Given a real function $f:\mathbb{R}^n \to \mathbb{R}$, denote by $\hat{f}$ its Fourier Transform. I have shown that $\hat{f}(\vec \omega)=(\hat{f}(-\vec \omega))^*$ where $^*$ denotes complex conjugation. We assume to be in an ideal situation, that the image function is proportional to the proton density of the object, which is thus a real non-negative function. Suppose you wish to scan an object, which measures $L \times L$, with a given spatial resolution $\Delta_x = \Delta_y$. Furthermore, you decide to scan only half $k$-space. The $k$-space trajectory you choose is the concentric half-circular trajectory, starting from the center.
Now I am wondering how many half circles do you need to be sure the image you get will not exhibit aliasing? And what is the distance between the circles, along the radial direction? And finally, can we write an analytical formula for the circular parts of the trajectory?