The MRI signal represents a scalar product of the object with a set of encoding functions ¹. The signal is complex because the encoding functions – in the simplest case, plane waves created by spin precession under field gradients – have real and imaginary components (the object itself can be made real by a spin echo sequence). Consequently, the phase of the signal is important for the reconstruction process and any uncontrolled phase changes, e.g. those induced by object motion result in image artefacts. We explore an alternative way of spatial encoding which produces real positive (phaseless) signals. Spurious phase components can be removed by taking the absolute value of the signal before the Fourier transform.

In a 2d spin-echo imaging sequence the phase encoding gradient pulse (y) has been moved from its usual position into a preparation sequence where it is flanked by two RF pulses of flip angle $$$\alpha$$$ and followed by a spoiler (Fig.1). Now, the excitation pulse hits a z-magnetization which is already spatially modulated by $$$ S \cos(2\pi k_y y) + C $$$ or by $$$ S \sin(2\pi k_y y) + C $$$, with the two $$$\alpha$$$-pulses having opposite or orthogonal phases, respectively, and where $$$ C=\cos^2(\alpha) $$$ and $$$ S=\sin^2(\alpha) $$$. Each phase encoding step is repeated with both phase shifts and the corresponding echoes are Fourier-transformed along the readout direction (x). The resulting profiles are

$$ s_a(x,k_y) = e^{i\phi_a} \int \rho (x,y) \left[ C\cos(2\pi k_y y)+S \right] dy $$

$$ s_b(x,k_y) = e^{i\phi_b} \int \rho (x,y) \left[ C\sin(2\pi k_y y)+S \right] dy $$

The uncontrolled phase shifts $$$\phi_a$$$ and $$$\phi_b$$$ may by different in each step and may also depend on x. Since the “object” $$$\rho(x,y)$$$ is real (spin echo), the function under the integral is real positive when $$$ \alpha < 45^\circ $$$, and we can remove the phase modulations by taking the absolute value. The profiles are thus combined:

$$ s(x,k_y)= \left| s_a(x,k_y) \right|+i\left| s_b(x,k_y) \right| $$

and Fourier transformed along y to give the final image

$$ \tilde{\rho}(x,y)=C\rho(x,y)+(1+i) S \delta(y)\int \rho(x,y)dy $$

The central line of this image contains a projection artefact, which can be removed by fitting a constant offset in high $$$k_y$$$ regions of each profile and subtracting it before the second FT.

An experimental proof of principle of the phaseless encoding was carried out on a phantom using a 3T MRI system (Achieva, Philips Healthcare, Netherlands). This system applies a random receiver phase cycle throughout encoding steps, which is normally demodulated by the reconstruction, and which was treated as unknown in our experiment. An acquisition with the classic phase encoding was compared with the proposed method. The classic phase-encoded data reconstructed by 2DFT shows ghosting due to the uncorrected receiver phase cycle (Fig. 2A). An attempt to remove the random phase by taking the absolute value after the FT along x and before the FT along y fails of course (Fig.2B), since the information contained in the phase is lost. However, the phaseless encoded data processed as described above produces a correct image with only a residual artefact along the central line (Fig.2C).Phaseless encoding makes the reconstruction independent of the global (and to limited extend, local) phase fluctuations between encoding steps. It may find applications in diffusion-weighted MRI, where such fluctuations arise due to object displacement in the presence of diffusion gradients ². It could also allow direct signal sampling without the burden of phase locking to the transmitter ³. The tradeoff is a reduction of the signal-to-noise ratio per scan time, which is four-fold in the optimal case of 45⁰ preparation pulses, as well as an enhanced dependence of image intensity on RF field amplitude (via the S coefficient). It should be noted that real-valued encoding functions such as Hadamard ⁴ or wavelets ⁵, which have been used in the past for the purpose of improved spatial response or scan time reduction, would not allow a phase-immune reconstruction from signal magnitude.