Some NMR experiments produce data with several of the initial points missing. The inverse discrete Fourier transform (IDFT) assumes these points are present so the data cannot be so transformed without artifact-ridden results. This problem is often particularly severe when projection imaging with free-induction decays (FIDs). This paper compares recent methods for obtaining a projection from incomplete data and elaborates on their strengths and limitations. One method is to write the transform that would take the desired projection to the truncated data set, and then solve the matrix equation by singular value decomposition. A second replaces the missing data with zeros, so that an IDFT produces a projection with unwanted artifacts. Then one solves the matrix equation that takes the desired projection to the artifact-ridden projection. A third uses the same artifact-ridden projection, but fits the region outside the bandwidth of the sample with as many sinusoidal functions as there are missing data. The coefficients of these functions are estimates of the missing data, and the projection is obtained by transforming the completed FID or subtracting the extrapolation of the fitted curve from the region containing the object. We show that when all three methods are applicable, they theoretically produce the same result. They differ by ease of implementation and possibly by computational errors. They give a result similar to that of the previous method that iteratively corrects the FID and projection after repeated IDFTs and DFTs. We find that one can obtain a projection despite missing a substantial number of data.
- Compact support
- Partial-data transform