITForestPCAAlgorithm CompressRatio 2 G 3 G Average MSEGaussianSalt & Pepper0.008433410.008853580.01333990.013447Average PSNRGaussianSalt & Pepper21.392721.137319.438719.3949Average Computation TimeGaussianSalt & Pepper0.01890510.01941920.02326190.0225446 *) *) . *) *) . 2 G 3 G ,- H /,. H -0 2 G 3 G 2 G 3 G ,- H /,. H 161ISBN: 1-60132-442-1, CSREA Press .0128229
114Int'l Conf. IP, Comp. Vision, and Pattern Recognition IPCV'16 Fig. 2 “Lena” image under Gaussian noise with mean 0 & variance 0.005Fig.3 Fragment of “Lena” image under Gaussian noise with mean 0 & variance 0.005ISBN: 1-60132-442-1, CSREA Press
Int'l Conf. IP, Comp. Vision, and Pattern Recognition IPCV'16 115Fig. 4 “Lena” image under “Salt & Pepper” noise with noise density 0.02Fig. 5 Fragment of “Lena” image under “Salt & Pepper” noise with noise density 0.025 ConclusionsExploring an image in different resolutions reveals itsdominant information in comparison to redundant one. Wepropose a novel multi-resolution DCT; based on our multiresolution DCT, we propose a novel algorithm for performingimage compression and denoising simultaneously. Ouralgorithm avoids blocking artifacts, has excellent energycompaction property, achieves multi-resolution analysis, and isideal for parallel computing. Compared to wavelet, ouralgorithm has good computation accuracy and efficiency.ISBN: 1-60132-442-1, CSREA Press
116Int'l Conf. IP, Comp. Vision, and Pattern Recognition IPCV'16 [1] Mark Nelson and Jean-Loup Gailly. “The DataCompression Book”. The second edition. M&T Books, 1995.“Lossy compression of images without visible distortions andits application”. IEEE 10th International Conference on SignalProcessing Proceedings, Beijing, China, pp. 698- 701, October,2010.[2] Rakesh Chalasani, Jose C. Principe andNaveenRamakrishnan . "A fast proximal method for convolutionalsparse coding". The 2013 International Joint Conference onNeural Networks (IJCNN), Dallas, TX, U.S., pp. 1-5, August,2013.[10] Sceuchin Chuah, Sorina Dumitrescu and Xiaolin Wu. "ℓ2optimized predictive image coding with ℓ bound". 2013IEEE International Conference on Acoustics, Speech andSignal Processing, Vancouver, BC, Canada, pp. 1315-1319,May, 2013.[3] Wang Yannan, Zhang Shudong and Liu Hui. "Study ofImage Compression Based on Wavelet Transform". 2013Fourth International Conference on Intelligent Systems Designand Engineering Applications, Zhangjiajie, China, pp. 575578, November, 2013.[11] John W. Woods and Sean D. O’Neil. "Subband coding ofimages". IEEE Transactions on Acoustics, Speech, and SignalProcessing, vol. 34, no. 5, pp. 1278-1288, October, 1986.6 References[4] Simone Milani and Pietro Zanuttigh, "Compression ofphoto collections using geometrical information". 2015 IEEEInternational Conference on Multimedia and Expo (ICME),Turin, Italy, pp. 1-6, July, 2015.[5] Guochao. Zhang, Shaohui Liu, Feng Jiang, Debin Zhaoand Wen Gao. "An improved image compression scheme withan adaptive parameters set in encrypted domain". IEEEConference on Visual Communications and Image Processing(VCIP), Kuching, Sarawak, Malaysia, pp. 1-6, November,2013.[6] Joaquin Zepeda, Christine Guillemot and Ewa Kijak."Image compression using the Iteration-Tuned and AlignedDictionary". 2011 IEEE International Conference onAcoustics, Speech and Signal Processing (ICASSP), Prague,Czech Republic, pp. 793-796, May, 2011.[7] Rane, Shantanu, Petros Boufounos, Anthony Vetro andYu Okada. "Low complexity efficient raw SAR datacompression". In SPIE Defense, Security, and Sensing, pp.80510W-80510W. International Society for Optics andPhotonics, May, 2011.[8] Rime Raj Singh Tomar and Kapil Jain. "Lossless ImageCompression Using Differential Pulse Code Modulation andits Application". 2015 Fifth International Conference onCommunication Systems and Network Technologies (CSNT),pp. 543-545, Gwalior, MP, India, April, 2015.[9] Vladimir V. Lukin, Mikhail S. Zriakhov, Nikolay N.Ponomarenko, Sergey S. Krivenko and Miao Zhenjiang.[12] Vivek K. Goyal. "Theoretical foundations of transformcoding". IEEE Signal Processing Magazine, vol. 18, no. 5, pp.9-21, September, 2001.[13] Meina Xu and Anthony Kuh "Image coding using featuremap finite-state vector quantization". IEEE Signal ProcessingLetters, vol. 3, no. 7, pp. 215-217, July, 1996.[14] Allen Gersho and Robert M. Gray. “Vector Quantizationand Signal Compression”. The first edition, Springer US, 1992.[15] N. Ahmed, T. Natarajan and K. R. Rao. "Discrete CosineTransform". IEEE Transactions on Computers, vol. C-23, no.1,pp. 90-93, January, 1974.[16] Zixiang Xiong, Kannan Ramchandran, Michael T.Orchard and Ya-Qin Zhang. "A comparative study of DCTand wavelet-based image coding". IEEE Transactions onCircuits and Systems for Video Technology, vol. 9, no. 5, pp.692-695, August, 1999.[17] Martin Vetterli. "Wavelets, approximation, andcompression". IEEE Signal Processing Magazine, vol. 18,no.5, pp. 59-73, September, 2001.[18] Bryan E. Usevitch. "A tutorial on modern lossy waveletimage compression: foundations of JPEG 2000". IEEE SignalProcessing Magazine, vol. 18, no. 5, pp. 22-35, September,2001.[19] Rafael C. Gonzalez and Richard E. Woods. “DigitalImage Processing”. The third edition, Pearson Prentice Hall,2008.ISBN: 1-60132-442-1, CSREA Press
The odd-even image tree and DCT tree are also ideal for parallel computing. We use Matlab function Our Image Compression and Denoising Algorithm Input: Image Output: Compressed and denoised image 4 Decompressed and denoised image 4 Part One: Encoding 1.1 Transform the image 7 into an odd-even image tree where
one for image denoising. In the course of the project, we also aimed to use wavelet denoising as a means of compression and were successfully able to implement a compression technique based on a unified denoising and compression principle. 1.2 The concept of denoising A more precise explanation of the wavelet denoising procedure can be given .
Denoising and Compression Using Wavelets Juan Pablo Madrigal Cianci Trevor Gianinni December 15, 2016 Abstract An explanation of the theory behind signal and image denoising and compression is presented. Di erent examples of image and signal denois-ing and image compression are implemented using MATLAB. Some of their characteristics are discussed.
2.2 Image Denoising. A typical application area for image reconstruction is image denoising, where the task is to remove noise to restore the original image. Here, we focus on image denoising tech-niques based on deep neural networks; for more detailed information about image denoising research, please refer to the following survey papers [9,11].
4 Image Denoising In image processing, wavelets are used for instance for edges detection, watermarking, texture detection, compression, denoising, and coding of interesting features for subsequent classifica-tion [2]. Image denoising by thresholding of the DWT coefficients is discussed in the following subsections. 4.1 Principles
In the recent years there has been a fair amount of research on wavelet based image denoising, because wavelet provides an appropriate basis for image denoising. But this single tree wavelet based image denoising has poor directionality, loss of phase information and shift sensitivity [11] as
Image Compression Model Image compression reduces the amount of data from the original image representation. There are two approaches to compress an image. These are: (a) Lossless compression (b) Lossy compression Fig.2.2 shows a general image compression model. Image data representation has redundancy (also called pixel
Matlab package for wavelet shrinkage image denoising process. As briefly discussed in Section 3, wavelet shrinkage is a powerful image denoising algorithm, and thus many researchers have proposed different modified versions of that algorithm. In this research, wavelet shrinkage is
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