Reducing excitation intensity or exposure time is employed to decrease the phototoxicity and photobleaching in structured illumination microscopy (SIM). However, the raw images obtained under this condition have a low signal-to-noise ratio, resulting in an error estimation of parameters and reconstruction artifacts. To improve the accuracy of parameter evaluation, some modified parameter evaluation algorithms have been proposed, including the prefiltering approach and iterative and noniterative parameter evaluation approaches. However, these approaches can only enhance the accuracy of the estimated parameters, but do not demonstrate how precise they are. In other words, these algorithms cannot assess whether there is a considerable deviation from the true value. However, to suppress the reconstruction artifacts in the reconstructed image, some reconstruction algorithms have been proposed, such as spectrum filtering, TV-SIM, and Hessian-SIM. These approaches usually reconstruct the super-resolution (SR) SIM image at the beginning, and then remove the artifacts. However, this reconstruction process will change the Poisson-Gaussian noise distribution in the images.
To address the above two issues, we first proposed a parameter evaluation approach based on empirical mode decomposition (EMD) in this research, which can precisely evaluate the initial phase and modulation depth. Measured with the proposed dispersion index, the accuracy of the estimated parameter can be given synchronously. Next, a denoising algorithm based on similar blocks was employed before SIM reconstruction in this study, which maintains the illumination pattern while suppressing the noise in the raw images. This predenoise process before reconstruction can enhance the accuracy of parameter evaluation and remove the artifacts.
This study employs empirical modal decomposition (EMD) to smooth out the frequency distribution of the estimated initial phase and modulation depth. The specific process is as follows. First, the distribution curve of the parameter estimate was decomposed with each inherent mode function component. Next, only the mode component below 4 was accumulated, which can be superimposed as a smooth fitting curve. Finally, we evaluated the initial phase and modulation depth by this superimposed smooth curve.
The dispersion index was proposed as a quantitative index that can be used to measure the evaluated initial phase and modulation depth accuracy in this study. This index primarily characterized the degree of concentration of the curve from the global and local aspects.
This study also suggested a new process of denoising the raw images first and then conducting SR reconstruction, which can enhance the accuracy of the parameter estimation and reduce the artifacts. First, we obtained the average of the raw images and concatenated it with the raw images. Next, a VST transformation was performed on the concatenated images, followed by a VBM3D-based denoising process. Finally, the denoised findings were subjected to a VST inverse transformation. After the complete denoising process, we performed the subsequent SR reconstruction.
We first imaged actin with 20 groups for comparison experiments. Each group contained 9 raw images, with three phases in every three directions, for a total of 180 images with 486 ms. These 20 groups of raw images can be directly reconstructed into 20 SR images by the Wiener reconstruction algorithm. Thereafter, we averaged the 20 SR images to one image that serves as the ground truth.
For the parameter evaluation, the frequency distribution of the parameter estimates is ideally similar to the shape of the impulse function [Fig. 1(b)]. However, because of the effect of noise, the actual distribution is a smooth curve with local jitter [blue solid line in Fig. 1(c)]. The EMD algorithm was then employed to fit this curve to make parameter estimation easier [Fig. 1(c) red dashed line].
To evaluate the accuracy of the estimated initial phase and modulation depth, we proposed a dispersion index and confirmed its effectiveness. First, we averaged the first 1, 3, 5, 7, 9, and 11 groups of the raw images to obtain six groups of images with various SNR. These six group images were employed as experimental data for the evaluation of the dispersion index. The experiment reveals that with the enhancement of the raw images' SNR, the EMD curve [Fig. 4(d)-(e) red dashed line] is more similar to the shape of the impulse function. Moreover, the dispersion index gradually decreases [Fig. 4(c)]. Therefore, the dispersion index characterized the relationship between the SNR of raw images and parameter evaluation accuracy.
The two benefits of the proposed denoising algorithm are also confirmed. First, we found that the EMD curve is more concentrated [Fig. 6(b)-(c)] and the dispersion index is low (Table 1 and Table 2), implying that the calculated parameters are more accurate when using the VST-VBM3D denoising algorithm. In the meantime, we discovered that the artifacts are suppressed [Fig. 7(d)] and the PSNR and SSIM values were high (Table 3) when using the proposed denoising algorithm.
This study proposes a parameter evaluation algorithm based on the EMD algorithm and corresponding dispersion index, which can accurately evaluate the initial phase and modulation depth from the raw images and simultaneously analyze the evaluation accuracy. Simultaneously, this study also proposes a VST-VBM3D-based denoising algorithm for SIM raw images, which may suppress the noise in the raw images before the SR reconstruction procedure. This predenoise approach not only enhances the accuracy of parameter evaluation but also reduces the artifacts in reconstructed SR images.
