IAENG International Journal of Applied Mathematics, 36:1, IJAM_36_1_9 ______________________________________________________________________________________ Haar Wavelet Based Approach for Image Compression and Quality Assessment of Compressed Image Kamrul Hasan TalukderI and Koichi HaradaII Abstract— With the increasing growth of technology and the The image is actually a kind of redundant data i.e. it entrance into the digital age, we have to handle a vast amount of contains the same information from certain perspective of information every time which often presents difficulties. So, the view. By using data compression techniques, it is possible to digital information must be stored and retrieved in an efficient remove some of the redundant information contained in and effective manner, in order for it to be put to practical use. images. Image compression minimizes the size in bytes of a Wavelets provide a mathematical way of encoding information in such a way that it is layered according to level of detail. This graphics file without degrading the quality of the image to an layering facilitates approximations at various intermediate unacceptable level. The reduction in file size allows more stages. These approximations can be stored using a lot less space images to be stored in a certain amount of disk or memory than the original data. Here a low complex 2D image space. It also reduces the time necessary for images to be sent compression method using wavelets as the basis functions and the over the Internet or downloaded from web pages. approach to measure the quality of the compressed image are The scheme of image compression is not new at all. The presented. The particular wavelet chosen and used here is the discovery of Discrete Cosine Transform (DCT) in 1974 [2] is simplest wavelet form namely the Haar Wavelet. The 2D discrete really an important achievement for those who work on image wavelet transform (DWT) has been applied and the detail compression. The DCT can be regarded as a discrete time matrices from the information matrix of the image have been estimated. The reconstructed image is synthesized using the version of the Fourier Cosine series. It is a close relative of estimated detail matrices and information matrix provided by the Discrete Fourier Transform (DFT), a technique for converting Wavelet transform. The quality of the compressed images has a signal into elementary frequency components. Thus DCT been evaluated using some factors like Compression Ratio (CR), can be computed with a Fast Fourier Transform (FFT) like Peak Signal to Noise Ratio (PSNR), Mean Opinion Score (MOS), algorithm of complexity O(nlog2 n). Unlike DFT, DCT is real- Picture Quality Scale (PQS) etc. valued and provides a better approximation of a signal with fewer coefficients. Index Terms— Fourier Transform, Haar Wavelet, Image There are a number of various methods in which image files Compression, Multiresolution Analysis. can be compressed. There are two main common compressed graphic image formats namely Joint Photographic Experts I. INTRODUCTION Group (JPEG, usually pronounced as JAY-pehg) [3] and The computer is becoming more and more powerful day by Graphic Interchange Format (GIF) for the use in the Internet. day. As a result, the use of digital images is increasing rapidly. The JPEG method established by ISO (International Standards Along with this increasing use of digital images comes the Organization) and IEC (International Electro-Technical serious issue of storing and transferring the huge volume of Commission) is more often used for photographs, while the data representing the images because the uncompressed GIF method is commonly used for line art and other images in multimedia (graphics, audio and video) data requires which geometric shapes are relatively simple. considerable storage capacity and transmission bandwidth. In 1992, JPEG established the first international standard Though there is a rapid progress in mass storage density, for still image compression where the encoders and decoders speed of the processor and the performance of the digital are DCT-based. The JPEG standard specifies three modes communication systems, the demand for data storage capacity namely sequential, progressive, and hierarchical for lossy and data transmission bandwidth continues to exceed the encoding, and one mode of lossless encoding. The capabilities of on hand technologies. Besides, the latest growth performance of the coders for JPEG usually degrades at low of data intensive multimedia based web applications has put bit-rates mainly because of the underlying block-based much pressure on the researchers to find the way of using the Discrete Cosine Transform (DCT) [4]. The baseline JPEG images in the web applications more effectively. Internet coder [5] is the sequential encoding in its simplest form. Fig. 1 teleconferencing, High Definition Television (HDTV), and 2 show the key processing steps in such an encoder and satellite communications and digital storage of movies are not decoder respectively for grayscale images. Color image feasible without a high degree of compression. As it is, such compression can be approximately regarded as compression of applications are far from realizing their full potential largely multiple grayscale images, which are either compressed due to the limitations of common image compression entirely one at a time, or are compressed by alternately techniques [1]. interleaving 8x8 sample blocks from each in turn. The DCT-based encoder can be thought of as essentially ICurrently a Phd student in the Department of Information Engineering of the compression of a stream of 8x8 blocks of image samples. Each Graduate School of Engineering, Hiroshima University, Japan (E-mail: 8x8 block makes its way through each processing step, and [email protected]). yields output in compressed form into the data stream. IIProfessor in the Graduate School of Engineering, Hiroshima University, Because adjacent image pixels are highly correlated, the Japan (E-mail: [email protected]). Forward DCT (FDCT) processing step lays the basis for (Advance online publication: 1 February 2007) gaining data compression by concentrating most of the signal as listed below, the top contenders in the JPEG-2000 standard in the lower spatial frequencies. For a typical 8x8 sample [8] are all wavelet-based compression algorithms. block from a typical source image, most of the spatial • Wavelet coding schemes at higher compression avoid frequencies have zero or near-zero amplitude and need not to blocking artifacts. be encoded. Generally, the DCT introduces no loss to the • They are better matched to the HVS (Human Visual source image samples; it merely transforms them to a domain System) characteristics. in which they can be more efficiently encoded. • Compression with wavelets is scalable as the transform process can be applied to an image as Original Quantizer Entropy many times as wanted and hence very high Image FDCT Compressed Encoder Image Data compression ratios can be achieved. • Wavelet based compression allow parametric gain control for image softening and sharpening. Quantization Huffman • Wavelet-based coding is more robust under Table Table (QT) transmission and decoding errors, and also facilitates progressive transmission of images. Fig.1: Encoder Block Diagram • Wavelet compression is very efficient at low bit rates. • Wavelts provide an efficient decomposition of Compressed signals prior to compression. Image Data Inverse Entropy Reconstructed Dequantizer DCT II. BACKGROUND Decoder Image Before we go into details of the method, we present some background topics of image compression which include the Quantization Huffman principles of image compression, the classification of Table (QT) Table compression methods and the framework of a general image coder and wavelets for image compression. Fig. 2: Decoder Block Diagram A. Principles of Image Compression After output from the Forward DCT (FDCT), each of the An ordinary characteristic of most images is that the 64 DCT coefficients is uniformly quantized in conjunction neighboring pixels are correlated and therefore hold redundant with a carefully designed 64-element Quantization Table information. The foremost task then is to find out less (QT). At the decoder, the quantized values are multiplied by correlated representation of the image. Two elementary the corresponding QT elements to pick up the original components of compression are redundancy and irrelevancy unquantized values. After quantization, all the quantized reduction. Redundancy reduction aims at removing coefficients are ordered into zig-zag sequence. This ordering duplication from the signal source image. Irrelevancy helps to facilitate entropy encoding by placing low frequency reduction omits parts of the signal that is not noticed by the non-zero coefficients before high-frequency coefficients. The signal receiver, namely the Human Visual System (HVS). In DC coefficient, which contains a significant fraction of the general, three types of redundancy can be identified: (a) total image energy, is differentially encoded. Spatial Redundancy or correlation between neighboring pixel Entropy Coding (EC) achieves additional compression values, (b) Spectral Redundancy or correlation between losslessly through encoding the quantized DCT coefficients different color planes or spectral bands and (c) Temporal more compactly based on their statistical characteristics. The Redundancy or correlation between adjacent