The JPEG Algorithm. The image is ®rst separated into 8 ́ 8 pixel subimages. The DCT of each subimage then is computed. These coef®cients then are quantized using a quantization table (for this illustration, each value is divided by 5). Finally, the quantized values are encoded from the upper left corner, with a `marker value' sent when there are no more nonzero values. 

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... JPEG (Joint Photographic Experts Group) compression standard is a widely used compression method that includes both revers- ible and irreversible techniques, and has been described in detail by Wallace. 2 Although JPEG was not designed for medical imagery (ie, it was not de®ned for 12-or 16-bit intensity scales), it has been adapted for radiologic images as de- scribed by Gillespy and Rowberg. 3 Figure 1 shows how the algorithm operates. It begins by dividing the image into 8 pixeí 8 pixel blocks. ...

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... STORAGE and image transfer requirements of medical images have hampered attempts to implement picture ar- chiving and communications systems (PACS) and teleradiology. Image compression recently has been explored as a means of reducing costs of managing large image data sets. Lossless compression methods use redundancy within an image to more eciently transmit image information while allowing perfect reconstruction, but these methods achieve only 2:1 to 4:1 re- duction for medical image. 1 Irreversible or ``lossy'' techniques can reduce images by arbi- trarily large ratios, but do not perfectly repro- duce the original image. However, the reproduction may be good enough that there is no perceptible image degradation nor compro- mised diagnostic value. This report reviews the application of image compression techniques to medical imagery, focusing on the irreversible methods, including the JPEG2000 standard. Following that is a review of measures for evaluating compression algorithm performance and some of the recent results for wavelet compression. Most irreversible image compression techniques involve 3 steps: transformation, quantization, and encoding. Transformation is a lossless step in which the image is transformed from grayscale values in the spatial domain to coecients in some other domain. One familiar transform is the Fourier transform used in re- constructing magnetic resonance images (MRI). Other transforms such as the discrete cosine transform (DCT) and discrete wavelet transform (DWT) are more commonly used for image compression. No loss of information occurs in the transformation step. Quantization is the step in which data integrity is lost. It attempts to minimize information loss by preferentially preserving the most important coecients, whereas less important coecients are roughly approximated, often as zero. Quantization may be as simple as converting ̄oating point values to integer values. Finally, these quantized coecients are encoded. This also is a lossless step in which the quantized coecients are com- pactly represented for ecient storage or transmission of the image. The JPEG (Joint Photographic Experts Group) compression standard is a widely used compression method that includes both revers- ible and irreversible techniques, and has been described in detail by Wallace. 2 Although JPEG was not designed for medical imagery (ie, it was not de®ned for 12- or 16-bit intensity scales), it has been adapted for radiologic images as described by Gillespy and Rowberg. 3 Figure 1 shows how the algorithm operates. It begins by dividing the image into 8 pixel ́ 8 pixel blocks. The DCT of each image block is computed, resulting in an 8 ́ 8 block of spectral coecients. Most of the information is concentrated in relatively few coecients in the upper left corner of this DCT image. Quantization is performed next. In this step, the coecients are approximated to values that are easy to represent in a small amount of space. There is an 8 ́ 8 table (called the quantization table), which contains the values by which corresponding coecients are to be divided. By using dierent values, spectral frequencies that are more important to the visual system can be preserved preferentially over less- important frequencies. The resulting values then are rounded o to the nearest integer. JPEG encodes the quantized coecients by reordering them in a zigzag pattern. This places the largest values ®rst, with long strings of zeros at the end, which can be eciently represented. Although the JPEG lossy algorithm is good for many types of images, it has some draw- backs when applied to radiographic images. It degrades ungracefully at high compression ratios, with prominent artifacts at block bound- aries, and it cannot take advantage of patterns larger than the 8 ́ 8 pixel blocks. Wavelet- based compression schemes generally outper- form JPEG in terms of image quality at a given compression ratio, and the improvement can be dramatic at high compression ratios. 4 The DWT of an image is computed 5 (Fig 2) using a pair of high- and low-pass ®lters with special mathematical properties. Many such ``wavelet'' ®lters exist, but many groups have adopted the 9-tap/7-tap bi-orthogonal ®lters of Antonini et al, 5 because they seem to work well in real-world application. 6 The 2 ®lters split the image into 2 components or subbands in each direction (each is half the original size). This produces 4 subband images, 1 containing the low-frequency information, 1 each for the high- frequency information in the X or Y direction, and 1 for high-frequency information in both X and Y. The process is repeated on the low- frequency component, breaking it up into ``high-low'' and ``low-low'' components. If this process is performed n times, an n -level discrete wavelet transform is created. A 5-level discrete wavelet transform of an MRI is shown in Fig 2. The DWT is eective for compression because it eectively concentrates the information into a few coecients, with most other coecients being zero or close enough to zero that they can be considered zero without degrading the image. Most wavelet compression algorithms compute a 4- or 5-level DWT, quantize the resulting coecients, and eciently encode the quantized coecients. The quantization is performed by dividing each coecient by a quantization parameter and rounding o to the nearest integer. Having a larger quantization parameter will result in more coecients that are zero, and hence, increases the compression ratio. Finally, encoding converts the coecients into values that can be stored or transmitted eciently. It is the way that the nonzero coecients are encoded that dierentiates the advanced wavelet compression techniques. Figure 2 graphically shows the hierarchical structure of the DWT; advanced techniques capitalize on this tree-based organization of the coecients. The most well known of these techniques is embedded zerotree coding, described by Shapiro, 7 and enhanced by Said and Pearlman. 8,9 The latter approach, termed set partitioning in hierarchical trees (SPIHT), was one of the early successful advanced wavelet techniquesÐit yielded signi®cantly better results than conven- tional wavelet compression with similar com- putational complexity. 9 In addition to resulting in ecient compression, it also transmitted the compressed bitstream in which approximations of the most important coecients (regardless of location) are transmitted ®rst. The values of these coecients are progressively re®ned, and the most important remaining informa- tionÐthat which yields the largest distortion reductionsÐis transmitted next. It can be shown that such a transmission scheme (with uniform weighting) is the optimal way to de- crease the root-mean-square (RMS) error in the reconstructed image. 9 Because JPEG was speci®ed for computers that existed over a decade ago, and because new technologies like wavelet had surpassed JPEG for many types of images, the JPEG group set out to update the standard, which is now known as JPEG2000. For this paper, JPEG will refer to the older compression method, wavelet will refer to the family of speci®c wavelet methods, and JPEG2000 will refer to the de- veloping standard. The JPEG2000 eort has been substantial. This group identi®ed a number of shortcomings of the JPEG standard that JPEG2000 would address. Among these were: 1. Better performance at high compression ratios 2. A single codestream that would support irreversible and lossless compression ...