This Communication concerns nonuniform omission of tions. We emphasize that the k y distributions thus obtained phase-encode values (k y ), aimed at reduction of Cartesian need not be considered as random or stochastic sampling. MRI scan times (1, 2). More precisely, it shows that a good Rather, a complete set of 10 26 distributions is represented by nonuniform k y distribution need not be limited to a set of a reduced, arbitrarily chosen set of one thousand. similar objects as implied in (1), but is, in fact, still applica-For each of the thousand distributions, the missing data ble to objects with other shapes. In addition, it shows that on omitted k y values were estimated by (i) a Bayesian (11) modifying the k y distribution need not spoil the quality of reconstruction method and (ii) two edge-preserving filter the reconstruction if certain rules are satisfied. This investimethods. The Bayesian reconstruction method (9, 12) regation is done for a symmetric nonuniform distribution yieldquires prior knowledge on MR images to estimate missing ing a scan time reduction of 30%. However, since nonunidata on omitted k y values. The prior knowledge we use conform scanning can be combined with one-sided scanning sists of (1) low intensity outside the perimeter of the object, (3, 4), this 30% reduction actually amounts to about 58%. (2) low intensity in the imaginary part of the complex image Note that nonuniform k y distributions require interpolation after phase correction, and (3) the Lorentzian distribution to retrieve the missing data. Uniform truncated k y distribuof edge intensities inside the object (2, 13, 14). Measured tions, on the other hand, require extrapolation (5-8). Since data (as opposed to missing data) are treated as best possible interpolation is more reliable than extrapolation, the image estimates and therefore remain unaltered. The resulting posquality resulting from nonuniform sampling supersedes that terior distribution is optimized using the conjugate gradient resulting from uniform truncated sampling. method (15). In the following, we rate 1000 different symmetric non-Edge-preserving filter methods do not require prior knowluniform 30% time-saving k y distributions according to ensuedge on MR images. They utilize only some qualitative proping image quality, using a single real-world scan and a erties like smooth regions with sharp changes. We have Bayesian reconstruction method (9). Subsequently, the qualchosen a sigma filter (6) and a Gaussian filter. The latter ity rating is tested on real-world scans of five other objects was originally developed to reduce noise in MR images using the same Bayesian reconstruction method and two (16, 17), but has also been used to reduce ringing artifacts edge-preserving filter methods. This test reveals that the de-( 18). Both filter reconstruction methods are iterative. In each tailed form of a good distribution is indeed not critical. Of iteration, the filter is applied to the real part of the phasecourse, each object has a unique optimal k y distribution but corrected complex-valued image. The imaginary part, supthe present work suggests that such a distribution will also posedly containing only noise, is set to zero. The real-valued be good for a large number of other objects.
filtered image is treated as an estimate of the wanted image. The number of symmetric nonuniform 30% time-saving k y
Putting back the phase information onto the filtered image distributions is prohibitively large: Taking into account that and Fourier transforming provides an estimate of the k-space the values k y ร
032, . . . , 32 are to be retained to enable data. Only estimates of missing data are retained while the phase correction (10) and that the image resolution equals measured data remain unaltered. Both filter methods contain 256 1 256, one arrives at [ 127 0 32 0.3 1 127 ] ร 10 26 possible distribua parameter which controls the rate of smoothing. In each tions. In order to keep the task practical, a mere thousand iteration, this parameter is adapted to enable the algorithm to distributions were selected from this large set with the aid of detect step edges which were lost in the previous smoothing a MATLAB random generator (The MathWorks Inc., Camoperation. In essence, the procedure just described is the bridge, Massachusetts). Starting from fully measured scans, data were then erased according to these thousand distribu-same as in (8). However, to reduce noise influence, we 70