Desirable Properties

The quality of geometric operations on images or volumes is very relevant in the context of biomedical data analysis. For example, the comparison of images taken with two different conditions requires that the geometric operation that aligns one with the other be high-quality in order to allow a detailed analysis of their differences. That is to say, it is not enough that the geometric alignment be correctly specified; it is also crucial that it be correctly performed. Typical instances of that class of problems involve functional magnetic resonance imaging (fMRI), where the relative amplitude of the difference between two conditions (e.g., active vs inactive) is very small. High quality (fidelity to the original) is also a desirable trait in common display operations such as the reslicing of anisotropic data or the change of scale and orientation of many pictorial representations. Typically, a physician will request that rotating or zooming (magnifying) a region of interest of an image introduce no spurious features, while keeping all the minute details he might be interested in.

The price to pay for high-quality interpolation is computa tion time. For this reason, it is important to select a synthesis function that offers the best trade-off. There are several aspects to consider. The most important deals with the support of or which is a measure of the interval in which we have that ^(x) ^ 0. The larger the support, the longer the computation time. Another important aspect is the quality of approximation inherent in the synthesis function. Often, the larger the support, the better the quality; but it should be noted that the quality of synthesis functions with identical support may vary. Other aspects involve the ease of analytical manipulation (when useful), the ease of computation, and the efficiency of the determination of the coefficients ck when ^ is used instead of Pinr

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