2.2 MRI: k-Space Preprocessing

"MRI does not acquire an image directly; it acquires the 'music' of the object in the frequency domain. Understanding k-space is learning the language of that music." — Wisdom of Medical Imaging Signal Processing

When radiologists look at a crisp MR image, few realize how complex the acquisition pipeline behind it is. From the moment the MR signal is captured by the receive coils, the data starts a journey in the frequency domain—from raw sampling in k-space, through noise estimation, preparation for parallel imaging, and motion-related corrections—before an inverse Fourier transform yields the image we see.

This chapter presents the complete MRI preprocessing workflow. Unlike CT (which focuses on projection-domain corrections), MRI preprocessing centers on the k-space domain. We explain the physical meaning of k-space, its noise statistics, the mechanics of parallel imaging, and how principled preprocessing improves image quality. These topics form the foundation of MRI reconstruction and are key to optimizing diagnostic performance.

k-Space Fundamentals (Frequency-Domain Representation)

MRI Signal Equation and Fourier Relationship

Under a 2D assumption with spatial coordinates (x, y), the received MR signal with gradient encoding can be written as the Fourier transform of the transverse magnetization (M_0(x,y)):

[ s(\mathbf{k}) = \iint M_0(x,y), e^{-i,2\pi,(\mathbf{k}\cdot\mathbf{r})}, d\mathbf{r},\quad \mathbf{r}=(x,y) ]

where (\mathbf{k}=(k_x,k_y)) is determined by gradients and time (frequency encoding and phase encoding). Once (s(k_x,k_y)) is acquired, the image (I(x,y)) is recovered by the inverse Fourier transform (IDFT in practice):

[ I(x,y) = \iint s(k_x,k_y), e^{+i,2\pi,(k_x x + k_y y)}, dk_x, dk_y. ]

Geometric Meaning and Sampling Trajectories

• k-space coordinates ((k_x,k_y)) represent spatial frequency content. Low-frequency samples near the center determine overall contrast; high-frequency samples at the periphery govern edges and fine details.
• The sampling interval (\Delta k) is inversely related to the field of view (FOV); k-space extent determines spatial resolution.
• Common trajectories: Cartesian (row-by-row), radial, spiral, EPI (single-shot or multi-shot). Non-Cartesian sampling requires regridding or NUFFT before reconstruction.

Typical Artifacts Related to k-Space

• Undersampling/aliasing (violating Nyquist): fold-over (wrap-around) artifacts.
• Truncation of high-frequency components: edge ringing (Gibbs ringing).
• Gradient delays/nonlinearities, B0/B1 inhomogeneity, eddy currents: k-space distortions causing ghosting and blurring.

Why k-Space Preprocessing Matters (for AI and Clinical Imaging)

• Image quality foundations: blur, unclear edges, and artifacts (aliasing, Gibbs) often originate from k-space sampling or inadequate compensation.
• Reconstruction and resampling strategy: understanding trajectories, signal models, and reconstruction is essential when customizing pipelines beyond standard DICOM images.
• Noise and artifact sources: motion, gradient imperfections, coil sensitivity changes manifest as abnormal frequency components; correcting them in k-space improves downstream segmentation/detection.
• AI compatibility: changes in sampling strategies (acceleration, parallel imaging, partial Fourier, non-Cartesian) alter noise textures/structure—account for these differences in training.

Noise Estimation and Denoising

Noise Statistics in MR Images

Magnitude MR images often exhibit Rician or, for multi-coil combinations, non-central chi (nc-χ) distributed noise. Noise can be spatially non-stationary due to coil sensitivity and parallel imaging reconstruction. Preprocessing operations (filtering, zero-padding, regridding) also alter noise mean and covariance.

Non-Local Means (NLM) Denoising

Core idea: leverage self-similarity of patches across the image, not only local neighborhoods, to preserve edges while suppressing noise. A generic form:

[ \tilde I(p) = \frac{1}{C(p)} \sum_{q} w(p,q), I(q), \quad w(p,q) = \exp!\Big( - \frac{|I(\mathcal{N}_p) - I(\mathcal{N}_q)|^2}{h^2} \Big). ]

NLM is widely used in MRI; parameters should account for non-Gaussian, spatially varying noise.

PCA-Based Denoising

Treat patches or voxels as high-dimensional vectors and perform principal component analysis (PCA). Suppress components dominated by noise (low-variance) and reconstruct patches from retained components. PCA denoising is effective for multi-echo, time-series (fMRI), or multi-coil data; model design must consider non-stationary noise and complex-valued signals.

Minimal k-Space Processing Pipeline

Key steps:

• Acquisition: gradient encoding and collecting k-space lines per phase-encode step.
• Preprocessing: zero-filling, windowing, regridding (for non-Cartesian), correcting gradient delays/nonlinearity, coil sensitivity preparation.
• Inverse FT: FFT/NUFFT to image space, followed by coil combination.
• Image-domain operations: denoising, bias-field correction, segmentation/detection.

Next:

• Reconstruction principles: Chapter 3 (/en/guide/ch03/01-analytic-recon)