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The JPEG-2000 Still Image Compression Standard.pdf

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    ISO/IEC JTC 1/SC 29/WG 1 N 2412 Date: September 1, 2001 ISO/IEC JTC 1/SC 29/WG 1 (ITU-T SG8) Coding of Still Pictures JBIG JPEG Joint Bi-level Image Joint Photographic Experts Group Experts Group TITLE: SOURCE: PROJECT: STATUS: The JPEG-2000 Still Image Compression Standard Michael D. Adams Dept. of Electrical and Computer Engineering University of British Columbia Vancouver, BC, Canada V6T 1Z4 E-mail: mdadams@ece.ubc.ca JPEG 2000 REQUESTED ACTION: DISTRIBUTION: Public Contact: ISO/IEC JTC1/SC29/WG1 Convener—Dr. Daniel Lee Yahoo!, 3420 Central Expressway, Santa Clara, California 95051, USA Tel: +1 408 992 7051, Fax: +1 253 830 0372, E-mail: dlee@yahoo-inc.com
    The JPEG-2000 Still Image Compression Standard∗ (Last Revised: September 1, 2001) Michael D. Adams Dept. of Elec. and Comp. Engineering, University of British Columbia Vancouver, BC, Canada V6T 1Z4 mailto:mdadams@ieee.org http://www.ece.ubc.ca/˜mdadams 1 Abstract—JPEG 2000, a new international standard for still image com- pression, is discussed at length. A high-level introduction to the JPEG-2000 standard is given, followed by a detailed technical description of the JPEG- 2000 Part-1 codec. Keywords—JPEG 2000, still image compression/coding, standards. I. INTRODUCTION Digital imagery is pervasive in our world today. Conse- quently, standards for the efficient representation and inter- change of digital images are essential. To date, some of the most successful still image compression standards have re- sulted from the ongoing work of the Joint Photographic Experts Group (JPEG). This group operates under the auspices of Joint Technical Committee 1, Subcommittee 29, Working Group 1 (JTC 1/SC 29/WG 1), a collaborative effort between the In- ternational Organization for Standardization (ISO) and Interna- tional Telecommunication Union Standardization Sector (ITU- T). Both the JPEG [1, 2] and JPEG-LS [3] standards were born from the work of the JPEG committee. For the last few years, the JPEG committee has been working towards the establish- ment of a new standard known as JPEG 2000 (i.e., ISO/IEC 15444). The fruits of these labors are now coming to bear, as JPEG-2000 Part 1 (i.e., ISO/IEC 15444-1 [4]) has recently been approved as a new international standard. In this paper, we provide a detailed technical description of the JPEG-2000 Part-1 codec, in addition to a brief overview of the JPEG-2000 standard. This exposition is intended to serve as a reader-friendly starting point for those interested in learning about JPEG 2000. Although many details are included in our presentation, some details are necessarily omitted. The reader should, therefore, refer to the standard [4] before attempting an implementation. The JPEG-2000 codec realization in the JasPer software [5–7] may also serve as a practical guide for imple- mentors. (See Appendix A for more information about JasPer.) The remainder of this paper is structured as follows. Sec- tion II begins with a overview of the JPEG-2000 standard. This is followed, in Section III, by a detailed description of the JPEG- 2000 Part-1 codec. Finally, we conclude with some closing re- marks in Section IV. Throughout our presentation, a basic un- derstanding of image coding is assumed. ∗This document is a revised version of the JPEG-2000 tutorial that I wrote which appeared in the JPEG working group document WG1N1734. The original tutorial contained numerous inaccuracies, some of which were introduced by changes in the evolving draft standard while others were due to typographical errors. Hopefully, most of these inaccuracies have been corrected in this revised document. In any case, this document will probably continue to evolve over time. Subsequent versions of this document will be made available from my home page (the URL for which is provided with my contact information). II. JPEG 2000 The JPEG-2000 standard supports lossy and lossless com- pression of single-component (e.g., grayscale) and multi- component (e.g., color) imagery. In addition to this basic com- pression functionality, however, numerous other features are provided, including: 1) progressive recovery of an image by fi- delity or resolution; 2) region of interest coding, whereby differ- ent parts of an image can be coded with differing fidelity; 3) ran- dom access to particular regions of an image without needing to decode the entire code stream; 4) a flexible file format with pro- visions for specifying opacity information and image sequences; and 5) good error resilience. Due to its excellent coding per- formance and many attractive features, JPEG 2000 has a very large potential application base. Some possible application ar- eas include: image archiving, Internet, web browsing, document imaging, digital photography, medical imaging, remote sensing, and desktop publishing. A. Why JPEG 2000? Work on the JPEG-2000 standard commenced with an initial call for contributions [8] in March 1997. The purpose of having a new standard was twofold. First, it would address a number of weaknesses in the existing JPEG standard. Second, it would provide a number of new features not available in the JPEG stan- dard. The preceding points led to several key objectives for the new standard, namely that it should: 1) allow efficient lossy and lossless compression within a single unified coding framework, 2) provide superior image quality, both objectively and subjec- tively, at low bit rates, 3) support additional features such as re- gion of interest coding, and a more flexible file format, 4) avoid excessive computational and memory complexity. Undoubtedly, much of the success of the original JPEG standard can be at- tributed to its royalty-free nature. Consequently, considerable effort has been made to ensure that minimally-compliant JPEG- 2000 codec can be implemented free of royalties1. B. Structure of the Standard The JPEG-2000 standard is comprised of numerous parts, several of which are listed in Table I. For convenience, we will refer to the codec defined in Part 1 of the standard as the baseline codec. The baseline codec is simply the core (or minimal func- tionality) coding system for the JPEG-2000 standard. Parts 2 and 3 describe extensions to the baseline codec that are useful for certain specific applications such as intraframe-style video 1Whether these efforts ultimately prove successful remains to be seen, how- ever, as there are still some unresolved intellectual property issues at the time of this writing.
    2 compression. In this paper, we will, for the most part, limit our discussion to the baseline codec. Some of the extensions pro- posed for inclusion in Part 2 will be discussed briefly. Unless otherwise indicated, our exposition considers only the baseline system. For the most part, the JPEG-2000 standard is written from the point of view of the decoder. That is, the decoder is defined quite precisely with many details being normative in nature (i.e., re- quired for compliance), while many parts of the encoder are less rigidly specified. Obviously, implementors must make a very clear distinction between normative and informative clauses in the standard. For the purposes of our discussion, however, we will only make such distinctions when absolutely necessary. III. JPEG-2000 CODEC Having briefly introduced the JPEG-2000 standard, we are now in a position to begin examining the JPEG-2000 codec in detail. The codec is based on wavelet/subband coding tech- niques [9,10]. It handles both lossy and lossless compression us- ing the same transform-based framework, and borrows heavily on ideas from the embedded block coding with optimized trun- cation (EBCOT) scheme [11,12]. In order to facilitate both lossy and lossless coding in an efficient manner, reversible integer- to-integer [13, 14] and nonreversible real-to-real transforms are employed. To code transform data, the codec makes use of bit- plane coding techniques. For entropy coding, a context-based adaptive binary arithmetic coder [15] is used—more specifi- cally, the MQ coder from the JBIG2 standard [16]. Two levels of syntax are employed to represent the coded image: a code stream and file format syntax. The code stream syntax is similar in spirit to that used in the JPEG standard. The remainder of Section III is structured as follows. First, Sections III-A to III-C, discuss the source image model and how an image is internally represented by the codec. Next, Sec- tion III-D examines the basic structure of the codec. This is fol- lowed, in Sections III-E to III-M by a detailed explanation of the coding engine itself. Next, Sections III-N and III-O explain the syntax used to represent a coded image. Finally, Section III-P briefly describes some extensions proposed for inclusion in Part 2 of the standard. (a) (b) Fig. 1. Source image model. (a) An image with N components. (b) Individual component. Fig. 2. Reference grid. be sampled at the same resolution. Consequently, the compo- nents themselves can have different sizes. For example, when color images are represented in a luminance-chrominance color space, the luminance information is often more finely sampled than the chrominance data. A. Source Image Model B. Reference Grid Before examining the internals of the codec, it is important to understand the image model that it employs. From the codec’s point of view, an image is comprised of one or more compo- nents (up to a limit of 214), as shown in Fig. 1(a). As illustrated in Fig. 1(b), each component consists of a rectangular array of samples. The sample values for each component are integer val- ued, and can be either signed or unsigned with a precision from 1 to 38 bits/sample. The signedness and precision of the sample data are specified on a per-component basis. All of the components are associated with the same spatial ex- tent in the source image, but represent different spectral or aux- iliary information. For example, a RGB color image has three components with one component representing each of the red, green, and blue color planes. In the simple case of a grayscale image, there is only one component, corresponding to the lu- minance plane. The various components of an image need not Given an image, the codec describes the geometry of the var- ious components in terms of a rectangular grid called the ref- erence grid. The reference grid has the general form shown in Fig. 2. The grid is of size Xsiz × Ysiz with the origin lo- cated at its top-left corner. The region with its top-left corner at (XOsiz, YOsiz) and bottom-right corner at (Xsiz − 1, Ysiz − 1) is called the image area, and corresponds to the picture data to be represented. The width and height of the reference grid can- not exceed 232 − 1 units, imposing an upper bound on the size of an image that can be handled by the codec. All of the components are mapped onto the image area of the reference grid. Since components need not be sampled at the full resolution of the reference grid, additional information is required in order to establish this mapping. For each com- ponent, we indicate the horizontal and vertical sampling period in units of the reference grid, denoted as XRsiz and YRsiz, re- Component 1Component 2...Component 0Component N−1Component i.........(Xsiz−1,Ysiz−1)(XOsiz,YOsiz)Xsiz−XOsizImage Area(0,0)XsizYsiz−YOsizYsizXOsizYOsiz
    TABLE I PARTS OF THE STANDARD Part 1 2 Title Core coding system Extensions 3 4 5 Motion JPEG-2000 Conformance testing Reference software Purpose Specifies the core (or minimum functionality) codec for the JPEG-2000 family of standards. Specifies additional functionalities that are useful in some applications but need not be supported by all codecs. Specifies extensions to JPEG-2000 for intraframe-style video compression. Specifies the procedure to be employed for compliance testing. Provides sample software implementations of the standard to serve as a guide for implementors. 3 XRsiz YRsiz YRsiz XRsiz , Xsiz YOsiz YRsiz − XOsiz XRsiz × Ysiz − YOsiz XOsiz spectively. These two parameters uniquely specify a (rectangu- lar) sampling grid consisting of all points whose horizontal and vertical positions are integer multiples of XRsiz and YRsiz, re- spectively. All such points that fall within the image area, con- stitute samples of the component in question. Thus, in terms of its own coordinate system, a component will have the size and its top-left sam- ple will correspond to the point . Note that the reference grid also imposes a particular alignment of sam- ples from the various components relative to one another. From the diagram, the size of the image area is (Xsiz − XOsiz)×(Ysiz−YOsiz). For a given image, many combinations of the Xsiz, Ysiz, XOsiz, and YOsiz parameters can be chosen to obtain an image area with the same size. Thus, one might won- der why the XOsiz and YOsiz parameters are not fixed at zero while the Xsiz and Ysiz parameters are set to the size of the image. As it turns out, there are subtle implications to chang- ing the XOsiz and YOsiz parameters (while keeping the size of the image area constant). Such changes affect codec behavior in several important ways, as will be described later. This behavior allows a number of basic operations to be performed efficiently on coded images, such as cropping, horizontal/vertical flipping, and rotation by an integer multiple of 90 degrees. Fig. 3. Tiling on the reference grid. C. Tiling In some situations, an image may be quite large in compar- ison to the amount of memory available to the codec. Conse- quently, it is not always feasible to code the entire image as a single atomic unit. To solve this problem, the codec allows an image to be broken into smaller pieces, each of which is inde- pendently coded. More specifically, an image is partitioned into one or more disjoint rectangular regions called tiles. As shown in Fig. 3, this partitioning is performed with respect to the ref- erence grid by overlaying the reference grid with a rectangu- lar tiling grid having horizontal and vertical spacings of XTsiz and YTsiz, respectively. The origin of the tiling grid is aligned with the point (XTOsiz, YTOsiz). Tiles have a nominal size of XTsiz × YTsiz, but those bordering on the edges of the image area may have a size which differs from the nominal size. The tiles are numbered in raster scan order (starting at zero). By mapping the position of each tile from the reference grid to the coordinate systems of the individual components, a parti- tioning of the components themselves is obtained. For example, suppose that a tile has an upper left corner and lower right corner with coordinates (tx0, ty0) and (tx1 − 1, ty1 − 1), respectively. Then, in the coordinate space of a particular component, the tile would have an upper left corner and lower right corner with co- ordinates (tcx0, tcy0) and (tcx1−1, tcy1−1), respectively, where Fig. 4. Tile-component coordinate system. (tcx0, tcy0) = (tx0/XRsiz ,ty0/YRsiz) (tcx1, tcy1) = (tx1/XRsiz ,ty1/YRsiz) . (1a) (1b) These equations correspond to the illustration in Fig. 4. The por- tion of a component that corresponds to a single tile is referred to as a tile-component. Although the tiling grid is regular with respect to the reference grid, it is important to note that the grid may not necessarily be regular with respect to the coordinate systems of the components. T7T0T1T2T3T4T5T6T8(XOsiz,YOsiz)(0,0)(XTOsiz,YTOsiz)XTOsizXTsizXTsizXTsizYTsizYTsizYTsizYTOsizYsizXsiztcx0tcy0tcx1tcy1(0,0)Tile−Component Data( , )( −1, −1)
    4 D. Codec Structure The general structure of the codec is shown in Fig. 5 with the form of the encoder given by Fig. 5(a) and the decoder given by Fig. 5(b). From these diagrams, the key processes associated with the codec can be identified: 1) preprocess- ing/postprocessing, 2) intercomponent transform, 3) intracom- ponent transform, 4) quantization/dequantization, 5) tier-1 cod- ing, 6) tier-2 coding, and 7) rate control. The decoder structure essentially mirrors that of the encoder. That is, with the excep- tion of rate control, there is a one-to-one correspondence be- tween functional blocks in the encoder and decoder. Each func- tional block in the decoder either exactly or approximately in- verts the effects of its corresponding block in the encoder. Since tiles are coded independently of one another, the input image is (conceptually, at least) processed one tile at a time. In the sections that follow, each of the above processes is examined in more detail. E. Preprocessing/Postprocessing The codec expects its input sample data to have a nominal dynamic range that is approximately centered about zero. The preprocessing stage of the encoder simply ensures that this ex- pectation is met. Suppose that a particular component has P bits/sample. The samples may be either signed or unsigned, leading to a nominal dynamic range of [−2P−1, 2P−1 − 1] or [0, 2P − 1], respectively. If the sample values are unsigned, the nominal dynamic range is clearly not centered about zero. Thus, the nominal dynamic range of the samples is adjusted by subtracting a bias of 2P−1 from each of the sample values. If the sample values for a component are signed, the nominal dy- namic range is already centered about zero, and no processing is required. By ensuring that the nominal dynamic range is cen- tered about zero, a number of simplifying assumptions could be made in the design of the codec (e.g., with respect to context modeling, numerical overflow, etc.). The postprocessing stage of the decoder essentially undoes the effects of preprocessing in the encoder. If the sample val- ues for a component are unsigned, the original nominal dynamic range is restored. Lastly, in the case of lossy coding, clipping is performed to ensure that the sample values do not exceed the allowable range. F. Intercomponent Transform In the encoder, the preprocessing stage is followed by the for- ward intercomponent transform stage. Here, an intercomponent transform can be applied to the tile-component data. Such a transform operates on all of the components together, and serves to reduce the correlation between components, leading to im- proved coding efficiency. Only two intercomponent transforms are defined in the base- line JPEG-2000 codec: the irreversible color transform (ICT) and reversible color transform (RCT). The ICT is nonreversible and real-to-real in nature, while the RCT is reversible and integer-to-integer. Both of these transforms essentially map im- age data from the RGB to YCrCb color space. The transforms are defined to operate on the first three components of an image, with the assumption that components 0, 1, and 2 correspond to the red, green, and blue color planes. Due to the nature of these transforms, the components on which they operate must be sampled at the same resolution (i.e., have the same size). As a consequence of the above facts, the ICT and RCT can only be employed when the image being coded has at least three com- ponents, and the first three components are sampled at the same resolution. The ICT may only be used in the case of lossy cod- ing, while the RCT can be used in either the lossy or lossless case. Even if a transform can be legally employed, it is not necessary to do so. That is, the decision to use a multicompo- nent transform is left at the discretion of the encoder. After the intercomponent transform stage in the encoder, data from each component is treated independently. The ICT is nothing more than the classic RGB to YCrCb color space transform. The forward transform is defined as V0(x, y) V1(x, y) V2(x, y)  =  0.299 0.5 −0.16875 −0.33126 0.587 0.114 0.5 −0.41869 −0.08131 U0(x, y) U1(x, y) U2(x, y)  (2) where U0(x, y), U1(x, y), and U2(x, y) are the input compo- nents corresponding to the red, green, and blue color planes, respectively, and V0(x, y), V1(x, y), and V2(x, y) are the output components corresponding to the Y, Cr, and Cb planes, respec- tively. The inverse transform can be shown to be U0(x, y)  = U1(x, y) U2(x, y) 1 0 1 −0.34413 −0.71414 1 −1.772 0 1.402 V0(x, y)  V1(x, y) V2(x, y) (3) The RCT is simply a reversible integer-to-integer approxima- tion to the ICT (similar to that proposed in [14]). The forward transform is given by 4 (U0(x, y) + 2U1(x, y) + U2(x, y)) 1 V0(x, y) = V1(x, y) = U2(x, y) − U1(x, y) V2(x, y) = U0(x, y) − U1(x, y) where U0(x, y), U1(x, y), U2(x, y), V0(x, y), V1(x, y), and V2(x, y) are defined as above. The inverse transform can be shown to be U1(x, y) = V0(x, y) − 1 4 (V1(x, y) + V2(x, y)) U0(x, y) = V2(x, y) + U1(x, y) U2(x, y) = V1(x, y) + U1(x, y) The inverse intercomponent transform stage in the decoder essentially undoes the effects of the forward intercomponent transform stage in the encoder. If a multicomponent transform was applied during encoding, its inverse is applied here. Unless the transform is reversible, however, the inversion may only be approximate due to the effects of finite-precision arithmetic. G. Intracomponent Transform Following the intercomponent transform stage in the encoder is the intracomponent transform stage. In this stage, transforms that operate on individual components can be applied. The par- ticular type of operator employed for this purpose is the wavelet transform. Through the application of the wavelet transform, a component is split into numerous frequency bands (i.e., sub- bands). Due to the statistical properties of these subband signals, (4a) (4b) (4c) (5a) (5b) (5c)
    5 h+ - y0[n] - s0 6 Qλ−1 6 Aλ−1(z) - - 6 - - ? Aλ−2(z) ? Qλ−2 ? h+ h+ - + − 6 Q1 - - ? A0(z) ? Q0 + − ?- h+ 6 A1(z) - -6 ··· ··· ··· ··· (a) ··· - ··· ··· ··· (b) y1[n] - s1 h+ ↑ 2 - x[n] - 6 z−1 6 ↑ 2 - Fig. 6. Lifting realization of 1-D 2-channel PR UMDFB. (a) Analysis side. (b) Synthesis side. wavelet decomposition is associated with R resolution levels, numbered from 0 to R − 1, with 0 and R − 1 corresponding to the coarsest and finest resolutions, respectively. Each sub- band of the decomposition is identified by its orientation (e.g., LL, LH, HL, HH) and its corresponding resolution level (e.g., 0, 1, . . . , R − 1). The input tile-component signal is consid- ered to be the LLR−1 band. At each resolution level (except the lowest) the LL band is further decomposed. For example, the LLR−1 band is decomposed to yield the LLR−2, LHR−2, HLR−2, and HHR−2 bands. Then, at the next level, the LLR−2 band is decomposed, and so on. This process repeats until the LL0 band is obtained, and results in the subband structure illus- trated in Fig. 7. In the degenerate case where no transform is applied, R = 1, and we effectively have only one subband (i.e., the LL0 band). As described above, the wavelet decomposition can be as- sociated with data at R different resolutions. Suppose that the top-left and bottom-right samples of a tile-component have co- ordinates (tcx0, tcy0) and (tcx1−1, tcy1−1), respectively. This Fig. 5. Codec structure. The structure of the (a) encoder and (b) decoder. (a) (b) ↓ 2 x[n] - - ? z ? - ↓ 2 h+ - 6 Q1 6 A1(z) -- 6 -- ? A0(z) ? Q0 ? h+ h+ - y0[n] - s−1 0 y1[n] - s−1 1 - + − 6 Qλ−1 6 Aλ−1(z) - 6 - - ? Aλ−2(z) ? Qλ−2 − ?- h+ - + the transformed data can usually be coded more efficiently than the original untransformed data. Both reversible integer-to-integer [13, 17–19] and nonre- versible real-to-real wavelet transforms are employed by the baseline codec. The basic building block for such transforms is the 1-D 2-channel perfect-reconstruction (PR) uniformly- maximally-decimated (UMD) filter bank (FB) which has the general form shown in Fig. 6. Here, we focus on the lifting realization of the UMDFB [20, 21], as it can be used to imple- ment the reversible integer-to-integer and nonreversible real-to- real wavelet transforms employed by the baseline codec. In fact, for this reason, it is likely that this realization strategy will be employed by many codec implementations. The analysis side of the UMDFB, depicted in Fig. 6(a), is associated with the for- ward transform, while the synthesis side, depicted in Fig. 6(b), is associated with the inverse transform. In the diagram, the {Ai(z)}λ−1 i=0 denote filter transfer functions, quantization operators, and (scalar) gains, respec- tively. To obtain integer-to-integer mappings, the {Qi(x)}λ−1 i=0 are selected such that they always yield integer values, and the {si}1 i=0 are chosen as integers. For real-to-real mappings, the {Qi(x)}λ−1 i=0 are simply chosen as the identity, and the {si}1 i=0 are selected from the real numbers. To facilitate filtering at sig- nal boundaries, symmetric extension [22] is employed. Since an image is a 2-D signal, clearly we need a 2-D UMDFB. By applying the 1-D UMDFB in both the horizontal and vertical di- rections, a 2-D UMDFB is effectively obtained. The wavelet transform is then calculated by recursively applying the 2-D UMDFB to the lowpass subband signal obtained at each level in the decomposition. i=0 , {Qi(x)}λ−1 i=0 , and {si}1 Suppose that a (R − 1)-level wavelet transform is to be em- ployed. To compute the forward transform, we apply the anal- ysis side of the 2-D UMDFB to the tile-component data in an iterative manner, resulting in a number of subband signals be- ing produced. Each application of the analysis side of the 2-D UMDFB yields four subbands: 1) horizontally and vertically lowpass (LL), 2) horizontally lowpass and vertically highpass (LH), 3) horizontally highpass and vertically lowpass (HL), and 4) horizontally and vertically highpass (HH). A (R − 1)-level PreprocessingForwardIntercomponentTransformForwardIntracomponentTransformQuantizationTier−1EncoderTier−2EncoderCodedImageRate ControlOriginalImageTier−2DecoderTier−1DecoderInverseIntracomponentTransformPostprocessingDequantizationInverseIntercomponentTransformCodedImageReconstructedImage
    6 of the original data.) As will be seen, the coordinate systems for the various resolutions and subbands of a tile-component play an important role in codec behavior. By examining (1), (6), and (7), we observe that the coordi- nates of the top-left sample for a particular subband, denoted (tbx0, tby0), are partially determined by the XOsiz and YOsiz parameters of the reference grid. At each level of the decompo- sition, the parity (i.e., oddness/evenness) of tbx0 and tby0 affects the outcome of the downsampling process (since downsampling is shift variant). In this way, the XOsiz and YOsiz parameters have a subtle, yet important, effect on the transform calculation. Having described the general transform framework, we now describe the two specific wavelet transforms supported by the baseline codec: the 5/3 and 9/7 transforms. The 5/3 transform is reversible, integer-to-integer, and nonlinear. This transform was proposed in [13], and is simply an approximation to a linear wavelet transform proposed in [23]. The 5/3 transform has an underlying 1-D UMDFB with the parameters: λ = 2, A0(z) = − 1 Q0(x) = −−x , Q1(x) = 2(z + 1), A1(z) = 1 x + 1 2 , 4(1 + z−1), s0 = s1 = 1. (8) The 9/7 transform is nonreversible and real-to-real. This trans- form, proposed in [10], is also employed in the FBI fingerprint compression standard [24] (although the normalizations differ). The 9/7 transform has an underlying 1-D UMDFB with the pa- rameters: λ = 4, A0(z) = α0(z + 1), A1(z) = α1(1 + z−1), A2(z) = α2(z + 1), A3(z) = α3(1 + z−1), (9) Qi(x) = x for i = 0, 1, 2, 3, α0 ≈ −1.586134, α1 ≈ −0.052980, α2 ≈ 0.882911, α3 ≈ 0.443506, s0 ≈ 1.230174, s1 = 1/s0. Fig. 7. Subband structure. being the case, the top-left and bottom-right samples of the tcy0/2R−r−1 tile-component at resolution r have coordinates (trx0, try0) and (trx1 − 1, try1 − 1), respectively, given by tcy1/2R−r−1 tcx0/2R−r−1 tcx1/2R−r−1 (trx0, try0) = (trx1, try1) = (6a) (6b) , , where r is the particular resolution of interest. Thus, the tile- component signal at a particular resolution has the size (trx1 − trx0) × (try1 − try0). Not only are the coordinate systems of the resolution levels important, but so too are the coordinate systems for the various subbands. Suppose that we denote the coordinates of the upper left and lower right samples in a subband as (tbx0, tby0) and (tbx1−1, tby1−1), respectively. These quantities are computed as (tbx0, tby0)   , , 2 tcx0 tcx0 tcx0 tcx0 tcx1 tcx1 tcx1 tcx1 2R−r−1 2R−r−1 − 1 2R−r−1 2R−r−1 − 1 2R−r−1 2R−r−1 − 1 2R−r−1 2R−r−1 − 1 2 2 2 , , = = (tbx1, tby1) , 2 tcy0 tcy0 tcy0 , tcy0 2R−r−1 2R−r−1 2R−r−1 − 1 2R−r−1 − 1 2R−r−1 − 1 2R−r−1 2R−r−1 − 1 2R−r−1 , tcy1 tcy1 tcy1 tcy1 2 2 2 , for LL band for HL band (7a) for LH band for HH band for LL band for HL band (7b) for LH band for HH band Since the 5/3 transform is reversible, it can be employed for either lossy or lossless coding. The 9/7 transform, lacking the reversible property, can only be used for lossy coding. The num- ber of resolution levels is a parameter of each transform. A typi- cal value for this parameter is six (for a sufficiently large image). The encoder may transform all, none, or a subset of the compo- nents. This choice is at the encoder’s discretion. The inverse intracomponent transform stage in the decoder essentially undoes the effects of the forward intracomponent transform stage in the encoder. If a transform was applied to a particular component during encoding, the corresponding in- verse transform is applied here. Due to the effects of finite- precision arithmetic, the inversion process is not guaranteed to be exact unless reversible transforms are employed. H. Quantization/Dequantization where r is the resolution level to which the band belongs, R is the number of resolution levels, and tcx0, tcy0, tcx1, and tcy1 are as defined in (1). Thus, a particular band has the size (tbx1 − tbx0)×(tby1−tby0). From the above equations, we can also see that (tbx0, tby0) = (trx0, try0) and (tbx1, tby1) = (trx1, try1) for the LLr band, as one would expect. (This should be the case since the LLr band is equivalent to a reduced resolution version In the encoder, after the tile-component data has been trans- formed (by intercomponent and/or intracomponent transforms), the resulting coefficients are quantized. Quantization allows greater compression to be achieved, by representing transform coefficients with only the minimal precision required to obtain the desired level of image quality. Quantization of transform co- efficients is one of the two primary sources of information loss LL0R−2HHLHR−2HLR−2HLR−1HHR−1LHR−1.........
    in the coding path (the other source being the discarding of cod- ing pass data as will be described later). Transform coefficients are quantized using scalar quantiza- tion with a deadzone. A different quantizer is employed for the coefficients of each subband, and each quantizer has only one parameter, its step size. Mathematically, the quantization pro- cess is defined as V (x, y) = |U(x, y)| /∆ sgn U(x, y) (10) where ∆ is the quantizer step size, U(x, y) is the input sub- band signal, and V (x, y) denotes the output quantizer indices for the subband. Since this equation is specified in an infor- mative clause of the standard, encoders need not use this precise formula. This said, however, it is likely that many encoders will, in fact, use the above equation. The baseline codec has two distinct modes of operation, re- ferred to herein as integer mode and real mode. In integer mode, all transforms employed are integer-to-integer in nature (e.g., RCT, 5/3 WT). In real mode, real-to-real transforms are em- ployed (e.g., ICT, 9/7 WT). In integer mode, the quantizer step sizes are always fixed at one, effectively bypassing quantization and forcing the quantizer indices and transform coefficients to be one and the same. In this case, lossy coding is still possible, but rate control is achieved by another mechanism (to be dis- cussed later). In the case of real mode (which implies lossy cod- ing), the quantizer step sizes are chosen in conjunction with rate control. Numerous strategies are possible for the selection of the quantizer step sizes, as will be discussed later in Section III-L. As one might expect, the quantizer step sizes used by the en- coder are conveyed to the decoder via the code stream. In pass- ing, we note that the step sizes specified in the code stream are relative and not absolute quantities. That is, the quantizer step size for each band is specified relative to the nominal dynamic range of the subband signal. In the decoder, the dequantization stage tries to undo the ef- fects of quantization. Unless all of the quantizer step sizes are less than or equal to one, the quantization process will normally result in some information loss, and this inversion process is only approximate. The quantized transform coefficient values are obtained from the quantizer indices. Mathematically, the de- quantization process is defined as U(x, y) = (V (x, y) + r sgn V (x, y)) ∆ (11) where ∆ is the quantizer step size, r is a bias parameter, V (x, y) are the input quantizer indices for the subband, and U(x, y) is the reconstructed subband signal. Although the value of r is not normatively specified in the standard, it is likely that many decoders will use the value of one half. I. Tier-1 Coding After quantization is performed in the encoder, tier-1 coding takes place. This is the first of two coding stages. The quantizer indices for each subband are partitioned into code blocks. Code blocks are rectangular in shape, and their nominal size is a free parameter of the coding process, subject to certain constraints, most notably: 1) the nominal width and height of a code block 7 Fig. 8. Partitioning of subband into code blocks. must be an integer power of two, and 2) the product of the nom- inal width and height cannot exceed 4096. Suppose that the nominal code block size is tentatively cho- sen to be 2xcb × 2ycb. In tier-2 coding, yet to be discussed, code blocks are grouped into what are called precincts. Since code blocks are not permitted to cross precinct boundaries, a re- duction in the nominal code block size may be required if the precinct size is sufficiently small. Suppose that the nominal code block size after any such adjustment is 2xcb’ × 2ycb’ where xcb’ ≤ xcb and ycb’ ≤ ycb. The subband is partitioned into code blocks by overlaying the subband with a rectangular grid having horizontal and vertical spacings of 2xcb’ and 2ycb’, respec- tively, as shown in Fig. 8. The origin of this grid is anchored at (0, 0) in the coordinate system of the subband. A typical choice for the nominal code block size is 64 × 64 (i.e., xcb = 6 and ycb = 6). Let us, again, denote the coordinates of the top-left sample in a subband as (tbx0, tby0). As explained in Section III-G, the quantity (tbx0, tby0) is partially determined by the refer- ence grid parameters XOsiz and YOsiz. In turn, the quantity (tbx0, tby0) affects the position of code block boundaries within a subband. In this way, the XOsiz and YOsiz parameters have an important effect on the behavior of the tier-1 coding process (i.e., they affect the location of code block boundaries). After a subband has been partitioned into code blocks, each of the code blocks is independently coded. The coding is per- formed using the bit-plane coder described later in Section III-J. For each code block, an embedded code is produced, comprised of numerous coding passes. The output of the tier-1 encoding process is, therefore, a collection of coding passes for the vari- ous code blocks. On the decoder side, the bit-plane coding passes for the var- ious code blocks are input to the tier-1 decoder, these passes are decoded, and the resulting data is assembled into subbands. In this way, we obtain the reconstructed quantizer indices for tbx0tby0( , )tbx1tby1( −1, −1)B0B2B5B4B3B1B6B7B82xcb’2xcb’2xcb’2ycb’2ycb’2ycb’2xcb’2ycb’(m , n )(0,0).....................
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