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Orientation identification of the power spectrum

Citation for published version (APA):

Rudnaya, M. E., Mattheij, R. M. M., Maubach, J. M. L., & Ter Morsche, H. G. (2011). Orientation identification of the power spectrum. Optical Engineering, 50(10), [103201]. https://doi.org/10.1117/1.3633333

DOI:

10.1117/1.3633333

Document status and date: Published: 01/12/2011

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Orientation identification of the power

spectrum

Maria E. Rudnaya

Robert M. M. Mattheij

Joseph M. L. Maubach

Hennie G. ter Morsche

(3)

Optical Engineering 50(10), 103201 (October 2011)

Orientation identification of the power spectrum

Maria E. Rudnaya

Robert M. M. Mattheij Joseph M. L. Maubach Hennie G. ter Morsche

Eindhoven University of Technology CASA

Department of Mathematics and Computer Science Den Dolech 2

Eindhoven, 5612AZ, The Netherlands E-mail: m.rudnaya@tue.nl

Abstract. The image Fourier transform is widely used for defocus and astigmatism correction in electron microscopy. The shape of a power spectrum (the square of a modulus of image Fourier transform) is directly related to the three microscope controls, namely, defocus and twofold (two-parameter) astigmatism. We propose a new method for power-spectrum orientation identification. The method is based on the three measures that are related to the microscope’s controls. The measures are derived from the mathematical moments of the power spectrum and is tested with the help of a Gaussian benchmark, as well as with the scan-ning electron microscopy experimental images. The method can be used as an assisting tool for increasing the capabilities of defocus and astigma-tism correction a of nonexperienced scanning electron microscopy user, as well as a basis for automated application.C2011 Society of Photo-Optical

Instrumentation Engineers (SPIE). [DOI: 10.1117/1.3633333]

Subject terms: Fourier transform; power spectrum; orientation identification; math-ematical moments; defocus/focus; astigmatism; electron microscopy.

Paper 101041RR received Dec. 8, 2010; revised manuscript received Aug. 11, 2011; accepted for publication Aug. 12, 2011; published online Sep. 29, 2011.

1 Introduction

For many practical applications in electron microscopy, both the defocus and twofold (two-parameter) astigmatism must be adjusted regularly during the continuous image-acquiring process. Possible reasons for change in defocus and twofold astigmatism are, for instance the instabilities of the electron microscope and environment, as well as the magnetic nature of some samples. Nowadays, an electron microscope still requires an expert operator to trigger recording of in-focus and astigmatism-free images using a visual feedback, which is a tedious task. In the future, the manual operation must be automated to improve the speed, quality, and repeatability of the measurements.

There are different ways of automated defocus and astig-matism correction in electron microscopy. One of them is optimization of an image-quality measure as a function of de-focus and twofold astigmatism (two-parameter astigmatism, which can be adjusted with x and y-stigmator controls in the electron microscope), i.e., three-parameter optimization.1

Existing image-quality measures are based on image derivative,2–4 variance,1,5 autocorrelation.6–8 Overviews of

existing-image quality measures can be found in.9–11In

gen-eral, it requires more image recordings than alternative group of methods (i.e., Fourier transform-based methods).

The image Fourier transform is important for image-quality improvement in electron microscopy as well as in other types of optical devices, such as telescopes, ophthal-moscopes and endoscopes.12 To this end, one needs, on the

one hand, a thorough analysis of the Fourier transform, and on the other hand, the analysis must be fast. The image’s power spectrum (i.e., the square of a modulus of its Fourier trans-form) is widely used for blind deconvolution procedures,13,14

for defocus and astigmatism correction in scanning electron microscopy (SEM),15–17as well as in other types of electron microscopes.18–23 A power spectrum is used for automated

defocus and astigmatism correction, as well as a visual

sup-0091-3286/2011/$25.00C2011 SPIE

port for a nonautomated correction performed by a human operator. Unfortunately, it is still hard for a nonexperienced human operator to correct defocus and astigmatism within a reasonable time, even with power-spectrum visualization.

In this paper, we discuss a new method for power-spectrum orientation identification. The power-power-spectrum model suggested in Ref. 13 is extended to a nonsymmet-rical case and is used for analytical derivations. Three func-tions corresponding to the three SEM parameters (defocus function, x-stigmator function, y-stigmator function) are in-troduced. The functions are related to the power-spectrum mathematical moments and are chosen to simplify defocus and astigmatism correction for a non-experienced human operator. The three real-valued measures (defocus measure, x-stigmator measure, y-stigmator measure) derived from de-focus/stigmator functions can be used as a basis for an auto-mated application.

Section 2 describes the image formation model, defo-cus, and astigmatism. Section3introduces power-spectrum model and its discretization (Sec.3.1). Sec.3.2discusses the relation of a power-spectrum orientation with defocus and astigmatism for the particular case of a Gaussian point-spread function. Section4explains the method of orientation iden-tification, which involves (i) computing defocus/stigmator functions, and (ii) computing defocus/stigmator measures. Sec.4.1discusses the particular case of a power spectrum modeled as a Gaussian function. Section5illustrates results of numerical experiments with Gaussian benchmarks and SEM experimental images. Section 6 provides discussion and conclusions.

2 Image Formation, Defocus and Astigmatism

According to a linear image-formation model24,25the

micro-scope image is

f (u, p) = [ f0(u)∗ h(u, p)](u)

:=   +∞

−∞ h(u

, p) f

(4)

Fig. 1 (a) Ray diagram for a lens without astigmatism, the lens has

one focal point. (b) Ray diagram for a lens with astigmatism; the lens has two focal points.

where u := [u, v]T ∈ R2 is a vector of spatial

coordi-nates, f0(u)≥ 0 is the object function that describes a

specimen’s geometry, h is the point-spread function, and

p := [d, σx, σy]T ∈ R3 is a vector of the microscope’s

parameters (controls), which corresponds to defocus and twofold astigmatism.

Astigmatism is a lens abberation caused by asymmetry of the lens. Figure1(a)shows a ray diagram for the astigmatism-free situation. The lens has one focal point F. The only ad-justable parameter is the current through magnetic lens; it changes the lens focal length and focuses the magnetic beam on the image plane.15 The current is adjusted with defocus

control d. Astigmatism implies that the rays traveling through a horizontal plane will be focused at a different focal point than the rays traveling through a vertical plane [Fig.1(b)]. Thus, the lens has two different focal points F1and F2 and

the image cannot be totally sharp. Because of the presence of astigmatism, the electron beam becomes elliptic.

In SEM the point-spread function can be approximated by a composition of Gaussian functions26 or, in the

sim-plest case, by one Gaussian function,5 as well as in light

microscopy27 h(u, p) = ¯G(u) := 1 2π ˆa2exp  −  u2 2 ˆa2 + v2 2 ˆa2  , ˆa > 0. (2) In Eq. (2) the Gaussian standard deviation (or the point-spread function width) ˆa is related to the defocus control d. The smaller ˆa is the better, the image f describes the object f0. Ideally, if we assume ˆa= 0, Gaussian point-spread

func-tion becomes aδ function and f = f0. However, in practice

the point-spread-function width is bounded by microscope’s physical limits ˆa= ˆamin> 0. Because of the presence of

astigmatism, the point-spread function becomes elliptic h(u, p) = G(u) := 1 2π ˆa ˆbexp  −  (u cos ˆα − v sin ˆα)2 2 ˆa2 +(u sin ˆα + v cos ˆα)2 2 ˆb2 , (3) ˆa> 0, bˆ> 0, −π 4 < ˆα ≤ π 4.

Figure2(a)visualizes the roles of parameters ˆa, ˆb, ˆα. When ˆa= ˆb, the value of ˆα does not play a role.

Fig. 2 (a) Roles of parameters in elliptic function and (b) typical for

SEM configuration of electrostatic stigmators (Ref.15).

For astigmatism correction in SEM, electrostatic or elec-tromagnetic stigmators are used. They produce electromag-netic field for correction of the ellipticity of the electron beam.28 A typical configuration of electrostatic stigmators

for SEM is shown in Fig.2(b). The elliptic electron beam is depicted in the middle of the scheme. Currents of the same magnitude go through coils A1, A2, C1, and C2, while

cur-rents of a different magnitude go through coils B1, B2, D1,

and D2. The field generated by A1, A2, C1, C2influences the

stretching of the electron beam along two orthogonal axes A and C. Similarly, the field generated by coils B1, B2, D1, D2 influences the stretching along two orthogonal axes C

andD.15 The angle between axesA and B is always π/4.

The magnitude and direction of the current through coils A1, A2, C1, C2are controlled by the stigmator control variable σx, and the magnitude and direction of the current through coils B1, B2, D1, D2are controlled by the stigmator control

variableσy. Thus, by adjusting stigmator controls σx and σy astigmatism is corrected.

For a defocus and astigmatism correction problem, we consider a vector of three microscope’s control variables

p := [d, σx, σy]T with the ideal parameter values (the ideal

parameter values correspond to the image of the highest pos-sible quality) denoted as p0:= [d0, σx0, σy0]

T.

3 Power-Spectrum Theory

For the vector of frequency coordinates x := [x, y]T

, an im-age f (u), its power spectrum p(x) is a square of a modulus of the image’s Fourier transform,

p(x) := |F[ f ]|2.

The image f is realvalued. Hence, one finds

p(x)= p(−x). (4)

In Fourier space, convolution becomes multiplication; thus, Eq.(1)can be rewritten as

F[ f ]= F[ f0]F[h]

or

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Rudnaya et al.: Orientation identification of the power spectrum

According to Refs. 13and29, a rotationally symmetric power spectrum can be modeled as a function

p(x)= ¯g(x) := A exp  −  x2 2a2 + y2 2a2 β , 0< β ≤ 1, a > 0, A > 0. (6) This is a consequence of the fact that in a variety of optical systems point-spread function h can be often approximated by a L´evy stable density with parameterβ, which is a property of an optical device (in our case, SEM). For β = 1, one obtains the Gaussian function and forβ = 1/2 – Lorentzian (or Cauchy) function. When β = 1, h has slim tails and finite variance. When 0< β < 1, h has fat tails and infinite variance.13

Because of the presence of astigmatism, power spectrum might lose its rotational symmetry. Therefore, we extend Eq.(6)to a nonsymmetric case,

p(x)= g(x) := A exp  −  (x cosα − y sin α)2 2a2 +(x sinα + y cos α)2 2b2 β , (7) a > 0, b > 0, −π 4 < α ≤ π 4. For polar coordinates

r := [r, ϕ]T ∈ {R+×[0, 2π)}, x = r cosϕ, y = r sinϕ, it becomes g(x)= A exp  −r2βcos2(ϕ + α) 2a2 + sin2(ϕ + α) 2b2 β . (8) 3.1 Discretization

We assume that the continuous power spectrum p(x) has a compact support inside

X := [−X, X]×[−X, X], which means

p(x)= 0, ∀x /∈ X.

In practice, images are always discrete and can be represented by matrices

F∈ R(2n+1)×(2n+1), n ∈ N.

A corresponding discrete power spectrum P := (Pi, j)2n+1i, j=1

can be computed with the fast Fourier transform method.30,31

We define frequency mesh points xi := ix,

yj:= jx, i, j = −n, . . . , n, where x := 2X (2n+ 1) − 1 = X n. (9) Then, we define Pi, j := p(xi, yj). (10)

(a) Non-windowed (b) Windowed

(c) Power spectrum non-windowed (d) Power spectrum windowed

Fig. 3 (a) SEM experimental image of gold-on-carbon (a

nonwin-dowed image), (b) the experimental image multiplied by the window function (a windowed image), (c) power spectrum of the experimen-tal nonwindowed image, and (d) power spectrum of the experimenexperimen-tal windowed image.

The power spectrum P has a high dynamic range: low frequencies [pixels with indices close to (i, j) = (0, 0)] have much higher values than high frequencies. A normal output graphic device does not have a sufficient dynamic range to display it simultaneously. It is suggested to use logarithmic scale for power-spectrum vizualization24

Pi(C), j = log(C + Pi, j), (11)

where C is a scaling constant for the contrast adjustments. In this paper, as well as in Ref.13, we use C= 0 for power-spectrum visualization.

The fast Fourier transform method is based on the assump-tion of funcassump-tion periodicity,31 which is usually not the case

for real-world images. Before the discrete power-spectrum computations, it is important to multiply the image by a win-dow function, for instance, with

W (x) :=  1+ cos πx|x| max  2, if |x| ≤ xmax, 0, elsewise, (12)

in order to avoid discontinuities on the boundary. For the discrete image, we make a choice of xmax= xn−1. Figure3

shows the difference between power spectrums of the non-windowed image and the non-windowed image. The nonwin-dowed power spectrum [Fig.3(c)] has two orthogonal lines crossing in the origin, whereas the windowed power spec-trum [Fig.3(d)] shows only elliptic distribution. The lines in the nonwidowed power spectrum [Fig. 3(c)] are results of discontinuity and can result in the errors during further power-spectrum analysis.

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f

0

u

v

−200 0 200 −200 0 200

PSF

h

u

−20 0 20 −20 0 20

f

u

||

−200 0 200 −200 0 200

PS of

f

0

x

y

−200 0 200 −200 0 200

PS of

h

x

×

−200 0 200 −200 0 200

PS of

f

x

||

−200 0 200 −200 0 200

Fig. 4 The upper row represents real space (spatial coordinates u := [u,v]T); from left to right: Experimental SEM image, which is considered to be the object function f0, Gaussian point spread functionh( ˆa= ˆb= 1, ˆα = 0), numerical result of their convolution f. The lower row represents

the three mentioned quantities in the Fourier space (frequency coordinates x := [x,y]T). PS denotes the power spectrum. PSF denotes the point-spread function.

3.2 Power-Spectrum Orientation in Relation to Defocus and Astigmatism

The power spectrum of a Gaussian point-spread function [Eq.(3)] is a Gaussian function

|F[G]|2 = ˆA exp{−[ˆa2(x cos ˆα − y sin ˆα)2

+ ˆb2(x sin ˆα + y cos ˆα)2]}, (13)

where ˆA> 0 is a constant. The rotation angle of the Gaussian power spectrum is equal to the rotational angle in real space, and the widths are inversely proportional to the width in real space.

We illustrate relation of power-spectrum orientation to de-focus and astigmatism with the help of three numerical

ex-amples. The left columns of Figs.4–6show the same exper-imental image of tin balls obtained with SEM and its power spectrum. For numerical experiment we consider this image to be an ideal image (i.e., the object function of a specimen f0). The power spectrum is nearly rotationally symmetric.

The image is convolved sequentially with a Gaussian point-spread function with parameters ˆa= ˆb = 1, ˆα = 0 (Fig.4), ˆa= ˆb = 5, ˆα = 0 (Fig.5), and ˆa= 5, ˆb = 3, ˆα = 0 (Fig.6). The upper row in Figs. 4–6 represents real space, and the bottom row represents Fourier space, where convolution be-comes multiplication. The influence of the point-spread func-tion’s parameters is visible in the result of the convolution in real space (image f is at the top right of Figs.4–6), as well as in the Fourier space.

f

0

u

v

−200 0 200 −200 0 200

PSF

h

u

−20 0 20 −20 0 20

f

u

||

−200 0 200 −200 0 200

PS of

f

0

x

y

−200 0 200 −200 0 200

PS of

h

x

×

−200 0 200 −200 0 200

PS of

f

x

||

−200 0 200 −200 0 200

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Rudnaya et al.: Orientation identification of the power spectrum

f

0

u

v

−200 0 200 −200 0 200

PSF

h

u

−20 0 20 −20 0 20

f

u

||

−200 0 200 −200 0 200

PS of

f

0

x

y

−200 0 200 −200 0 200

PS of

h

x

×

−200 0 200 −200 0 200

PS of

f

x

||

−200 0 200 −200 0 200

Fig. 6 Similar to Fig.4numerical computation, but with the Gaussian point-spread-function parameters ˆa= 5, ˆb= 3, ˆα = 0.

When we consider a rotationally symmetric point-spread function with a relatively small width (Fig.4) the final image f does not deviate much from original f0. Its power spectrum

is rotationally symmetric and has, as well as the power spec-trum of the point-spread function, a relatively large width. When we consider a rotationally symmetric point-spread function with a larger width (Fig.5), its power spectrum has a smaller width, as well as the power spectrum of f (power spectrum’s intensity decreases), and the image f itself looks more blurred than in Fig.4. Furthermore, when we consider a nonsymmetric point-spread function, the power spectrum of f becomes non-symmetric as well (Fig.6).

The knowledge of the above described power-spectrum behavior is used by a human operator during defocus and astigmatism correction, when adjusting the controls p. For the amorphous object with a rotationally symmetric power spectrum, the operator tries to obtain an image with a power spectrum as intense as possible without stretching in any direction. The difficult situation is when the power spectrum of the object f0is not rotationally symmetric (the object has

a strong preferential direction). In this case, it is important to compare the difference between power spectrums obtained for different p and to find a center of symmetry.15

It is often difficult for a nonexperienced human operator to understand, which of the controls p and in which directions are to be adjusted solely from observing an elliptic power spectrum. In Sec. 4we provide a methodology to simplify the correction for a human. Also, this methodology can be used as a basis for an automated defocus and astigmatism correction method.

4 Orientation Identification Method

With the power spectrum p(x)= p(r cos ϕ, r sin ϕ), we as-sociate the three functions

h0(r ) :=  2π 0 p(r cosϕ, r sin ϕ)dϕ, (14) h1(r ) :=  2π 0

p(r cosϕ, r sin ϕ) cos 2ϕdϕ, (15)

h2(r ) :=

 2π 0

p(r cosϕ, r sin ϕ) sin 2ϕdϕ. (16) The function h0(defocus function) is related to the defocus d.

It is clear that h0(r )≥ 0 [p(x) ≥ 0 by definition]. The

func-tions h1(x-stigmator function) and h2(y-stigmator function)

are related to the stigmatorsσx, σy respectively. In Eqs.(15)

and(16), cos 2ϕ and sin 2ϕ play the roles of weight func-tions. Later, we it will show, for particular examples, how they help to obtain information about the signs of the stig-matorsσx and σy. Because of the power-spectrum symmetry [Eq.(4)], h0(r )= 2  π 0 p(r cosϕ, r sin ϕ)dϕ, (17) h1(r )= 2  π 0

p(r cosϕ, r sin ϕ) cos 2ϕdϕ, (18)

h2(r )= 2

 π

0

p(r cosϕ, r sin ϕ) sin 2ϕdϕ. (19) Equations(17)–(19)make further numerical computations faster.

The defocus and stigmator functions [Eqs. (14)–(16)] are related to the mathematical moments of the power spectrum mk,l :=   +∞ −∞ x k ylp(x)dx =  0 rk+l+1  2π 0

p(r cosϕ, r sin ϕ) coskϕ sinlϕdϕdr, which are widely used in different applications for orientation identification and other purposes.32–34The 0 moment is m0,0=



0

(8)

The symmetry of the power spectrum [Eq.(4)] leads to the fact that the first moments are equal to zero

m1,0=  0 r2  2π 0

p(r cosϕ, r sin ϕ) cos ϕdϕdr = 0,

m0,1=  0 r2  2π 0

p(r cosϕ, r sin ϕ) sin ϕdϕdr = 0. The Second moments are

m2,0=  0 r3  2π 0

p(r cosϕ, r sin ϕ) cos2ϕdϕdr = cos2ϕ = (1/2) + (1/2) cos 2ϕ 1 2  0 r3[h0(r )+ h1(r )]dr, (21) m0,2=  0 r3  2π 0

p(r cosϕ, r sin ϕ) sin2ϕdϕdr = sin2ϕ = (1/2) − (1/2) cos 2ϕ 1 2  0 r3[h0(r )− h1(r )]dr, (22) m1,1=  0 r3  2π 0

p(r cosϕ, r sin ϕ) cos ϕ sin ϕdϕdr = 1

2 

0

r3h2(r )dϕdr. (23)

On the basis of Eqs. (21)–(23), we introduce the three measures

sq :=



0

r3hq(r )dr, q = 0, 1, 2, (24)

related to the second mathematical moments as follows from Eqs.(21)–(23): m2,0= 12(s0+ s1), m0,2= 1 2(s0− s1), m1,1= 1 2s2.

For the particular case of the function [Eq.(8)], it follows that: s0 = Aπab  1+ 2 β  (b2+ a2), (25) s1 = Aπab  1+ 2 β  (b2− a2) cos 2α, (26) s2 = Aπab  1+ 2 β  (b2− a2) sin 2α, (27) where is the Gamma function

(z) :=  +∞

0

tz−1e−tdt, z > 0.

For the particular case of a Gaussian function, (3) = 2. Con-sider the power-spectrum function [Eq.(8)]. Then, parameter

α can be directly estimated from numerically computed val-ues of s1and s2 α = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1 2arctan( s2 s1 ), if s1 = 0, π 4, if s1= 0, s2 = 0, ∀, if s1= 0, s2= 0, (28)

and the ratio R := a/b

R= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩  s0 cosα + s1 s0 cosα − s1 , if s1 = 0,  s0 sinα + s2 s0 sinα − s2, if s1= 0, s2 = 0, 1, if s1= 0, s2 = 0. (29)

4.1 Gaussian Power Spectrum

Consider the power spectrum modeled as a Gaussian function [β = 1 in Eq.(7)]. Then, one can rewrite the expressions for defocus and stigmator functions [Eqs.(14)–(16)as follows:

h0(r )=  2π 0 e−r2  cos2 (ϕ+α) 2a2 + sin2 (ϕ+α) 2b2  dϕ = 2e(a2+b2)r24a2 b2 π I0  1 4  1 b2 − 1 a2  r2  , h1(r )=  2π 0 e−r2(cos2 (2a2ϕ+α)+ sin2 (ϕ+α) 2b2 ) cos 2ϕdϕ = 2e(a2+b2)r24a2 b2 π I1  1 4  1 b2 − 1 a2  r2  cos 2α, h2(r )=  2π 0 e−r2(cos2 (2a2ϕ+α)+ sin2 (ϕ+α) 2b2 ) sin 2ϕdϕ = 2e(a2+b2)r2 4a2 b2 π I1  1 4  1 b2 − 1 a2  r2  sin 2α,

where Ik(z) is the modified Bessel function of the first kind,

that can be expressed as35

Ik(z) := 1 π  π 0 er cosϕ cos kϕdϕ,

which is related to the Bessel function of the first kind as35

Ik(z) := i−kJk(iz).

Figure7shows Bessel functions of the first kind and modified Bessel functions of the first kind for k= 0, 1.

From these observations, it is clear that for the particular case of a Gaussian function for (−π/4) < α ≤ (π/4) sgn(h1)= sgn(b2− a2), (30)

sgn(h2)= sgn(b2− a2) sgn(sin 2α). (31)

This means that the directions of stigmator controls variables can be easily identified from the stigmator functions sgn(σx− σx0)= sgn(h1), (32) sgn(σy− σy0)= sgn(h2). (33)

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Rudnaya et al.: Orientation identification of the power spectrum −10 −5 0 5 10 −0.5 0 0.5

z

J

k Bessel functions of the first kind

J 0 J 1 −5 0 5 −20 −10 0 10 20

z

I

k

Modified Bessel functions of the first kind

I

0

I1

Fig. 7 Bessel functions of the first kind and modified Bessel functions of the first kind fork= 0, 1.

4.2 Discretization

We define the radial mesh points rk= kr + r/2, k =

1, . . . , N, where r := X/N, and the angular mesh points ϕl= lϕ + ϕ/2, l = 1, . . . , M, where ϕ := π/M.

Fur-thermore, we fill in the matrix values ˜P∈ RN×M

x(k,l) := rk cosϕl, y(k,l):= rk sinϕl. (34)

Each point [x(k,l), y(k,l)] is bounded xi−1≤ x(k,l)≤ xi, yj−1 ≤ y(k,l)≤ yj,

i∈ {−n + 1, . . . , n}, j ∈ {−n + 1, . . . , n}.

We compute values of ˜Pk,lwith a linear interpolation (see

Fig.8), ˜ Pk,l =   1− δx x  pi−1, j−1+ δx xpi, j−1  1− δy x  +  1− δx x  pi−1, j+ δx xpi, j  δ y x, where δx := x(k,l)− xi−1, δy:= y(k,l)− yi−1.

For k∈ {1, . . . , N}, we approximate the values of h0(rk), h1(rk), and h2(rk) with the midpoint numerical

in-Fig. 8 Linear interpolation of power-spectrum values.

tegration rule h0(rk) .= 2ϕ M  l=1 ˜ Pk,l, (35) h1(rk) .= 2ϕ M  l=1 ˜ Pk,l cos 2ϕl, (36) h2(rk) .= 2ϕ M  l=1 ˜ Pk,l sin 2ϕl. (37)

Then, the measures are also approximated with the midpoint rule sq .= r N  k=1 rk3hq(rk), q = 1, 2, 3. (38) 5 Numerical Experiments

5.1 Numerical Experiments for a Gaussian Function We consider a discrete Gaussian function with dimen-sions (2n+ 1) × (2n + 1) = 101×101 pixels. For further nu-merical computations of defocus and stigmator functions h0, h1, h2, the number of data points for discretization of

po-lar radius r is chosen N = n = 50 and for discretization of polar angleϕ is chosen M = 2(2n + 1) = 202. We consider x = 1 in Eq.(9), thus X = n. Each row of Fig.9shows a Gaussian functions with particular parameters values a, b, α and corresponding h0, h1, h2 functions. The rotation angle α and the ratio R are numerically estimated according to Eqs. (28) and (29) and indicated by two orthogonal lines plotted above each of the Gaussian functions.

Here, we provide an overview of illustrated numerical experiments:

1. Figures9(a)–9(d): Gaussian is rotationally symmetric (a= b = 20). For this reason, the computed values of h1and h2are equal to zero.

2. Figures9(e)–9(h): Gaussian is rotationally symmet-ric (a= b = 10). The values of h1 and h2 are equal

to zero. The value of Gaussian width is smaller than in Fig.9(a). The integral of the function h0 in

Fig.9(f) decreases in comparison with the previous experiment [Fig.9(b)]. The goal of a human operator in this case is to find the SEM parameters such that the Gaussian width and, as a consequence, the integral

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(a) a = 20, b = 20, α = 0 10 20 30 40 0.5 1 1.5 2 x 10−3 r h0 (b) 10 20 30 40 −5 0 5 x 10−4 r h1 (c) 10 20 30 40 −5 0 5 x 10−4 r h2 (d) (e) a = 10, b = 10, α = 0 10 20 30 40 2 4 6 8 x 10−3 r h0 (f) 10 20 30 40 −5 0 5 x 10−4 r h1 (g) 10 20 30 40 −5 0 5 x 10−4 r h2 (h) (i) a = 20, b = 10, α = 0 10 20 30 40 1 2 3 4 x 10−3 r h0 (j) 10 20 30 40 −5 0 5 x 10−4 r h1 (k) 10 20 30 40 −5 0 5 x 10−4 r h2 (l) (m) a = 10, b = 20, α = 0 10 20 30 40 1 2 3 4 x 10−3 r h0 (n) 10 20 30 40 −5 0 5 x 10−4 r h1 (o) 10 20 30 40 −5 0 5 x 10−4 r h2 (p) (q)a = 20, b = 10, α =π 4 10 20 30 40 1 2 3 4 x 10−3 r h0 (r) 10 20 30 40 −5 0 5 x 10−4 r h1 (s) 10 20 30 40 −5 0 5 x 10−4 r h2 (t) (u) a = 20, b = 10, α = −π6 10 20 30 40 1 2 3 4 x 10−3 r h0 (v) 10 20 30 40 −5 0 5x 10 −4 r h1 (w) 10 20 30 40 −5 0 5x 10 −4 r h2 (x)

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Rudnaya et al.: Orientation identification of the power spectrum (a) (b) 50 100 150 200 100 120 140 160 180 r h0 (c) 50 100 150 200 −5 0 5 r h1 (d) 50 100 150 200 −5 0 5 r h2 (e)

Fig. 10 From left to right, top to bottom: SEM experimental image, logarithmic scale of its power spectrum, numerically computed functions

h0(r), h1(r), and h2(r).

of h0 is as large as possible. In order to estimate h0

intensity and to minimize the defocus in automated applications the measure based on the zero mathe-matical moment [Eq.(20)] is often used in electron microscopy10 s0,0:=  rmax rmin r  2π 0 p(r cosϕ, r sin ϕ)dϕdr,

where rmin and rmaxare the low- and high-frequency

bands, parameters given as an input by a user. 3. Figures9(i)–9(l): Elliptic Gaussian. The widths a> b

and, as a consequence, all the values of h1 are < 0. The values of h2 are equal to zero, because α = 0.

4. Figures 9(m)–9(p): Elliptic Gaussian. The widths b> a and as a consequence, all the values of h1

(a) (b) 50 100 150 200 180 200 220 240 260 r h0 (c) 50 100 150 200 −2 0 2 r h1 (d) 50 100 150 200 −2 0 2 r h2 (e)

Fig. 11 From left to right, top to bottom: SEM experimental image, logarithmic scale of its power spectrum, numerically computed functions

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(a) (b) 50 100 150 200 180 200 220 240 260 r h0 (c) 50 100 150 200 −5 0 5 r h1 (d) 50 100 150 200 −5 0 5 r h2 (e)

Fig. 12 From left to right, top to bottom: SEM experimental image, logarithmic scale of its power spectrum, numerically computed functions

h0(r), h1(r), and h2(r).

are >0. The values of h2 are equal to zero, because α = 0.

5. Figures 9(q)–9(t): Elliptic Gaussian with a > b, α = π/4. As a consequence, h1= 0, h2 < 0.

6. Figures 9(u)–9(x): Elliptic Gaussian with a> b, α = −(π/6). As a consequence, h1< 0, h2> 0.

The numerical results correspond to the analytical obser-vations [Eqs.(30)and(31)] and can guide an unexperienced

human operator, giving a visual suggestion about a proper choice of stigmator control and its sign [Eqs.(32)and(33)].

5.2 Numerical Experiments for Scanning Electron Microscopy Images

In this section the functions h0, h1, and h2are computed for

power spectrums of SEM experimental images. Each of Fig-ures10–14show the SEM experimental image; Logarithmic

(a) (b) 50 100 150 200 120 140 160 180 r h0 (c) 50 100 150 200 −2 −1 0 1 2 r h1 (d) 50 100 150 200 −2 −1 0 1 2 r h2 (e)

Fig. 13 From left to right, top to bottom: SEM experimental image, logarithmic scale of its power spectrum, numerically computed functions

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Rudnaya et al.: Orientation identification of the power spectrum (a) (b) 50 100 150 200 160 180 200 220 240 r h0 (c) 50 100 150 200 −5 0 5 r h1 (d) 50 100 150 200 −5 0 5 r h2 (e)

Fig. 14 From left to right, top to bottom: SEM experimental image, logarithmic scale of its power spectrum, numerically computed functions

h0(r), h1(r), and h2(r).

scale of its power spectrum; functions h0, h1, h2 computed

with Eqs.(35)–(37).

The size of each experimental image is (2n+ 1)×(2n + 1) = 441×441 pixels. For each function h0, h1, h2 the

measures s0, s1, s2 are computed with Eq. (38), and the

following values chosenx = 1, N = n = 220, M = 2(2n + 1) = 882. The rotation angle α and the ratio R are nu-merically estimated using Eqs.(28)and(29). Two orthogo-nal lines above each power spectrum visualize the observed value of α. The functions h0, h1, h2 shown in Figs. 10–14

are computed for the logarithmic scale of power spectrum for the reason of convenience of visualization. However, the angle α displayed by the two orthogonal lines is found from defocus and stigmator functions before the logarithmic scale.

Figure10shows the experiment for gold-on-carbon stig-matic image. Changing bothσxandσyis needed to improve

the quality. The signs of the curves h1and h2indicate the

di-rection of change. Figure11shows an in-focus astigmatism-free image of tin balls. Both h1 and h2 are numerical noise

around zero. Figure12shows a defocused stigmatic image of the same sample. We can see how the image quality de-creases and power-spectrum shape changes. The y-stigmator function suggests the change ofσy, whereas adjustment ofσx

does not seem to be necessary. Figure13shows a magnified image of tin balls. The image does not have many details, and as a consequence, its power spectrum has only a few values of low frequencies different from noise. Because of the lack of information in Fourier space of the image, it is difficult to analyse this power spectrum and to draw the conclusions about the presence of astigmatism in the image. In the case of this type of samples, the method of orientation identifica-tion as well as other Fourier transform-based methods might fail. Figure14shows one more defocused and stigmatic im-age. We can clearly see that correction of bothσxandσyis

needed.

6 Discussion and Conclusions

The method for power-spectrum orientation identification was proposed and tested on Gaussian benchmarks and on SEM experimental images. The method involves computing of defocus/stigmator functions and defocus/stigmator real-valued measures, which as far as we know have not been used before. For power spectrum modeled as a L´evy stable density, the defocus/stigmator measures are expressed via the function. For power spectrum modeled as a Gaussian function (the particular case of a L´evy stable density), the defocus/stigmator functions are expressed via the modified Bessel functions of the first kind. The method can be used for increasing the capabilities of defocus and astigmatism cor-rection for nonexperienced SEM users (via defocus/stigmator functions). From defocus/stigmator functions, we compute defocus/stigmator real-valued measures. The method could be used as a basis for automated defocus and astigmatism correction in SEM if defocus/stigmator measures are applied and compared for images of the same sample obtained with different microscope settings.

The alternative approach to extracting power spectrum pa-rameters could be simply fitting it with a continuous model13

by minimizing least-squares difference of continuous and experimental data with the help of an iterative method (for instance, the Newton method). In Ref.13this is done for a one-dimensional case under the assumption that power spec-trum is rotationally symmetric. For a one-dimensional case, this would take much longer computational time than com-puting of defocus/stigmator functions, which is noniterative. Another approach for extracting defocus/stigmator infor-mation from the power spectrum could be based on a nonit-erative fitting of discrete power spectrum with a set of basis functions, for instance, via the projection method.36 In this case, the defocus/stigmator function could be precomputed analytically for the given set of basis functions. However, for the set of basis functions explored thus far, the approach

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is still slower than direct computation of the focus/stigmatic function described in this paper.

Acknowledgment

We thank Seyno Sluyterman (FEI, The Netherlands) for ideas, support, thoughtful discussions, and processing the re-sults. We also are thankful to Max Otten, Rob van Vucht (FEI, The Netherlands) Dirk Van Dyck, and Wouter van den Broek (EMAT, Belgium) for thoughtful discussions. We kindly acknowledge Rob van Vucht and Michael Janus (FEI, The Netherlands) for assistance with obtaining the experi-mental data. This work has been carried out as a part of the Condor project at FEI Company under the responsibilities of the Embedded Systems Institute (ESI). This project is par-tially supported by the Dutch Ministry of Economic Affairs under the BSIK program.

Appendix: Extension for Other Types of Aberrations

In this paper, we have considered defocus and twofold astig-matism aberrations. For the higher orders of astigastig-matism, the cos 2ϕ in Eq.(15)must be replaced with cos 2kϕ. In order to generalize our observations and extend them for other types of aberration, we define a function

I (a, b, α, k, p) : =  2π 0 e−ikϕ  0 rp × e−r2β[a2cos2(ϕ+α)+b2sin2(ϕ+α)]β dr dϕ. (39) The function is related to the measures [Eq.(24)] applied to the Levy stable density power spectrum [Eq.(8)], as follows: I  1 √ 2a, 1 √ 2b, α, 2, 3  = s1+ is2. Property 1:

For p= 2n − 1 in(39)it follows that for F (a, b, 2k, 2n − 1) := 2π 4n−1  a− b a+ b k (−1)k (a− b)2n−1 × n−1  l=0  n+ k − 1 l   2n− l − 2 n− 1  × (a − b)2n−2l−2(4ab)l, I (a, b, α, 2k, 2n − 1) = e2ikα 2β F (a, b, 2k, 2n − 1)  n β  . Proof. From the definition of the function, we have 

0

rpe−r2β(a2cos2ϕ+b2sin2ϕ)βdr

= 1

2β(a2 cos2ϕ + b2 sin2ϕ)p+1/2

 0 rp+1/2β−1e−rdr    [p+1/2β] .

For the function [Eq. (39)] it follows that I (a, b, α, k, p) = eikαI (a, b, 0, k, p). Then, for

F (a, b, k, p) :=  2π 0 e−ikϕ (a2 cos2ϕ + b2 sin2ϕ)( p+1)/2dϕ, (40) I (a, b, α, k, p) = eikα 2β  p+ 1 2β  F (a, b, k, p). For p= 2n − 1, we rewrite the function [Eq.(40)] as

F (a, b, 2k, 2n − 1) =  2π 0 e−2ikϕ (a2 cos2ϕ + b2 sin2ϕ)ndϕ =  π −π e−2ikϕ (a2 cos2ϕ + b2 sin2ϕ)ndϕ = 2  π 0 e−2ikϕ (a2 cos2ϕ + b2sin2ϕ)ndϕ =  2π 0 eikϕ a2 cos2ϕ/2 + b2 sin2ϕ/2ndϕ =  2π 0 eikϕ

[(a2+b2)/2+(a2−b2)/2 cos ϕ]ndϕ.

We substitute z := eiϕand, for a = b, obtain

F = 1 i( 4 a2− b2) n ×  |z|=1 zn+k−1 z2+ 2[(a2+ b2)/(a2− b2)]z+ 1ndz = 1 i  4 a2− b2 n |z|=1 zn+k−1 (z− z1)(z− z2) dz, z1= b− a a+ b, z2= 1/z1. Then, F = n−1  l=0  n+ k − 1 l  (−1)n−l−1 z n+k−1−l 1 (n− l − 1)! ×n(n+ 1), . . . , (2n − l − 2) (z1− z2)2n−l−1 , zn1+k−1−l (z1− z2)2n−l−1 = ( b− a a+ b) n+k−1−l(a2− b2)2n−l−1 (4ab)2n−l−1 = (−1)n+k−1−l(a− b)k+3n−2l−2 (4ab)2n−l−1 ,

and the statement of the property is straightforward. Further-more, the analysis could be expanded based on the work on a diffraction-free Bessel beam, which has been done

before.37 

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