Linnik point spread functions, time-reversed logarithmic diffusion equations, and blind deconvolution of electron microscope imagery

TL;DR

Addresses sharpening of SEM/HIM images under a blind deconvolution framework where the blur is modeled by a Linnik optical transfer function with parameters $(\lambda,\gamma)$ estimated by least squares from a preconditioned image. The method performs deconvolution by marching backward in time along the diffusion equation $w_t = -\lambda\{\log(I+\gamma(-\Delta))\}w$ in Fourier space, resulting in a partially deconvolved image at $t=\bar{t}$ with $0<\bar{t}<1$. It shows improvements over Lévy-based PSFs for microscopy imagery and relies on adaptive histogram equalization preconditioning to enable robust parameter estimation. The approach is demonstrated on SEM and HIM data and is positioned for broader applicability in biomedical, pharmaceutical, and semiconductor imaging where sub-nanometer details matter.

Abstract

A non iterative direct blind deconvolution procedure, previously used successfully to sharpen Hubble Space Telescope imagery, is now found useful in sharpening nanoscale scanning electron microscope (SEM) and helium ion microscope (HIM) images. The method is restricted to images $g(x,y)$, whose Fourier transforms $\hat{g}(ξ,η)$ are such that $log~|\hat{g}(ξ,0)|$ is globally monotone decreasing and convex. The method is not applicable to defocus blurs. A point spread function in the form of a Linnik probability density function is postulated, with parameters obtained by least squares fitting the Fourier transform of the preconditioned microscopy image. Deconvolution is implemented in slow motion by marching backward in time, in Fourier space, from $t = 1$ to $t = 0$, in an associated logarithmic diffusion equation. Best results are usually found in a partial deconvolution at time $\bar{t}$, with $0 < \bar{t} < 1$, rather than in total deconvolution at $t=0$. The method requires familarity with microscopy images, as well as interactive search for optimal parameters.

Figures (12)

• Figure 1.1: As shown in carIM, the use of Linnik probability densities, rather than Lévy stable densities, produces better results in deblurring astronomical imagery. Above Whirlpool Galaxy image was obtained at Kitt Peak National Observarory. Successful Linnik deblurring of several Hubble Space Telescope color images may also be found in carIM. Horizontal Field Width (HFW) in above images is approximately 53,000 light years.
• Figure 2.1: Adaptive Histogram Equalization (ADHE), reveals valuable information while generating significant noise that must be smoothed out. This is reflected in the respective Fourier spectra. Above images are $1 \mu m$ HFW secondary electron images.
• Figure 2.2: In the smoothed ADHE image $g(x,y)$, least squares fitting $log |\hat{g}(\xi,0)|$ with the expression $-\lambda(log(1+4 \pi^2 \gamma \xi^2) -1.5$, leads to parameter values $\lambda=0.969,~\gamma=1.64$, for $\hat{h}(\xi,\eta)$ in Eq. (\ref{['eq:1.01']})
• Figure 2.3: Partial deconvolution sequence with $s=0.001,~K=300$. Best image is found at some $~\bar{t}~$ lying between $t=0.75$ and $t=0.8$. At smaller $t$ values, $\parallel w(.,t_n) \parallel_{tv}$ increases rapidly as serious noise develops. Above images are $1 \mu m$ HFW secondary electron images.
• Figure 4.1: $1 \mu m$ HFW images. Left: Secondary electron image of an etched glass sample showing weak surface details. Right: rich details after Linnik processing. Parameters in Eqs. (\ref{['eq:1.01']}) and (\ref{['eq:2.01']}): $\lambda=0.969$, $\gamma=1.64$, $s=0.001,~K=300$, $\bar{t}=0.77$.