3.4 | Fourier analysis to decipher effects of blurring in the images
In general, we observed that the outcomes of U-Net are blurry compared to the ground truth despite significant removal of background noise. To understand the effects of blurring for U-Net outcomes in more details, we analyzed the scenario in frequency domain. The 2D frequency domain images with magnitude spectrum are shown in 2nd, and 4th rows corresponding to the spatial images depicted in 1st, and 3rd rows, respectively of Fig. 7. The 1st row represents graphite lead phantom, and 3rd row depicts subcutaneous mouse tumor image. The inner dotted circular cores mainly cover the low frequencies while the outside of the inner cores indicate high frequencies. According to O’ Hare et al [58], the amplitude reduction at high spatial frequency zone indicates a degree of perceived blurriness of an image.
We believe that noise is generally distributed within the outer dotted circular region which is reduced (substantiated by lower spectrum magnitude than low no. of frame averaging) by both high no. of frame averaging and U-Net. As the magnitude spectrum of the high frequency region between two dotted circles (indicated by red arrows in Fig. 7) is attenuated more by both U-Net algorithm than the high no. of frame averaging, the U-Net outcomes are comparatively blurrier than high no. of frame averaging images. The inset zoomed-in spatial domain images in white boxes in 3rd row pictorially shows the blurring effects. Lower frequencies for both the cases are somewhat preserved. The U-Net outcomes also show small edge effects which the high no. of frame averaging images does not possess. Another important fact to note is the high diagonal frequencies of the high no. of frame averaging and U-Net outcomes generate less magnitude than low no. of frame averaging. It suggests that the averaging process degrades the high diagonal frequencies. We tried to quantify the degree of blurring by relative decay of sharpness in the U-Net outcomes compared to the high no. of frame averaging ones as mentioned in [59], and found a 25.5 ± 4.99 % sharpness decrease in U-Net with respect to the high no. of frame averaging. The representative tumor images are provided in supplementary figure S4.