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where C is the CFL number. The Godunov scheme is stable so long as t satis es equation (7.26) with C < 1. This CFL condition is not the only stability constraint that t must satisfy, however. In particular, a consequence of our explicit treatment of the Lagrangian equations of motion is that t must satisfy an additional constraint of the form t = O(h2max ) or even t = O(h4max ), depending on the de nition of F.5 Consequently, t typically satis es equation (7.26) for C 1, so that in a single timestep, each curvilinear mesh node may move at most C 1 fractional meshwidths per timestep in any coordinate direction. Over the course of 1/C timesteps, each curvilinear mesh node will move less than one Cartesian meshwidth in any coordinate direction. By constructing the locally re ned Cartesian grid according to the foregoing tagging criteria, it suf ces to rebuild the patch hierarchy every 1/C timesteps. For instance, by ensuring that t satis es equation (7.26) for C 0.1, it suf ces to reconstruct the locally re ned Cartesian grid every 10 timesteps.

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Figure 4.12. Atmospheric Pressure Chemical Ionization. Sample is vapourized by heat and nebulizing gas. The vapour contains both sample and solvent. Solvent is ionized by a corona discharge from the nearby discharge electrode. Sample molecules are in turn ionized by the excited solvent molecules in a CI process. The vapour is sampled and subsequently analysed by MS.

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of the evidence of subperiosteal resorption based on the reconstructed images with the predetermined compression ratio will be as good as that made (by this observer) from the original images In other words, this compression ratio is acceptable for making the diagnosis on hand radiographs with bone re-absorption In doing ROC analysis, the statistical power of the study is important: the higher the power, the more con dence can be placed in the result The statistical power is determined by the number of images and the number of observers used in the study A meaningful ROC analysis often requires many images (100 or more) and ve to six observers to determine one type of image with a given disease Although performing an ROC analysis is tedious, time- consuming, and expensive, this method is acceptable by the radiology community for determination of the quality of the reconstructed image.

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For example, we describe a study based on work by Sayre et al (1992) entailing the analysis of 71 hand radiographs, of which 45 were normal and 26 had subperiosteal resorption The images were digitized to 2K 2K 12-bit resolution and printed on lm (14 17in) The digitized images were compressed to 20:1 using the full-frame cosine transform method and printed on lm of the same size Figures 69 and 610 show the results from using ROC analysis with ve observers Figure 69A, B , and C show an original hand radiograph with evidence of subperiosteal resorption, 20:1 compression ratio reconstructed image, and the difference image, respectively Figure 6.

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10 shows the ROC curves of the original and the reconstructed images from the ve observers: statistics demonstrate that there is no signi cant difference between using the original or the reconstructed images with the 20:1 compression ratio for the diagnosis of subperiosteal resorption from hand radiographs 66 661 THREE-DIMENSIONAL IMAGE COMPRESSION Background.

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