The Essential Guide to Image Processing- P16 ppt

17.11 Performance and Extensions 457 If the segmentation symbol is not decoded properly, the data in the corresponding bit plane and of the subsequent bit planes in the code-block should be discarded. Finally, resynchronization markers, including the numbering of packets, are also inserted in front of each packet in a tile. 17.11 PERFORMANCE AND EXTENSIONS The performance of JPEG2000 when compared with the JPEG baseline algorithm is briefly discussed in this section. The extensions included in Part 2 of the JPEG2000 standard are also listed. 17.11.1 Comparison of Performance The efficiency of the JPEG2000 lossy coding algorithm in comparison with the JPEG baseline compression standard has been extensively studied and key results are sum- marized in [7, 9, 24]. The superior RD and error resilience performance, together with features such as progressive coding by resolution, scalability,and region of interest, clearly demonstrate the advantages of JPEG2000 over the baseline JPEG (with optimum Huff- man codes). For coding common test images such as Foreman and Lena in the range of 0.125-1.25 bits/pixel, an improvement in the peak signal-to-noise ratio (PSNR) for JPEG2000 is consistently demonstrated at each compression ratio. For example, for the Foreman image,an improvement of 1.5 to 4 dB is observed as thebits per pixel are reduced from 1.2 to 0.12 [7]. 17.11.2 Part 2 Extensions Most of the technologies that have not been included in Part 1 due to their complexity or because of intellectual property rights (IPR) issues have been included in Part 2 [14]. These extensions concern the use of the following: ■ different offset values for the different image components; ■ different deadzone sizes for the different subbands; ■ TCQ [23]; ■ visual masking based on the application of a nonlinearity to the wavelet coefficients [44, 45]; ■ arbitrary wavelet decomposition for each tile component; ■ arbitrary wavelet filters; ■ single sample tile overlap; ■ arbitrary scaling of the ROI coefficients with the necessity to code and transmit the ROI mask to the decoder; 458 CHAPTER 17 JPEG and JPEG2000 ■ nonlinear transformations of component samples and transformations to decorrelate multiple component data; ■ extensions to the JP2 file format. 17.12 ADDITIONAL INFORMATION Some sources and links for further information on the standards are provided here. 17.12.1 Useful Information and Links for the JPEG Standard A key source of information on theJPEG compression standard is the book by Pennebaker and Mitchell [28]. This book also contains the entire text of the official committee draft international standard ISO DIS 10918-1 and ISO DIS 10918-2. The official standards document [11] contains information on JPEG Part 3. The JPEG committee maintains an official website http://www.jpeg.org, which con- tains general information about the committee and its activities, announcements, and other useful links related to the different JPEG standards. The JPEG FAQ is located at http://www.faqs.org/faqs/jpeg-faq/part1/preamble.html. Free, portable C code for JPEG compression is available from the Independent JPEG Group (IJG). Source code, documentation, and test files are included. Version 6b is available from ftp.uu.net:/graphics/jpeg/jpegsrc.v6b.tar.gz and in ZIP archive format at ftp.simtel.net:/pub/simtelnet/msdos/graphics/jpegsr6b.zip. The IJG code includes a reusable JPEG compression/decompression library, plus sample applications for compression, decompression, transcoding, and file format conversion. The package is highly portable and has been used successfully on many machines ranging from personal computers to super computers. The IJG code is free for both noncommer- cial and commercial use; only an acknowledgement in your documentation is required to use it in a product. A different free JPEG implementation, written by the PVRG group at Stanford,is available from http://www.havefun.stanford.edu:/pub/jpeg/JPEGv1.2.1.tar.Z. The PVRG code is designed for research and experimentation rather than production use; it is slower, harder to use, and less portable than the IJG code, but the PVRG code is easier to understand. 17.12.2 Useful Information and Links for the JPEG2000 Standard Useful sources of information on the JPEG2000 compression standard include two books published on thetopic [1,36]. Further information on the different parts of the JPEG2000 standard can be found on the JPEG website http://www.jpeg.org/jpeg2000.html. This website provide links to sites from which various official standards and other documents References 459 can be downloaded. It also provides links to sites from which software implementations of the standard can be downloaded. Some software implementations are available at the following addresses: ■ JJ2000 software that can be accessed at http://www.jpeg2000.epfl.ch. The JJ2000 software is a Java implementation of JPEG2000 Part 1. ■ Kakadu software that can be accessed at http://www.ee.unsw.edu.au/taubman/ kakadu. The Kakadu software is a C++ implementation of JPEG2000 Part 1. The Kakadu software is provided with the book [36]. ■ Jasper software that can be accessed at http://www.ece.ubc.ca/mdadams/jasper/. Jasper is a C implementation of JPEG2000 that is free for commercial use. REFERENCES [1] T. Acharya and P S. Tsai. JPEG2000 Standard for Image Compression. John Wiley & Sons, New Jersey, 2005. [2] N.Ahmed,T. Natrajan, and K. R. Rao. Discrete cosine tr ansform. IEEE Trans. Comput.,C-23:90–93, 1974. [3] A. J. Ahumada and H. A. Peterson. Luminance model based DCT quantization for color image compression. Human Vision, Visual Processing, and Digital Display III, Proc. SPIE, 1666:365–374, 1992. [4] A. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies. Image coding using the wavelet transform. IEEE Trans. Image Process., 1(2):205–220, 1992. [5] E. Atsumi and N. Farvardin. Lossy/lossless region-of-interestimage coding based on set partitioning in hier archical trees. In Proc. IEEE Int. Conf. Image Process., 1(4–7):87–91, October 1998. [6] A. Bilgin, P. J. Sementilli, and M. W. Marcellin. Progressive image coding using trellis coded quantization. IEEE Trans. Image Process., 8(11):1638–1643, 1999. [7] D. Chai and A. Bouzerdoum. JPEG2000 image compression: an overview. Australian and New Zealand Intelligent Information Systems Conference (ANZIIS’2001), Perth, Australia, 237–241, November 2001. [8] C. Christopoulos, J. Askelof, and M. Larsson. Efficient methods for encoding regions of interest in the upcoming JPEG2000 still image coding standard. IEEE Signal Process. Lett., 7(9):247–249, 2000. [9] C. Christopoulos, A. Skodras, and T. Ebrahimi. The JPEG 2000 still image coding system: an over- view. IEEE Trans. Consum. Electron., 46(4):1103–1127, 2000. [10] K. W. Chun, K. W. Lim, H. D. Cho, and J. B. Ra. An adaptive perceptual quantization algorithm for video coding. IEEE Trans. Consum. Electron., 39(3):555–558, 1993. [11] ISO/IEC JTC 1/SC 29/WG 1 N 993. Information technology—digital compression and coding of continuous-tone still images. Recommendation T.84 ISO/IEC CD 10918-3. 1994. [12] ISO/IEC International standard 14492 and ITU recommendation T.88. JBIG2 Bi-Level Image Compression Standard. 2000. [13] ISO/IEC International standard 15444-1 and ITU recommendation T.800. Information Technology—JPEG2000 Image Coding System. 2000. 460 CHAPTER 17 JPEG and JPEG2000 [14] ISO/IEC International standard 15444-2 and ITU recommendation T.801. Information Technology—JPEG2000 Image Coding System: Part 2, Extensions. 2001. [15] ISO/IEC International standard 15444-3 and ITU recommendation T.802. Information Technology—JPEG2000 Image Coding System: Part 3, Motion JPEG2000. 2001. [16] ISO/IEC International standard 15444-4 and ITU recommendation T.803. Information Technology—JPEG2000 Image Coding System: Part 4, Compliance Testing. 2001. [17] ISO/IEC International standard 15444-5 and ITU recommendation T.804. Information Technology—JPEG2000 Image Coding System: Part 5, Reference Software. 2001. [18] N. Jayant, R. Safranek, and J. Johnston. Signal compression based on models of human perception. Proc. IEEE, 83:1385–1422, 1993. [19] JPEG2000. http://www.jpeg.org/jpeg2000/. [20] L. Karam. Lossless Image Compression, Chapter 15, The Essential Guide to Image Processing. Elsevier Academic Press, Burlington, MA, 2008. [21] K. Konstantinides and D. Tretter. A method for variable quantization in JPEG for improved text quality in compound documents. In Proc. IEEE Int. Conf. Image Process., Chicago,IL, October 1998. [22] D. Le Gall and A. Tabatabai. Subband coding of digital images using symmetric short kernel filters and arithmetic coding techniques. In Proc. Intl. Conf. on Acoust., Speech and Signal Process., ICASSP’88, 761–764, April 1988. [23] M. W. Marcellin and T. R. Fisher. Trellis coded quantization of memoryless and Gauss-Markov sources. IEEE Trans. Commun., 38(1):82–93, 1990. [24] M. W. Marcellin, M. J. Gormish, A. Bilgin, and M. P. Boliek. An overview of JPEG2000. In Proc. of IEEE Data Compression Conference, 523–541, 2000. [25] N. Memon, C. Guillemot, and R. Ansari. The JPEG Lossless Compression Standards. Chapter 5.6, Handbook of Image and Video Processing. Elsevier Academic Press, Burlington, MA, 2005. [26] P. Moulin. Multiscale Image Decomposition and Wavelets, Chapter 6, The Essential Guide to Image Processing. Elsevier Academic Press, Burlington, MA, 2008. [27] W. B. Pennebaker, J. L. Mitchell, G. G. Langdon, and R. B. Arps. An overview of the basic principles of the q-coder adaptive binary arithmetic coder. IBM J. Res. Dev., 32(6):717–726, 1988. [28] W. B. Pennebaker and J. L. Mitchell. JPEG Still Image Data Compression Standard. Van Nostrand Reinhold, New York, 1993. [29] M. Rabbani and R. Joshi. An overview of the JPEG2000 still image compression standard. Elsevier J. Sig nal Process., 17:3–48, 2002. [30] V. Ratnakar and M. Livny. RD-OPT: an efficient algorithm for optimizing DCT quantization tables. IEEE Proc. Data Compression Conference (DCC), Snowbird, UT, 332–341, 1995. [31] K. R. Rao and P. Yip. Discrete Cosine Transform—Algorithms, Advantages, Applications. Academic Press, San Diego, CA, 1990. [32] P. J. Sementilli, A. Bilgin, J. H. Kasner, and M. W. Marcellin. Wavelet tcq: submission to JPEG2000. In Proc. SPIE, Applications of Digital Processing, 2–12, July 1998. [33] A. Skodras, C. Christopoulos, and T. Ebrahimi. The JPEG 2000 still image compression standard. IEEE Sig nal Process. Mag., 18(5):36–58, 2001. [34] B. J. Sullivan, R. Ansari, M. L. Giger, and H. MacMohan. Relative effects of resolution and quanti- zation on the quality of compressed medical images. In Proc. IEEE Int. Conf. Image Process., Austin, TX, 987–991, November 1994. References 461 [35] D. Taubman. High performance scalable image compression with ebcot. IEEE Trans. Image Process., 9(7):1158–1170, 1999. [36] D. Taubman and M.W. Marcellin. JPEG2000: Image Compression Fundamentals: Standards and Practice. Kluwer Academic Publishers, New York, 2002. [37] R. VanderKam and P. Wong. Customized JPEG compression for grayscale printing. In Proc. Data Compression Conference (DCC), Snowbird, UT, 156–165, 1994. [38] M. Vetterli and J. Kovacevic. Wavelet and Subband Coding. Prentice-Hall, Englewood Cliffs, NJ, 1995. [39] G. K. Wallace. The JPEG still picture compression standard. Commun. ACM, 34(4):31–44, 1991. [40] P. W. Wang. Image Quantization, Halftoning, and Printing. Chapter 8.1, Handbook of Image and Video Processing. Elsevier Academic Press, Burlington, MA, 2005. [41] A. B. Watson. Visually optimal DCT quantization matrices for individual images. In Proc. IEEE Data Compression Conference (DCC), Snowbird, UT, 178–187, 1993. [42] I. H. Witten, R. M. Neal, and J. G. Cleary. Arithmetic coding for data compression. Commun. ACM, 30(6):520–540, 1987. [43] World Wide Web Consortium (W3C). Extensible Markup Language (XML) 1.0, 3rd ed., T. Bray, J. Paoli, C. M. Sperberg-McQueen, E. Maler, F. Yergeau, editors, http://www.w3.org/TR/REC-xml, 2004. [44] W. Zeng, S. Daly, and S. Lei. Point-wise extended visual masking for JPEG2000 image compression. In Proc. IEEE Int. Conf. Image Process., Vancouver, BC, Canada, vol. 1, 657–660, September 2000. [45] W. Zeng , S. Daly, and S. Lei. Visual optimization tools in JPEG2000. In Proc. IEEE Int. Conf. Image Process., Vancouver, BC, Canada, vol. 2, 37–40, September 2000. CHAPTER 18 Wavelet Image Compression Zixiang Xiong 1 and Kannan Ramchandran 2 1 Texas A&M University; 2 University of California 18.1 WHAT ARE WAVELETS: WHY ARE THEY GOOD FOR IMAGE CODING? During the past 15 years, wavelets have made quite a splash in the field of image compression. The FBI adopted a wavelet-based standard for fingerprint image com- pression. The JPEG2000 image compression standard [1], which is a much more efficient alternative to the old JPEG standard (see Chapter 17), is also based on wavelets. A natural question to ask then is why wavelets have made such an impact on image compression. This chapter will answer this question, providing both high-level intuition and illustra- tive details based on state-of-the-art wavelet-based coding algorithms. Visually appealing time-frequency-based analysis tools are sprinkled in generously to aid in our task. Wavelets are tools for decomposing signals, such as images,into a hierarchy of increas- ing resolutions: as we consider more and more resolution layers, we get a more and more detailed look at the image. Figure 18.1 shows a three-level hierarchy wavelet decom- position of the popular test image Lena from coarse to fine resolutions (for a detailed treatment on wavelets and multiresolution decompositions, also see Chapter 6). Wavelets can be regarded as “mathematical microscopes” that permit one to “zoom in” and“zoom out” of images at multiple resolutions. The remarkable thing about the wavelet decom- position is that it enables this zooming feature at absolutely no cost in terms of excess redundancy: for an M ϫ N image, there are exactly MN wavelet coefficients—exactly the same as the number of original image pixels (see Fig. 18.2). As a basic tool for decomposing signals, wavelets can be considered as duals to the more traditional Fourier-based analysis methods that we encounter in traditional under- graduate engineering curricula. Fourier analysis associates the very intuitive engineering concept of “spectrum” or “frequency content” of the signal. Wavelet analysis, in con- trast, associates the equally intuitive concept of “resolution” or “scale” of the signal. At a functional level, Fourier analysis is to wavelet analysis as spectrum analyzers are to microscopes. As wavelets and multiresolution decompositions have been described in greater depth in Chapter 6, our focus here will be more on the image compression application. Our goal is to provide a self-contained treatment of wavelets within the scope of their role 463 464 CHAPTER 18 Wavelet Image Compression Level 0 Level 1 Level 2 Level 3 FIGURE 18.1 A three-level hierarchy wavelet decomposition of the 512 ϫ 512 color Lena image. Level 1 (512 ϫ 512) is the one-level wavelet representation of the original Lena at Level 0; Level 2 (256 ϫ 256) shows the one-level wavelet representation of the lowpass image at Level 1; and Level 3 (128 ϫ 128) gives the one-level wavelet representation of the lowpass image at Level 2. 18.1 What Are Wavelets: Why Are They Good for Image Coding? 465 FIGURE 18.2 A three-level wavelet representation of the Lena image generated from the top view of the three- level hierarchy wavelet decomposition in Fig. 18.1. It has exactly the same number of samples as in the image domain. in image compression. More importantly, our goal is to provide a high-level explanation for why they are well suited for image compression. Indeed, wavelets have superior properties vis-a-vis the more traditional Fourier-based method in the form of the discrete cosine transform (DCT) that is deployed in the old JPEG image compression standard (see Chapter 17). We will also cover powerful generalizations of wavelets, known as wavelet packets, that have already made an impact in the standardization world: the FBI fingerprint compression standard is based on wavelet packets. Although this chapter is about image coding, 1 which involves two-dimensional (2D) signals or images, it is much easier to understand the role of wavelets in image coding using a one-dimensional (1D) framework, as the conceptual extension to 2D is straight- forward. In the interests of clarity, we will therefore consider a 1D treatment here. The story begins with what is known as the time-frequency analysis of the 1D signal. As mentioned, wavelets are a tool for chang ing the coordinate system in which we represent the signal: we transform the signal into another domain that is much better suited for processing, e.g., compression. What makes for a good transform or analysis tool? At the basic level, the goal is to be able to represent all the useful signal features and impor tant phenomena in as compact a manner as possible. It is important to be able to compact the bulk of the signal energy into the fewest number of transform coefficients: this way, we can discard the bulk of the transform domain data without losing too much information. For example, if the signal is a time impulse, then the best thing is to do no transforms at 1 We use the terms image compression and image coding interchangeably in this chapter. 466 CHAPTER 18 Wavelet Image Compression all! Keep the signal information in its original and sparse time-domain representation, as that will maximize the temporal energy concentration or time resolution. However, what if the signal has a critical frequency component (e.g., a low-frequency background sinusoid) that lasts for a long t ime duration? In this case, the energy is spread out in the time domain, but it would be succinctly captured in a single frequency coefficient if one did a Fourier analysis of the signal. If we know that the signals of interest are pure sinusoids, then Fourier analysis is the way to go. But, what if we want to capture both the time impulse and the frequency impulse with good resolution? Can we get arbitrarily fine resolution in both time and frequency? The answer is no. There exists an uncertainty theorem (much like what we learn in quantum physics), which disallows the existence of arbitrary resolution in time and frequency [2]. A good way of conceptualizing these ideas and the role of wavelet basis functions is through what is known as time-frequency“tiling”plots, as shown in Fig . 18.3, which shows where the basis functions live on the time-frequency plane: i.e., where is the bulk of the energy of the elementary basis elements localized? Consider the Fourier Time Frequency (a) Time Frequency ( b ) FIGURE 18.3 Tiling diagrams associated with the STFT bases and wavelet bases. (a) STFT bases and the tiling diagram associated with a STFT expansion. STFT bases of different frequencies have the same resolution (or length) in time; (b) Wavelet bases and tiling diagram associated with a wavelet expansion. The time resolution is inversely proportional to frequency for wavelet bases. 18.1 What Are Wavelets: Why Are They Good for Image Coding? 467 case first. As impulses in time are completely spread out in the frequency domain, all localization is lost with Fourier analysis. To alleviate this problem, one typically decom- poses the signal into finite-length chunks using windows or so-called short-time Fourier transform (STFT). Then, the time-frequency tradeoffs will be determined by the win- dow size. An STFT expansion consists of basis functions that are shifted versions of one another in both time and frequency: some elements capture low-frequency events localized in time, and others capture high-frequency events localized in time, but the resolution or window size is constant in both time and frequency (see Fig. 18.3(a)). Note that the uncertainty theorem says that the area of these tiles has to be nonzero. Shown in Fig. 18.3(b) is the corresponding tiling diagram associated with the wavelet expansion. The key difference between this and the Fourier case, which is the critical point, is that the tiles are not all of the same size in time (or frequency). Some basis elements have short time windows; others have short frequency windows. Of course, the uncertainty theorem ensures that the area of each tile is constant and nonzero. It can be shown that the basis functions are related to one another by shifts and scales as this is the key to wavelet analysis. Why are wavelets well suited for image compression? The answer lies in the time- frequency (or more correctly, space-frequency) characteristics of typical natural images, which turn out to be well captured by the wavelet basis functions shown in Fig. 18.3(b). Note that the STFT tiling diagram of Fig. 18.3(a) is conceptually similar to what com- mercial DCT-based image transform coding methods like JPEG use. Why are wavelets inherently a better choice? Looking at Fig . 18.3(b), one can note that the wavelet basis offers elements having good frequency resolution at lower frequency (the short and fat basis elements) while simultaneously offering elements that have good time resolution at higher frequencies (the tall and skinny basis elements). This tradeoff works well for natural images and scenes that are t ypically composed of a mixture of important long-term low-frequency trends that have larger spatial duration (such as slowly varying backgrounds like the blue sky, and the surface of lakes) as well as important transient short duration high-frequency phenomena such as sharp edges. The wavelet representation turns out to be par ticularly well suited to capturing both the transient high-frequency phenomena such as image edges (using the tall and skinny tiles) and long spatial duration low-frequency phenomena such as image backgrounds (the short and fat tiles). As natural images are dominated by a mixture of these kinds of events, 2 wavelets promise to be very efficient in capturing the bulk of the image energy in a small fraction of the coefficients. To summarize, the task of separating transient behavior from long-term trends is a very difficult task in image analysis and compression. In the case of images, the difficulty stems from the fact that statistical analysis methods often require the introduction of at least some local stationarity assumption, i.e., the image statistics do not change abruptly 2 Typical images also contain textures; however, conceptually, textures can be assumed to be a dense concentration of edges, and so it is fairly accurate to model typical images as smooth regions delimited by edges. [...]... error (MSE) or peak signal -to- noise ratio (PSNR)3 between the original and compressed versions These fundamental compression performance bounds are called the theoretical RD bounds for the source: they dictate the minimum rate R needed to compress the source if the tolerable distortion level is D (or alternatively, what is the minimum distortion D subject to a bit rate of R) These bounds are unfortunately... 18.3 The Transform Coding Paradigm framework, the goal of operational RD theory is to minimize the rate R subject to a distortion constraint D, or vice versa The message of Shannon’s RD theory is that one can come close to the theoretical compression limit of the source if one considers vectors of source symbols that get infinitely large in dimension in the limit; i.e., it is a good idea not to code the. .. typical statistical class of images This class is well suited to the characteristics of the chosen fixed transform This raises the natural question; is it possible to do better by being adaptive in the transformation so as to best match the features of the transform to the specific attributes of arbitrary individual images that may not belong to the typical ensemble? To be specific, the wavelet transform is... but the above formula can be generalized pretty easily to account for this As another extension of the results given in the above example, it can be shown that the necessary condition for optimal bit allocation is that all subbands should incur the same distortion at optimality—else it is possible to steal some bits from the lower distortion bands to the higher distortion bands in a way that makes the. .. spatial region of the original image (e.g., the eye of Lena) Arrows identify the parent-children dependencies structure representing the eye region of Lena Arrows in Fig 18.9(b) identify the parentchildren dependencies in a tree The lowest frequency band of the decomposition is represented by the root nodes (top) of the tree, the highest frequency bands by the leaf nodes (bottom) of the tree, and each... transform’s ability to concentrate the image energy disparately in the different frequency bands, with the lower frequency bands having a much higher energy density What these coders failed to exploit was the very definite spatial characterization of the wavelet representation In fact, this is even apparent to the naked eye if one views the wavelet decomposition of the Lena image in Fig 18.1, where the spatial... coder After the encoded bitstream of an input image is transmitted over the channel (assumed to be perfect), the decoder undoes all the functionalities applied in the encoder and tries to reconstruct a decoded image that looks as close as possible to the original input image, based on the transmitted information A block diagram of this transform image paradigm is shown in Fig 18.5 For the sake of simplicity,... library adaptively to the individual image In order to make this feasible, there are two requirements First, the library must contain a good representative set of entries (e.g., it would be good to include the conventional wavelet decomposition) Second, it is essential that there exists a fast way of searching through the library to find the best transform in an image- adaptive manner Both these requirements... used to set all coefficients in the set to zero; otherwise, the set is partitioned into subsets (or child sets) for further significance tests After all coefficients are tested in one pass, the threshold is halved before the next pass The underlying assumption of the zerotree coding framework is that most images can be modeled as having decaying power spectral densities That is, if a parent node in the. .. the four found in the first pass, the set of significant coefficients now becomes {63, Ϫ34, 49, 47, Ϫ31, 23} The reconstruction result at the end of the second pass is shown in Fig 18.11(b) The above encoding process continues from one pass to another and can stop at any point For better coding performance, arithmetic coding [16] can be used to further compress the binary bitstream out of the SPIHT encoder . representation of the Lena image generated from the top view of the three- level hierarchy wavelet decomposition in Fig. 18.1. It has exactly the same number of samples as in the image domain. in image compression discard the bulk of the transform domain data without losing too much information. For example, if the signal is a time impulse, then the best thing is to do no transforms at 1 We use the terms image. dictate the minimum rate R needed to compress the source if the tolerable distortion level is D (or alternatively, what is the minimum distortion D subjecttoabitrateofR). These bounds are unfortunately