MRI demodulation frequency changes provide different information
Quang Minh Tieng, Gary Cowin, David C. Reutens, Graham J. Galloway, Viktor Vegh · Magnetic Resonance in Medicine · 2011
Uecker and coworkers claim that our conclusion in Ref.1 was supported by neither the theoretical arguments nor the experimental data presented. They purport that image improvements were due to an incorrect restoration of the low-resolution signal as a consequence of the interpolation method used, and high-frequency noise causes an illusion of resolution enhancement. We used the zero-filling approach to implement sinc interpolation (2), which is identical to their method. We tested our method with and without the addition of noise, and showed that improvements were present in both cases (see Figs. 4 and 5 in Ref.1). Therefore, image enhancement was not an illusion caused by high-frequency noise components. Furthermore, comparison of Fig. 4(a) to (b) in Ref.1 shows that enhancement is most observable at object features, where dimensions are near the effective width of the point spread function (PSF). We conducted a point source simulation to explain why image shifts provide different information to sinc interpolation. According to sampling theory, if a continuous signal is sampled at the Nyquist rate, then it can be recovered exactly using the sinc function over an infinite number of sample points (3). In practice, however, sampling is finite. Therefore, we considered a finite number of samples and different sampling locations with respect to point source location. In Fig. 1, a point source is represented as a sinc function, corresponding to the shape of the PSF. The solid line represents the real data, sampled according to the Nyquist rate at locations of circles. The dot-dashed line is the interpolation obtained by zero-filling of the Fourier transform of the discrete samples. We are interested in the magnitude of the main lobe of the interpolated data, because it defines point source strength. We ignore side lobes, as they contribute to image background noise. Figure 1 shows that, when point source and sampling location misalign, the interpolated main lobe peak is smaller than that of the solid line. Here, we found the ratio between the height of the peak of the interpolated and solid curves, when the point source is at a half distance between sampling points, to be 0.92, 0.96, and 0.97 for 10, 20, and 30 samples, respectively. Therefore, the difference reduces as a number of samples increase. A single point source represented by a continuous sinc function (solid line), sample points (circles) along the sinc function, and the interpolated signal (dot-dashed) obtained from these sample points. Figure 2 depicts two point sources, one aligned with a sampling point and another shifted by two and a half distances between sampling points. The result shows that even with a finite number of samples, interpolation can capture point source strength when sampling location and point source align. Otherwise, the value is smaller than expected. This specific simulation demonstrates that using many differently positioned point sources with a common intensity gives different interpolated values, which depend on point source location. The different values result in image blurring, or signal distortion. Two point sources represented by summation of two sinc functions (solid line). The first point source is at a sampling position and the other is shifted by two and half pixels with respect to the first one. Circles denote sampling points along the solid line. The dot-dashed line is the interpolated signal derived from the discrete samples. To remove distortions, the point source needs to be aligned with the sampling location. In MRI, alignment can be achieved by changing the instrument demodulation frequency. The result is linear phase shift in the frequency domain and sample shift with respect to the spatial domain field-of-view. The effect depends on whether the linear phase shift is applied before or after echo signal digitization. The original data shifts if linear phase shift is used before digitization. A postdigitization phase change results in a shift in interpolated data, and the true value of point source strength cannot be recovered. The interpolation tends toward the original data when the number of samples is large; hence the two approaches yield a similar outcome. Uecker and coworkers have not provided sufficient information about the phantom used in their experiment nor the size of the PSF. This information is important where phantom detail is small, otherwise improvements cannot easily be observed. From our analysis of their Fig. 2, we conclude that some phantom features could be around the size of the PSF, particularly the width of the dot-dashed vertical line. However, the line aligns with the sampling grid; hence, distortions and improvements are not seen. Furthermore, the multiple shifts used are orthogonal to the width of the dot-dashed line; thus, a line width improvement cannot be achieved. Figure 2 also shows a sample with small features surrounded by a large background. Our simulation verifies that when the number of side lobes of the PSF is increased, or image background is enlarged, the interpolated data tend toward the original data. We believe this is the reason why a difference between the combined and interpolated images was difficult to visualize.