Noise estimation for off-axis electron holography

Ultramicroscopy 144 (2014) 32–42

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Noise estimation for off-axis electron holography Falk Röder a,n, Axel Lubk a, Daniel Wolf a, Tore Niermann b a b

Triebenberg Labor, Institut für Strukturphysik, Technische Universität Dresden, D-01062 Dresden, Germany Institut für Optik und Atomare Physik, Technische Universität Berlin, Straße des 17. Juni 135, 10623 Berlin, Germany

art ic l e i nf o

a b s t r a c t

Article history: Received 13 January 2014 Received in revised form 3 April 2014 Accepted 11 April 2014 Available online 26 April 2014

Off-axis electron holography provides access to the phase of the elastically scattered wave in a transmission electron microscope at scales ranging from several hundreds of nanometres down to 0.1 nm. In many cases the reconstructed phase shift is directly proportional to projected electric and magnetic potentials rendering electron holography a useful and established characterisation method for materials science. However, quantitative interpretation of experimental phase shifts requires quantitative knowledge about the noise, which has been previously established for some limiting cases only. Here, we present a general noise transfer formalism for off-axis electron holography allowing to compute the covariance (noise) of reconstructed amplitude and phase from characteristic detector functions and general properties of the reconstruction process. Experimentally, we verify the presented noise transfer formulas for two different cameras with and without objects within the errors given by the experimental noise determination. & 2014 Elsevier B.V. All rights reserved.

Keywords: Electron holography Holographic reconstruction Noise transfer Noise spread function

1. Introduction Gabor's idea of lens-free imaging [1] to circumvent unavoidable aberrations in electron optics [2] instigated new imaging techniques summarised by the term “holography”. The speciality they have in common is to retrieve (at least in principle) the phase of a scattered electron wave by interference, hence to solve the phase problem. One realisation among more than 20 forms [3] is (image plane) off-axis electron holography [4]. The Möllenstedt biprism [5] is used to interfere two coherent electron waves forming a detectable interference pattern. There, the phase shift manifests as local bending of interference fringes being the observable in all reconstruction schemes for off-axis electron holography, e.g. Refs. [6–8]. The phase shifted by electric fields allows statements about mean inner potentials [10–12], functional potentials, e.g. at pn-junctions [13–15] or in the case of dark-field holography about strain fields [16–18]. The phase shifted by magnetic fields is suited for studying local magnetism in, e.g. thin films [19,20], patterned nano-structures [21] or in particle assemblies [22]. Furthermore, quantities like the total charge [23] or dipole moments [24] can be retrieved using Maxwell's equations underlining the quantitative character of this method. The projection property of the phase further allows for reconstructing the complete 3D object potential using tomographic approaches [25], which have been developed to an automatised holographic technique for electric potentials

n

Corresponding author. E-mail address: [email protected] (F. Röder).

http://dx.doi.org/10.1016/j.ultramic.2014.04.002 0304-3991/& 2014 Elsevier B.V. All rights reserved.

[26]. However, the complete 3D reconstruction of the magnetic vector potential remains still challenging [27]. For a more comprehensive overview about applications of off-axis electron holography we recommend [9,28,29]. Quantitative statements retrieved from holographic measurements require the computation of errorbars for phase and amplitude. To date the signal transfer obtained from reconstruction processes is known, but the noise transfer is unknown and depends on the detection and reconstruction process in general. Principally, the noise can be determined through statistical hologram series evaluation for each detector pixel as demonstrated, e.g. in [30]. In contrary, statistical analysis of fluctuations within a certain region of interest of the reconstructed images as conducted in the past [31–33] is valid only, if


the region of interest is large compared to the correlation length of the correlated noise and


illumination, object and detector are homogeneous therein. Both conditions show that noise measurement using a region of interest is not applicable for strongly varying signals like in images of atomic structures. However, in many cases, the acquisition of a series of holograms is critical because of various instrumental instabilities (drifting aberrations, object or biprism) or radiation damages especially in rather time consuming procedures like electron holographic tomography [26]. For those applications, there is a need of determining the noise, i.e. the local errorbars and the statistical correlations between neighboured pixels, in the reconstructed amplitude and phase

F. Röder et al. / Ultramicroscopy 144 (2014) 32–42

images directly from individual holograms depending on detection and reconstruction processes. This gave the impetus for this work. Furthermore, the noise transfer from the reconstructed wave through possible subsequently applied image processing steps requires the knowledge of noise correlation in the reconstructed data, which was not conducted in the past. Noise correlation also plays an important role, e.g. for model-based quantification of experimental results [34]. First considerations on noise in off-axis electron holography were conducted in [35]. The transfer of noise from the hologram into amplitude and phase was described by the Lenz formulas [36]. Later de Ruijter et al. investigated the influence of further unknown parameters like the carrier frequency of the fringes pattern (qc ) [37]. Both considered the restricted case of uncorrelated Poissonian noise in the hologram, i.e. shot noise. Additionally, the resulting noise transfer formulas were only valid for a special real-space reconstruction scheme [7] and are therefore not applicable for general reconstruction apertures. Experimental attempts to verify these formulas resulted in deviations empirically described by coefficients like “transfer efficiency” (TE) [31,32,37] or by the “detective quantum efficiency” (DQE) [45,35,38,39]. But all these correction factors do not address the problem of correlated noise in the hologram through the detection process correctly. Even the deconvolution of the modulation transfer function (MTF) attempting to decorrelate detector pixels [40] fails because of the fundamental difference of noise and signal transfer [41,43,45]. Individual scattering paths of different highly energetic electrons in the scintillator lead to broadening of the incoming signal and noise resulting in statistical correlations of adjacent pixels. Furthermore, the stochastic character of the scattering processes spans an additional probability space affecting the transferred noise only. A general treatment of noise transfer of the detector addressing the mentioned problems was recently presented [43,44]. In this paper, the following improvements with respect to previous works are

33

2. Theory In the following subsection we summarise the state of the art reconstruction procedure for off-axis electron holograms. Then we address the noise transferred through this procedure and derive the according formulas. The next subsection demonstrates that the Lenz formulas are a special case of these transfer formulas and subsequently discuss the effect of wave normalisation by “empty holograms”. Finally, we incorporate the noise properties of the detector.

2.1. The standard holographic reconstruction An off-axis electron hologram emerges from the coherent superposition of both object and reference wave, inclined towards each other by a Möllensted biprism [5], and subsequent detection, e.g. by a CCD-camera (Fig. 1a). The acquired intensity distribution then results in a harmonic fringe pattern (see inset of Fig. 1b) with a defined carrier frequency qc proportional to the inclination angle. This pattern encodes both the centre band intensity ICB and the sideband intensity ISB and phase φ by IðrÞ ¼ I CB ðrÞ þ 2I SB ðrÞ cos ð2π qc r þ φðrÞÞ:

ð1Þ


general noise transfer through the holographic reconstruction process for arbitrary holographic reconstruction apertures,


reproduction of the Lenz model as a special case,
calculation of covariances instead of variances only,
estimation of the covariance matrix of a measured intensity


distribution (considering the correlation of noise) based on a rigorous noise transfer theory for the detector, discussion of noise in reconstructed amplitude and phase images of real objects at medium and atomic resolution.

In the following section we describe the standard reconstruction scheme for off-axis electron holography and derive the corresponding noise transfer formulas for an acquired correlated signal (hologram) describing the correlated noise of the reconstructed amplitude and phase. Furthermore, we reproduce the known formulas by Lenz [36] as a special case. Finally, we discuss the detector influence by the noise spread function [43] and present a suitable approximation for the covariance matrix, i.e. the correlated noise, for an off-axis electron hologram. In Section 3 we outline the procedure for measuring the noise using pixel-wise statistical series evaluation. In addition, we describe the measurement of required detector noise properties summarised in a quantity called covariance-to-signal ratio (CSRh ). In Section 4 we present the measured CSRh for two different cameras and determine the noise for empty and locally also for object holograms for an arbitrary reconstruction aperture, which exceeds the Lenz approximation. The comparison with experimentally measured noise yields good agreement within the errors and verifies the presented noise transfer formulas. We discuss the influence of the local object signal on the noise distribution and confirm qualitative aspects of the Lenz formulas.

Fig. 1. Acquisition and reconstruction procedure for off-axis electron holography. (a) Path of object (yellow) and reference wave (vacuum, green) interfering in the image plane by means of a Möllenstedt biprism. (b) Acquired fringe pattern (so-called hologram) of an atomic SrTiO3 lattice in [001] zone axis orientation. The magnified inset shows the bending of the fringes by the phase shift through the atomic potentials. (c) Fourier transformation (FT) of the hologram consisting of two sidebands and one centre band. (d) Wave reconstruction by inverse Fourier transformation (FT  1) of the isolated sideband and normalisation by a reference measurement reveals the complex wave at the image plane in terms of amplitude and phase (wrapped). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this paper.)

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F. Röder et al. / Ultramicroscopy 144 (2014) 32–42

Note that I SB ¼ μpc AA0 , where A is the amplitude of the object wave, A0 is the amplitude of the reference wave and μpc the absolute degree of partial coherence [9]. The aim of a holographic reconstruction is to separate centre band and sideband (wave) contributions from each other. A commonly used method is based on Fourier filtering (Fig. 1c and d) originally performed on an optical bench [4,6]. This procedure becomes obvious when considering the Fourier transformation (FT) of (1) FT½IðrÞ ¼ FT½I CB ðrÞ  δðqÞ þ FT½I SB ðrÞ expð þ iφðrÞÞ  δðq  qc Þ þ FT½I SB ðrÞ expð  iφðrÞÞ  δðq þ qc Þ

ð2Þ

The application of the convolution theorem yields three terms, the first is called centre band and the other two sidebands (Fig. 1c). The centre band contains the intensity ICB and the sidebands I SB expð 7 iφÞ. The cosine function introduces a frequency shift in the Fourier space by qc separating centre band and sidebands from each other described by convolution with corresponding Dirac delta functions. A complete separation is achieved, if the carrier frequency obeys jqc j 4 3q0 with q0 being the maximal spatial frequency in the sideband spectrum [8]. The centring of one sideband, the subsequent low-pass filtering by an aperture function BðqÞ of defined radius (Fig. 1c) and the final inverse Fourier transformation (Fig. 1d) comprise the reconstruction procedure. The radius of the aperture function thereby defines the spatial resolution of the reconstructed sideband and influences also the noise transferred from the hologram into the reconstructed data. The following equation summarises these reconstruction steps I SB ðrÞ expðiφðrÞÞ ¼ ðIðrÞ expð  2π iqc rÞÞ  KðrÞ:

ð3Þ

The multiplication with the phase wedge in position space is equivalent to the centring of the holographic sideband in Fourier space. The convolution with KðrÞ is equivalent to the abovementioned low-pass Fourier filtering of the spectral sideband by an aperture function BðqÞ. This aperture function relates to KðrÞ via the convolution theorem BðqÞ ¼ FT½KðrÞ:

ð4Þ

To finally obtain the normalised wave ψ we have to normalise the reconstructed sideband (3) by the reconstructed sideband of a reference hologram without object I 0SB expðiφ0 Þ, i.e. I SB ðrÞ expðiðφðrÞ  φ0 ðrÞÞÞ I 0SB ðrÞ AðrÞ expðiðφðrÞ  φ0 ðrÞÞÞ: ¼ A0 ðrÞ

ψ ðrÞ ¼

ð5Þ

Eqs. (3) and (5) provide direct access to the wave encoded in an off-axis electron hologram. For the further discussion, real quantities rather than complex ones are more suitable. For the following equations we use discrete notation adapted to the discrete nature of the sampled holograms and the ensuing reconstructed quantities. In the following we therefore consider the real XðrÞ and imaginary parts YðrÞ of the reconstructed sideband (3). The assumption of a symmetric reconstruction aperture (BðqÞ ¼ Bð  qÞ) implies a real valued KðrÞ; consequently the real and imaginary parts directly follow from Eq. (3) as 0

0

0

XðrÞ ¼ ∑Iðr ÞKðr  r ÞCðr Þ r0

YðrÞ ¼ ∑Iðr0 ÞKðr  r0 ÞSðr0 Þ r0

ð6Þ

with the abbreviations CðrÞ ¼ cos ð 2π qc rÞ

ð7Þ

SðrÞ ¼ sin ð  2π qc rÞ

ð8Þ

and the definitions: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I SB ðrÞ ¼ XðrÞ2 þ YðrÞ2

ð9Þ

φðrÞ ¼ atanðYðrÞ=XðrÞÞ:

ð10Þ

The question is, how the noise is transferred from the hologram into the reconstructed quantities. In the next subsection, we derive the respective transfer formulas. 2.2. Noise transfer through the standard holographic reconstruction The spread, i.e. the noise, of a single random variable X around its mean value E½X, is characterised by the second centralised moment, the variance varðXÞ ¼ E½ðX  E½XÞ2 . When considering the noise in vectorial data as produced by a pixel detector one generalises the former to the covariance matrix covðX; YÞ ¼ E½ðX E½XÞðY E½YÞ:

ð11Þ

It contains the variances (s ) on the main diagonal and allows us to quantify the correlation between two random variables (e.g. two pixels) from the off-diagonal elements. The covariance also forms the basis for computing the error of quantities composed of multiple random variables, such as the total average of a pixel image. The corresponding law of error propagation states that a linear transformation g ¼ Lf of the vectorial random variable f (e.g. the pixel image) translates into a simple transformation of the corresponding covariance matrix according to covg ¼ L covf LT . The error propagation rules readily apply to the formulas (6) since they are linear in the experimental values IðrÞ. One obtains the following transfer of the experimental noise in the hologram covII0 ¼ covðIðrÞ; Iðr0 ÞÞ to the noise of real and imaginary parts of the reconstructed waves 2

covXX 0 ¼ ∑ covI″I‴ Kðr  r″ÞKðr0  r‴ÞC″C‴ r″;r‴

covXY 0 ¼ ∑ covI″I‴ Kðr r″ÞKðr0  r‴ÞC″S‴ r″;r‴

covYY 0 ¼ ∑ covI″I‴ Kðr  r″ÞKðr0 r‴ÞS″S‴: r″;r‴

ð12Þ

Here, we introduced the following short-hand notation C 0 ; S″; … for Cðr0 Þ; Sðr″Þ; …. Accordingly, the subscript priming is related to the priming of the position vector r associated to the corresponding function. In Section 2.5 we construct a model for the required covariance matrix of the hologram in dependence on specific noise properties of the detector. Since the sideband is normally represented in terms of sideband intensity and phase rather than real and imaginary parts, we additionally derive the covariance matrices of sideband intensity and phase. The application of the error propagation rules to the relations (9) and (10) directly yields the final expressions for the correlated noise in the sideband intensity ISB and phase images φ 1 covISB ISB 0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ðXX 0 covXX 0 þ YY 0 covYY 0 2 2 X þY X 02 þ Y 02 þ XY 0 covXY 0 þYX 0 covYX 0 Þ

1  ðYY 0 covXX 0 þ XX 0 covYY 0 ðX 2 þ Y 2 ÞðX 02 þY 02 Þ YX 0 covXY 0  XY 0 covYX 0 Þ:

covφφ0 ¼

ð13Þ

The covariance between sideband intensity and phase (covISB φ0 ) can be derived in the same way, but this will not be discussed in this work. Eqs. (12) and (13) describe the noise transfer from an experimental hologram with the given covariance matrix of the measured intensity (covII0 ) into the reconstructed sideband intensities and phases. The noise transfer depends in general on the reconstruction aperture BðqÞ. For a certain choice of B and special assumptions for the noise in the hologram, simple analytical expressions for the covariances between sideband intensity, phase

F. Röder et al. / Ultramicroscopy 144 (2014) 32–42

35

and centre band intensity can be derived which are referred to as the Lenz model [36] in the following. To date, this model is commonly used for the estimation of noise in the reconstructed images [9,31,32,47].

obtains

2.3. The Lenz model

Due to lateral homogeneity we omit the arguments (r) of the reconstructed quantities. The according expressions for the variances of sideband intensity and phase result from inserting (19) into (13). We use I~ rec ¼ N x Ny I~CB for the total intensity contributing to the reconstructed pixel and express the reconstructed sideband intensity I SB ¼ μ~ I~CB =2 in terms of the fringe contrast μ. Thus, the variances of sideband intensity and phase finally read

Since we concentrate on the discussion of sideband intensity and phase noise, we derive the corresponding Lenz model formulas only. This will also serve as a verification for the general transfer formulas (12) and (13) given above. The Lenz model applies to the real space reconstruction technique [8], which corresponds to a special KðrÞ determined by a rectangularshaped window function KðrÞ ¼ H Nx ðxÞH Ny ðyÞ

ð14Þ

with the function H Nx ðxÞ (and H Ny ðyÞ) defined by ( 1 1 for jxjr N x =2 H Nx ðxÞ≔ Nx 0 for jxj4 N x =2:

ð15Þ

The parameters Nx and Ny determine the number of detector pixels contributing to one reconstructed pixel in the respective lateral dimension. The normalisation is chosen such that ∑r KðrÞ ¼ 1, i.e. the reconstructed quantities correspond to mean values within the reconstructed pixel. The reconstruction kernel (14) separates sideband from centre band, if two assumptions are fulfilled. First, the fringe pattern has to be commensurable to the detector pixel size. Second, the centre band ICB and the sideband may only slowly vary in the area of the reconstructed pixel, i.e. are sufficiently band limited. The latter facilitates the following approximation of the hologram intensity: IðrÞ  I~ CB ðrÞ þ 2X~ ðrÞCðrÞ þ2Y~ ðrÞSðrÞ:

ð16Þ

The notation using tilde denotes the invariance of these quantities within the area of the reconstructed pixel given by KðrÞ. The reason for the second condition becomes clear when inserting (16) and (14) into (3). Then, the reconstruction kernel KðrÞ restricts the summation over the set of detector pixels in (6) to the area of the reconstructed pixel. Then, only those terms containing cos 2 or sin 2 remain, which are directly related to the real or imaginary part of the sideband. In cases of strongly varying signals, i.e. the invariance of X~ , Y~ and I~CB within the reconstructed pixel is not given, the application of the reconstruction kernel (14) would mix contributions from the centre band, the real and the imaginary part into the reconstructed quantities. Then, a separation of the sideband from the centre band is not possible using the window kernel. The third important condition for the rigorous validity of the Lenz model is the assumption of Poissonian distributed (uncorrelated) noise in the acquired hologram, i.e. covðIðrÞ; Iðr0 ÞÞ ¼ δr;r0 varðIðrÞÞ ¼ δr;r0 IðrÞ:

ð17Þ

Here, the Kronecker symbol δr;r0 exhibits the uncorrelated character of the noise, which considerably simplifies the covariances of the reconstructed quantities in (12) to 0

0 2

covX~ X~ ¼ ∑I Kðr  r Þ C

02

r0

covX~ Y~ ¼ ∑I 0 Kðr  r0 Þ2 C 0 S0 r0

covY~ Y~ ¼ ∑I 0 Kðr  r0 Þ2 S02 : r0

ð18Þ

Inserting the commensurable (condition 1) window shaped kernel (14) and the hologram parametrisation (16, condition 2), the covariances between real and imaginary parts vanish and one

s2I~ I~ CB ¼ CB 2N x N y 2 ¼ 0:

covX~ X~ ¼ covY~ Y~ ¼ covX~ Y~ ¼ covY~ X~

covI~SB I~SB ¼ s2I~ ¼ SB

covφ~ φ~ ¼ s2φ~ ¼

I~ CB 2N x N y I~CB

2 2N x Ny I~ SB

¼

2 : I~ rec μ~ 2

ð19Þ

ð20Þ

Consequently, the variances reduce with increasing size (N x N y ) of the reconstructed pixel KðrÞ. The variance of the phase coincides with the relation found by Lenz. The agreement of the sideband intensity variance can be shown using expressions for the variance of μ in [36] 2  μ~ 2 s2μ~ ¼ ~ I rec

ð21Þ

and the variance of the centre band intensity (see Eq. (19), not derived here). Applying error propagation to the relation I SB ¼ μI CB =2 and inserting (19), (21) yields 1 4

s2I~SB ¼ ðs2μ~ I~CB þ s2I~CB μ~ 2 Þ ¼ 2

I~CB : 2Nx N y

ð22Þ

We have thus shown that the Lenz model represents a special case of our general noise transfer formalism. Consequently, if at least one of the underlying assumptions is violated the Lenz model fails to describe the noise in reconstructed holographic quantities like fringe contrast and total intensity quantitatively. Indeed, the noise properties of the detector deviate from an ideal Poissonian noise and other reconstruction kernels than the rectangular window are frequently used. Both effects will be considered within the generalised transfer theory further below. 2.4. The effect of sideband normalisation by empty holograms To remove artefacts from an experimental electron hologram [48] and to obtain the correctly normalised image wave ψ, the reconstructed sidebands are normalised according to (5). Here, we investigate the effect of sideband normalisation on the noise in the normalised amplitude and phase images in the frame of the Lenz model. We consider two experimental empty holograms with I SB ¼ I 0SB and φ ¼ φ0 (see Eq. (5)). The noise for the corresponding 0 normalised reconstructed amplitude A~ N ¼ I~ SB =I~SB and the normal~N ¼φ ~ φ ~ 0 can be determined by error propagation ised phase φ rules applied to (20). It turns out that the variances in amplitude and phase are equal, i.e. ~ N; φ ~ NÞ ¼ covðA~ N ; A~ N Þ ¼ covðφ

4 I~rec μ~ 2

ð23Þ

and larger compared to the not normalised values. For instance, in the case of normalisation of an empty sideband with another pffiffiffi empty sideband the phase noise increases by a factor of 2. To take the sideband normalisation into account for the noise propagation in the following, we will propagate the noise through the reconstruction process of object and empty holograms separately and determine the final noise of the normalised amplitude and phase images by further error propagation.

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F. Röder et al. / Ultramicroscopy 144 (2014) 32–42

one obtains the relation

2.5. Detector influence The noise transfer properties of the detector determine the noise in the experimentally acquired hologram and therefore also in the reconstructed data. In this work we used scintillatorcoupled CCD cameras, where the various parts of the detector (in particular the scintillator) cause noise spread and noise amplification due to stochastic scattering and transformation of both electrons and photons. Commonly used quantities like the detective quantum efficiency (DQE) [39] or noise transfer function (NTF) [41] describe only partial aspects of the noise transferred through such an image detector. They represent useful figures of merit for comparing different cameras as demonstrated in, e.g. [42]. However, a one-to-one relation of these figures to the hologram's covariance matrix required for computing the covariances of the reconstructed quantities (see Eq. (12)) does not exist. Indeed, the notion of a four-dimensional noise spread function (NSF) is required to compute the modified covariance of the detected image intensities in comparison to the ideal Poissonian shot noise [43]. Such a NSF can be derived within a generalised linear transfer theory for the covariance similar to the established notion of a point spread function for the average signal [43]. Moreover, the NSF can be measured with methods similar to those developed for the PSF measurement [44]. According to the general transfer theory the covariance transfer to the detector output (i.e. the image) reads Z 2 covðIðrÞ; Iðr0 ÞÞ ¼ I in ðr″ÞNSFðr r″; r0 r″Þ d r″: ð24Þ The function I in ðrÞ denotes the position dependent average incoming intensity of the incident electron beam which is equal to the variance of the Poissonian shot noise at the detector input. Since state-of-the-art microscopes are sufficiently stable we do not discuss additional input noise [43] here (e.g. beam flickering). If the variations in the average input intensity are sufficiently small one can exclude the input intensity from the integration in (24) to obtain Z 2 covðIðrÞ; Iðr0 ÞÞ  I in ðrÞ NSFðr r″; r0 r″Þ d r″: ð25Þ In the case of oscillating signals, like in a holographic fringe pattern, this approximation overestimates the noise at intensity maxima and underestimates it at the minima. Based on the same assumption of small input contrast we can further express the input intensity according to IðrÞ ¼ PSFðrÞ  I in ðrÞ  gI in ðrÞ;

ð26Þ

where the constant g denotes the conversion rate between incident electrons and pixel values in arbitrary digital units. The PSFðrÞ, i.e. the inverse Fourier transformation of the MTF times the camera's conversion rate, denotes the point spread function of the detector. That substitution finally yields 1 covðIðrÞ; Iðr0 ÞÞ  IðrÞ FT  1 ½NPSðr  r0 Þ: g

ð27Þ

where we have introduced the noise power spectrum (NPS) [45] defined by the Fourier transformation of the integrated NSF [43], i.e. Z 2 FT  1 ½NPSðr  r0 Þ ¼ NSFðr  r″; r0  r″Þ d r″: ð28Þ Note that these approximations are empirical and will be proven sufficiently accurate for practical noise determination of amplitude and phase images in the next section. In the case of a homogeneously illuminated camera, i.e. IðrÞ ¼ I h ¼ const, (26) and (27) hold exactly true. Consequently,

covðI h ðrÞ; I h ðr0 ÞÞ Ih

1 ¼ FT  1 ½NPSðΔrÞ≕CSRh ðΔrÞ: g

ð29Þ

Here we define the covariance-to-signal ratio (CSR h ) for a homogeneously illuminated camera, which is equal to the gain normalised inverse Fourier transformation of the known noise power spectrum (NPS). The relation (29) implies that the region of interest for determining the noise has to be large compared to the correlation length given by the width of the covariance matrix for correctly sampling the NPS. Note that the covariance of a homogeneously illuminated signal depends only on the distance Δr ¼ r  r0 between the pixels. The normalisation of the covariance by the mean intensity keeps the CSRh constant over a wide intensity range. However, additive contributions to the covariance, like dark current noise, become dominating for small intensities. Therefore the CSRh increases with decreasing intensities below a certain limit. For the cameras investigated here, this effect was observed to become critical below 200 digital counts per pixel. Accordingly, the covariance matrix for the measured intensity can be approximated using Eq. (25) by covðIðrÞ; Iðr0 ÞÞ  IðrÞCSRh ðr  r0 Þ:

ð30Þ

The CSRh can be computed from a series of homogeneously illuminated images and a subsequent evaluation using the Wiener–Khinchin-theorem as conducted in the next section. Within this approximation the conversion rate g is not explicitly required for the estimation of the covariance matrix. That is advantageous since determining the conversion rate usually requires a calibration device (e.g. a Faraday cup) which is not available at every microscope.

3. Experiment In the following subsection we present the experimental determination of noise conducted in this work. The subsequent subsection shows how to measure the necessary noise properties of the detector. 3.1. Experimental noise determination In order to verify the transfer formulas (12) and (13), we compare the covariance obtained from the transfer theory to that estimated directly from a series of reconstructed quantities. That requires a repetition of the experiment Ns times under equal conditions. To this end, we acquire a hologram series from the same object position and reconstruct all in the same way using (6). For the correction of artificial fringe bending due to the camera or projective lens distortions, we additionally use empty holograms as references [48]. After the acquisition of Ns object holograms the same number of empty holograms is recorded. The reconstruction of the hologram series is conducted by the Triebenberg Holography Software package developed for Gatan DigitalMicrograph™. The reconstructed sidebands are normalised by the corresponding sidebands reconstructed from a series of empty holograms of same length. From the series of reconstructed normalised waves, we determine pixel-wise the sample (co-)variances of real and imaginary parts

s2X ðrÞ ¼

Ns 1 ∑ ðX ðrÞ  X ðrÞÞ2 N s 1 i ¼ 1 i

ð31Þ

s2Y ðrÞ ¼

Ns 1 ∑ ðY ðrÞ  Y ðrÞÞ2 Ns  1 i ¼ 1 i

ð32Þ

F. Röder et al. / Ultramicroscopy 144 (2014) 32–42

covXY ðrÞ ¼

Ns 1 ∑ ðX i ðrÞ  X ðrÞÞðY i ðrÞ  Y ðrÞÞ Ns  1 i ¼ 1

ð33Þ

The quantities X and Y denote the mean real and imaginary parts of the wave calculated from the Ns images Xi and Yi as defined through X ðrÞ ¼ Y ðrÞ ¼

k

Ns

1 ∑ Y ðrÞ Ns i ¼ 1 i

ð34Þ

Here we note that the complex valued averages of the reconstructed wave series must be calculated as the mean values of real and imaginary parts. Averaging amplitude and phase instead leads to artefacts, whenever the amplitude is not constant or phase wraps occur as demonstrated, e.g. in [30,33]. We underline that even the sideband of a single hologram corresponds in a certain sense to a complex average of the exit-waves over the exposure time of the hologram [49]. The subsequent application of (13) finally yields the variances of amplitude and phase images. However, the experimentally determined variances contain two contributions, one from the detection process and the other from various instrumental instabilities between individual hologram acquisitions over the time of the series. The latter result, e.g. from object displacements, unstable aberrations as well as moving and charging of the biprism. The latter introduce fluctuating phase offsets and tilts. The hologram series were corrected for object displacements, phase offsets and phase tilts as follows: In the case of empty holograms, we numerically fit a two-dimensional linear function to the reconstructed phase to each member of the series and subtract these results from the corresponding experimental images. In the case of object holograms, we first correct object displacements: By comparing two reconstructed images, we evaluate the displacements. Subsequently, we shift these images to remove the corresponding displacements with the precision of one pixel. Afterwards, we correct phase-offset and tilt as in the case of the empty holograms by using a vacuum reference region included in the object hologram. The aligned wave images are used to determine the variances of real and imaginary parts according to Eqs. (31)–(33). In the case of high-resolution holograms the stability of aberrations becomes particularly important for the series acquisition. For reconstructed waves, however, the a posteriori aberration correction is feasible to further reduce the fraction of systematic errors, which is, however, beyond the scope of this paper. 3.2. Measurement of detector properties The covariance-to-signal ratio (29) for homogeneous illumination determines the noise properties of the detector important for this investigation. Experimentally this quantity can be determined using the relation (29) to

Np

∑

m;n

normalisation by means of the total intensity (or variance) renders this expression independent from incoming signal. The CSRh only depends on the difference between the pixel positions. Using the Wiener–Khinchin-theorem the spatial convolution in CSRh may be efficiently evaluated in the Fourier space CSRh ðΔm; ΔnÞ "

1 Ns ∑ X ðrÞ Ns i ¼ 1 i

CSRh ðΔm; ΔnÞ ¼

37

1 k

p s I^ ∑m;n ∑N k ¼ 1 mn

N

! Ns k k 1 Ns ^ k ^ k0 ∑ I^mn I^m þ Δm;n þ Δn  ∑ I mn I m þ Δm;n þ Δn : N s k;k0 ¼ 1 k¼1

ð35Þ

Here, k is the series index, (m,n) index positions in the images k running from 1 to Np (number of pixel per dimension) and I^ mn denotes the intensity measured in the pixel (m,n) of the k-th image 0 of the series. The inner summation over k and k provides the fourdimensional covariance matrix for the homogeneously illuminated camera. The summation over all lateral pixel coordinates (m,n) leads to an average over each covariance matrix per pixel. The

s FT  1 ∑N jFT½I^ m;n j2  k

¼

  2 #  1  Ns ^k  FT ∑ I k mn  Ns 

k Np s I^ ∑m;n ∑N k ¼ 1 mn

ð36Þ

by means of Fast Fourier Transformation (FFT) algorithms. In the following we determine the CSRh for two different cameras. Stable illumination conditions have to be used to avoid global intensity variations among the acquired images, which would cause nonzero backgrounds in the determined CSRh .

4. Results and discussion We first determine the CSRh matrices for two different cameras. Subsequently, we calculate the noise of empty holograms using the derived transfer formulas and compare the results to the experimentally measured noise. In the next subsection we investigate the noise in the presence of objects at medium and high spatial resolution.

4.1. The experimental determination of CSRh Series of almost 100 empty flat-field corrected images are acquired for two different cameras. Both are Gatan 1024  1024 CCD cameras of model MSC 794 equipped with different scintillators. One camera is mounted beneath a Philips CM200 TEM and the other under a FEI TECNAI F20. Both TEMs are operated at 200 kV. The covariance-to-signal ratios are shown in Fig. 2 serve as input for Eq. (30) estimating the covariance of the measured hologram. These one-dimensional profiles of the twodimensional functions show very similar shapes. The non-zero off-diagonal elements (jΔmj4 0) indicate the correlation between adjacent pixels separated by Δm. Note that the Poissonian shot noise would lead to zero for jΔmj 4 0 and one for Δm ¼ 0. The main difference between the CM200 and the F20 camera is a scaling factor. This factor may be explained by different gain values, which might be of a deterministic (i.e. digital) or stochastic (e.g. due to different scintillator materials) origin [43]. The latter contribution can be affirmed by considering that the MTFs for the two cameras (not shown here) are different. For ultimately clarifying the difference a gain measurement is required. We finally note that the matrices considered here do not depend on intensity under normal holographic imaging conditions [50]. Nevertheless, non-linearities in the detection process may introduce an intensity dependence over a larger range of current densities than considered here.

4.2. Noise determination for empty holograms For each of both mentioned cameras a series of almost 100 empty holograms were acquired and reconstructed. The mean holographic parameters for the two acquisitions are summarised in Table 1.

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F. Röder et al. / Ultramicroscopy 144 (2014) 32–42

Fig. 2. One-dimensional profiles of covariance-to-signal ratios (CSR h , Eq. (29)) for two different cameras at 200 kV: (a) Camera at Philips CM200 and (b) Camera at FEI TECNAI F20.

Table 1 Holographic parameters (mean centre band intensity I CB in ADUs, mean fringe contrast μ and fringe spacing s (pixel)) of a series of Ns empty holograms, acquired on two different cameras. Microscope

Ns

I CB

μ

s

Philips CM200 FEI TECNAI F20

96 87

281 1162

0.215 0.188

5.06 5.00

The following holograms were reconstructed with a Butterworth filter: BðqÞ ¼

1  2n q  1 þ   q0

ð37Þ

of order n ¼5 and with the cut-off frequency q0 ¼ jq0 j of integer fractions of the carrier frequency jqc j. The reconstructed sidebands are normalised by their subsequent series neighbours (5). In Fig. 3 we present the sample variances of amplitude and phase in dependence on the aperture size. Integer fraction of the carrier frequency was chosen as different mask radii. At large aperture sizes the variances are larger than at small aperture sizes in agreement with the predictions by the simple Lenz model (20). Smaller apertures sizes providing smaller variances, however, lead to broader correlations lengths and smaller spatial resolution compared to larger apertures (not shown here). The experimental values (red) are determined through pixel-wise statistical evaluation of the reconstructed and normalised amplitude and phase

Fig. 3. Sample variance (logarithmic scale) of amplitude (a) and phase (b) depending on the size of reconstruction aperture q0 as integer fraction of the fixed carrier frequency qc . Experimental results depicted in dashed red and calculated values in solid blue. Results for the F20 camera are indicated by squares and for CM200 camera by circles. With increasing ratio the diameter of the aperture becomes smaller, hence the noise is reduced. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this paper.)

values (Eqs. (31)–(33) and (13)). Their errorbars are determined by analysing the fourth statistical momentum μ4 (see Appendix A) of the sample. The variance s2s2 of the sample variance s2 is then determined by [46]

s2s2 ¼

μ4  s4 Gauss s2 ; 2Ns 4N s s2

ð38Þ

which simplifies in the case of Gaussian distributed noise. The calculation of μ4 for amplitude and phase from real and imaginary parts is outlined in Appendix A. Note that the variance of the variance is not equal to the error of the variance. Instead, the square root of the variance of variance s2s2 has to be interpreted as the error of the standard deviation. Consequently, the errors of the sample variance, i.e. the red errorbars in Fig. 3, are given by error propagation: pffiffiffiffiffiffiqffiffiffiffiffiffiffiffi

Δs2 ¼ 2 s2 s2s2 :

ð39Þ

For both measurements we obtain a relative error for the sample variances of about 16%. This is in agreement with the values obtained under the assumption of Gaussian distributed noise with the corresponding values for Ns in Table 1. Therefore we will use the last part of (38) for the error estimation of the errors in the case of object holograms in the next subsection.

F. Röder et al. / Ultramicroscopy 144 (2014) 32–42

For each of the Ns hologram pairs error propagation (Eq. (12), (13), and (30)) with the corresponding CSRh from Fig. 2 was applied leading to sample variances of normalised amplitude and phase images. These variances were pixel-wise determined and averaged over the same detection area of the experiment leading to the blue coloured values in Fig. 3. Their errorbars are equal to the pixel-wise standard deviation of the calculated variance series. The calculated variances for amplitude and phase (blue) for different cameras in dependence on different aperture radii are within the errorbars of experimental results (red). Thus, the presented noise propagation formulas correctly predict the variances for different reconstruction apertures. However, it is obvious that the experimental values are systematically larger than the calculated. This is presumably because of unavoidable disturbances of the biprism or illumination, leading to additional fluctuations of the fringe contrast. This influence increases the experimental noise (red). A further reason could be the limited validity of approximation (27) for high spatial frequencies present in the fringe pattern. This influence would affect the calculated noise (blue). Finally, the deviations are small compared to the errorbars indicating a sufficient predictive power of the presented transfer formulas. In extension to the above evaluation, we present in Fig. 4 experimentally determined elements of the covariance matrix for amplitude and phase (red squares for F20 and red circles for CM200 camera) using a Butterworth filter with a fixed radius of q0 ¼ qc =2. Furthermore, we compare these results to the theoretically values (blue) obtained for the given empty holograms using the respective CSRh of the considered cameras. The errorbars of the experimentally determined (red) covariances are approximated by the relative error of the sample variances of about 16% (see above) times the measured absolute covariance. The errors of the calculated variances are determined analogous to Fig. 3. Obviously, also the off-diagonal elements (Δm a0) of the covariance matrix of amplitude and phase can be computed from the derived noise propagation formulas. Although the phase covariances systematically show a slight underestimation (due to the same reasons as for the deviations in Fig. 3), the calculated values are within the errorbars of the experiment for two different cameras. The width of the covariances (correlation length) in Fig. 4 is broader compared to Fig. 2 due to additional correlations introduced by the reconstruction aperture (12). Although a Butterworth mask (Lenz model not valid) is chosen, amplitude and phase covariances are equal within the errorbars. These results verify the here derived noise transfer formulas for the holographic reconstruction process taking into account detector properties. 4.3. Noise determination for object holograms Finally, we apply the noise transfer formulas to two different example object holograms acquired with the two considered cameras. The calculated noise is compared to experimentally measured noise, which is obtained by statistical evaluation of hologram series. Here, we are interested in mean values of the series, hence on deviations of the mean values. Consequently, the noise, i.e. the standard deviation, of the mean amplitude as given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi sAn ¼ covðA; AÞ= Ns : ð40Þ and of the mean phase pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi sφn ¼ covðφ; φÞ= N s

ð41Þ pffiffiffiffiffiffi are considered. The additional division with Ns indicates the reduction of noise for increasing number of holograms per series (Ns). For estimating the error of the experimental mean value standard deviations, we use (38) under the assumption of Gaussian distributed noise, which was verified above for the empty holograms.

39

Fig. 4. Sample covariances for amplitude (a) and phase (b) reconstructed from of empty holograms with q0 ¼ qc =2 (dashed red) compared to values calculated using the noise propagation formulas (dotted blue). The results for the F20 camera are depicted as squares and for the CM200 camera as circles (Table 1). The abscissa shows the position of the off-diagonal elements with respect to the diagonal elements (index 0). The diagonal elements correspond to the left most values in Fig. 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this paper.)

4.3.1. Medium resolution The first comparison is conducted with medium resolution holograms of a PbZr0.2Ti0.8O3/La0.7Sr0.3MnO3 (PZT/LSMO) bilayer system epitaxially grown on an SrTiO3 substrate [51]. The holograms were acquired at the Philips CM200 in Lorentz-mode (objective lens switched off). For the object holograms we obtain 800 mean counts and 7.9% mean fringe contrast and for the empty holograms 974 mean counts and 14.4% mean fringe contrast. The corresponding detector noise properties are depicted in Fig. 2a. The elastic electron scattering on this bilayer system leads to different centre band, amplitude and phase modulations, which influence also the local noise of the amplitude and phase images (Fig. 5a and c). These images represent mean values of the reconstructed data averaged over 10 reconstructed waves according to Eq. (34). Here a Butterworth filter of order 5 is chosen with a cut-off radius of q0 ¼ 0:1 nm  1 . According to (38) we expect a relative error of the experimentally determined standard deviations of about 23%, i.e. larger deviations to the calculated noise will be attributed as systematic errors. Artefacts due to the drift of object and fringe pattern as well as biprism charging occurring during series acquisition are removed before averaging. The reconstructed waves are divided by corresponding empty waves reconstructed from an empty hologram series of same length (acquired subsequently) to remove artefacts reported in [48]. However, influences due to

40

F. Röder et al. / Ultramicroscopy 144 (2014) 32–42

Fig. 5. Comparison between experimental determined and theoretically expected noise at medium resolution example. The mean amplitude image of a PZT/LSMO/STO bilayer system is shown in (a). The amplitude noise represented in (b) is determined experimentally (black) and theoretically (grey) along the black arrow in (a). The mean phase image of a PZT/LSMO/STO bilayer system is shown in (c). The phase noise represented in (d) is determined experimentally (black) and theoretically (grey) along the black arrow in (c). The profiles show good agreement. Deviations at the object rim larger than 23% are due to uncorrectable systematic errors.

illumination drift and Fresnel fringe displacements could not be removed completely, hence will remain as residual systematic errors. Especially in the case of non-parallel illumination conditions like in Lorentz mode, an illumination drift induces illumination tilt, which consequently induces drifting aberrations. The experimentally and theoretically determined noise distributions (errorbars) along the line profiles indicated by the black arrows in Fig. 5 are depicted in the profiles (b) and (d). As mentioned above, in the vacuum region the noise of normalised amplitude and phase coincides. In the object the phase noise increases whereas the amplitude noise decreases, which is qualitatively expected for decreasing centre band intensity according to the Lenz formulas (20). The calculated noise is for the most areas within the errors of the experimentally determined noise (23%). However, the edges exhibit much higher deviations caused by residual uncorrectable systematic errors mentioned above. Especially the transition area between vacuum and PZT (filled with glue) may additionally alter due to radiation damage during series acquisition inducing further systematic errors related to the object. This comparison yields quantitative verification of the transfer formulas in the case of object holograms at medium resolution. Furthermore, we are interested in the validity of the transfer formulas for strongly varying wave modulations.

4.3.2. High resolution The second example is a high-resolution electron hologram. Here, we investigate a BaTiO3 (BTO) thin film epitaxially grown on SrTiO3 (STO). The series acquisition of 10 holograms is performed as described above at the TECNAI F20. For the object holograms we obtain 3506 mean counts and 10.3% mean fringe contrast and for the empty holograms 3604 mean counts and 14.0% mean fringe contrast. Here, we expect again a relative error of the experimentally determined standard deviations of about 23%. For the reconstruction we again use a Butterworth filter of order 5 with a radius yielding a spatial resolution of 0.2 nm in the reconstructed wave. In Fig. 6, we compare the measured and calculated noise along two different lines indicated by arrows in the images of the holographic mean values (centre band (a), amplitude (b), and phase (c)). For calculating the noise by the derived transfer formulas we use the corresponding CSRh (Fig. 2b). The two line profiles are different in the arrangement of local amplitude and phase extrema due to different scattering properties of the different object positions. For position 1 located in BTO the phase maxima coincide with the amplitude minima leading to maximal phase noise at the phase maxima (dashed lines in (b) and (h)). For position 2 located in STO the phase maxima coincide with the amplitude maxima (c) leading to minimal phase noise at the phase maxima (dashed lines in (c) and (i)). These qualitative findings as predicted by the Lenz formulas (20) are reproduced by both experimentally and theoretically determined noise.

F. Röder et al. / Ultramicroscopy 144 (2014) 32–42

41

Fig. 6. Comparison of the experimentally measured noise and the predicted noise using the noise transfer formulas for a high resolution example consisting of BaTiO3 grown on SrTiO3. The according holograms are acquired at FEI TECNAI F20 with a CSRh of the used camera depicted in Fig. 2b. Mean values from holographic reconstruction are depicted in (a), (d) and (g). Noise analysis is conducted at two different line profiles indicated by the arrows. The data of profile 1 (BaTiO3) is shown in (b) and the data of profile 2 (SrTiO3) in (c). (e) and (f) show the standard deviations of the mean amplitudes (d). (h) and (i) show the standard deviations of the mean phases (g). The dashed vertical lines in (b, e, h) denote the position of a phase maximum (dashed–dotted blue) and amplitude minimum (dotted red). Here the phase noise has a maximum. The dashed vertical lines in (c, f, i) denote maxima in amplitude and phase. The phase noise here is minimal. Deviations larger than 23% are due to systematic errors. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this paper.)

The amplitude noise does not show strong modulations, because the amplitude noise is mainly determined by the centre band intensity (grey curves in Fig. 6b and c) as predicted by Lenz (20). In contrast the phase noise exhibits strong variation because of the dependency on both centre band and sideband amplitudes (20). However, the theoretical noise slightly underestimates the experimental noise of the mean amplitudes (e) and (f) and of the mean phases (h) and (i). Any changes of the hologram pattern, due to either instabilities of the microscope (e.g. drift of aberrations during series acquisition) or dynamic processes of the crystal lattice in the presence of the electron beam, will manifest in additional terms in the hologram's covariance matrix (24) [43] and consequently will also appear as additive terms in the covariances of

amplitude and phase. This shows that the presented noise transfer formulas provide at least lower bounds for the noise in the reconstructed images and reproduces qualitative features predicted by Lenz quantitatively. In Fig. 6 (h) and (i) the noise propagation yields differences up to a factor of 2 between phase noise maxima and minima, the experimental noise even up to a factor between 4 and 5.

5. Conclusions In this work a reliable method for the determination of the noise in amplitude and phase from the hologram intensity is presented.

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F. Röder et al. / Ultramicroscopy 144 (2014) 32–42

We derived noise transfer formulas quantitatively describing the noise and correlations in amplitude and phase images reconstructed from off-axis electron holograms using arbitrary reconstruction apertures and incorporating realistic detector noise. The latter is based on a suitable adaption of our recently published general noise transfer theory for image detectors. This general description includes the well known special case proposed by Lenz. In particular, we use the covariance-to-signal ratio CSRh measured for homogeneously illuminated cameras to approximate the covariance of the recorded holograms. We verified the transfer formalism within the errors of the experimental noise determination. We used hologram series of both vacuum and object regions at medium and high resolution recorded at two different detectors. Only in the case of object holograms the experimentally measured errors overestimate significantly the calculated ones, which is therefore ascribed to additional object changes due to drift and radiation damage influencing the experimental noise determination. We show that these systematic errors especially due to radiation damage can be indeed large and even exceed the shot/camera noise. A quantification of radiation damage by statistical evaluation of time series could be possible in future, if the noise transfer through the camera and reconstruction process is properly taken into account. From these results we furthermore expect a much stronger phase noise dependency on object thickness and defocus than predicted in [47]. Thus, noise optimizing aberrations for taking holograms remains still an open topic and should be addressed in future. Acknowledgments We acknowledge stimulating and fruitful discussions with Prof. Dr. H. Lichte. We gratefully thank Prof. Dr. D. Hesse and Dr. I. Vrejoiu from Max Planck Institute for Microstructure Physics in Halle (Germany) for providing the PZT/LSMO/STO sample and Dr. R. Dittmann and Dr. D. Park from Forschungszentrum Jülich (Germany) for providing the BTO/STO sample. The research leading to these results has received funding from the European Union Seventh Framework Programme under Grant Agreement 312483— ESTEEM2 (Integrated Infrastructure Initiative—I3). T.N. acknowledges support from the DFG within the SFB 787. Appendix A. Computation of the fourth statistical momenta of amplitude and phase The derivation of the propagation of the fourth momenta is in accordance with the propagation of the (co-)variances (second momenta). For a given series of Ns amplitude (A) and phase (φ) images, the fourth statistical moments of the samples are defined by (see, e.g. [46])

μ4 ðφÞ 

 4 Ns 1 ∂φ ∂φ ∑ ðX i  X Þ þ ðY i  Y Þ N s  1 i ¼ 1 ∂X ∂Y

The fourth momenta are important for estimating the variance of the sample variance according to Eq. (38).

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μ4 ðAÞ ¼

Ns 1 ∑ ðA  AÞ4 Ns  1 i ¼ 1 i

ðA:1Þ

μ4 ðφÞ ¼

Ns 1 ∑ ðφi  φ Þ4 Ns  1 i ¼ 1

ðA:2Þ

[43] [44]

The index i enumerates the corresponding images in the series and the bared variables denote the mean values of the series. As amplitude and phase are functions of real (X) and imaginary parts (Y) (Eqs. (9) and (10)), we can approximate for small deviations of the samples from the mean values the differences by the total differential:  4 Ns 1 ∂A ∂A ðX i  X Þ þ ðY i  Y Þ ∑ μ4 ðAÞ  N s  1 i ¼ 1 ∂X ∂Y

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