The Fourier Radial Error Spectrum Plot (ESP): Code and Examples

This page provides sample code for a basic Matlab implementation of the Fourier Radial Error Spectrum Plot (ESP), as described in:

T. H. Kim, J. P. Haldar.
The Fourier Radial Error Spectrum Plot: A more nuanced quantitative evaluation of image reconstruction quality.
IEEE International Symposium on Biomedical Imaging, Washington, DC, 2018. In Press.

and

T. H. Kim, J. P. Haldar.
Assessing MR image reconstruction quality using the Fourier Radial Error Spectrum plot.
Joint Annual Meeting ISMRM-ESMRMB, Paris, 2018. In Press.

The ESP is a new approach to quantitatively measuring image quality in the presence of a gold standard reference image, and overcomes the limitations of some of the traditional scalar-valued image quality metrics like mean-squared error (MSE) or the structural similarity index (SSIM), which are widely used in the modern imaging literature.

We illustrate the benefits of ESP over scalar measures like MSE with the simple example shown below.

In this example, we are given a gold standard reference image, and have generated two corrupted versions of this original image.

 

Gold Standard Corrupted Image 1 Corrupted Image 2
Error Image 1 (Scaled 2×) Error Image 2 (Scaled 2×)

As we can see in this example, the two corrupted images have very different error characteristics — in the first image, errors are concentrated near edge locations and the image appears blurry, while the second image appears noisy with errors uniformly distributed throughout the field-of-view.  However, it turns out that they both have identical MSE values (in each case, the mean-squared error is roughly 23% of mean-squared value of the gold standard).  This illustrates the limitation of scalar-valued measures like MSE, which fail to distinguish between these two cases.

The ESP provides a more nuanced characterization, as shown below for these same two images:

In this case, the ESP clearly shows us that the first image has lower error than the second image at high-resolution spatial scales (i.e., at high spatial frequencies), while the opposite is true at low spatial frequencies.  We can clearly see that neither of these images is uniformly better than the other, and are much better able to capture the positive and negative aspects of each image. This type of information can be very useful when, e.g., comparing two different MRI reconstruction methods.  While one method may appear to be “better” than another when only looking at parameters like MSE, the ESP may provide deeper insights.

Software to generate ESPs can be downloaded using the form below:

Download

This code is licensed under the CC Attribution-Noncommercial-Share Alike 3.0 License. You are required to cite our ISBI paper (or a later journal paper if one comes out in the interim) if you use this code or its derivatives in your own work.

T. H. Kim, J. P. Haldar.
The Fourier Radial Error Spectrum Plot: A more nuanced quantitative evaluation of image reconstruction quality.
IEEE International Symposium on Biomedical Imaging, Washington, DC, 2018. In Press.

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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