# Image Quality Assessment for Gibbs Ringing Reduction
## 2 Test Images  
In this Section, we introduce the three test images, depicted in Figure 1. These range from the minimalist rectangle function, which provides a discontinuity and little else to confound analysis, to an example slice of an MRI for a more realistic test. This variety of test images makes our analysis more robust. The first test image is a \(400\times 400\) image of a 2D rectangle function shown in Figure 1(\(a_{1}\)). The central 200 \(\times\) 200 pixels have value 1 and the remaining pixels are zeros. One of the advantages of this test image is that its Fourier transform can easily be calculated mathematically: it is a 2D sinc function. This means that we know the space and Fourier domain samples exactly, as opposed to other examples where we know only one domain exactly and must determine the other numerically at the cost of aliasing. Figure 1: The three test images before and after introducing ringing. (\(\mathbf{a_{1}}\)) Original \(400\times 400\) numerically simulated square. (\(\mathbf{b_{1}}\)) Original \(400\times 400\) Shepp–Logan phantom. (\(\mathbf{c_{1}}\)) Original \(512\times 512\) MR image. (\(\mathbf{a_{2}}\)-\(\mathbf{c_{2}}\)) Spectra of (\(\mathbf{a_{1}}\)-\(\mathbf{c_{1}}\)). Fourier coefficients highlighted in red are set to zero, simulating zero padding of a reduced number of measured Fourier coefficients. (\(\mathbf{a_{3}}\)–\(\mathbf{c_{3}}\)) Fourier reconstruction using (\(\mathbf{a_{2}}\)–\(\mathbf{c_{2}}\)) showing visible ringing. \[g(x,y) =\text{rect}\Big{(}\frac{x}{4}\Big{)}\text{rect}\Big{(}\frac{y}{4} \Big{)} \tag{1}\] \[G(u,v) =16*\text{sinc}(4\pi u)\text{sinc}(4\pi v) \tag{2}\]  
\(G(u,v)\) was sampled with 400 \(\times\) 400 sampling points up to a cutoff frequency of 10,000 lines/mm in both \(x\) and \(y\). The spectrum is truncated (in Figure 1(\(a_{2}\)), truncation of 80% of Fourier coefficients is shown) and inverse Fourier transformed. The resulting image (Figure 1(\(a_{3}\))) exhibits Gibbs ringing and can be compared with \(g(x,y)\) sampled at 20,000 lines/mm. The second test image is shown in Figure 1(\(b_{1}\)) is a 400 \(\times\) 400 Shepp-Logan phantom. It was developed as a test image for MRI reconstruction algorithms, resembling an MRI head section, and is a widely used test image in Gibbs ringing reduction studies in MRI [6; 9; 14]. The Fourier coefficients can be approximated by discrete Fourier transform of the phantom. We know the ground truth in the space domain, but do not know the Fourier domain samples exactly. Finally, the third test image is a 512 \(\times\) 512 pixel MRI slice obtained from the Brain Tumor Progression dataset [21] of The Cancer Imaging Archive [22]. We will refer to this as MR image. It is shown in Figure 1(\(c_{1}\)). As before, we can use a discrete Fourier transform of the image to estimate the Fourier coefficients but, as with the Shepp-Logan phantom, we have a reduced reference. There is a small difference: the phantom is defined in the space domain, whereas the MR image was reconstructed from Fourier samples in the first place. These three test images have increasing complexity: the rectangle function is a highly simplified test image; the phantom has edges that are not aligned with the \(x\) or \(y\) axis, which are not straight and which overlap; and the MRI is a real image with intricate details which lacks such clearly defined edges. We also have different knowledge about the ground truth of each image. For all three images, we introduce ringing the same way. In the plots up to Figure 7, the central 20% of Fourier data remained on both the \(x\) and \(y\) axis. The magnitude of the Fourier data is shown in Figure 1(\(a_{2}\)-\(c_{2}\)). Fourier coefficients that are highlighted in red are set to zero. Adjusting the fraction of coefficients which are set to zero allows us to control the amount of ringing but maintain the number of samples to simplify comparisons. Figure 1(\(a_{3}\)-\(c_{3}\)) show the Fourier reconstruction based on the Fourier coefficients after truncation in the Fourier domain.