Sub-pixel shifting of in vivo perfusion studies was shown to change the appearance of Gibbs artifacts. This effect was visible in the original uninterpolated images, and in the post-FFT interpolated images. The same shifted data interpolated by pre-FFT zero-filling exhibited much less variability in the Gibbs artifact. The in vivo findings were confirmed by phantom imaging and numerical simulations.
Unless pre-FFT zero-filling interpolation is performed, Gibbs artifacts are very dependent on the position of the subendocardial wall within the pixel. By introducing sub-pixel shifts relative to the endocardial border, some of the variability of the dark rim artifacts in different myocardial segments, in different patients and from frame to frame during first-pass perfusion due to cardiac and respiratory motion can be explained. Image interpolation by zero-filling can be used to minimize this dependency.
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This work showed that the DRA variability is reduced by zero-filling the raw-data before FFT and reconstructing a larger matrix image, since this technique upsamples (i.e. places more interpolated pixels on each cycle of) the Gibbs signal oscillations and thereby reduces the variability of Gibbs appearance; a factor higher than 2 in the interpolation would make the zero-filled data even less dependent on the edge position [12, 17]. On the other hand, if interpolation in the image space is used, the sub-pixel shift variability is the same as the original data. It is important to point out that zero-filling does not reduce the artifacts in any way; it only reduces their variability by reducing their dependency on the endocardial border position. Nevertheless a less variable artifact may be preferable, especially if quantification methods are used.
Motion artifacts are more complicated than Gibbs artifacts, in that their signal oscillation wavelength increases with distance from the sharp edge. Considering a simplification of a constant motion and a normal Cartesian k-space filling, the signal oscillations will have a first lobe width of approximately 2 pixels for a typical perfusion sequence protocol [6]. Motion artifacts alone are therefore wider and probably less prone to be affected by sub-pixel shifts, but motion artifacts may occur in combination with Gibbs artifacts, since the latter are always present (assuming no image filtering).
It was shown that the contribution made by Gibbs artifacts to DRAs in perfusion studies is very dependent on the position of the subendocardial wall inside the pixel in the absence of zero-filled pre-FFT interpolation. Position variations between patient studies and from frame to frame in a typical ECG-gated perfusion study can explain some of the variability often seen in DRAs. This work also showed that image-based pixel interpolation does not reduce this source of DRA variability. However, interpolation by zero-filling prior to FFT makes the DRA appearance less variable, i.e. reduces the artifact's ringing dependency on the position of the subendocardial wall in relation to image pixels.
A MVT was defined as the peak torque sustained for 200 ms observed across any one of the 3 trials for that primary joint direction. The corresponding secondary torques exerted along the other joint axes during the 200 ms MVT window were also identified. For example, during maximum voluntary knee flexion exertions, secondary torques consisted of those generated along the ankle flexion-extension axis, hip flexion-extension axis, and hip abduction-adduction axis. Secondary torques generated during all trials were normalized to the MVT measured for that particular joint direction. Cases where a secondary torque exceeded 100% MVT indicated that the subject generated less torque while attempting to maximize that particular direction than when they were trying to maximize a different direction.
A single factor ANOVA was used to compare the means of the chronic stroke subjects to the control subjects for each of the eight primary joint torque directions. A single factor ANOVA was used to compare the mean secondary torques, as well as the mean EMGs, between the stroke and control groups. An independent Student's t-test was used to identify secondary torques that were significantly greater than zero (P
The maximum voluntary primary torques for the eight joint directions are shown in figure 2. The stroke group was significantly weaker (p
Figures 3 through 6 show the normalized secondary torque patterns as well as the normalized EMG activity for all control subjects and all but one stroke subject during the eight different primary directions. EMG data for one stroke subject was improperly collected and has hence been omitted. The stick figure diagrams illustrate the secondary torque generation that was significantly greater than zero (P
As illustrated in Figure 3, during ankle flexion, both controls and stroke subjects generated knee extension and hip flexion secondary torques. While generating maximal ankle flexion, the stroke subjects had significantly less tibilias anterior activity but significantly greater gastrocnemius, biceps femoris, gluteus maximus, and gluteus medius activity. During maximal ankle extension exertions, the stroke subjects generated a knee flexion secondary torque that was significantly higher than the control subjects (p
For each group the secondary torques significantly greater than zero for the eight primary joint directions (figures 3 through 6) are summarized in Table 2. For each primary joint direction listed on the left, the secondary torques significantly greater than zero are marked with an 'X'. Additionally, significant correlations (p
The interesting finding that in control subjects, hip flexion secondary torques were greater than 100% MVT might be explained by the activity of the rectus femoris. During hip flexion MVT control subjects seamed to rely on moderate levels of both rectus femoris (42% maximum) and adductor longus (52% maximum) to achieve hip flexion torques. But during knee extension MVT the rectus femoris activity of the control subjects was higher (54% maximum). A recalculation of hip co-contraction index during knee extension MVT shows that there is less co-contraction about the hip during voluntary knee extension (3.58 +/- 1.39 stdv) than during voluntary hip flexion (2.73 +/- 0.68 stdv). However these findings are not significantly different and had more muscles been recorded from a better understanding for the observed behavior could be explained.
The Kakwani index for federal taxes is not statistically different from zero for all the cantons apart from canton Basel Land (BL), which has a slightly progressive value. This means that federal taxes are concentrated quite proportionally among the population. This result could be easily explained if we think that the tax amount considered here is the sum of a progressive federal direct tax (based on income) and an indirect tax (VAT) based on consumption that is normally regressive. Canton Basel Land is the only one in which the effect of the progressive direct tax more than offsets the effect of the regressive VAT.
The Kakwani indexes for the cantonal and municipal taxes are more difficult to explain. While we would expect a highly progressive value, most of them are not statistically different from zero. Only Solothurn (SO), Ticino (TI), and Basel Land (BL) present a slightly positive value of the Kakwani index, which means a slightly progressive tax. These values are smaller than the results expected, but there are at least two reasons that could explain them. 2ff7e9595c
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