Noise Amplification in Linear Systems

Application to SENSE Parallel Imaging in MRI

Noise Samples:
Noise Variance:
Noise Covariance:
Coil Noise Margins
Noise Samples

This interactive visualization demonstrates how noise is transformed in a linear system, for a geometric perspective on noise behaviour, in a toy example R=2 SENSE parallel imaging reconstruction. The visualization consists of two panels: on the left, coil vectors, a target voxel and its projections onto the coil vectors are shown in the abstract 2D space representing the possible values of the voxels (A,B). These illustrate how 2 measurement vectors constrain the solution of a 2D unknown (A,B). Also shown are noise characteristics that result from the application of these constraints (by inverting the system), and illustrates how input noise is transformed and amplified through solving linear reconstruction problems. On the right, locations of the voxels (A,B), and schematic coil elements and sensitivities representing a physical interpretation of the sensitivity vectors is shown.

The interactive components of this visualization are:

Noise Samples slider
controls how many simulated noise samples to show
Noise Variance slider
controls the scaling of the diagonal entries of the 2x2 coil noise covariance matrix
Noise Covariance slider
controls the scaling of the off-diagonal entries of the 2x2 coil noise covariance matrix
Coil Noise Margins checkbox
toggles the full-width at tenth maximum coil noise margins
Noise Samples checkbox
toggles whether noise information is shown on the visualization
Black dot
draggable, represents the true value of the voxels (A,B) in the 2D voxel space
Coil vectors
draggable, thin solid blue or orange lines representing the sensitivity vectors for coils 1 and 2
Coil elements
draggable, thick “coils” rotating around a simulated FOV on the right-hand panel

This simulation allows you to independently control coil sensitivities and coil noise covariance characteristics, which are not typically independent degrees of freedom in realistic MR recieve coil arrays. However, with this toy example, we can provide some intuition and insight into some features of parallel imaging reconstructions: