Volume Elements or Density Compensation

What are Volume Elements? Basic Usage

Volume elements act as weights for each sampling point in k-space, enabling accurate image reconstruction. For non-Cartesian distributions, the sampling density is not constant in k-space. This means some regions are sampled more densely than others. Without proper compensation, this uneven distribution can cause severe artifacts in the reconstructed image.

A useful intuition: the local point density is inversely proportional to the volume element:

\[\text{Density at a point} \propto \frac{1}{\Delta k_p}.\]

This explains the term density compensation commonly used in MRI literature.

To compute the volume elements in Monalisa, use the following function:

ve = bmVolumeElement(trajectory, type_of_trajectory, optional_arguments)

Here the parameters are: - trajectory: The k-space sampling positions (see the Trajectory section in the docs). - type_of_trajectory: A string that determines which model is used to compute the volume elements. Carefully choose from the table below:

Supported Trajectory Types

type_of_trajectory	Required Parameters	Use Case
voronoi_center_out_radial2	None	2D radial trajectory with center-out spokes (typical for 2D UTE)
voronoi_center_out_radial3	None	3D radial trajectory with center-out spokes (typical for 3D UTE)
voronoi_full_radial2	None	2D full radial trajectory with diametric spokes (center sampled twice)
voronoi_full_radial3	None	3D full radial trajectory with diametric spokes (center sampled twice)
voronoi_full_radial2_nonUnique	nAvg: number of averages	2D full radial with duplicate samples, including multiple center acquisitions
voronoi_full_radial3_nonUnique	None	3D full radial with duplicate samples, including multiple center acquisitions
voronoi_box2	None	Generic 2D layout without duplicates; fallback when no other type fits
voronoi_box3	None	Generic 3D layout without duplicates; computationally expensive, avoid for large data
cartesian2	None	2D Cartesian grid (uniform sampling); assumes center point is at index N/2 + 1
cartesian3	None	3D Cartesian grid (uniform sampling); assumes center point is at index N/2 + 1
randomPartialCartesian2_x	None	2D Cartesian trajectory with randomly missing samples
randomPartialCartesian3_x	None	3D Cartesian trajectory with randomly missing samples
center_out_radial3	None	Fast approximate estimation for 3D center-out radial (non-Voronoi)
full_radial3	None	Fast approximate estimation for 3D full radial (non-Voronoi)
imDeformField2	deformationField: deformation vector field	2D volume elements corrected for deformation fields (motion/distortion)
imDeformField3	deformationField: deformation vector field	3D volume elements corrected for deformation fields (motion/distortion)

You don’t see your usecase, or you don’t know which to pick? Consider opening an issue on Monalisa’s GitHub page.

Volume elements are typically computed using Voronoi parcellation, which naturally estimates how much space each point “owns” in k-space. Each Voronoi cell contains all points closer to a given sample than to any other.

Important notes:

• Volume elements depend on binning. If you divide your acquisition into bins (e.g., for motion correction), recompute the volume elements after binning.

• It is strongly recommended to set a ve_max threshold to avoid numerical instability.

• Output ve is a [1, nPt] double-precision vector.

• The function does not normalize or rescale values.

% Set maximum volume element to avoid instability
ve_max = 10 * dKu_x * dKu_y;  % For 2D
ve_max = 10 * dKu_x * dKu_y * dKu_z;  % For 3D

Ok, but why are there Volume Elements?

Volume elements originate from the need to approximate integrals using discrete data. For example, consider the goal of computing:

\[\int_{\mathbb{R}^3} f(\mathbf{k})\,d^3\mathbf{k}.\]

With sampled k-space points \(\{\mathbf{k}_p\}_{p=1}^N\), we approximate:

\[\int_{\mathbb{R}^3} f(\mathbf{k})\,d^3\mathbf{k} \approx \sum_{p=1}^{N} \Delta k_p\, f(\mathbf{k}_p),\]

where \(\Delta k_p\) is the volume element for point \(\mathbf{k}_p\).

In Cartesian sampling, all \(\Delta k_p\) are constant, so we write:

\[\int_{\mathbb{R}^3} f(\mathbf{k})\,d^3\mathbf{k} \approx \Delta k \sum_{p=1}^{N} f(\mathbf{k}_p).\]

And therefore can be ignored since anyways raw data are usually normalized. However, in non-Cartesian sampling, the density of points varies across space, hence we need to estimate the \(\Delta k_p\) for each point to correctly approximate the integral. For example Radial sampling oversamples the center. Historically, this concept is referred to as “density compensation” in MRI, originating from the transition from uniform trajectories, where the density is constant and can be neglected, to non-uniform trajectories. Although non-uniform density was once viewed as a problem to be “compensated”, it is in fact the general case, with uniform sampling being a special scenario.

Need More Help?

• Open an issue on GitHub

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