# Noise Amplification in Linear Systems

## Application to SENSE Parallel Imaging in MRI

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:

- Why the co-linearity of coil sensitivities can lead to poor performance and massive noise amplification
- We can see why, even though both voxels alias onto one another, g-factor noise amplification is often asymmetric, affecting one voxel more than the other
- Even when the input coil noise has no off-diagonal covariance (i.e. independent noise on each channel), voxel-wise correlations in the noise will result (when the coil sensitivities are not orthogonal)
- It is possible to have significantly improved noise performance in a system with high input noise covariance and highly co-linear coils (better than a system with no input off-diagonal covariance and orthogonal coils!)