Undersampling

Data undersampling or subsampling is a common technique in MRI to reduce the amount of data acquired or in other words to accelerate the acquisition process. This is achieved by reducing the number of k-space lines acquired. The missing k-space lines are then estimated using different techniques.

The forward model of accelerated MRI is given by \(y = P F x + n\), where \(y\) is the acquired data, \(x\) is the image, \(F\) is the FFT operator, \(P\) is the undersampling operator and \(n\) is the noise from the acquisition process.

\(P\) can either applied prospectively or retrospectively. In prospective undersampling, the k-space is undersampled in the scanner, during the acquisition process. In retrospective undersampling, the k-space is undersampled after the acquisition process, meaning that we acquire fully sampled k-space and then we apply the undersampling operator to the acquired data.

ATOMMIC can handle both prospective and retrospective undersampling via the mask_args of the train_ds, validation_ds and test_ds functions. The mask_args is a dictionary that contains the following keys:

• type:

  The type of the undersampling mask. It can be one of the following Equispaced1DMaskFunc, Equispaced2DMaskFunc, Gaussian1DMaskFunc, Gaussian2DMaskFunc, Poisson2DMaskFunc, Random1DMaskFunc, or none (no undersampling).

• accelerations:

  The acceleration factors to be used. It can be a list of integers or a single integer. If it is a list of integers, then the undersampling mask is randomly selected from the list. If it is a single integer, then the undersampling mask is randomly selected from the list of acceleration factors that are less than or equal to the given integer. For example, if accelerations is [4, 8, 12], then the undersampling mask is randomly selected from [4, 8, 12]. If accelerations is 4, then the undersampling mask is selected as 4.

• center_fractions:

  The center fractions to be used. It can be a list of floats or a single float. If it is a list of floats, then the undersampling mask is randomly selected from the list. If it is a single float, then the undersampling mask is randomly selected from the list of center fractions that are less than or equal to the given float. For example, if center_fractions is [0.08, 0.04, 0.02], then the undersampling mask is randomly selected from [0.08, 0.04, 0.02]. If center_fractions is 0.08, then the undersampling mask is selected as 0.08.

  Note

  The term center fraction is basically used to define the number of k-space lines that are acquired in 1D undersampling. In case of 2D undersampling, the center fraction refers for example to Full-Width-at-Half-Maximum (FWHM) in case of Gaussian undersampling or the radius of the circle in case of Poisson-Disk undersampling. It is kept as center fraction for consistency reasons, but it is not necessarily the fraction of the center of the k-space that is acquired. Should be renamed in the future.

• shift_mask:

  Whether to shift the undersampling mask or not. If True, then the undersampling mask is shifted by a random amount. If False, then the undersampling mask is not shifted. It is useful when data are shifted to the center of the image.

• use_seed:

  Whether to use a seed for the random number generator or not. For example, it should be se to True in validation, but it should be set to False in training.

Here is an overview example on the CC359 dataset, with the available undersampling options in ATOMMIC. The top row presents the undersampled images as the coil-combined Root-Sum-of-Squares (rss()) and the bottom row presents the undersampling masks.

PF refers to Partial Fourier, while the last column presents the default mask in the CC359 dataset.

Prospective Undersampling

In prospective undersampling, the type in mask_args should be set to none, data are then assumed to be undersampled in the scanner. This is useful for performing inference on undersampled data with a pre-trained model.

Retrospective Undersampling

The following example shows how to do retrospective undersampling in ATOMMIC:

train_ds:
    type: random1d
    accelerations: [4, 8]
    center_fractions: [0.08, 0.04]
    shift_mask: false
    use_seed: false

validation_ds:
    type: random1d
    accelerations: [4, 8]
    center_fractions: [0.08, 0.04]
    shift_mask: false
    use_seed: true

test_ds:
    type: none
    shift_mask: false
    use_seed: false

type can also be set to none in train_ds and validation_ds in retrospective undersampling if you want to load pre-computed undersampling masks. In that case, you need to set the mask_path in the train_ds and validation_ds to the path of the undersampling masks. Otherwise, mask_path should be set to none.