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Modelling PSF of Scanning Electron Microscopes for Image Restoration

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Modelling PSF of Scanning Electron Microscopes for Image Restoration Mo delling PSF of Scanning Electron Microscop es for Image Restoration J. Swindel ls M. Razaz K. Tovey* Scho ol of Information Systems University of East Anglia Norwich England *Scho ol of Environmental Sciences University of East Anglia Norwich England email: [email protected], [email protected] The image collection and mo delling pro cedures for the ABSTRACT PSF, as well as some typical exp erimental results based A pro cedure to determine the p oint spread function (PSF) on the SEM, are presented and discussed. ofatwo-dimensional SEM imaging system based on ex- p erimental data is presented. The sp ecimen required 2 Measurement Pro cedure and Mo delling PSF for capture by the imaging system is simple to manu- The p oint spread function (PSF) of a SEM is equal to facture, and only requires a sharp edge to b e of use. An the microscop e unit impulse resp onse. This resp onse overview is given of caveats that exist at each stage of can b e measured by collecting the observed image when the pro cess, in addition to a breakdown of the pro cess the input to the microscop e is a small source (of energy) itself. Results based on a Scanning Electron Microscop e whose dimensions are b elow the resolution of the micro- are shown. The use of a PSF estimated in this fashion scop e. However, in practice emulating a p oint source is shown to result in much improved restorations, when would b e very dicult to implement for high resolution compared to its theoretical equivalent. instruments such as SEM, as the source needs to be small enough (b elow the resolution of the microscop e) to show the PSF, yet still b e visible. Its inherently sub- 1 Intro duction resolution size also makes the PSF vulnerable to noise. Despite these diculties this typ e of PSF measurement The Scanning Electron Microscope (SEM) is an instru- is p ossible for certain typ es of imaging such as optical ment used to capture two-dimensional images of mate- microscopy, and has b een successfully carried out be- rials. SEM imaging has many applications in material fore for wide- eld and confo cal microscopy using uo- science, biology,environmental sciences and so on. Im- rescently lab elled sub-resolution b eads [RRP94][SR91]. ages pro duced by SEM are invariably blurred by the p oint spread function of the imaging instrument, and We describ e here a practical approachinvolving mea- they are also noisy to a degree. These degradations surement of step resp onse of SEM, and then deriving the reduce b oth resolution and signal to noise ratio of the PSF mo del from the measured data. This exp erimen- collected (observed) images [Cas96][Jan84]. tally mo delled PSF gives a far closer approximation to the true PSF, thus considerably increasing the quality Restoration of SEM images is necessary to remove of restoration p ossible. these degradations and recover the original image from the knowledge of the p oint spread function(PSF) and We now brie y describ e the pro cedure used for col- noise. It is essential in this pro cess to have an accurate lecting image data. The SEM wehave used can op erate mo del of the PSF. Even in the cases of blind restoration either in secondary-electron mo de which gives a view of [RKH96], some knowledge ab out the PSF would b e nec- the top ography of the sp ecimen, or back-scattered elec- essary. No attempt has b een made in the past to mo del tron mo de (BSE) which is used for highlighting di erent accurately the PSF of a SEM. Often very approximate chemical comp ositions. In this work, we used the SEM analytical expressions have b een in the literature, for in BSE mo de with the sample normal to the b eam. The example see [TH95]. sample is aligned using a reference mark. Using a mag- ni cation factor of 2000x (0.11 mm pixel spacing), each These mo dels can not prop erly emulate the "true" pixel is typically captured with adwell time of 51 ms. PSF of the complex SEM imaging system. We present The op erating voltage is 18 kV, with working distance in this pap er a new and accurate PSF mo del of SEM set to 15 mm. The resulting image is 512x512 pixels based on exp erimental results. We have applied this with 256 grey levels. mo del to restoration of SEM images, have obtained re- There are a numb er of steps in generating a 2D mo del stored images with signi cantly improved resolution and for the SEM, as shown in Figure 1. The rst step is to signal-to-noise ratio. collect the raw step resp onse data . A step resp onse is the result of viewing a sharp edge, which is a sudden from a high-intensity to low-intensity area. transition StepFunction This is achieved using a glass slide, emb edded in resin, as a sp ecimen: glass has the prop erty of b eing high in- tensity, and the resin has no luminescent prop erties. The second step is to numerically di erentiate the step resp onse. The image collected from this sp ecimen sampled p erp endicular to the direction of the edge, is ImagingSystem with the aid of statistical, Fourier and image enhance- ment techniques [FW93][Cas96]. To calculate the line spread function (LSF) from the step resp onse, this data is numerically di erentiated. Such an op eration is e ec- tively a high-pass ltering, which means that the high y noise present are likely to overwhelm frequencies of an StepResponse the relatively low frequency step resp onse. Hence the collection of edge samples is averaged to minimise noise p erturbations, whilst preserving the overall shap e of the step resp onse. In addition the data is smo othed using a cubic interp olation scheme [Num88]. The interp olation a cubic spline, such that the original curve is mo d- is Differentiation elled by a series of cubic functions knotted together. The more knots there are in the spline, the more closely the spline ts the original data. In this case, a prede- termined numb er of knots is used to achieve the desired amount of smo othing. The e ect of the interp olation ve all lo cal spikes in the data, however small, is to remo LineSpread while completely retaining the larger-scale shap e of the Function edge resp onse. The nal step is to derive the PSF from the line spread function data obtained in step 2. There is a gen- eralised relationship between the LSF and the optical function (OTF), the Fourier transform of the transfer Integration PSF [Tat65]. More recently the technique has b een ap- plied to particular optical instruments by Pieralli [Pie94] and Weickenmeier et al [WNM95]. Since the PSF of SEM is radially symmetric, we can numerically integrate the LSF data in a sp eci c manner to directly nd the PSF as. PointSpread Z 1 1 d (x)r:dx A Function p C (r )= (1) 2 2 dr x x r r where C(r) is the PSF with resp ect to r (the distance from its centre), and A(x) is the LSF. This pro cedure is Figure 1: Derivation of a PSF from a step function similar to that used in [Mar64]. Space variance may be comp ensated for by keeping each cross-section from the step image separate. Each x2000. Figure 2(c) is the PSF derived from this image, of these cross-sectional lines is the LSF at a particular by solution of equation 1. Although the technique has lo cation in the viewable region of the microscop e, and b een applied in this instance to an electron microscop e, placing the edge sp ecimen at di erent p ositions should it is equally suitable for any imaging system where a yield a set of LSFs, each of whichmaybeevaluated in suitable step function may b e approximated. the usual way to determine the PSF at each lo cation. Figure 3(b) shows typical restoration results using the theoretical and mo delled PSFs. Figure 3(a) is the ob- 3 Results served image from a clay sample emb edded in resin, and is viewed at x2000 in back-scatter mo de. The restora- Figure 2 shows an example of the unit step resp onse tion technique used is an iterative deconvolution metho d data collected in backscatter mo de at a magni cation of (a) Original step resp onse data (b) 3-D view of step resp onse (c) PSF derived from from unit step resp onse data Figure 2: Results of Estimation b e applied to any imaging system - even 3D - provided that simultaneously removes noise and blurring, and is a suitable sp ecimen can be formed. In many applica- describ ed in [RRP92]. tions it is highly dicult or imp ossible to isolate a sp ec- The theoretical PSF used in the restoration of the imen that approximates a p oint source of energy, so this image in gure 3(a) [TH95] is larger than the actual metho d based on a comparatively large ob ject maybe PSF of the microscop e. This intro duces ringing, as g- more desirable. For a 3D system the sp ecimen must have ure 3(b) shows. Comparison of gures 3(b) and 3(c) a at base that lies within the fo cus of the microscop e; shows clearly that the use of our mo delled PSF has con- the image capture may then extend from an energetic siderably improved the quality of the restored image in region to a non-energetic region in the direction of the terms of clarity and resolution enhancement.
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