RESPONSE MRI Demodulation Frequency Changes Provide Different Information Quang M. Tieng, Gary J. Cowin, David C. Reutens, Graham J. Galloway, and Viktor Vegh* 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 iden- tical to their method. We tested our method with and with- out 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, compar- ison 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 interpo- lation. According to sampling theory, if a continuous sig- nal 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. Fig- ure 1 shows that, when point source and sampling loca- tion 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. Figure 2 depicts two point sources, one aligned with a sampling point and another shifted by two and a half dis- tances between sampling points. The result shows that FIG. 1. 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 sam- ple points. FIG. 2. Two point sources represented by summation of two sinc functions (solid line). The first point source is at a sampling posi- tion 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. Centre for Advanced Imaging, The University of Queensland, Brisbane, Australia. *Correspondence to: Viktor Vegh, Ph.D., Centre for Advanced Imaging, The University of Queensland, Research Road, Brisbane Q4072, Australia. E-mail: v.vegh@uq.edu.au Received 10 May 2011; revised 21 June 2011; accepted 23 June 2011. DOI 10.1002/mrm.23099 Published online 23 August 2011 in Wiley Online Library (wileyonlinelibrary. com). Magnetic Resonance in Medicine 66:1513–1514 (2011) V C 2011 Wiley Periodicals, Inc. 1513