Quantitative Validation

To quantitatively measure the performance of the phase unwrapping algorithm with respect to signal to noise ratio (SNR), a set of simulated images were unwrapped. The results were then compared with the original (ground truth) phase and the mis-classification ratio (MCR) calculated. In this context the MCR is the number of voxels that were incorrectly unwrapped (that is, had the incorrect multiple of $2\pi$ added to them) divided by the total number of voxels.

The base image used was a $64 \times 64 \times 25$ complex image with unit magnitude at all voxels and a quadratic phase function centred in the middle of the image (see figure 10). The maximum phase gradient in the base image was $\pi$ radians between neighbouring voxels, which occurs at the outermost edge of the image. In the interior of the image the phase gradient changes linearly with position, being zero at the centre of the image.

Various amounts of Gaussian noise was added to the real and imaginary parts of the base image to generate a set of synthetic images, each with a different SNR. That is, $I_{synth} = I_{base} + \sigma (n_R + j n_I)$, where $n_R$ and $n_I$ are independent, unit variance, Gaussian noise images. The SNR of this image is given by the signal amplitude (unity in this case) divided by the total noise amplitude, giving SNR $= 1/\sqrt{2}\sigma$. Some of these synthetic images are shown in figure 11. The added noise induces phase errors which can be measured by comparing the phase of $I_{synth}$ with the phase of $I_{base}$, the true phase, thus allowing the standard deviation of the induced phase change to be calculated.

The unwrapping results are shown in table 1 as well as some examples which are displayed in figure 12. It can be seen that the the algorithm was extremely accurate and robust, with zero MCR for SNR greater than 5, and only a single voxel erroneously unwrapped with an SNR of 5. It is only at very low SNR (1 and 2), where the standard deviation of the phase exceeds $20^\circ$, that a significant number of voxels were incorrectly unwrapped. In fact, even with a SNR of 1, the unwrapping was largely successful in the central part of the image (where the phase gradient was less than $\pi/2$ radians per voxel).

For most MR imaging methods the SNR is significantly better than this. For example, a SNR of 50 (equivalent to a phase error of approximately $1^\circ$) a B0 mapping sequence is typical at 3T. This is an order of magnitude better than required for this algorithm to work well and therefore this method should be both accurate and robust for phase images acquired with typical MR sequences.

Table 1: Results of simulated tests for phase unwrapping

SNR	Noise level	std. dev. phase	MCR %
1	1	50.2$^\circ$	80.0
2	0.5	22.5$^\circ$	7.3
5	0.2	8.18$^\circ$	0.001
10	0.1	4.05$^\circ$	0.000
20	0.05	2.03$^\circ$	0.000
50	0.02	0.81$^\circ$	0.000
100	0.01	0.40$^\circ$	0.000
200	0.005	0.20$^\circ$	0.000
500	0.002	0.081$^\circ$	0.000
1000	0.001	0.040$^\circ$	0.000

Figure 10: Synthetic phase image

Figure 11: Some of the synthetic phase images with added Gaussian noise (central slice shown).

Figure 12: Some of the unwrapping results of the synthetic phase images with added Gaussian noise (central slice shown). Note that the leftmost image, included for comparison, is the true phase which was used to generate the synthetic images.