Our goal in this work was to use phase to enhance contrast between tissues with different susceptibilities. This can be accomplished in several steps. First, we employ the high-pass filter described in Ref. 8 to remove the low-spatial-frequency components of the background field. In the work shown here, we use a 64 × 64 low-pass filter and divide this into the original phase image (512 × 512) to create a high-pass filter effect.
Second, this “corrected” phase image is used to create a “phase” mask that is used to multiply the original magnitude image to create novel contrasts in the magnitude image. The phase mask is designed to suppress those pixels that have certain phases. It is usually applied in the following manner: If the minimum phase of interest is, for example, –π, then the phase mask is designed to be f(x) = (φ(x) + π)/π for phases < 0, and to be unity otherwise, where φ(x) is the phase at location x. That is, those pixels with a phase of –π will be completely suppressed and those with a value between –π and zero phase will be only partly suppressed. This phase mask (f(x)) then takes on values that lie between zero and unity. We will refer to it as the negative phase mask. It can be applied any number of times (integer m) to the original magnitude image (ρ(x)) to create a new image fm(x)ρ(x) with different contrasts (11, 13, 22). Another mask might be defined to highlight positive phase differences:
If the maximum phase of interest is, for example, π, then the phase mask is designed to be g(x) = (π–φ(x))/π for phase > 0, and unity otherwise. We will refer to this as the positive phase mask.
Alternatively, if echo times (TEs) are so long that they cause difficulties, or if it is desirous to calculate phase from very short TEs without any RF penetration phase effects, an interleaved double-echo scan can (29) be acquired to simulate the equivalent phase of a short-TE scan. The complex data from the first echo are then divided into those of the second echo to create an equivalent phase image to that for a TE of ΔTE. That is, the phase in the complex division becomes –γΔBΔTE. We use this concept to create an effective TE = 2 ms image from an interleaved TE = 8 ms and TE = 10 ms data set.
All sequences used in this study were high-resolution, 3D gradient-echo scans. In-plane resolution varied from 0.5 mm × 0.5 mm to 1 mm × 1 mm with slice thicknesses of 0.7–2 mm. Except for the interleaved double-echo experiment for highlighting fat described above, the experiments were run with TE = 40 ms. These experiments were all performed at 1.5T except for one case in which the data were obtained at 3.0T.
Phase Mask Multiplication: Theoretical Considerations
Phase masks are created to enhance the contrast in the original magnitude images. Depending on the constructs used to create the filter, the number of multiplications needed to optimize the contrast-to-noise ratio (CNR) in the SW images will vary. We consider the positive phase mask case below. The results for the negative phase mask follow by letting φ go to –φ. The first step is to write an expression for the CNR between two tissue types. Consider first the example in which all tissues have the same signal S0 with Gaussian noise, and contrast is generated only by the phase images. The contrast in the magnitude image is therefore zero. Contrast appears only after multiplication by the phase mask has been performed. We create a function that is dependent on m, the number of multiplications that are performed with the phase mask. The goal is to optimize m or, equivalently, find the point at which CNR(m) is maximized. The region of the object where there is a phase difference will then change its signal after multiplication with the phase mask. For the positive phase mask considered here, the multiplication factor of the signal will become (1–φ/π)m in the positive-phase region, while that in the negative-phase region remains unity. The inherent contrast that develops will then be 1–(1–φ/π)m times S0 of the object in the original magnitude image. The noise in the new image must take into account the noise in the original image plus the noise generated from the multiplications. For a high signal-to-noise ratio (SNR) in the magnitude images (>4:1), the variance of the final image after m multiplications is given by
where σo is the standard deviation (SD) of the Gaussian noise in the original magnitude image (see the Appendix for a full derivation). Therefore, the functional form for CNR(m) is the contrast divided by the noise, and is given by
where SNR0 is the original SNR (i.e., S0/σo). When the exponential decay of the MR signal is included in our analysis, the CNR(m) becomes:
where we have assumed that TE/T2* is unity when φ = π (since it is just the increase in signal for shorter TE and hence smaller m that we are after here). The more general form can be obtained by replacing exp(–φ/π) with exp((–φ/π)T2*/TE).
However, T2* plays a role in the signal decay, and we can not arbitrarily choose a long TE to get the phase to be π without a great loss in SNR. Thus, when the phase is π, there will be circumstances in which the number of multiplications required will be >1 in order to show the optimal contrast in the images. This raises an interesting question. If one is willing to spend a fixed amount of time imaging (specifically, to acquire a long enough TE to ensure that the phase is π), then it is not clear whether a shorter TE with more phase multiplications might not do just as good a job. That is, it might be better to look into reducing the TE (and hence the TR), collecting the data with a shorter TE (i.e., with a higher signal), performing more multiplications, and averaging over several acquisitions in order to obtain an optimal CNR.
For the case of the same volume coverage as well as the same imaging time, one then considers the efficiency CNR*sqrt(number of slices)/sqrt(time) rather than the CNR. This introduces another factor of sqrt(π/φ) into the right-hand side of Eq.  such that CNR is proportional to (although we ignore the need to increase the read gradient strength when TR becomes too short):
This term arises because the only way to collect these data is to either increase the overall time by acquiring the data a second time or use a segmented echo-planar-like approach, which in turn requires that the gradient strength be increased accordingly. When the gradient strength is increased (such that both the sampling time and TE are decreased), then a factor of sqrt(φ) loss in SNR occurs, and Eq.  reduces once again to Eq. .
If one is interested in comparing images with circular objects of radius p pixels, (Fig. 1) then one can look at visibility instead of CNR (29), where
In the plots for Figs. 2 and 3, we have taken p = 2.