Spatial Resolution and K-Space Coverage

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The broader the coverage in k-space, the better the MR
Spatial resolution.
A reduced pixel or voxel size should lead to improved or
Higher resolution.
Reconstructing images from windows size 64 x 64 (larger pixel size), 128 x 128, 256 x 256, and 512 x 512 (smaller pixel size) show that the coverage of a larger window in k-space improves the definition of the small holes or
'resolution elements'. In particular, the Gibbs ringing pattern becomes less noticeable as voxel size decreases.
An additional effect is that a halving of voxel size in each direction by doubling the sampling points leads to a
Square root of 2 loss in SNR.
An interesting enhancement of the Gibbs artifact occurs when two or more edges from a series of objects in a row come close enough together that the ringing from each edge coherently combines with that from other edges. In this instance, the ringing extends significantly past the edge of the outermost object. The row where this occurs shifts as a function of
Resolution.
Resolution phantom images reconstructed from filtering a 512 x 512 matrix size acquisition in a fixed FOV of 256 mm. The reconstructions are from the reduction of the full window down to a 64 x 64 window, 128 x 128 window, 256 x 256 window, and also the full 512 x 512 window. The improved resolution and reduction in Gibbs ringing Effects are evident for each increase in the
Window size, which implies a decrease in the pixel size.
Sampling and windowing are a natural part of the data processing used in MR imaging. Signal variations as a function of time due to T2* decay during data sampling in the read direction can be determined from the exponential expressions found earlier for transverse relaxation. In particular, such innat filtering imposes limits on the best spatial resolution which can be achieved in an experiment. Even if infinite time were available for sampling, and continuous data could be acquired, the signal would still be
Filtered by the transverse decay envelope leading to a limited resolution.
The Hanning filter represents a simple model of symmetric windowing. Calculating the image resolution in the presence of the Hanning filter when the filter is applied to k-space data, the image resolution is
Reduced by a factor of 2.
The spatial resolution may be defined in terms of 'width' of the
Point spread function.
Inverse Fourier transformation of the MR spatial resolution with filter due to the T2* effect leads to the point spread function h filter (x) whose full width at half maximum gives an estimate of the spatial extent of the
Blur caused by the filter. The FWHM may be a resonable first-order guess of the spatial resolution. This FWHM represents an additional blur over and above that caused by the usual finite sampling window.
SNR per pixel in the reconstructed image is proportional to the
Square root of the sampling time, so SNR is yet another criterion that limits sampling time. Nevertheless, one might want to increase sampling time, at the expense of a loss in resolution, in order to increase SNR.
One method of interpolating incomplete k-space data is to fill out the data with additional zeros such that the pixel size after image reconstruction meets the desired
Interpolation. A discussion on the Effects of the point spread function on the interpolated images shows that up to a 36% signal loss can be obtained if a point-like object is not centered within a reconstructed voxel. Sub-voxel shifts performed by using the shift theorm can be used to recover this signal loss.
The shift theorem allows us to overcome this worst-case 36% underestimation of the signal by centering the object within a
Reconstructed voxel. However, recovering the 'missing' 36% from a shifted object does not imply that the image is resolved any more accurately.
This property of the Fourier transform shift theorem, used to recover the reconstructed signal loss, can be taken advantage of in a number of practical situations. For example, zero filled interpolation has been in use for a long time in NMR spectroscopy for improved
Spectral peak estimation. The improved peak estimation leads to improved estimates of the spectral position. Extending the same ideas to imaging, it has een increasingly used for improved object signal estimation by overcoming the partial volume averaging effect due to sharing of the object by two neighboring voxels. This is particularly helpful for MR angiographic applications and another reason why the shift theorem plays an important role in visualizing vessels.
In certain cases, the number of data points on one side of the k-space origin is much more than the number of points on the other side (ie., the data are highly asymmetric). In some applications where minimizing imaging time is of the essence, k-space is covered highly asymmetrically in the phase encoding direction. For example, the numer of phase encoding lines required by the Nyquist condition is collected only in one half of k-space, while the other half (often the negative half of k-space) is covered only partially. Since phase encoding lines are usually separated in time by TR, collecting fewer phase encoding lines implies a shortening of the total
Imaging time. The amount of time saved is determined by the degree of asymmetry. In other applications such as MR angiography where shorter field echoes provide less sensitivity to rapid flow, very short echo times can be achieved. One of the most popular means to achieve this is to obtain asymmetric echoes in the readout direction. These methods of asymmetric k-space coverage are generically classified as partial Fourier imaging methods.