Local Perturbation Responses: Code and Examples

This page provides sample MATLAB code for computing local perturbation responses (LPRs), as described in:

C.-C. Chan, J. P. Haldar.
Local Perturbation Responses and Checkerboard Tests: Characterization tools for nonlinear MRI methods.
Magnetic Resonance in Medicine. In Press.
[link]

LPRs can provide useful insights into the sensitivity, spatial resolution, and aliasing characteristics of arbitrary MRI estimation methods (e.g., for image reconstruction, denoising, and parameter mapping). These characterizations can be used to help gauge the amount of trust/confidence that can be placed in imaging results obtained when using advanced estimation methods.

LPRs are computed by applying controlled perturbations to measured data, and then examining how well those perturbations are preserved after being passed through the image estimation method. LPRs are compatible with arbitrary datasets, arbitrary forward models, and arbitrary (possibly highly nonlinear) image estimation methods.

In this code, we provide an illustrative example of using LPRs in the context of image reconstruction from undersampled k-space data. Specifically, we compute LPRs for simple zero-filled Fourier reconstruction of k-space data obtained from a numerical phantom, producing the results shown below. This is not a very good reconstruction method, and the LPR result shows that this particular reconstruct should be expected to have limited spatial resolution (as evident from the blurring of the checkerboard pattern) in addition to aliasing artifacts.


Ground Truth
Reconstruction
(no perturbation)

Perturbation		Reconstruction
(with perturbation)
LPR

Although this example is intentionally quite simple, the code has been designed so that it is easy to use it together with whichever datasets, forward models, and image estimation methods that the user desires. As described in the LPR journal paper (citation given above), we believe that this approach may be of particular interest for evaluating complicated nonlinear reconstruction methods, for which the visual appearance of the image may be misleading. For example, the images shown below (modified from the aforementioned journal article) demonstrate that while the U-Net machine learning reconstruction approach may have good quantitative performance metrics and good visual appearance, an LPR-based characterization suggests that the method may have limited spatial resolution in some spatial regions and limited sensitivity to novel image features. In contrast, the LORAKS reconstruction (see LORAKS 2.0: Implementation and Examples) may have worse performance metrics and may look visually less appealing, although its LPR suggests that it should have much better spatial resolution characteristics and sensitivity to novel image features.

Ground TruthLORAKS ReconstructionU-Net Reconstruction
PerturbationLORAKS LPRU-NET LPR

The software can be downloaded using the form below:

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    Permission to use, copy, modify, and distribute this software and its documentation for educational, research and non-profit purposes, without fee, and without a written agreement is hereby granted, provided that the above copyright notice, this paragraph and the following paragraph appear in all copies. If you use this code or its derivatives in your own work, you are required to cite the original LPR journal article (C.-C. Chan, J. P. Haldar. Local Perturbation Responses and Checkerboard Tests: Characterization tools for nonlinear MRI methods. Magnetic Resonance in Medicine. In Press. DOI:10.1002/mrm.28828).

    This software program and documentation are copyrighted by the University of Southern California. In no event shall the University of Southern California, the Authors, or the Distributors be liable to any party for direct, indirect, special, incidental, or consequential damages, including lost profits, arising out of the use of this software, its documentation, or any derivatives thereof, even if the authors have been advised of the possibility of such damage. The University of Southern California, the Authors, and the Distributors specifically disclaim any warranties, including, but not limited to, the implied warranties of merchantability, fitness for a particular purpose, and non-infringement. This software is provided on an "as is" basis, and the authors and distributors have no obligation to provide maintenance, support, updates, enhancements, or modifications. This software is for research purposes only and has not been approved for clinical use.

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