Recovery of Background Structures in Nanoscale Helium Ion Microscope Imaging

This paper discusses a two step enhancement technique applicable to noisy Helium Ion Microscope images in which background structures are not easily discernible due to a weak signal. The method is based on a preliminary adaptive histogram equalization, followed by ‘slow motion’ low-exponent Lévy fractional diffusion smoothing. This combined approach is unexpectedly effective, resulting in a companion enhanced image in which background structures are rendered much more visible, and noise is significantly reduced, all with minimal loss of image sharpness. The method also provides useful enhancements of scanning charged-particle microscopy images obtained by composing multiple drift-corrected ‘fast scan’ frames. The paper includes software routines, written in Interactive Data Language (IDL),1 that can perform the above image processing tasks.


Introduction
The advantages of Helium ion microscopy (HIM) over conventional scanning electron microscopy (SEM) are explored in [9] and [10]. HIM imaging achieves higher resolution with higher contrast and greater depth of field than is possible with SEM. In many cases, HIM images reveal greater surface detail. The secondary electrons that carry the sample's surface detail information are generated in higher quantities by He irradiation than is the case with electron beams. As a result, low probe currents, on the order of one picoampere, can be used effectively to acquire HIM images. The use of small He ion beam currents is also necessary to minimize damage to the sample. However, such low current HIM images are often noisy, and important background structures may not be easily discernible due to the weak signal.
This paper discusses a two-step enhancement technique that can significantly improve this situation. As shown in Fig. 1, this technique results in a companion enhanced image in which deep background structural details are recovered, while noise is significantly reduced with minimal loss of image sharpness. Previous work on denoising HIM imagery [1], did not address the background recovery problem. While limited success was achieved in more recent work on forensic latent fingerprint enhancement [2], the present results, involving an entirely different class of images, could not be anticipated and had to be discovered independently.
The two stage approach used here involves a preliminary image-specific adaptive histogram equalization of the given HIM image [11]. Such equalization enhances background information while significantly magnifying noise, and is not generally advisable with noisy data. However, at the second stage, an effective and easy to use progressive denoising technique, based on low exponent Lévy fractional diffusion smoothing, can be successfully applied to this histogram equalized image with magnified noise. As may be inferred from Figs. 2 and 3, such Lévy smoothing can be fine-tuned interactively so as to preserve the detailed surface morphology of the sample. Despite many parameter trials, comparable fidelity to surface detail was not found feasible with some other better-known denoising techniques, such as methods based on minimizing the image 'total variation' norm [6], [8], or methods based on thresholding curvelet transforms [13], [14]. This is illustrated in Figure 4. The ability to explore efficiently in parameter space and visually select the best result, is the key to successful digital image enhancement. There is no single set of parameters that will be useful in all cases, nor is there an automatic way of selecting the best set of parameters. Only a trained and experienced human analyst can locate useful parameter values, and determine the proper amount of fine tuning that best displays the information sought. The paper includes software routines, written in Interactive Data Language (IDL) [7], that can be used to perform the above two separate image processing tasks.

Preliminary Enhancement of HIM Image Using 'IDLAdapt.pro'
Histogram equalization is a useful enhancement technique for images where significant information is suspected of being hidden in dark regions. A complete discussion of this technique is given in [11], together with examples illustrating the advantages of the more sophisticated adaptive formulation. However, in practice, usefully recovered background information is often obscured by the amplification of the accompanying noise, and the resulting improvements may not be particularly helpful without additional intervention.
The routine IDLAdapt.pro listed in Sec. 7, is to be applied to an 8 bit greyscale TIFF image, which may be rectangular. The routine identifies and prints the dimensions of that image, performs adaptive histogram equalization, and displays the equalized image. Next, the equalized image is embedded in a zero padded larger square array of even dimension specified by the user. That dimension should typically be between fifteen and twenty percent greater than the larger side of the input image. The zero padded equalized 8 bit greyscale TIFF image is returned in the file 'idlsharp.tiff'. This process is illustrated in Fig. 5.

Progressive Low Exponent Lévy Fractional Diffusion Smoothing
Given a noisy image f (x, y), the smoothing procedure results from solving an initial value problem for a special type of diffusion equation, with the image f (x, y) as initial data. Such smoothing is applied to the whole image, and not just to a selected portion of the image. With fixed p with 0 < p ≤ 1, consider the linear fractional diffusion initial value problem in L 2 (R 2 ), where ∆ denotes the 2D Laplacian. This reduces to the classical heat conduction equation when p = 1. However, our smoothing procedure uses values of 1 p  , such as p = 0.2, for example. Define the 2D Fourier transform of the image f (x, y) by Eq. (1) has the unique Fourier domain solution As is evident from Eq. (4), w (x, y, t) becomes increasingly smoother as t increases. However, for small p, and over a short time interval, the smoothed image may be expected to retain many of the essential features present in the initial data f (x, y). In Eq. (3), the function 2 2 ( , , ) = exp{ (2 ) (2 ) }, > 0, is the Fourier transform of the Green's function for the linear fractional diffusion equation in Eq. (1). For each fixed t > 0, the function in Eq. (5) is also the Fourier transform of an isotropic Lévy stable probability density function with exponent 2p, [12]. When p = 1, Eq. (5) corresponds to a Gaussian distribution. For 1 p  , Eq. (5) corresponds to a heavy-tailed density in physical (x, y) space. Unlike a Gaussian, that Lévy density is not known in closed form in the physical variables (x, y), and it has infinite mean and infinite variance.
Smoothing an image by convolution with a Gaussian is equivalent to using p = 1, and solving the heat conduction equation in Eq. (1). The significance of Lévy stable fractional diffusion smoothing with 1 p  can be inferred from Eq. (3). Clearly, attenuation of high frequency information, corresponding to large ( | | | | x η + ), is dramatically more severe when p = 1, than it is when p = 0.1 for example.

FFT Lévy Fractional Diffusion Smoothing Using 'IDLLevy.pro'
As illustrated in Fig. 5, the smoothing software routine IDLLevy.pro, listed in Sec. 7, assumes a zero padded square input image of even dimension, and returns a square smoothed image of the same size. The smoothed image can subsequently be cropped to the original size.
Given the 2N × 2N pixel image f (x, y) as initial data, 'IDLLevy.pro' computes the solution w (x, y, t) in Eq. (1) at any given t > 0, by using the forward and inverse FFT to implement the operations in Eq. (3) and Eq. (4) respectively. In order to render mathematical formulae more transparent, we use the same notation, ˆ( , ) f x η , for both discrete and continuous Fourier transforms. In the discrete FFT case, the frequencies 2px and 2pη are understood to be integer-valued and to range from −N to N. After selecting a tentative maximum smoothing time T max at which to terminate the smoothing process, Eq. (4) can be evaluated at finitely many intermediate times

Some Examples of HIM Image Enhancement
Figures 6 through 9 illustrate the kind of improvements that can be obtained using the above IDL software routines. Note the small values of t in these figures.

Drift-Corrected Composition of Multiple Fast Scan Frames
A NIST-developed image composition technique for scanning charged-particle microscopy is discussed in [3][4][5]. This methodology is based on super-fast acquisition of a large number of image frames. Due to the small beam currents generally used in scanning particle beam microscopy, these individual frames are inherently very noisy, but exhibit significantly less drift-related distortions. The drift correction takes place after finding the center of each frame, properly aligning these frames, and adding them together into a single image. Such composed images contain much less noise and exhibit significantly less blur and deformation than do images obtained by traditional slow scan methods, or images obtained by simply adding together fast image frames without compensating for drift. To improve repeatability of HIM and SEM images, this technique must be used with the minimum number of fast images, which limits the achievable SNR. In many cases, the resulting composed image is still somewhat noisy and some type of noise processing may be beneficial.
As shown in Fig. 10, useful improvements of such composed SEM imagery are possible using the techniques in the present paper. Additionally, improvements to composed HIM imagery are also possible as shown in Fig. 11. The left image in Fig. 11 is a composition of 10 fast image frames. The NIST processed right image in Fig. 11 shows new details, such as augmented faint greyscale transitions at the tops of the gold grains, as well as very fine structures near these tops. Such details can hardly be seen initially in the original left image; however, they become discernible in the original image after studying the companion NIST-enhanced image on the right.
Improvements to single raw HIM image frames are also possible as shown in Fig. 12. One can recognize features in the enhanced right image in Fig. 12 that might easily have been dismissed as noise in the raw image, rather than actual information already present in that raw image.
Because the very finely focused helium ion beam in HIM imaging readily mills the gold particles in Fig. 12, long image acquisition times are not possible without substantially altering the sample. The best solution is the acquisition of a set of fast images that can be composed into a single image, yet one that does not show significant sample modification. This often leads to inherently noisy images. However, this can be remedied using the two-step process discussed in this paper, which results in less noisy images with more perceptible fine details.

Two IDL Routines for HIM Image Enhancement
The two software routines listed below were used to produce the enhanced images shown in the figures.