Discrete Wavelet Transform for Image Processing

International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume...
Author: Todd Booth
19 downloads 0 Views 523KB Size
International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 3, March 201.5

Discrete Wavelet Transform for Image Processing Dipalee Gupta1, Siddhartha Choubey2 In order to compress the image, redundancies must be exploited, for example, areas where there is little or no change between pixel values. Therefore images having large areas of uniform colour will have large redundancies, and conversely images that have frequent and large changes in colour will be less redundant and harder to compress[3]. The objective of image compression is to reduce redundancy (i.e. Coding redundancy, Interpixel redundancy & psycho visual redundancy) of the image data in order to be able to store or transmit data in an efficient form. Data compression is achieved when one or more of these redundancies are reduced or eliminated. There are several methods of image compression available today that is lossless and lossy image compression. In lossless compression , every single bit of data that was originally in the file remains after the file is uncompressed. All of the information is completely restored. Medical imaging, technical drawings, and astronomical observations typically use lossless compression techniques. The Graphics Interchange File (GIF) is an image format used on the Web that provides lossless compression. In lossy compression reduces a file by permanently eliminating certain information, especially redundant information. When the file is uncompressed, only a part of the original information is still there. Pictures and videos from digital cameras are examples of digital files that are commonly compressed using lossy methods. A simple method of lossy image compression is to reduce the color space to a smaller set of colors. The JPEG image file, commonly used for photographs and other complex still images on the Web, is an image that has lossy compression[4-5]. Wavelets have been first found in the literature in works of Grossmann and Morlet. Wavelet analysis is a mathematical technique used to represent data or functions[6]. The wavelet transform is an emerging signal processing technique that can be used to represent real-life non- stationary signals with high efficiency. A wavelet is a waveform of effectively limited duration that has an average value of zero. The term ‗wavelet‘ comes from the fact that they integrate to zero; they wave up and down across the axis. Many wavelets also display a property ideal for compact signal representation: orthogonality[7].

Abstract— Image compression is a method through which we can reduce the storage space of images which will helpful to increase storage and transmission process’s performance. In this paper, we present the comparison of the performance of discrete wavelets like Haar Wavelet and Daubechies Wavelet for implementation in a still image compression system. The performances of these transforms are compared in terms of Mean squared error (MSE) and Energy Retained (ER) etc. The main objective is to investigate the still image compression of a gray scale image using wavelet theory. This is implemented in software using MATLAB Wavelet Toolbox and 2D-DWT technique. The experiments and results is carried out on .jpg format images. These results provide a good reference for application developers to choose a good wavelet compression system for their application. Keywords—Image Compression, Discrete Wavelet Transform, Haar Wavelet, Daubechies Wavelet , Wavelet Toolbox (WT)

I. INTRODUCTION Compression is one of the major image processing techniques. It is one of the most useful and commercially successful technologies in the field of digital image processing. Image compression is the representation of an image in digital form with as few bits as possible while maintaining an acceptable level of image quality[1]. Increasingly images are acquired and stored digitally or various film digitizers are used to convert traditional raw images into digital format. Data$ compression is the technique to reduce the redundancies in data representation in order to decrease data storage requirements and hence communication costs. Reducing the storage requirement is equivalent to increasing the capacity of the storage medium increase the speed of transmission and hence communication bandwidth[2]. The efficient ways of storing large amount of data and due to the bandwidth and storage limitations, images must be compressed before transmission and storage.At some later time, the compressed image is decompressed to reconstruct the original image or approximation of it. Redundancy is that portion of data that can be removed when it is not needed or can be reinserted to interpret the data when needed. Most often, the redundancy is reinserted in order to generate the original data in its original form. An image can be thought of as a matrix of pixel values.

598

International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 3, March 201.5 Discrete wavelet transforms is the most popular transformation technique adopted for image compression. So the proposed methodology of this paper is to achieve high compression ratio in images through implementing Haar Wavelet Transform and daubachies wavelet transform using software tools MATLAB. Through experimental results we compare performance of both the transforms. We works in the MATLAB computing environment, the Wavelet Toolbox offers some significant advantages. The amount of information retained by an image after compression and decompression is known as the energy retained, and this is proportional to the sum of the squares of the pixel values. If the energy retained is 100% then the compression is known as lossless. This occurs when the threshold value is set to zero, meaning that the detail has not been changed. If any values are changed then energy will be lost and this is known as lossy compression. Ideally, during compression the no of zeros and the energy retention will be as high as possible. However, as more zeros are obtained more energy is lost, so a balance between the two needs to be found. This article is structured as follows: Section 2 describes the Discrete Wavelet Transform, Section 3 illustrates the Methodology , Section 4 presents the result and discussion. Finally, Section 6 concludes the paper.

The signal to be analyzed is passed through filters with different cut-off frequencies at different scales. It is easy to implement and reduces the computation time and resources required[9]. A 2-D DWT can be seen as a 1-D wavelet scheme which transform along the rows and then a 1-D wavelet transform along the columns,. The 2-D DWT operates in a straight forward manner by inserting array transposition between the two 1-D DWT. The rows of the array are processed first with only one level of decomposition. This essentially divides the array into two vertical halves, with the first half storing the average coefficients, while the second vertical half stores the detail coefficients. This process is repeated again with the columns, resulting in four sub-bands (see Fig. 1) within the array defined by filter output. Fig. 1 shows a three- level 2D DWT decomposition of the image.

II. DISCRETE WAVELET TRANSFORM The wavelet transform has gained widespread acceptance in signal processing and image compression. Recently the JPEG committee has released its new image coding standard, JPEG-2000, which has been based upon DWT. Wavelet transform decomposes a signal into a set of basis functions. These basis functions are called wavelets. Wavelets are obtained from a single prototype wavelet called mother wavelet by dilations and shifting[8]. The DWT has been introduced as a highly efficient and flexible method for sub band decomposition of signals. The 2DDWT is nowadays established as a key operation in image processing .It is multi-resolution analysis and it decomposes images into wavelet coefficients and scaling function. In Discrete Wavelet Transform, signal energy concentrates to specific wavelet coefficients. This characteristic is useful for compressing images[9]. Wavelets convert the image into a series of wavelets that can be stored more efficiently than pixel blocks. Wavelets have rough edges, they are able to render pictures better by eliminating the ―blockiness‖. In DWT, a timescale representation of the digital signal is obtained using digital filtering techniques.

Fig 1. Three level decomposition for 2D – DWT

Image consists of pixels that are arranged in two dimensional matrix, each pixel represents the digital equivalent of image intensity. In spatial domain adjacent pixel values are highly correlated and hence redundant. In order to compress images, these redundancies existing among pixels needs to be eliminated. DWT processor transforms the spatial domain pixels into frequency domain information that are represented in multiple sub-bands, representing different time scale and frequency points. One of the prominent features of JPEG2000 standard, providing it the resolution scalability , is the use of the 2D-DWT to convert the image samples into a more compressible form. The JPEG 2000 standard proposes a wavelet transform stage since it offers better rate/distortion (R/D) performance than the traditional DCT.

599

International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 3, March 201.5 A. Haar Transform Haar functions have been used from 1910 when they were introduced by the Hungarian mathematician Alfred Haar[10]. Haar wavelet is discontinuous, and resembles a step function. It represents the same wavelet as Daubechies db1. Haar used these functions to give an example of an orthonormal system for the space of square-integrable function on the unit interval [0, 1]. Fig 3. Daubechies Wavelet

The Daubechies wavelet transforms are defined in the same way as Haar wavelet transform by computing the running averages and differences via scalar products with scaling signals and wavelets the only difference between them consists in how these scaling signals and wavelets are defined[13]. This wavelet type has balanced frequency responses but non-linear phase responses. Daubechies wavelets use overlapping windows, so the high frequency coefficient spectrum reflects all high frequency changes. Therefore Daubechies wavelets are useful in compression and noise removal of audio signal processing.

Fig 2. Haar Wavelet

For an input represented by a list of numbers, the Haar wavelet transform may be considered to simply pair up input values, storing the difference and passing the sum. This process is repeated recursively, pairing up the sums to provide the next scale, finally resulting in differences and one final sum. The Haar Wavelet Transformation is a simple form of compression which involves averaging and differencing terms, storing detail coefficients, eliminating data, and reconstructing the matrix such that the resulting matrix is similar to the initial matrix.[11-12]. A Haar wavelet is the simplest type of wavelet. In discrete form, Haar wavelets are related to a mathematical operation called the Haar transform. The Haar transform serves as a prototype for all other wavelet transforms. Like all wavelet transforms, the Haar transform decomposes a discrete signal into two sub-signals of half its length. One sub-signal is a running average or trend; the other subsignal is a running difference or fluctuation.

III. METHODOLOGY Following steps are performed for compression : a) Load the image which is compressed. b) Applying the transform-The compression algorithm starts by transforming the image from data space to wavelet space. This is done on several levels. c) Chossing the threshold- neglect all the wavelet coefficients that fall below a certain threshold. We select our threshold in such a way as to preserve a certain percent of the total coefficients - this is known as ‖quantile‖ thresholding. d) Perform compression at different transform IV. RESULT In this paper, we compared Haar and Daubechies wavelet of Discrete wavelet transform (DWT). The quality of a compression method could be measured by the traditional distortion measures such as Mean square error (MSE) and energy retained(ER). We compared the performance of these transforms on image ―Koala.jpg‖(512x512).

B. Daubechies Transform Ingrid Daubechies, one of the brightest stars in the world of wavelet research, invented what are called compactly supported orthonormal wavelets - thus making discrete wavelet analysis practicable. The names of the Daubechies family wavelets are written dbN, where N is the order, and db the ―surname‖ of the wavelet. The db1 wavelet, as mentioned above, is the same as Haar wavelet. Here is the wavelet functions psi of the next nine members of the family. 600

International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 3, March 201.5

(b)

Fig 4. Original Image

Fig 6. Compressed Imge (a) Haar (b) Daubechies

(a)

. Fig 5. Decomposition al level 3

(b) Where

(a) Fig 7. Graph shows the retained energy and zeros in % (a) Haar Wavelet (b) Daubechies Wavelet

601

International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 3, March 201.5 In lossy compression the information is loss, so we not restore all the information. Ideally, during compression the no of zeros and the energy retention will be as high as possible. In Daubechies wavelet retained energy is 98.03% and in haar transform it is 97.70%. So from the results, it has been concluded that Daubechies wavelet transform shows the best results in terms of Energy Retain as compare to Haar wavelet transform. REFERENCES [1] (a)

[2] [3]

[4]

[5] (b) [6]

[7]

[8]

[9] (c) Fig 8. More on Residuals for wavelet 2D-compression (a) Residuals (b) Histogram (c) Cumulative Histogram

[10]

V. CONCLUSION

[11]

Compression of image is an important field in Digital signal processing. In this paper, comparison of various transforms based image compression method is described. If the energy retained is 100% then the compression is known as lossless means we successfully restore all the information useful for user. This occurs when the threshold value is set to zero, meaning that the detail has not been changed. If any values are changed then energy will be lost and this is known as lossy compression.

[12]

[13]

602

Priyanka Singh, Priti Singh, Rakesh Kumar Sharma, 2011, ―JPEG Image Compression based on Biorthogonal, Coiflets and Daubechies Wavelet Families‖ International Journal of Computer Applications , Volume 13– No.1. Maneesha Gupta, Dr.Amit Kumar garg, Mr.Abhishek Kaushik, 2011, ―Review: Image Compression Algorithm‖ IJCSET , Vol 1. B.B.S.Kumar, Dr.P.S.Satyanarayana , 2013, ―Image Analysis Using Biorthogonal Wavelet‖ Published in International Journalof Innovative Research And Development, Vol 2. Ms. Sonam Malik and Mr. Vikram Verma , 2012, ―Comparative analysis of DCT, Haar and Daubechies Wavelet for Image Compression‖ Published in International Journal of Applied Engineering Research, Vol.7-No.11. Sanjeev Singla, Abhilasha Jain, 2013, ―Improved 2-D DCT Image Compression Using optimal compressed value‖ Sanjeev et al. / IJAIR, Vol. 2. Daljeet Kaur Khanduja & M.Y.Gokhal ―Time Domain Signal Analysis Using Modified Haar and Modified Daubechies Wavelet Transform‖ Signal Processing-An International Journal (SPIJ), Volume (4). M. Mozammel Hoque Chowdhury and Amina Khatun, 2012, ―Image Compression Using Discrete Wavelet Transform‖, IJCSI International Journal of Computer Science, Vol. 9- No 1. A.Alice Blessie, J. Nalini and S.C.Ramesh, 2011, ―Image Compression Using Wavelet Transform Based on the Lifting Scheme and its Implementation‖ IJCSI International Journal of Computer Science Issues, Vol. 8-No. 1. Ms.Yamini S.Bute, Prof. R.W. Jasutkar, 2012, ―Implementation of Discrete Wavelet Transform Processor For Image Compression‖ International Journal of Computer Science and Network (IJCSN), Vol. 1. Radomir S. Stankovic, Bogdan J. Falkowski, 2003, ―The Haar wavelet transform: its status and achievements‖ Computers and Electrical Engineering, 25–44 P. Raviraj and M.Y. Sanavullah, 2007, ―The Modified 2D-Haar Wavelet Transformation in Image Compression‖. Middle-East Journal of Scientific Research, 2 (2): 73-78. Kamrul Hasan Talukder and Koichi Harada,2007, ―Haar Wavelet Based Approach for Image Compression and Quality Assessment of Compressed Image‖ Published in IAENG International Journal of Applied Mathematics, 36:1. Mohamed I. Mahmoud, Moawad I. M. Dessouky, Salah Deyab, and Fatma H. Elfouly. 2007, ―Comparison between Haar and Daubechies Wavelet Transformions on FPGA Technology‖ Proceedings Of World Academy Of Science, Engineering And Technology, VOLUME 20.