Compressed sensing for scanning tunnel microscopy imaging of defects and disorder

Compressed sensing (CS) is a valuable technique for reconstructing measurements in numerous domains. CS has not yet gained widespread adoption in scanning tunneling microscopy (STM), despite potentially offering the advantages of lower acquisition time and enhanced tolerance to noise. Here we applied a simple CS framework, using a weighted iterative thresholding algorithm for CS reconstruction, to representative high-resolution STM images of superconducting surfaces and adsorbed molecules. We calculated reconstruction diagrams for a range of scanning patterns, sampling densities, and noise intensities, evaluating reconstruction quality for the whole image and chosen defects. Overall, we find that typical STM images can be satisfactorily reconstructed down to 30% sampling—already a strong improvement. We furthermore outline limitations of this method, such as sampling pattern artifacts, which become particularly pronounced for images with intrinsic long-range disorder, and propose ways to mitigate some of them. Finally, we investigate compressibility of STM images as a measure of intrinsic noise in the image and a precursor to CS reconstruction, enabling a priori estimation of the effectiveness of CS reconstruction with minimal computational cost.

Figure 1

STM images of FeSe (a), molecular film of TCNQ/Ag(111) (b), and that of ${\mathrm{C}}_{60}/\mathrm{Ag}\left(111\right)$ (c), with representative defects magnified in each inset. (d) The distribution of normalized constant-current STM height for each image.

Figure 2

Discrete cosine transform of the STM images in Fig. 1, correspondingly for (a) FeSe, (b) TCNQ, and (c) ${\mathrm{C}}_{60}$. (d) The intensity of the diagonal coefficients for each DCT, as well as the diagonal of the DCT for an array of random Gaussian noise, which reveals sparsity of STM images.

Figure 3

The path of the rotated line pattern is shown in (a), with simulated start and end points denoted by green and red circles. Despite sparse sampling of the image (b), very good CS reconstruction is achieved (c). The same process is also shown for Lissajous trajectory (d)–(f). Reconstructions in this figure performed for 20% sampling density and 100 iterations.

Figure 4

Reconstructed images for ten-, five-, and twofold undersampling for TCNQ (a)–(c), ${\mathrm{C}}_{60}$ (d)–(f), and FeSe (g)–(i), with magnified defects in insets. All reconstructions were performed for 100 iterations using the rotated line sampling pattern.

Figure 5

Reconstruction diagrams for TCNQ, ${\mathrm{C}}_{60}$ and FeSe for the rotated line (d)–(f) and Lissajous (m)–(o) patterns, with relevant reconstructions shown above and below the diagram for each sample. The diagram plots SSIM of the reconstructed image versus intensity of noise perturbation intensity ($\delta$) and sampling density ($\rho$). The parameters used for the reconstructions in the top row are marked by black squares in the respective diagrams; the bottom row parameters are marked by white circles.

Figure 6

Compressibility of STM images (a), along with random noise (b) and an ordered lattice (c) via DCT transform. The MSE curves as a function of compressed size were normalized against the peak value for each curve. Compressibility of TCNQ for varying levels of Gaussian noise applied to TCNQ (d) (curves increase in noise level from bottom to top). (e) Parametric plot of compression error versus CS reconstruction error with added Gaussian noise as a parameter. The curves were calculated for three values of compression (0.04, 0.1, and 0.2).

Figure 7

Reconstruction diagrams of CS reconstruction of TCNQ [(a), (d)], ${\mathrm{C}}_{60}$ [(b), (e)], and FeSe [(c), (f)] calculated for rotated line (top row) and Lissajous (bottom row) sampling patterns using MSE quality metric. Insets show diagrams for defects. All reconstruction diagrams have been normalized to their respective maximum MSE.

Figure 8

(a) A wide-variance DCT weight model used to reconstruct ${\mathrm{C}}_{60}$ for Lissajous (b) and rotated line (c) patterns using 300 iterations and a 1% threshold on the number of nonzero coefficients. The low-frequency corner snippet of a low-variance model is shown in the inset of (a), and the diagonal coefficients for both models and each sample DCT are displayed in (d). Reconstructions using the low-variance model are shown for Lissajous (e) and rotated line (f).

Figure 9

SSIM evaluated for CS reconstructions of TCNQ [(a), (c)] and ${\mathrm{C}}_{60}$ [(b), (d)] across varying levels of $\sigma$ (the width of the variance in the DCT weight model) and threshold ratio (the relative number of of nonzero coefficients used by the reconstruction algorithm). The top and bottom rows, respectively, correspond to reconstructions performed using Lissajous and rotated line sampling patterns. All reconstructions performed with sampling density $\rho =0.2$.