Not all clinical MR images are created equal.*

After the using the Fourier transform to transform our measured k-space data into image space, the image data is of complex type. This image data is then manipulated for different clinical utility. For example, a magnitude image is used to maximise the signal-to-noise ratio (SNR). Phase images are used to measure flow. Let’s look at how our MR signal is recorded and how these image types are calculated.

The changing magnetic field which is the source of the signal measured in MRI is a vector which we represent using complex notation. This is quadrature detection, which refers to the detection of a circularly polarised magnetic field, and results in two data streams with a 90° phase difference. The digitised values from these signals become the real part and the imaginary part of each complex data point in k-space. The ultimate purpose of quadrature detection is to increase SNR by a factor of √2.

Now, the two signals are the real and imaginary channels which are sometimes denoted I (for in phase, the “real” data) and Q (for quadrature phase, the “imaginary” data). The imaginary data is not imaginary in the colloquial sense; it is a measured quantity. These signals are corrupted by “white” noise, which has a Gaussian probability distribution. After the inverse Fourier transform of the complex data, the noise in the complex image data is still white (Gaussian). However, we don’t generally work with the real or imaginary components of the image data (i.e. calculate images using only the real data, or images using only the imaginary data). To use both parts of the complex data values, we calculate magnitude images and phase images, which have physical meaning (proton density and flow, respectively, ignoring contrast weighting and background phase variation for the moment).

Magnitude images are the real and the imaginary parts combined, calculated after the Fourier transform as √(Real²+Imag²), for the complex data point at each image pixel.

Phase images are calculated after the Fourier transform as tan^-1(Imag/Real) for the complex data point at each image pixel. Phase is also known as the complex argument of a complex number.

After making the calculation of a magnitude image, the noise probability distribution is no longer white, and becomes Rician (tending to a Rayleigh distribution as the SNR goes to zero). This sounds concerning, since most MR quality assurance (QA) programs rely on SNR as the primary parameter of choice as a daily-QA metric. However, it seems that if the SNR is above a very low value (<2), the noise probability distribution is approximately Gaussian again anyway. And note that an MR image with an SNR of 2 would not be practically useful.

*Apologies to Philip Mazzei.

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