Using magnetic field gradients in the three spatial dimensions (G_x, G_y, G_z) identical nuclei experience locally different magnetic fields and therefore different energy level spacings. As a consequence, the resonance frequency becomes a function of the position within the probe. Again, the resulting signal (the FID) is an overlay of individual periodic functions and, after Fourier transformation, the intensities represent the number of water protons present in a definite volume element. The varying intensities are displayed as respective grey scale values in the final MR image.

The difficulty of this method is to individually encode and decode the spatial information for the three orthogonal directions. In 2D imaging (tomography) we have to distinguish between slice selection, frequency encoding, and phase encoding which are introduced in the following. Although the respective axes are sometimes labeled with x, y and z, they do not necessarily coincide with the cartesian axes. In principle, the choice of axes is arbitrary, because any set of orthogonal gradient axes can be produced by linear combination of the G_x,G_y,G_z.

The total magnetic field at a position z_SS (SS = slice selection) during application of G_z is given by B₀+z_SS·G_z and the spatially selective excitation energy or frequency, respectively, can be easily calculated using the Larmor equation.

In tomographic MR imaging slice selection allows for reduction of the 3D problem to only two dimensions.

In addition to this descriptive depiction, frequency encoding can also be explained using the k-space formalism (analogous to phase encoding). This is briefly introduced in the following section.

Phase changes of a magnetization vector depend on the gradient strength and its length. In a so-called pulse sequence (see below), RF pulses and the acquisition window are displayed together with the gradient lobes defining amplitude and length of the gradient pulses along the three orthogonal directions. As an example, the magnitude of k_x in Fig. 10 is proportional to the gradient duration τ_x as well as to the (constant) gradient amplitude G_x.

As a consequence, we need N acquisition cycles with G_PE modified by a constant ΔG_PE per time step (Fig. 11, left). Using a constant gradient duration τ, k_PE changes equidistantly by a value Δk_PE.

By analogy, we can also explain the effect of readout gradients by using the k-space formalism. In reality, the G_RO amplitude during data acquisition is identical for the entire experiment. However, a continuous variation of k_RO is realized by acquiring the data points at different time points i·Δt (i=1..N) (Fig. 11, right).

In the upper part of the figure, the variable k_RO is drawn as a function of time. As can be seen, k_RO changes from minimal to maximal value within the acquisition interval (between the time points A and B). The negative start value at A is produced by switching on a negative gradient before the acquisition. This changes the magnetization of the probe. The change is different for different positions on the readout axis thereby producing a net dephasing of the original M₀ value.

Subsequently switching a positive gradient reverses these phase manipulations until, for equal areas of negative and positive gradients, the initial phases are recovered for the whole probe. At this point k_RO equals zero and the echo is at its maximum. Further application of the readout gradient again produces net dephasing and therefore a continuously decreasing signal.

A positive gradient means that for x<0, a magnet field anti-parallel to B₀ is overlayed, correspondingly a field parallel to B₀ for x>0. Hence, the magnetization vector is static at x=0, whereas neighboring spins on both sides are moving symmetrically but in different directions within the xy plane.

A negative gradient (relations to B₀ are given by the arrows) inverts the moving direction of all spins.