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Fractional Diffusion, Low Exponent Levy Stable Laws, and Slow Motion Denoising of Helium Ion Microscope Nanoscale Imagery



Andras Vladar, Alfred S. Carasso


Helium ion microscopes (HIM) are capable of acquiring images with better than 1nm resolution, and HIM images are particularly rich in morphological surface details. However, such images are generally quite noisy. A major challenge is to denoise these images while preserv- ing delicate surface information. This paper presents a powerful slow motion denoising technique, based on solving linear fractional diffusion equations forward in time. The method is easily imple- mented computationally, using fast Fourier transform (FFT) algorithms. When applied to actual HIM images, the method is found to reproduce the essential surface morphology of the sample with high fidelity. In contrast, such highly sophisticated methodologies as Curvelet Transform denoising, and Total Variation denoising using split Bregman iterations, are found to eliminate vital fine scale information, along with the noise. Image Lipschitz exponents are a useful image metrology tool for quantifying the fine structure content in an image. In this paper, this tool is applied to rank order the above three distinct denoising approaches, in terms of their texture preserving properties. In several denoising experiments on actual HIM images, it was found that fractional diffusion smoothing performed noticeably better than split Bregman TV, which in turn, performedslightly better than Curvelet denoising.
Journal of Research (NIST JRES) - 117.006
Report Number


HIM images, image denoising, image texture, image metrology, total variation, curvelet transform, low exponent L evy stable laws, image Lipschitz exponents, surface morphology.


Vladar, A. and Carasso, A. (2012), Fractional Diffusion, Low Exponent Levy Stable Laws, and Slow Motion Denoising of Helium Ion Microscope Nanoscale Imagery, Journal of Research (NIST JRES), National Institute of Standards and Technology, Gaithersburg, MD, [online],, (Accessed April 21, 2024)
Created February 21, 2012, Updated October 14, 2021