This post explains how to unfold the undersampled, aliased image to the indices in the reconstruction matrix in SENSE.

## In short

I sketched how to place the mapping window for acceleration factors 2, 3 and 4. You can see the wrap-around of aliasing windows in the even acceleration factors.

## SENSE

SENSE (SENSitivity Encoding)[1] is an image-domain parallel imaging technique in MRI. For more information, you can visit the following links:

## Naive implementation

When you write your implementation of SENSE, you might be inclined to split the target image in two windows side by side and map to the corresponding index in each window as shown below.

However, the above mapping is incorrect.

Below I overlaid the original unwrapped image with such mapping. We can see that, even though the point in the undersampled image is aliased, one of the unwrapped points is pointing at the background.

## Correct implementation

We can see that the aliased image still contains the phantom at the center of the image, just like the fully-sampled image. Therefore, we must relocate one of the mapping windows at the center of the reconstruction matrix. The second window naturally wraps around from the end of the image to the beginning of the image as shown below.

We can see that now we found the corresponding locations from the aliased pixel.

In fact, this phenomenon happens whenever the acceleration factor is an even number. In the bottom figure I sketched how to place the mapping window for acceleration factors 2, 3 and 4.

The take-home lessons are the following:

- When using the acceleration factor of R, if R is an even number, one of the windows splits in half and wraps around in the target matrix.
- If we start counting the windows from the center one (index 0), the window which splits in half has index R/2.

## References

[1] SENSE: Sensitivity Encoding for Fast MRI. K. P. Pruessmann, M. Weiger, M. B. Scheidegger and P. Boesiger. Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 42.5 (1999): 952-962.