How the methods work¶

For all descriptions, let \(I(\mathbf{x})\) be the MR brain image under consideration where \(\mathbf{x} \in [0,N] \times [0,M] \times [0,L] \subset \mathbb{N}^3\), and let \(B \subset I\) be the brain mask (the set of voxels in the brain).

Z-score¶

Z-score normalization uses the brain mask \(B\) to determine the mean and standard deviation of the intensities inside the brain:

\[ \mu = \frac{1}{|B|} \sum_{\mathbf{b} \in B} I(\mathbf{b}) \quad \text{and} \quad \sigma = \sqrt{\frac{\sum_{\mathbf{b} \in B} (I(\mathbf{b}) - \mu)^2}{|B| - 1}} \]

Then the z-score normalized image is

\[ I_{\text{z-score}}(\mathbf{x}) = \frac{I(\mathbf{x}) - \mu}{\sigma}. \]

Z-score is not brain-specific and works for any anatomy.

Fuzzy C-means (FCM)¶

FCM normalization scales the image so the mean of a chosen tissue class equals a constant. Let \(T \subset B\) be the tissue of interest (CSF, GM, or WM). The tissue mean is

\[ \mu = \frac{1}{|T|} \sum_{\mathbf{t} \in T} I(\mathbf{t}) \]

and the normalized image is

\[ I_{\text{fcm}}(\mathbf{x}) = \frac{c \cdot I(\mathbf{x})}{\mu} \]

where \(c \in \mathbb{R}_{>0}\) is a constant (norm_value, default 1). We use three-class fuzzy c-means over the brain mask of a T1-w image to get soft tissue memberships, sorted ascending by cluster center so the classes map to CSF/GM/WM, and the tissue "mean" is the membership-weighted average. For other contrasts of the same subject, the memberships from the co-registered T1-w image are reused.

Kernel Density Estimation (KDE)¶

KDE-based normalization estimates the empirical probability density of the foreground intensities and scales the image so the mode of the tissue of interest (white matter for T1-w) equals a constant:

\[ \hat{p}(x) = \frac{1}{n\,\delta} \sum_{i=1}^{n} K\left(\frac{x - x_i}{\delta}\right) \]

where \(K\) is a Gaussian kernel, \(\delta\) the bandwidth, and \(n\) the number of foreground voxels (subsampled deterministically for speed; the mode is statistically unchanged). The smooth density makes mode-finding robust. Which local maximum is the tissue of interest depends on the modality: the last mode for T1-w, the largest for T2-w/FLAIR, the first for PD/MD. If \(p\) is the tissue mode,

\[ I_{\text{kde}}(\mathbf{x}) = \frac{c \cdot I(\mathbf{x})}{p}. \]

Piecewise Linear Histogram Matching (Nyúl & Udupa)¶

Histogram matching [1] learns a standard histogram for a set of images of one contrast and maps each image's intensities onto it, demarcated by landmark percentiles (default: \(1, 10, 20, \ldots, 90, 99\); values outside \([1\%, 99\%]\) are treated as outliers) [2].

Learning the standard histogram¶

Let \(\mathbf{I} = \{I_1, I_2, \ldots, I_K\}\) be \(K\) images of one contrast. For each image \(I_i\), compute the 1% and 99% intensity values \(m_1^i, m_{99}^i\) and linearly map the image to a standard range \([m_{\text{min}}^s, m_{\text{max}}^s]\) (default \([1, 100]\)):

\[ \tilde{I}_i(\mathbf{x}) = \left(I_i(\mathbf{x}) - m_1^i + m_{\text{min}}^s\right) \frac{m_{\text{max}}^s}{m_{99}^i}. \]

Then compute the landmark percentiles of \(\tilde{I}_i\) and average them across the set — the mean of each corresponding landmark is the learned standard scale:

\[ m_n^s = \frac{1}{K} \sum_{i=1}^{K} \tilde{m}_n^i \quad \text{for } n \in \{1, 10, \ldots, 90, 99\}. \]

Normalizing new images¶

For a new image, compute its landmark percentiles \(\{m_1, m_{10}, \ldots, m_{99}\}\) and piecewise-linearly interpolate each decile onto the standard scale:

\[ I_{\text{nyul}}(\mathbf{x}) = \frac{I(\mathbf{x}) - m_i}{m_j - m_i}\left(m_j^s - m_i^s\right) + m_i^s \quad \text{for } I(\mathbf{x}) \in [m_i, m_j). \]

Because the mapping is a fixed piecewise-linear function after fitting, the saved transform applies identically to any later image.

WhiteStripe¶

WhiteStripe [3] performs a z-score-like standardization within the normal-appearing white matter (NAWM). Smooth the foreground histogram (KDE) and take the tissue mode \(\mu\) (last mode for T1-w). The "white stripe" is the intensity band within a quantile window around the mode: with \(F\) the empirical CDF of the foreground and \(\tau = 5\%\) (width),

\[ \Omega_\tau = \left\{ I(\mathbf{x}) \mid F^{-1}\left(F(\mu) - \tau\right) < I(\mathbf{x}) < F^{-1}\left(F(\mu) + \tau\right) \right\}. \]

Let \(\hat\mu\) and \(\hat\sigma\) be the mean and standard deviation over \(\Omega_\tau\). Then

\[ I_{\text{ws}}(\mathbf{x}) = \frac{I(\mathbf{x}) - \hat\mu}{\hat\sigma}. \]

Least Squares (LSQ)¶

LSQ pulls the CSF/GM/WM tissue means of every image toward a common standard in a least-squares sense. Tissue means are computed with fuzzy c-means memberships (as in FCM). The standard tissue means \(\mathbf{m}^s \in \mathbb{R}^3\) are learned from a reference image (scaled so its CSF mean equals norm_value). Each image \(I\) with tissue means \(\mathbf{m}\) is then scaled by the factor minimizing \(\lVert \mathbf{m}/s - \mathbf{m}^s \rVert^2\):

\[ s = \frac{\mathbf{m}^\top \mathbf{m}}{\mathbf{m}^\top \mathbf{m}^s}, \qquad I_{\text{lsq}}(\mathbf{x}) = \frac{c \cdot I(\mathbf{x})}{s}. \]

RAVEL¶

RAVEL [4] improves on WhiteStripe by removing unwanted technical variation (e.g., scanner effects). It assumes a population of WhiteStripe-normalized images of one contrast follows the additive model

\[ V = \alpha 1^\top + \beta X^\top + \gamma Z^\top + R \]

where \(V\) is the image matrix (rows are voxels, columns are images), \(\alpha 1^\top\) the average scan, \(\beta X^\top\) known clinical covariates, \(\gamma Z^\top\) unknown unwanted factors, and \(R\) residuals. Using voxels where clinical covariates are assumed absent — CSF control voxels — the unwanted factors are identifiable: with \(V_c\) the control-voxel matrix,

\[ V_c = \gamma Z^\top + R = U \Sigma W^\top \]

via the SVD. The first \(b\) right singular vectors \(W_b\) form an orthogonal basis for the unwanted factors \(Z\) [5]. Voxel-wise linear regression gives the coefficients \(\gamma\), and

\[ I_{\text{ravel}}(\mathbf{x}) = I_{\text{ws}}(\mathbf{x}) - \gamma_{\mathbf{x}} Z^\top. \]

Following the original paper, the default is \(b = 1\); the first right singular vector is highly correlated (>95%) with the mean CSF intensity. RAVEL requires same-shape, co-registered images and corrects the batch it was fit on — it is not a fit-once/apply-later method.

References¶

L. G. Nyúl, J. K. Udupa, and X. Zhang, "New Variants of a Method of MRI Scale Standardization," IEEE Trans. Med. Imaging, vol. 19, no. 2, pp. 143–150, 2000.
M. Shah et al., "Evaluating intensity normalization on MRIs of human brain with multiple sclerosis," Med. Image Anal., vol. 15, no. 2, pp. 267–282, 2011.
R. T. Shinohara et al., "Statistical normalization techniques for magnetic resonance imaging," NeuroImage Clin., vol. 6, pp. 9–19, 2014.
J. P. Fortin et al., "Removing inter-subject technical variability in magnetic resonance imaging studies," NeuroImage, vol. 132, pp. 198–212, 2016.
J. T. Leek and J. D. Storey, "Capturing heterogeneity in gene expression studies by surrogate variable analysis," PLoS Genet., vol. 3, no. 9, pp. 1724–1735, 2007.