Uncompressed multimedia (graphics, audio and video) data requires considerable storage capacity and transmission bandwidth. Despite rapid progress in mass-storage density, processor speeds, and digital communication system performance, demand for data storage capacity and data-transmission bandwidth continues to outstrip the capabilities of available technologies. The recent growth of data intensive multimedia-based web applications have not only sustained the need for more efficient ways to encode signals and images but have made compression of such signals central to storage and communication technology.For still image compression, the `Joint Photographic Experts Group' or JPEG[19] standard has been established by ISO (International Standards Organization) and IEC (International Electro-Technical Commission). The performance of these coders generally degrades at low bit-rates mainly because of the underlying block-based Discrete Cosine Transform (DCT)[20] scheme. More recently, the wavelet transform has emerged as a cutting edge technology, within the field of image compression. Wavelet-based coding[29] provides substantial improvements in picture quality at higher compression ratios. Over the past few years, a variety of powerful and sophisticated wavelet-based schemes for image compression, as discussed later, have been developed and implemented. Because of the many advantages, the top contenders in the upcoming JPEG-2000 standard[14] are all wavelet-based compression algorithms.The goal of this article is two-fold. First, for readers new to compression, we briefly review some basic concepts on image compression and present a short overview of the DCT-based JPEG standard and the more popular wavelet-based image coding schemes. Second, for more advanced readers, we mention a few sophisticated, modern, and popular wavelet-based techniques including one we are currently pursuing. The goal of the upcoming JPEG-2000 image compression standard, which is going to be wavelet-based, is briefly presented. For those who are curious, a number of useful references are given. There is also abundance of information about image compression on the Internet.Background Why do we need compression?
The figures in Table 1 show the qualitative transition from simple text to full-motion video data and the disk space, transmission bandwidth, and transmission time needed to store and transmit such uncompressed data.Table 1 Multimedia data types and uncompressed storage space, transmission bandwidth, and transmission time required. The prefix kilo- denotes a factor of 1000 rather than 1024.The examples above clearly illustrate the need for sufficient storage space, large transmission bandwidth, and long transmission time for image, audio, and video data. At the present state of technology, the only solution is to compress multimedia data before its storage and transmission, and decompress it at the receiver for play back. For example, with a compression ratio of 32:1, the space, bandwidth, and transmission time requirements can be reduced by a factor of 32, with acceptable quality.What are the principles behind compression?A common characteristic of most images is that the neighboring pixels are correlated and therefore contain redundant information. The foremost task then is to find less correlated representation of the image. Two fundamental components of compression are redundancy and irrelevancy reduction. Redundancy reduction aims at removing duplication from the signal source (image/video).The Fourier transform decomposes a signal in the time domain into a sum of sines and cosines of specific frequencies and amplitudes. The sines and cosines are basis functions that span the entire time interval, oscillating indefinitely - i.e. they are not localized in time. The Fourier transform of a signal characterized by bursts over a short period of time, generally requires a very broad spectrum of frequencies. Thus to get a reasonably faithful reconstruction of a signal from its Fourier components may require more information than is needed to describe the original signal.The wavelet transform decomposes a signal into a set of wavelet basis functions,  just ``wavelets'' for short, that are localized in time. Therefore signals with short bursts can be reconstructed with a much smaller set of wavelet basis functions. Thus they can be more economical than a Fourier transform. Wavelet basis functions are typically designed so that one can readily separate smooth components and detailed components. This can help in image compression where the detailed components may be discarded without serious degradation of the image.We illustrate the wavelet decomposition using the Haar basis. This is the simplest example of a wavelet transform.The Fourier transform decomposes a signal in the time domain into a sum of sines and cosines of specific frequencies and amplitudes. The sines and cosines are basis functions that span the entire time interval.Uncompressed multimedia (graphics, audio and video) data requires considerable storage capacity and transmission bandwidth. Despite rapid progress in mass-storage density, processor speeds, and digital communication system performance, demand for data storage capacity and data-transmission bandwidth continues to outstrip the capabilities of available technologies. The recent growth of data intensive multimedia-based web applications have not only sustained the need for more efficient ways to encode signals and images but have made compression of such signals central to storage and communication technology.For still image compression, the `Joint Photographic Experts Group' or JPEG[19] standard has been established by ISO (International Standards Organization) and IEC (International Electro-Technical Commission). The performance of these coders generally degrades at low bit-rates mainly because of the underlying block-based Discrete Cosine Transform (DCT)[20] scheme. More recently, the wavelet transform has emerged as a cutting edge technology, within the field of image compression. Wavelet-based coding[29] provides substantial improvements in picture quality at higher compression ratios. Over the past few years, a variety of powerful and sophisticated wavelet-based schemes for image compression, as discussed later, have been developed and implemented. Because of the many advantages, the top contenders in the upcoming JPEG-2000 standard[14] are all wavelet-based compression algorithms.The goal of this article is two-fold. First, for readers new to compression, we briefly review some basic concepts on image compression and present a short overview of the DCT-based JPEG standard and the more popular wavelet-based image coding schemes. Second, for more advanced readers, we mention a few sophisticated, modern, and popular wavelet-based techniques including one we are currently pursuing. The goal of the upcoming JPEG-2000 image compression standard, which is going to be wavelet-based, is briefly presented. For those who are curious, a number of useful references are given. There is also abundance of information about image compression on the Internet.Background Why do we need compression?
The figures in Table 1 show the qualitative transition from simple text to full-motion video data and the disk space, transmission bandwidth, and transmission time needed to store and transmit such uncompressed data.Table 1 Multimedia data types and uncompressed storage space, transmission bandwidth, and transmission time required. The prefix kilo- denotes a factor of 1000 rather than 1024.The examples above clearly illustrate the need for sufficient storage space, large transmission bandwidth, and long transmission time for image, audio, and video data. At the present state of technology, the only solution is to compress multimedia data before its storage and transmission, and decompress it at the receiver for play back. For example, with a compression ratio of 32:1, the space, bandwidth, and transmission time requirements can be reduced by a factor of 32, with acceptable quality.What are the principles behind compression?A common characteristic of most images is that the neighboring pixels are correlated and therefore contain redundant information. The foremost task then is to find less correlated representation of the image. Two fundamental components of compression are redundancy and irrelevancy reduction. Redundancy reduction aims at removing duplication from the signal source (image/video).The Fourier transform decomposes a signal in the time domain into a sum of sines and cosines of specific frequencies and amplitudes. The sines and cosines are basis functions that span the entire time interval, oscillating indefinitely - i.e. they are not localized in time. The Fourier transform of a signal characterized by bursts over a short period of time, generally requires a very broad spectrum of frequencies. Thus to get a reasonably faithful reconstruction of a signal from its Fourier components may require more information than is needed to describe the original signal.The wavelet transform decomposes a signal into a set of wavelet basis functions,  just ``wavelets'' for short, that are localized in time. Therefore signals with short bursts can be reconstructed with a much smaller set of wavelet basis functions. Thus they can be more economical than a Fourier transform. Wavelet basis functions are typically designed so that one can readily separate smooth components and detailed components. This can help in image compression where the detailed components may be discarded without serious degradation of the image.We illustrate the wavelet decomposition using the Haar basis. This is the simplest example of a wavelet transform.The Fourier transform decomposes a signal in the time domain into a sum of sines and cosines of specific frequencies and amplitudes. The sines and cosines are basis functions that span the entire time interval.