# - Acharya T.

ISBN 0-471-48422-9

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76 JPEG: STILL IMAGE COMPRESSION STANDARD

(a) (b)

Fig. 3.7 Progressive encoding: (a) spectral selection; (b) successive approximation.

imation offers better reconstructed quality in the earlier scans compared to the spectral selection method.

Eight DCT-based progressive coding methodologies have been defined in JPEG standard [1].

3.5 HIERARCHICAL MODE

In hierarchical mode, JPEG provides a progressive coding with increasing spatial resolution in a number of stages. This is particularly suitable for applications where a higher-resolution image is viewed on a lower-resolution display device and other similar kinds of applications. The pyramidal mul-

SUMMARY 77

tiresolution approach for implementation of the hierarchical mode of coding is shown in Figure 3.8.

0\

Fig. 3.8 Hierarchical multiresolution encoding.

In this mode, the original image is filtered and down-sampled by required multiples of two for the target resolution and the lower-resolution image is encoded using any of the other three JPEG modes. The compressed lowerresolution image is then decoded and interpolated for upsampling by the same interpolation method that will be used at the decoder. The interpolated image is then subtracted from the next-higher-resolution image. The difference is then encoded by one of the other three JPEG modes (lossless or lossy). This procedure of the hierarchical encoding process is continued until it encodes all the resolutions.

Fourteen different methods for encoding the difference images in the hierarchical mode have been explained in greater detail in JPEG standard [1].

3.6 SUMMARY

In this chapter, we described the JPEG standard for still image compression. We discussed the details of the algorithm for lossless JPEG. We also discussed in great detail the principles and algorithms for the baseline JPEG standard. Baseline JPEG is the most widely used algorithm among all different modes in the JPEG standard for still image compression. We presented some results of baseline JPEG and compared them with the new JPEG2000 standard. The features, concepts, and principles behind the algorithms for the JPEG2000

78 JPEG: STILL IMAGE COMPRESSION STANDARD

Standard will be elaborated on in great details in Chapters 6, 7, 8, and 10. We also summarized the progressive mode and hierarchical mode of operation of the JPEG standard in this chapter with examples.

REFERENCES

1. W. B. Pennebaker and J. L. Mitchell, JPEG Still Image Data Compression Standard. Chapman & Hall, New York, 1993.

2. T. Acharya and A. Mukherjee, “High-Speed Parallel VLSI Architectures for Image Decorrelation,” International Journal of Pattern Recognition and Artificial Intelligence, Vol. 9, No. 2, pp. 343-365, 1995.

3. H. Lohscheller, “A Subjectively Adapted Image Communication System,” IEEE Transactions on Communications, COM-32, Vol. 12, pp. 1316— 1322, 1984.

4. The independent JPEG Group, “The Sixth Public Release of the Independent JPEG Group’s Free JPEG Software,” C source code of JPEG Encoder research 6b, March 1998 (ftp://ftp.uu.net/graphics/jpeg/).

5. K. R. Rao and P. Yip, Discrete Cosine Transform—Algorithms, Advantages, Applications. Academic Press, San Diego, 1990.

6. B. G. Lee, “FCT—A Fast Cosine Transform,” Proc. of the Intl. Conf. on Acoustics, Speech, and Signal Processing, pp. 28A.3.1-28A.3.3, San Diego, March 1984.

7. N. I. Cho and S. U. Lee, “Fast Algorithm and Implementation of 2D Discrete Cosine Transform,” IEEE Trans, on Circuits and Systems, Vol. 38, pp. 297-305, March 1991.

8. “Information Technology—JPEG2000 Image Coding System,” Final Committee Draft Version 1.0, ISO/IEC JTC 1/SC 29/WG 1 N1646R, March 2000.

4

Introduction to Discrete Wavelet Transform

4.1 INTRODUCTION

Although the “wavelet” has become very popular and is widely used as a versatile signal analysis function, its concepts were hidden in the works of mathematicians even more than a century ago. In 1873, Karl Weierstrass mathematically described how a family of functions can be constructed by superimposing scaled versions of a given basis function [1]. Mathematically a “wave” is expressed as a sinusoidal (or oscillating) function of time or space. Fourier analysis expands an arbitrary signal in terms of infinite number of sinusoidal functions of its harmonics and has been well studied by the signal processing community for decades. Fourier representation of signals is known to be very effective in analysis of time-invariant (stationary) periodic signals. In contrast to a sinusoidal function, a wavelet is a small wave whose energy is concentrated in time. The term wavelet was originally used in the field of seismology to describe the disturbances that emanate and proceed outward from a sharp seismic impulse [2]. In 1982, Morlet et al. first described how the seismic wavelets could be effectively modelled mathematically [3]. In 1984, Grossman and Morlet extended this work to show how an arbitrary signal can be analyzed in terms of scaling and translation of a single mother wavelet function (basis) [4, 5]. Properties of wavelets allow both time and frequency analysis of signals simultaneously because of the fact that the energy of wavelets is concentrated in time and still possesses the wave-like (periodic) characteristics. As a result, wavelet representation provides a versatile mathematical tool to analyze transient, time-variant (nonstationary) signals that

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