Total Variation-Based Image Decomposition and Denoising for Microscopy Images

TL;DR

This work addresses noise and unwanted signals in rapidly acquired microscopy images by introducing a total variation (TV) based image decomposition workflow that performs background subtraction and denoising. It systematically compares three TV-based formulations—TV-$L^1$, Huber-ROF, and TGV-$L^1$—to assess their suitability for background extraction and feature preservation in AFM, STM, SEM, and STEM images. The study finds that Huber-ROF is the most flexible background extractor, while TGV-$L^1$ offers superior denoising with smoother edges; TV-$L^1$ lags in background subtraction but can help texture smoothing. The proposed workflow, implemented in AiSurf, demonstrates broad applicability to microscopy image restoration and is designed for integration into automated experimental pipelines, with future work focusing on automated parameter selection.

Abstract

Experimentally acquired microscopy images are unavoidably affected by the presence of noise and other unwanted signals, which degrade their quality and might hide relevant features. With the recent increase in image acquisition rate, modern denoising and restoration solutions become necessary. This study focuses on image decomposition and denoising of microscopy images through a workflow based on total variation (TV), addressing images obtained from various microscopy techniques, including atomic force microscopy (AFM), scanning tunneling microscopy (STM), and scanning electron microscopy (SEM). Our approach consists in restoring an image by extracting its unwanted signal components and subtracting them from the raw one, or by denoising it. We evaluate the performance of TV-$L^1$, Huber-ROF, and TGV-$L^1$ in achieving this goal in distinct study cases. Huber-ROF proved to be the most flexible one, while TGV-$L^1$ is the most suitable for denoising. Our results suggest a wider applicability of this method in microscopy, restricted not only to STM, AFM, and SEM images. The Python code used for this study is publicly available as part of AiSurf. It is designed to be integrated into experimental workflows for image acquisition or can be used to denoise previously acquired images.

Figures (6)

• Figure 2: Flowchart of the proposed workflow. Starting from the raw image f, its background b is extracted via TV minimization and u is calculated via subtraction. If b contains signal associated to prominent features' contrast v (e.g. from terraces or different terminations), the latter gets extracted from b and added to u. If u is affected by noise, the signal can be smoothed, obtaining $u'$. If both cases are present, one must first perform the contrast retrieval and then proceed with the smoothing. The red circles highlight the output from each TV minimization.
• Figure 3: Comparison between the background subtraction carried out with TV-$L^1$, TGV-$L^1$, and Huber-ROF on ncAFM image of K ions on mica presented in Fig. \ref{['fig:exp_images']}a. The Raw image and a zoomed area are compared with the Restored ones obtained from the background subtraction of the raw image; different backgrounds extracted via TV minimization are displayed in the right column. It is evident how differently the three methodologies perform. Huber-ROF is the most flexible method among the three, applicable to all the cases shown in this study, and it performs best in background extraction. The output for TV-$L^1$ has been obtained with $\lambda=0.2$, the one for TGV-$L^1$ with $\lambda=0.1$, and for Huber-ROF with $\lambda=0.5$. A median filter with a $1 \times 3$ kernel has been applied to the restored images for all the three methods.
• Figure 4: Background and restored image as a function of $\lambda$ using Huber-ROF. Raw image in Fig. \ref{['fig:exp_images']}a. Increasing $\lambda$ allows to filter smaller-sized features. The optimal $\lambda$ value lies between 0.5 and 1, where the background is removed without compromising the contrast. $\lambda=10$ shows that setting this parameter too high excessively reduces the contrast between K ions and the background. This might be beneficial for some tasks that require a clear imaging of the atoms or structures, but for pure restoration purposes it is advised to preserve the original features as much as possible. A median filter with a $1 \times 3$ kernel has been applied to the restored images.
• Figure 5: Background subtraction of the STM image of LSMO(110) presented in Fig. \ref{['fig:exp_images']}b via Huber-ROF method, with $\lambda=0.05$. The image is corrupted by visible bright, horizontal lines. (a) raw image overlayed with a dashed line and the region highlighted in panel (d); (b) background extracted via TV minimization overlayed with a dashed line; (c) restored image obtained via background subtraction of the raw image, overlayed with a dashed line and the region highlighted in panel (e); (d), (e) zoomed regions of panels (a), (c) respectively; (f) line profiles relative to the lines overlaying panels (a), (b), and (c), each one normalized to the maximum signal intensity of the image they refer to, for comparison purposes. A median filter with a $1 \times 3$ kernel has been applied to the restored image.
• Figure 6: Restoration of the STM image of a (2×5)-reconstructed SrTiO$_3$(110) surface displayed in Fig. \ref{['fig:exp_images']}c, following the steps described in the workflow subsection, using the Huber-ROF method with $\lambda=0.05$ for both background extraction and background blurring. These steps are required when the image contains different terminations, terraces, or in general topographically equivalent areas with different contrasts. (a) raw image; (b) flat image obtained with background subtraction; (c) background extracted with TV minimization; (d) smooth background, obtained by applying TV denoising on the background shown in (c); (e) line profiles relative to raw, flat and denoised images; each line profile is normalized to the maximum signal intensity of the image they refer to, for comparison purposes; (f) restored image.