Tiêu chuẩn iso ts 24597 2011

Reference numberISO/TS 24597:2011EFirst edition2011-06-15 Microbeam analysis — Scanning electron microscopy — Methods of evaluating image sharpness Analyse par microfaisceaux — Microsco

General

For effective SEM image acquisition, it is crucial to adjust the microscope settings as outlined in Annex B of ISO 16700:2004 Image sharpness relies on several factors, including the specimen characteristics, the smoothness of the foreground and background, brightness and contrast, and the contrast-to-noise ratio (CNR) To evaluate image sharpness accurately, follow the procedures detailed in sections 4.2 to 4.10 for all three methods discussed Special emphasis should be placed on optimizing the electron probe current and focusing conditions to achieve the best brightness, contrast, and CNR, as specified in sections 4.6 and 4.7.

Specimen

As of the publication date, there is no certified reference material (CRM) available However, acceptable results can be achieved using a specimen prepared according to the method outlined in Annex G It is important to select a specimen with a smooth, flat surface, and for image sharpness evaluations, choose an area containing circular particles on the substrate Capture the desired images at the specified magnification as per section 4.4.

Specimen tilt

Set the specimen tilt angle at 0° (non-tilting condition)

NOTE Errors within ±3° in the tilt angle of the specimen will not affect the evaluation of the image sharpness.

Selection of the field of view

To ensure optimal image sharpness, select a field of view that encompasses a flat and smooth surface, as image clarity is affected by the surface's evenness Figures 1 a) and b) illustrate acceptable and unacceptable fields of view, respectively It is recommended to choose particles that span several tens of pixels, as shown in Figure 1 a).

Figure 1 — SEM images with a) acceptable and b) unacceptable structured foreground images

Selection of the pixel size

Before evaluating the image sharpness, it is necessary to calibrate the image magnification and/or the scale marker in accordance with ISO 16700

4.5.2 Determination of the pixel size from a field of view

The pixel size L p (in nm) is determined from the following equation: p FOV p

L FOV is the horizontal width of the field of view on an SEM image, in nm;

N p is the number of pixels covering the horizontal width of the field of view

Copyright International Organization for Standardization

4.5.3 Determination of the pixel size from a scale marker

The pixel size L p (in nm) is calculated by using a scale marker as follows: scale p scale

L scale is the “indicator” value (e.g the nominal value, in nm) of the scale marker;

N scale is the number of pixels covering the length of the scale marker

4.5.4 Conversion of the pixel size

The image sharpness as derived by the methods described herein (R PX ) is in pixels Converted to nanometres, the image sharpness R L is then given by the expression:

R =L ×R where L p is the pixel size

To achieve optimal image sharpness, set the pixel size to approximately 40% of the anticipated sharpness value For instance, if the expected image sharpness is 2 nm, adjust the pixel size to 0.8 nm.

Brightness and contrast of the image

For optimal image quality, the signal intensity must be evenly distributed Figures 2 a), b), c), and d) illustrate examples of images with both acceptable and unacceptable brightness and contrast levels To aid visual understanding, line profiles corresponding to the dotted lines at the same vertical position in each image are provided Specifically, image a) demonstrates acceptable quality, while image b) is deemed unacceptable due to over-saturation.

Figure 2 — SEM images with acceptable and unacceptable brightness and contrast

Contrast-to-noise ratio of the image

The contrast-to-noise ratio (CNR) of the image must be 10 or greater, where CNR is defined as the ratio of the image contrast (C image) to the standard deviation of the image noise (σn).

A procedure for the determination of the CNR ratio is given in Annex A

Figure 4 shows the simulated appearance of images with CNRs of 5, 10 and 50

Figure 5 shows examples of SEM images with different CNRs of about 4 and 30

To achieve high-quality SEM images with optimal contrast-to-noise ratio (CNR), it is essential to adjust the probe current and image acquisition time It is important to note that any changes to these parameters can significantly influence the evaluation of image sharpness.

Figure 3 — Intensity profile of an image a) CNR = 5 b) CNR = 10 c) CNR = 50

Figure 4 — Simulated images with different contrast-to-noise ratios

Figure 5 — SEM images with different contrast-to-noise ratios

Focus and astigmatism of the image

Focus the electron beam as well as possible Use an image that is as free of astigmatism as possible.

Interference from external factors

External factors like mechanical vibrations and magnetic field distortions, as outlined in Annex B of ISO 16700:2004, can significantly impact image sharpness It is essential to minimize the influence of these factors to ensure the clarity of the images used.

Erroneous contrast

Make sure that the images do not contain erroneous contrast (e.g contrast due to charging of the specimen).

SEM image data file

The image data, which is directly saved from an SEM, shall be in digital format, with the grey scale at least

8 bits deep The data file of the image shall be in an uncompressed graphics-file format, e.g uncompressed bitmap or uncompressed TIF

Do not use the data obtained from a printed SEM image

5 Acquisition of an SEM image and selection of an area within the image

The procedure outlined in this clause is standard across all methods in this Technical Specification (refer to Clause 6) First, utilize a specimen prepared according to the guidelines in section 4.2 Next, capture an image while adhering to the instructions provided in sections 4.3 to 4.10 Finally, choose a square area within the SEM image, henceforth referred to as the image, ensuring it encompasses a minimum specified size.

256 × 256 pixels The area shall not have any superimposed extraneous data (e.g magnification display, scale marker, characters, arrows, etc.)

Choose an area containing images of preferably non-overlapping particles c) Store the selected SEM image in a data file in an uncompressed graphics-file format specified in 4.11

General

The evaluation methods outlined in sections 6.3 to 6.5 assume a Gaussian profile for the electron beam, which means the results may not accurately reflect the true beam size, as noted in Clause E.4 Additionally, Figure 6 presents a flow chart that illustrates the general procedure for evaluating an SEM image, including the standard method for assessing the Contrast-to-Noise Ratio (CNR) detailed in Clause 5.

To achieve optimal image sharpness in SEM imaging, follow these essential steps: First, select an SEM image in accordance with Clause 5 Next, assess the Contrast-to-Noise Ratio (CNR) of the chosen image, ensuring it meets or exceeds a value of 10 Finally, calculate the sharpness factor, denoted as 2σ, for the SEM image in either frequency or real space, based on the method employed The sharpness is derived from a convolution of the binary SEM image with a two-dimensional Gaussian profile characterized by the sharpness factor 2σ, representing a twofold standard deviation.

NOTE The calculation procedure depends on the method used d) The image sharpness is defined as k × 2σ, where k =1/ 2

Load an SEM image Start

Execute the FT method, or the CG method or the DR method

Figure 6 — General flow chart for the evaluation of an SEM image

Contrast-to-noise ratio

The contrast-to-noise ratio (CNR), initially developed for medical imaging, is crucial for evaluating selected SEM images Only images with a CNR of 10 or higher will proceed to the next phase of image sharpness assessment A flow chart illustrating the CNR evaluation process is provided in Figure 7, with detailed procedures outlined in Annex A.

If the value of CNR is < 10, discard the SEM image Acquire a new SEM image with lower noise and carry out the evaluation again

Load an SEM image a) Calculation of the CNR value

Figure 7 — Flow chart for the evaluation of the CNR

Fourier transform (FT) method

The Fourier transform (FT) method is employed to assess image sharpness by analyzing the spatial frequency components derived from a scanning electron microscope (SEM) image These components are compared to those of images generated through the convolution of the binarized SEM image with Gaussian profiles, utilizing various sharpness factors of 2σ Detailed procedures for implementing the FT method can be found in Annex B.

NOTE The signal intensity of an image I m is expressed as I m (i, j), and the coordinates i and j are chosen as

0, 1, …, L − 1 for an image with x- and y-size L (= 256, 512, …) However, the coordinates i and j are treated as integers ranging from −L/2 to (L/2) − 1 for the FT pattern

Figure 8 — (a) a selected SEM image I O (i, j) with image size L = 256, (b) the binarized image I B (i, j), (c) and (d) the convoluted images I C (i, j; 2σ) with 2σ = 4 pixels and I C (i, j; 2σ) with 2σ = 6 pixels, respectively

Figure 9 — Averaged and smoothed FT curves plotted as common logarithms: F OH (f j ) for the selected

SEM image I O (i, j), and F CH (f j ; 2σ) and F CH (f j ; 2σ OH ) for the convoluted images I C (i, j; 2σ) and

1) Generate a filtered image I OF (i, j), processed by the 3 × 3 median filter, of a selected SEM image

2) Produce a histogram H(S) of I OF (i, j) and then obtain a smoothed histogram H s (S) by using the moving averages of 9 points Then calculate h s (S) = log 10 [H s (S) + 1]

3) Determine S L and S H that correspond to the intensities of the substrate and the particles, respectively, and determine a threshold level (S L + S H )/2 by using h s (S)

4) Produce a binarized image I B (i, j) by using (S L + S H )/2

5) Add the white noise to the selected image I O (i, j) by setting SNR p (signal-to-noise ratio for particles) to 30 for the signal intensity S = 192

To generate convoluted images \( I_C(i, j; 2\sigma) \), convolve the binarized image \( I_B(i, j) \) with two-dimensional Gaussian profiles that have varying sharpness factors \( 2\sigma = 2\sigma(N) \) Start with \( 2\sigma(1) = 1 \), where each \( \sigma \) represents the standard deviation of the Gaussian distribution, and \( N \) denotes the step number, taking values \( N = 1, 2, \ldots \).

7) Adjust the intensity of the various convoluted images I C (i, j; 2σ) so that the maximum and the minimum intensities are S H and S L , respectively b) Generation of curves of FT patterns

1) Carry out the FT for the selected SEM image I O (i, j) and the various convoluted images I C (i, j; 2σ)

G O (f i , f j ) and G C (f i , f j ; 2σ) represent the FT patterns corresponding to I O (i, j) and I C (i, j; 2σ), respectively

To calculate the curves \( F_{OHA}(f_j) \) and \( F_{OVA}(f_i) \), first obtain the horizontally and vertically averaged-and-smoothed values of \( \lvert \text{Re}[G_O(f_i, f_j)] \rvert \) Then, apply the common logarithm to these values.

NOTE Re[…] denotes the real part and ⎪…⎪ denotes the absolute value

3) Obtain the averaged curves of F OH (f j ) and F OV (f i ) by applying the moving averages of 5 points along the horizontal f j and the vertical f i directions for the curves F OHA (f j ) and F OVA (f i ), respectively

4) Obtain the averaged curves F CHB (f j ; 2σ) and F CVB(f i ; 2σ) for G C(f i , f j ; 2σ) in a similar manner c) Calculation of temporary image sharpness R PXO

1) Determine the noise areas for both of the curves F OH (f j ) and F OV (f i ) and then obtain the respective noise functions F NH (f j ) and F NV (f i ) in the noise areas by linear approximation

2) Calculate the corrected curves F CH (f j ; 2σ) and F CV(f i ; 2σ) from the averaged curves F CHB (f j ; 2σ) and

F CVB (f i ; 2σ) by using the signal and noise intensities at the origin of (f i , f j )

3) Obtain the value f i = f jC by using F OH (f j ), F NH (f j ) and a specified constant C N and then calculate the horizontal coordinate f jH from f jC by linear interpolation

4) From the functions obtained, determine the coordinates of three points, P 1H [on the curve F OH (f j )],

P 2H [on the line F NH (f j )] and P 3H [on the curve F CH (f j ; 2σOH)], lying on a vertical line with horizontal coordinate f j as shown in Figure 9

5) Determine the coordinates of the three points P 1V [on the curve F OV (f i )], P 2V [on the line F NV (f i )] and

P 3V [on the curve F CV (f i ; 2σOV)] in a similar manner

6) Obtain the sharpness factors 2σOH and 2σOV by linear interpolation of 2σ(N) by increasing the step number N

7) Calculate the sharpness factor 2σ O from 2σ O = (2σ OH + 2σ OV )/2

8) Calculate the temporary image sharpness R PXO from R PXO =2σ O / 2 d) Calculation of the image sharpness R PX

1) Calculate the coefficient C F from the sharpness factor 2σ O used for calibration

2) Obtain the calibrated sharpness factor 2σ C by using the coefficient C F

3) Evaluate the image sharpness R PX from R PX =2σ C / 2

Calculation of temporary image sharpness R PXO

Calculation of image sharpness R PX

Figure 10 — Brief flow chart of processes in the FT method

Contrast-to-gradient (CG) method

The contrast-to-gradient (CG) method involves extracting the intensity gradient at each pixel by fitting a quadratic surface to a 3 × 3 area centered on that pixel The sharpness of the CG image, denoted as R CG, is inversely related to the weighted harmonic mean of these gradients.

CG image sharpness R CG is converted to the image sharpness R ES using standard images with various sharpness factors 2σ

`,,```,,,,````-`-`,,`,,`,`,,` - © ISO 2011 – All rights reserved 13 a) Original SEM image b) Depth image corresponding to the original image, showing a typical quadratic surface fitted to the 3 × 3 area centred at a pixel point

Figure 11 — Original SEM image and the fitting of a quadratic surface to the 3 × 3 area centred at each pixel point of the corresponding depth image

Image sharpness is minimally affected by noise and is assessed using the Contrast-to-Noise Ratio (CNR) Figure 12 presents a flow chart illustrating the steps of the CG method, which includes routines a) to d), with further details available in Annex C The first step involves calculating the CG image sharpness, denoted as R CG, for the original image.

Reduced images are generated using reduction factors \( r \) of 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, and 20, with each image labeled as a (1/r)-size image (e.g., 1/2-size, 1/4-size) The (1/r)-size image for \( r = 1 \) represents the original image This image reduction process effectively decreases image noise, albeit at the expense of image-sampling frequency In the subsequent routine, four types of sharpness are calculated: local sharpness, directional sharpness, directionally averaged sharpness, and another unspecified measure.

The sharpness of CG images is assessed through three calculated types for each reduced image Additionally, the final type of sharpness, which defines the overall image quality, is derived from the curves of \( R \) and \( \Delta R/R \) plotted against \( r \).

In each image, the local sharpness at any pixel (i, j) is calculated as follows:

∆C is the threshold contrast; g(i, j; θ) is the local gradient with directional information θ

The local gradient is found by fitting a quadratic surface over a 3 × 3 pixel area centred at each pixel (i, j) The fitting error ∆g provides the fluctuation in R p , i.e ∆R p

The directional sharpness \( R_k \) is determined as the weighted harmonic mean of local sharpness within the \( k \)th sector of the azimuth angle \( \theta \) in the image Additionally, the ratios \( \frac{\Delta R_k}{R_k} \) are computed using \( \frac{\Delta R_p}{R_p} \).

The directionally averaged sharpness R, defined as the root mean square of R k , is calculated The values of ∆R/R are also calculated using ∆R k /R k

The CG image sharpness, denoted as \( R_{CG} \), is determined by analyzing graphs of \( R \) and \( \Delta R/R \) against the reduction value \( r \) For all reduced images, \( R \) and \( \Delta R/R \) are evaluated at \( r = 1 \) The minimum reduction value \( r_{min} \) is identified where \( \Delta R/R \) reaches its lowest point, and \( R_{CG} \) is defined as \( R \) at \( r = r_{min} \) This measure of sharpness is deemed reliable due to the minimized \( \Delta R/R \), although it is also affected by noise levels Additionally, standard images are generated to facilitate the calculation of their corresponding CG image sharpness \( R_{CG} \).

Standard images are generated by blurring a binary Scanning Electron Microscope (SEM) image using Gaussian profiles with varying sharpness factors, denoted as 2σ Additionally, Gaussian random noise is introduced to ensure that the contrast-to-noise ratio of the standard image matches that of the original SEM image Calibration of the conversion constants A and B is also essential in this process.

The conversion constants A and B are influenced by the structure, size, and noise of the SEM image Therefore, these constants must be calibrated for each evaluated SEM image using standard images with varying known sharpness factors of 2σ This process involves converting the R CG value to the image sharpness R ES by applying the calibrated constants A and B.

2σ is the sharpness factor, given by

Generation of the standard images and calculation of their CG image sharpness R CG

Calculation of the CG image sharpness R CG

Calibration of the conversion constants A and B

Conversion of the R CG value to the image sharpness R ES using the calibrated constants A and B

Figure 12 — Brief flow chart of the CG method

Derivative (DR) method

The derivative method involves extracting edge profiles and fitting error functions to assess edge sharpness, which is linked to the Rayleigh-Abbe criterion By modeling edge profiles as error functions, particularly under the assumption of a Gaussian point-spread function, the method approximates edges in SEM images The fitted error functions for all extracted profiles allow for the calculation of an average sharpness factor, directly related to the overall image sharpness.

Figure 13 — Basic concept of the derivative (DR) method

Figure 14 shows a brief flow chart of the DR method composed of the following routines a) to d) Details of the routines are given in Annex D a) Generation of a binary mask image M(x, y)

1) The gradient magnitude G M (x, y) is computed by convoluting the original image with first-order derivative Gaussian profiles of standard deviation σ equal to 2 pixels

2) A binary image B(x, y) is computed from G M (x, y), based on a two-mean threshold

A binary mask image M(x, y) is created by applying a one-iteration binary-closing operation to clean up B(x, y) Subsequently, object pixels near the image borders are set to zero, and small objects are removed This process leads to the generation of an edge position map E(x, y).

1) An edge location image P L (x, y) is computed by convoluting the original image with first- and second- order derivative Gaussian profiles of standard deviation σ

2) A binary mask M 1 (x, y) is computed from the maximum value of [P L (x, y) − ⎪P L (x, y)⎪], based on a two-mean threshold within M(x, y)

3) An initial binary edge map image E 1 (x, y) is computed by skeletonization from the result of the one- iteration binary-closing operation carried out on M 1 (x, y)

4) An edge position map E(x, y) is computed from E 1 (x, y) by considering only positions along a contour that are separated from each other by a distance of at least 10 pixels

1) The normalized gradient G N (x, y) is computed for all positions of E(x, y), based on the normalization of G M (x, y)

2) The sub-pixel profile positions P Si (x, y) are calculated from the initial edge positions given by E(x, y) along both directions of G N (x, y) for a total of 41 positions with a pitch of 0,5 pixels

3) The sub-pixel intensity values P j (x, y) at P Si (x, y) are retrieved from the original image at the profile positions by cubic interpolation

4) An error function is fitted to each P j (x, y) and the edge sharpness s j is calculated and stored d) Calculation of image sharpness R DR

1) The overall edge sharpness s is calculated as the average of the edge sharpnesses of all the edge slopes determined

2) The image sharpness R DR is calculated asR DR = 2s

Extraction of the edge profiles P j (x, y) and fitting of error function

Calculation of image sharpness R DR

Generation of a binary mask image M(x, y)

Generation of an edge position map E(x, y)

Figure 14 — Brief flow chart of the DR method

General

The test report prepared by the laboratory shall be accurate, clear and unambiguous, and in accordance with the specific instructions in the evaluation methods described in this Technical Specification

The evaluation results must include the information specified in section 5.10.2 of ISO/IEC 17025:2005 These results can be presented in a simplified format if there is a written agreement with an external client or a mutual understanding with internal clients Any information from section 5.10.2 of ISO/IEC 17025:2005 that is not provided to the client must be readily accessible in the laboratory that conducted the tests.

Contents of test report

The test report must encompass essential elements, including a title, laboratory name and address, and a unique identification number It should also list the client's name and address when applicable, identify the testing method used (such as ISO/TS 24597 or FT method), and provide details about the manufacturer, model, and serial number of the instrument Additionally, the report should specify the reference materials utilized, operating values like accelerating voltage, working distance, and magnification, along with any pertinent information such as imaging mode and scan speed It must include original SEM images, their sizes, and associated data files, as well as the evaluator's name, date and time of evaluation, and signatures of authorized personnel Finally, if applicable, a statement clarifying that the results pertain solely to the tested items should be included.

The data files of the original SEM images and the selected SEM images used in obtaining the reported results shall be kept for a specified mandatory period

Laboratories issuing a test report shall specify that the report shall only be reproduced in full and with the written permission of the laboratory

Details of contrast-to-noise ratio (CNR)

This annex provides details of the evaluation of contrast-to-noise ratio (CNR) A flow chart of the CNR evaluation is given in Figure A.1

NOTE The explanation applies to an image with L = 512 or 256 and 8 bits in the grey scale for ease of understanding

Produce a three-time median- filtered image for the SEM image

Evalutate the standard deviation n for the noise of the SEM image

The CNR evaluation process begins with the creation of a three-time median-filtered image, achieved by applying unweighted 3 × 3 median filtering sequentially three times to the SEM image, utilizing a 3 × 3 filter matrix.

The 3 × 3 median filtering technique involves sorting the pixel intensities within a 3 × 3 square and replacing the intensity of the central pixel I(i, j) with the median value, which is the fifth pixel intensity in the ordered list.

NOTE 2 Any pixel position of the image is expressed as (i, j), where i (and j) = 0, 1, 2, …, and i max (and j max )

NOTE 3 Both i max and j max = 511 (or 255) (depending on the pixel size of the original SEM image, namely

NOTE 4 Edge pixels are processed in a special way for median filtering (see the end of this annex)

Figure A.2 — The principle of 3 × 3 median filtering: the figure shows the pixel (i, j) concerned and its neighbouring pixels in the 3 × 3 square b) Evaluate the standard deviation σn of the image noise:

(A.1) where I med (i, j) and I(i, j) are the pixel intensities at pixel position (i, j) of the three-time median-filtered and SEM images, respectively

NOTE The denominator (i max + 1) × (j max + 1) corresponds to the total number of pixels in the median-filtered region (see Figure A.3)

Figure A.3 — Intensities of a three-time median-filtered image

1) Divide the three-time median-filtered SEM image into nine (or 3 × 3) segment-images as shown in Figure A.4

The i (or j) region is categorized into three ranges: 1 to 170, 171 to 340, and 341 to 510 for a maximum value of i (or j) equal to 511 For an image with a maximum value of i (or j) set at 255, the ranges are divided into 1 to 84, 85 to 169, and 170 to 254.

2) For each segment s, make a sequence of increasing or decreasing intensity as shown in Figure A.5

For each segment \( s \), calculate \( z_{\text{max,av},s} \) by averaging the first \( q \) elements of the descending intensity sequence, and determine \( z_{\text{min,av},s} \) by averaging the first \( q \) elements of the ascending intensity sequence The value of \( q \) is an integer representing 0.2% of the total number of elements in the sequence, with a minimum value of 1.

1 0,2 % of the number of elements in the sequence

Figure A.5 — Intensity sequence for segment-image s

4) Determine the threshold intensity z threshold,s-av as follows: z threshold,s-av= [Maximum (z max,av,0 , z max,av,1 , …, z max,av,8 )

+ Minimum (z min,av,0 , z min,av,1 , …, z min,av,8 )]/2 (A.2)

5) Determine the averages Avz max,av,s and Avz min,av,s from the following equations:

= Average (only for z max,av,s > z threshold,s-av) of (z max,av,0 , z max,av,1 , …, z max,av,8 ) (A.3) Avz min,av,s

= Average (only for z min,av,s < z threshold,s-av) of (z min,av,0 , z min,av,1 , …, z min,av,8 ) (A.4)

6) Calculate the temporary contrast C temp of the image from the following equation:

C temp = Avz max,av,s − Avz min,av,s (A.5)

7) Determine the image contrast C image by correcting C temp using a correction term k corr × σ n , where k corr = 1,38 (empirical value), as follows:

C image = C temp − k corr ×σ n (A.6) d) Evaluate the CNR value of the SEM image as follows:

The values of Avz max,av,s and Avz min,av,s shall be in the ranges 245 W Avz max,av,s W 170 and

80 W Avz min,av,s W 10 (for 8 bits in the grey scale), respectively If the values are not in their corresponding ranges, discard the SEM image

F OUT (i, j) = M ED [F IN (i − 1, j − 1), F IN (i − 1, j), F IN (i − 1, j + 1), F IN (i, j − 1), F IN (i, j), F IN (i, j + 1),

F IN (i, j) is the input image data for 511 W i, j W 0 (for an SEM image with 512 × 512 pixels);

F OUT (i, j) is the 3 × 3 median-filtered data

The median-filtering function \( M_{ED}(a_1, a_2, \ldots, a_N) \) organizes the values \( a_n \) in ascending order and determines the median for odd \( N \) or the average of the \( (N/2) \)th and \( [(N/2) + 1] \)th values for even \( N \).

NOTE If i − 1 < 0, i + 1 > 511, j − 1 < 0 or j + 1 > 511, discard the corresponding F IN (…, …) and carry out M ED [……]

Details of the Fourier transform (FT) method

This annex provides details of the procedure used for the Fourier transform (FT) method

Figure B.1 shows an example of an SEM image

Figure B.1 — Example of an SEM image

To generate convoluted images, first prepare a filtered image \( I_{OF}(i, j) \) by applying a three-time sequential unweighted 3 × 3 median filter to a selected SEM image \( I_O(i, j) \) Next, create a histogram \( H(S) \) for the filtered image, where \( S \) ranges from 0 to 255 Subsequently, smooth the histogram \( H(S) \) to obtain \( H_s(S) \) using a nine-point window, and calculate \( h_s(S) = \log_{10}[H_s(S) + 1] \) Finally, extract two signal intensities \( S_L \) and \( S_H \) from the smoothed histogram to determine a threshold value.

1) Find the maximum values of h s (S), h s (S 1 ) and h s (S 2 ), in the intervals [0, 127] and [128, 255], respectively, which satisfy the following conditions: h s (S 1 − 16) < h s (S 1 ) and h s (S 1 + 16) < h s (S 1 ); h s (S 2 − 16) < h s (S 2 ) and h s (S 2 + 16) < h s (S 2 );

Then set S 1 to S L and set S 2 to S H and go to step 3) Otherwise, go to step 2)

If S 1 − 16 < 0 or 255 < S 2 + 16, use h s (0) or h s (255) instead of h s (S 1 − 16) or h s (S 2 + 16), respectively

To determine the maximum value \( S_A \) and the minimum value \( S_B \) of the signal intensity \( S \), ensure that the sum of the histogram intensity \( H_s(S) \) within the intervals \([0, S_A - 1]\) and \([S_B + 1, 255]\) is as close to, but less than, 0.2% of \( L_2 \) Subsequently, calculate the two signal intensities \( S_L \) and \( S_H \) accordingly.

3) Calculate the threshold value S T (see Figure B.2) as follows:

S = S +S e) Obtain a binarized image I B (i, j) by applying the threshold value S T (see Figure B.3)

Figure B.2 — Example of a smoothed histogram

Figure B.3 — Example of a binarized image I B (i, j) f) Add the white noise to the selected image I O (i, j) so that the effect of weak correlated noise is neglected, as follows:

1) Set SNR p (signal-to-noise ratio for particles) to 30 for the signal intensity S = 192 and calculate the noise intensity s n (i, j) for the selected image intensity I O (i, j) as follows:

2) Obtain the intensity I ON (i, j) of the noisy image as follows:

I i j =I i j +s i j r⋅ where r G is a random value which obeys the normal distribution with a mean value of 0 and a standard deviation of 1

NOTE This is done by setting I ON (i, j; 2σ) to 0 if I ON (i, j; 2σ) < 0 and setting I ON (i, j; 2σ) to 255 if

3) Set I ON (i, j) to I O (i, j) g) Generate convoluted images I C (i, j; 2σ) by using the convolution of the binarized image I B (i, j) with two- dimensional Gaussian profiles I G (i, j; 2σ) having various sharpness factors 2σ, given by

⎣ ⎦ where σ is the standard deviation of the Gaussian distribution

1) Set the sharpness factor 2σ(N = 1) to 1 as the initial step During the evaluation process, 2σ(N) is increased in the following way: if 1u uN 8, then 2 ( )σ N =N ; if 9uN , then 2 ( ) 2σ N = Q + 1 +2 Q − 1 ⋅(N−4 )Q

NOTE N is the step number The maximum values of N and 2σ(N) are 24 + 4[(log 2 L) − 8] and L/2, respectively

2) Compute the Fourier transform pattern G B (f i , f j ) of the binarized image I B (i, j)

3) Compute the Fourier transform pattern G G (f i , f j ; 2σ) of the Gaussian profile I G (i, j; 2σ) with a sharpness factor 2σ equal to 2σ(N) for the Nth step in a similar manner

4) Calculate the product of G B (f i , f j ) and G G (f i , f j ; 2σ):

5) Obtain the image I BG (i, j; 2σ) from G BG(f i , f j ; 2σ) by applying the inverse Fourier transform

6) Obtain the convoluted images I C (i, j; 2σ) (see Figure B.4) as follows:

= − + where the mathematical symbol ⎪ … ⎪ means “the absolute value of” and max[…] means “the maximum value of” a) 2σ= 2 pixels b) 2σ= 4 pixels c) 2σ= 6 pixels

Figure B.4 — Examples of convoluted images I C (i, j; 2σ) with image size L = 256

B.3 Generation of curves of FT patterns

The procedures a), c), d), and e) are executed once for the selected SEM image I O (i, j) when the step number N = 1 First, the Fourier transform pattern G O (f i , f j ) of the image I O (i, j) is computed Next, the Fourier transform pattern G C (f i , f j ; 2σ) of the convoluted image I C (i, j; 2σ) is determined The real part Re[G O (f i , f j )] of G O (f i , f j ) is then extracted, followed by calculating the absolute value ⎪Re[G O (f i , f j )]⎪ Finally, the vertically and horizontally averaged values of ⎪Re[G O (f i , f j )]⎪ are obtained, and their common logarithms are calculated.

⎩ ∑ ⎭ where ε is taken as 10 −20 to avoid log 10 0 occurring;

L is the image size e) Calculate a smoothed horizontal curve F OH (f j ) from F OHA (f j ) by applying the procedures in B.6.2, using a window of five points in the interval [−L/2, (L/2) − 1] of f j

To derive a smoothed vertical curve F OV (f i ) from F OVA (f i ), first extract the real part Re[G C (f i , f j ; 2σ)] of G C (f i , f j ; 2σ) Next, compute the absolute value of this real part, denoted as ⎪Re[G C (f i , f j ; 2σ)]⎪ Subsequently, calculate both the vertically and horizontally averaged values of ⎪Re[G C (f i , f j ; 2σ)]⎪, and determine their common logarithms.

⎩ ∑ ⎭ h) Calculate a smoothed horizontal curve F CHB (f j ; 2σ) from F CHA (f j ; 2σ) by applying the procedures in B.6.2, using a window of five points in the interval [−L/2, (L/2) − 1] of f j

Calculate a smoothed vertical curve F CVB (f i ; 2σ) from F CVA (f i ; 2σ) in a similar manner

B.4 Calculation of temporary image sharpness R PXO

The procedures a) to f) are executed once for the chosen SEM image I O (i, j) when the step number N = 1 Initially, the slope m H and the intercept b H are calculated using the least-squares method outlined in B.6.3, resulting in a linear function that approximates the smoothed curve F OH (f j ) within the interval [−L/2, −(L/4) − 1] of f j.

Calculate the slope m V and the intercept b V of the smoothed curve F OV (f i ) in the interval [−L/2, −(L/4) − 1] of f i in a similar manner b) Determine the noise functions as follows:

F f =m ⋅ +f b c) Calculate the corrected curves F CH (f j ; 2σ) and F CV(f i ; 2σ), using the signal and noise intensities at the origin of (f i , f j ), as follows:

To ensure accurate computation, it is advisable to plot the graphs of \( F_{OH}(f_j) \) and \( F_{CH}(f_j; 2\sigma) \) for the horizontal direction, alongside the graphs of \( F_{OV}(f_i) \) and \( F_{CV}(f_i; 2\sigma) \) for the vertical direction, as illustrated in Figure B.5.

Figure B.5 — Examples of averaged and smoothed curves for the FT patterns in the horizontal and the vertical directions

`,,```,,,,````-`-`,,`,,`,`,,` - d) Obtain the horizontal coordinate f j = f jH as follows:

1) Set the parameters A and B as

A=F f and B=F NH (f j )+C N where C N is the contribution factor determined from the convoluted image in the Fourier space and is given by

2) Set f j = − L/2 as the initial value, then increase f j until the condition A < B changes to A W B Set f j = f jC for this change

3) Calculate the horizontal coordinate f jH as follows:

To find the coordinates of point \( P_{1H} \) on the curve \( F_{OH}(f_j) \) for the original image, identify point \( P_{2H} \) on the linear function \( F_{NH}(f_j) \) representing the noise, and locate point \( P_{3H} \) on the curve \( F_{CH}(f_j; 2\sigma_{OH}) \) for the convoluted image.

P f F f + a e) Obtain the vertical coordinate f i = f iV using f i = f iC in a similar manner

To find the coordinates of point P1V on the curve FOV(fi) for the original image, point P2V on the linear function FNV(fi) for the noise, and point P3V on the curve FCV(fi; 2σOV) for the convoluted image, set fi = fiV as illustrated in Figure B.6.

Figure B.6 — Graphs showing the points P 1H , P 2H , P 3H , P 1V , P 2V and P 3V f) Calculate the values of F CH (f jH ; 2σ) and F CV(f iV ; 2σ) at f j = f jH and f i = f jV , respectively, using linear interpolation, as follows:

F f σ = F f σ −F f − σ ⋅ f − f − +F f − σ g) Find the step numbers N = N OH and N = N OV for the sharpness factors 2σ(N OH) = 2σHL,

2σ(N OH − 1) = 2σ HU , 2σ(N OV ) = 2σ VL and 2σ(N OV − 1) = 2σ VU as follows:

1) Stop the evaluation if either of the following inequalities is satisfied for the initial convoluted image with 2σ(N = 1) = 1 Otherwise, go to step 2)

CH( j H; 1) NH( j H) log10 Nor CV( i V; 1) NV( i V) log10 N

A termination message should be generated when the sharpness factor, either 2σ OH or 2σ OV, of the selected image I O (i, j) falls below 1 pixel or if the image is irregular.

2) Find the step numbers N = N OH and N = N OV which satisfy the following conditions by increasing the step number N and then repeat the procedures from Clause B.2 f) 1) to the present step

CH( j H; 2 HL) NH( j H) log10 N CH( j H; 2 HU)

CV( i V; 2 VL) NV( i V) log10 N CV( i V; 2 VU)

`,,```,,,,````-`-`,,`,,`,`,,` - h) Calculate 2σ by linear interpolation as follows:

NOTE The values of 2σ OV and 2σ OH are similar in magnitude to each other for an image with a low level of astigmatism i) Obtain the sharpness factor 2σ O as follows:

2σO = (2σOH + 2σOV)/2 j) Obtain the temporary image sharpness R PXO before calibration as follows:

B.5 Calculation of image sharpness R PX a) Calculate the coefficient C F by using the following formulae:

C =c σ +c σ +c σ +c where c 3 = 1,489 79 × 10 − 4 , c 2 = −6,646 10 × 10 − 3 , c 1 = 9,638 83 × 10 − 2 and c 0 = 5,456 65 × 10 − 1 b) Obtain the calibrated sharpness factor 2σC as follows:

Tiêu đề	Technical Specification Iso/ts 24597 2011
Trường học	International Organization for Standardization
Chuyên ngành	Microbeam Analysis
Thể loại	Tiêu chuẩn
Năm xuất bản	2011
Thành phố	Geneva

Định dạng
Số trang	94
Dung lượng	18,34 MB