Symmetry-guided nonrigid registration: the case for distortion correction in multidimensional photoemission spectroscopy
SSymmetry-guided nonrigid registration:the case for distortion correction inmultidimensional photoemission spectroscopy
R. Patrick Xian , Laurenz Rettig , and Ralph Ernstorfer Department of Physical Chemistry, Fritz Haber Institute of the Max Planck Society,Faradayweg 4-6, 14195 Berlin, Germany
Abstract
Image symmetrization is an effective strategy to correct symmetry distortion in experimental data for whichsymmetry is essential in the subsequent analysis. In the process, a coordinate transform, the symmetrizationtransform, is required to undo the distortion. The transform may be determined by image registration (i.e.alignment) with symmetry constraints imposed in the registration target and in the iterative parameter tuning,which we call symmetry-guided registration. An example use case of image symmetrization is found in electronicband structure mapping by multidimensional photoemission spectroscopy, which employs a 3D time-of-flightdetector to measure electrons sorted into the momentum ( k x , k y ) and energy ( E ) coordinates. In reality, imperfectinstrument design, sample geometry and experimental settings cause distortion of the photoelectron trajectoriesand, therefore, the symmetry in the measured band structure, which hinders the full understanding and use ofthe volumetric datasets. We demonstrate that symmetry-guided registration can correct the symmetry distortionin the momentum-resolved photoemission patterns. Using proposed symmetry metrics, we show quantitativelythat the iterative approach to symmetrization outperforms its non-iterative counterpart in the restored symmetryof the outcome while preserving the average shape of the photoemission pattern. Our approach is generalizableto distortion corrections in different types of symmetries and should also find applications in other experimentalmethods that produce images with similar features. The use of symmetrization appears in various scientific contexts to assist data analysis. Generally speaking,symmetrizing an object requires a coordinate transform that enhances its symmetry with minimal alteration of theshape (e.g. curvature, area or volume) [1]. In solid state physics, the symmetry of the electronic band structure(EBS) of materials is often revealed in photoemission measurements with resolution of the photoelectron momenta[2, 3]. Thanks to recent technological advances, characterization of the EBS in momentum k x , k y , and energy1 a r X i v : . [ phy s i c s . d a t a - a n ] A p r coordinates can now be achieved in en bloc measurements using time-of-flight electron detectors, a techniquenamed multidimensional photoemission spectroscopy (MPES) [4, 5]. The processing of MPES data often utilizestheir symmetry for signal averaging, and for extracting and comparing the underlying physical quantities, such asthe transition matrix elements [3, 6] and the electron self-energy [3], or to reconstruct the electronic orbitals ofsolid state materials [7, 8]. However, systematic deviations in experimental data from the symmetry expected forthe EBS, called symmetry distortion (see Fig. 1a-b), are often present due to various sources of imperfectionsin the instrument design and experimental setup, such as mechanical inaccuracies, incompletely shielded straymagnetic fields, sample geometry and surface quality, experimental conditions, including the lens alignment andvoltage settings, the photon beam shape, etc. While some of the hardware imperfections may be calibrated usingsamples with a well-known band structure, the complication of the numerous factors varying by the sample andby the experiment renders a complete calibration of symmetry distortion impossible. Nevertheless, since thesenonidealities result in perturbations to the trajectories of the photoemitted low-energy electrons, and subsequently,the deviation from perfect symmetry in the detected photoemission pattern, a post-processing algorithm to correct(see Fig. 1a) and quantify imperfect symmetries using coordinate transforms [9, 10] and image features is of greatpractical use. In established angle-resolved photoemission experiments measuring the 2D k − E dependence usinghemispherical analyzers, rescaling of the image coordinates may be used to minimize the symmetry distortion.For the distortion correction of 3D EBS mapping data measured by time-of-flight detectors or the deflector modeof the hemispherical analyzers, a symmetrization transform needs to be applied to at least the two momentumdimensions, k x and k y , simultaneously. We describe an algorithm that achieves this.Several symmetrization methods have been previously developed in different scientific contexts. For rotationallysymmetric scenarios, a circularization procedure using directional rescaling [11] was developed for cryoelectrontomography. Alternatively, the basis expansion approach such as decomposition into Fourier series [12] or sphericalharmonics [13] can also be used to symmetrize image patterns. In the studies on translation-symmetric systems,the strain field mapping technique used in high-resolution transmission electron microscopy [14] to characterizethe local atomic misarrangements, and the stage drift correction algorithms [15, 16] may also be regarded assolving symmetrization problems. However, these approaches have limited applicability to MPES data due tothe differences in their image features. Typical MPES data possess discrete rotational symmetry with anisotropicintensity modulations resulting from the matrix element effect [3, 6], which should be considered in the algorithmdesign and adaptation for the purpose of image symmetrization.Despite the complex and often unknown origins, the distortion in the photoemission pattern may be classifiedby visual inspection (see examples in Fig. 1a-b). We recognize the major types of distortions as (1) biaxial shearingand (2) nonuniform scaling, along with (3) higher-order nonlinear distortion (including perspective distortion andoff-centeredness), which contribute in less amounts. Undistorting the image generally requires changing thedistances between points by applying a nonrigid coordinate transform, which may be determined by matching thecorresponding features in the distorted image (reference) and the undistorted one (target), similar to the techniquecalled nonrigid image registration (i.e. alignment) [10, 17]. However, in image symmetrization, the target imagedoesn’t exist, so the problem is constrained only on the reference image side, whereas image registration isconstrained on both the reference and the target sides. Previously, tuning of a single shearing parameter usingpoint feature correspondences has been shown to improve the symmetry of the photoemission pattern [5], but itssimplicity doesn’t always guarantee sufficient symmetrization, hampering subsequent detailed data analysis. Ourapproach, the symmetry-guided nonrigid registration, rests on the observations that the outcome of symmetrization2
1) (2) (3)(4) (2a)(2b)(2d)(2c)
Distortedimage
Correctedimage
Featurecoordinates
Coordinatemapping c d distorted undistortedlandmark localization e center-vertexsymmetry error vertex-vertexsymmetry erroroff-centeredness +- b biaxial shearing nonuniform scalinghigher-ordernonlinear distortionundistorted image rigid translationrigid rotation a P P P P P P C P P ~ P ~ P ~ P ~ P ~ P ~ P C ~ L (x, y) k x k y Figure 1: (a) A cartoon example of distorted (left) and corrected (right) hexagonal pattern measured in photoe-mission spectroscopy. The dashed lines are for guiding the eye. (b) Comparison of undistorted image with thoseafter rigid and nonrigid distortions. (c) Schematic of the proposed procedure for symmetry distortion correction.(1) Extraction of coordinates ˜ P of the symmetry-related features from the distorted image ˜ I . The coordinates ˜ P are used as inputs to the optimization loop in (2), which comprises of four parts: (2a) generation of targetlandmarks; (2b)-(2c) the landmark registration process; (2d) update of the generator G under the symmetry con-straints f . More details are given in the text. The loop in (2) yields (3), the optimized pixel coordinate transform L opt , which is then (4) applied to all pixels to correct the image distortion. Illustrations of (d) the origin of thelandmark localization errors and (e) the geometrical representations of the error terms (Eq. 4-6) as red arrows inthe optimization cost function. The distorted (blue) and undistorted (black) patterns share the center. The plusand minus signs next to the red arrows indicate whether the length should be added or subtracted, respectively.3sing image registration is affected by (1) the degrees of freedom in the coordinate transform of choice and (2)the quality of the feature correspondences such as the uncertainties in their exact positions, also known as thelocalization errors (LE, see Fig. 1d) [18, 19]. To improve the former aspect, we resort to nonrigid transforms[9, 10, 17] that account for more types of distortions (see Fig. 1b). For the latter, we use an iterative frameworkto explore more spatial configurations of the feature correspondences to compensate for the LE and optimize formore symmetry in the outcome.In the following, we first present the definition and characterization of the iterative symmetrization framework,followed by the mathematical description of its components in an example case of a polygon-type pattern withdiscrete rotational symmetry. The description of alternative cases are presented in the Supplementary Materialsection S1. We then compare the iterative and the non-iterative approaches using experimental MPES data andquantify their performances using our proposed symmetry metrics independent of the symmetrization algorithm.In addition, we provide an open-source software package, symmetrize [20], written in Python that implements theprocedures described in this work for public use. Symmetrization of MPES data makes use of the perceived symmetry of the photoemission pattern in the k x − k y plane (see Fig. 1a). The pattern’s geometric shape relates to the symmetry of the bulk or surface Brillouin zoneof the material under study, and is generally more robust than the intensity symmetry against modulations by thematrix element effect in the photoemission process [21] and the experimental configuration, which, in combination,create complex anisotropic modifications to the measured intensities in the momentum space [6]. The geometricpattern is defined by point landmarks. Those that are relevant for the perceived symmetry may be used for imageregistration based on landmarks (i.e. landmark registration) [17]. Typically, landmark registration employs a largeamount of point correspondences between the reference and the target images obtained by a feature selector.The correspondences are putative and often have varied accuracies in their locations and relations such that arobust fitting algorithm is required to determine the transform parameters [22]. For symmetrization, with onlyconstraints on the reference side, it’s possible to produce point correspondences using symmetry relations anda target (feature) generator with high consistency (i.e. genuine instead of putative correspondence). However,the position inaccuracies in the landmarks selection, also known as the landmark localization errors (LLE, seeFig. 1d) [18, 19], affects the symmetrization more severely with a small amount of correspondences, which ischaracteristic of MPES data symmetrization. Since LLE only exists in the reference for image symmetrization,we can compensate for it by relaxing the symmetry in the generated target landmarks. In practice, we iterativelyoptimize the parameters of the target generator to tune the amount of symmetry relaxation in order to improvethe symmetrization outcome. The transform determined with the iterative approach improves on the result fromusing a fixed set of symmetric target landmarks. On the other hand, the optimization allows the use of constraintsdesigned to simultaneously minimize the size change of the pattern, which is important for maintaining the imageresolution (or electron momentum resolution in the case of MPES data). Moreover, the iterative approach isapplicable to various types of coordinate transforms. We formulate the iterative symmetrization procedure asfollows, Given a set of symmetry-related point landmarks ˜ P sampled from the distorted reference image ˜ I , find atarget point set P = G ( α ; ˜ P ) , related to ˜ P by the generator function G with parameters α . ˜ P and P together etermines a parametrized transform L , such that the transformed point set, P = L ˜ P , satisfies the constraints f . The iterative symmetrization is at the center of the overall distortion correction procedure, shown schematicallyin Fig. 1c. In the present work, we demonstrate the effectiveness of the iterative approach using two types oftransform functions as L . The first is the eight-parameter 2D projective transform (also known as a perspectivetransform or homography [22, 23]), which corrects most of the biaxial shearing and nonuniform scaling. For amore complete distortion correction, we use the thin plate spline (TPS) [24, 25] transform. In addition to these,other similar types of parametric transforms are also feasible [9, 10]. The constraints f are formulated usinggeometric relations as a cost function for optimization. The point features used here may be generalized to othersalient symmetry-equivalent image features such as lines, curves, etc.The procedure in Fig. 1c starts with (1) a point set extracted using a landmark detection algorithm (e.g.2D peak detection) or manual selection, it is ordered clockwise or counterclockwise using the opening anglebetween a fixed axis unit vector (e.g. along the image axis x or y ) and the vector connecting each point andthe center or centroid. The ordered point set represents the vertices of a polygon, and we adopt the termvertex and center from now on. For a pattern without a clearly defined center, the centroid position of thevertices may be used. We denote the ordered reference and target point sets as ˜ P and P , respectively, asshown in the example in Fig. 1a. The intermediate point set P relates to P exactly through the transform L . Note that P = { P i } , ˜ P = { ˜ P i } , P = { P i } , excluding their respective centers ˜ P C , P C and P C . All pointpositions are expressed in the homogeneous coordinate system [22] for convenience of calculation, therefore, ˜ P , P , P ∈ R , with P i = ( x i , y i , T , ˜ P i = (˜ x i , ˜ y i , T , P i = ( x i , y i , T ( i = 1 , , ... ) . The first two dimensionsin the homogeneous coordinates are the 2D Cartesian coordinates. For distortion correction using the projectivetransform, these point sets are related to one another in the optimization loop (see Fig. 1c) in the following ways. ( ) P i = G i ( α i ; ˜ P i ) = s i R ( θ i ) ˜ P i (1) ( ) L = DLT ( ˜
P , P ) (2) ( ) P i = L ˜ P i (3)In Eq. 1, P i and ˜ P i are related by a scaling factor s i and a 2D rotation matrix R ( θ i ) . The collection of scalingand rotations constitutes the generator G in Fig. 1c, with s i , θ i being the set of parameters α . The transform L (in case of projective transform) is determined by the established method of direct linear transformation (DLT)[23], and the single-pass solution P i is the current best approximate to P i , as expressed in Eq. 2. The labels(2a)-(2c) before the equations refer to the corresponding steps in Fig. 1c. The subsequent step (2d) refers toupdating the parameters { s i , θ i } in the generator G under the constraints or cost function f . The steps (2a)-(2d)lead to the sequential update ( G → P → L → P → G ) of the quantities on the four corners in the optimizationloop in Fig. 1c at every iteration.The cost function f in Fig. 1c is formulated using the sum of squared errors between the actual positionsregistered to ( P ) and the ideal positions represented by the average distances as expected for a symmetricpattern, which will be discussed next. For a pattern with discrete even-order rotational symmetry (e.g. hexagon),it contains three error terms. The off-centeredness f centeredness is calculated by the distance between the actualcenter ( ˜ P C , if present), usually the Γ point in the band structure mapping data, and the center determined by thevertices. The symmetry error comprises of two parts, one for the center-vertex symmetry f cvsym , calculated usingall vertices, and the other for the vertex-vertex symmetry f vvsym , calculated using the nearest-neighboring vertices.5he mathematical expressions of the geometric errors are given in the following for the case of a regular (i.e.equiangular and equilateral) polygon pattern with an even number of vertices (even-order rotational symmetry), f centeredness = 2 n n/ X i =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P i + P i + n − ˜ P C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (4) f cvsym = 1 n n X i =1 (cid:12)(cid:12)(cid:12) k P i − P C k − k ˜ P i − ˜ P C k (cid:12)(cid:12)(cid:12) (5) f vvsym = 1 n X NN (cid:12)(cid:12)(cid:12) k P i − P j k − k ˜ P i − ˜ P j k NN (cid:12)(cid:12)(cid:12) (6)In Eq. 4-6, n denotes the order of the rotation symmetry, e.g. n = 6 for a hexagonal pattern. We use k·k to denotethe Euclidean norm and | · | the absolute value of a scalar. The term k ˜ P i − ˜ P C k represents the average center-vertex distance, and k ˜ P i − ˜ P j k NN the average nearest-neighbor (NN) vertex-vertex distance, both calculated inthe coordinate system of the distorted image. For the hexagonal pattern, the average NN vertex-vertex distanceis calculated from an average of k ˜ P − ˜ P k , k ˜ P − ˜ P k , k ˜ P − ˜ P k , k ˜ P − ˜ P k , k ˜ P − ˜ P k and k ˜ P − ˜ P k . Theformulation using average distances penalizes simultaneously the deviations of both the symmetry and the sizechange of the pattern. Extensions of the geometric errors based on Eq. 4-6 to other scenarios are discussed inthe Supplementary Material (see Section S1). The overall cost function f is a numerical combination of all threeterms described above, f = f ( P , ˜ P ) = f centeredness + f cvsym + f vvsym (7)As the amount of distortion can vary with experimental conditions, one may introduce a weight to each contributorof the multi-objective cost function in Eq. 7 in order to balance the contributors and steer the optimization process.The weighted cost function is, therefore, f w = w c f centeredness + w cv f cvsym + w vv f vvsym (8)The weights may be estimated by the amount of distortion in a particular term. To solve for the optimal targetpoint set P opt that compensates for the LLE in the reference image, we minimize f (or f w ) with respect to theparameters { s i , θ i } in the generator G , { s opt i , θ opt i } = argmin { s i ,θ i } f ( P , ˜ P ) (9)Then, P opt can be determined using { s opt i , θ opt i } and Eq. 1. It’s not a strictly rotation-symmetric point set (seered crosses in Fig. 2b, 2d) and the deviation accounts for the LLE from the selection of ˜ P . The steps (3) and(4) in Fig. 1c correspond to determining the transform L opt using Eq. 2 and applying it to all pixels in theimage, respectively. Likewise, the transform may also be applied to the single photoelectron event data directlyrecorded by the time-of-flight 3D detector. Extension of the iterative symmetrization approach to more complexfeatures only requires to update the geometric constraints in Eq. 4-6 using feature-specific metrics [17, 26]. Inthe Supplementary Material, we have provided a walk-through (see section S2) of the steps shown in Fig. 1cusing the relevant functions of the symmetrize software package.To quantify the level of symmetry within the image using point positions, we propose to use metrics differentfrom the error terms in Eq. 4-6. In this way, the metrics will also allow for comparison with future iterations of6imilar algorithms. In this context, the continuous symmetry measure (CSM) [27] adapted to rotationally sym-metric objects [28, 29] serves the purpose. The measure is invariant under similarity transformation (translation,rotation and uniform scaling) [22] and bounded within the range [0 , , with representing complete symmetry, for complete asymmetry, and the values in between for various degrees of distorted symmetry. In addition,we introduce the area retainment measure (ARM), tanh (cid:12)(cid:12)(cid:12) − AA (cid:12)(cid:12)(cid:12) , to quantify the area change before and afterthe distortion correction. A and A represent the areas of the polygonal region of interest in the uncorrectedand corrected images, respectively. The hyperbolic tangent function is used to squash an unbounded function in [0 , + ∞ ) down to a bounded function in [0 , . Similar to CSM, the closer the ARM is to zero, the less the areachange (or the higher the area retainment), and the closer it is to one, the more area change. We define thesymmetry recovery score (SRS) S in the outcome as the mean of the CSM and the ARM, S = 12 (cid:18) CSM + tanh (cid:12)(cid:12)(cid:12)(cid:12) − AA (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (10)The SRS reflects the balance between the level of symmetry and the amount of shape perturbation in thesymmetrization process [1]. Further details on the computation of the proposed symmetry scores are given in theSupplementary Material (see section S3). The band structure mapping measurement was carried out using a commercial electron momentum microscope(METIS 1000, SPECS GmbH) and a home-built table-top high harmonic generation-based pulsed extreme UVphoton source at 21.7 eV [30]. Single crystalline tungsten diselenide (WSe ) samples, which possess a sixfoldrotational symmetry in its Brillouin zone, were used for the meausurements. The crystals were purchased fromHQ Graphene and downsized and glued to a sample holder using conductive epoxy before measurements. Upontransferring to the measurement location, the sample was cleaved in vacuum using a cleaving pin. The photo-electron events recorded by the momentum microscope were converted to volumetric data via multidimensionalhistogramming in post-processing [31]. Energy slices from the uncorrected data are shown in Fig. 2a-d andFig. 3a-c. The horizontal and the vertical axes shown in each image are the photoelectron momenta k x and k y ,respectively. Methods CSM ARM SRSUncorrected . × − . × − Non-iterative projective . × − . × − . × − Iterative projective . × − . × − . × − Thin plate spline . × − . × − . × − Iterative TPS . × − . × − . × − Table 1: Symmetry scores before and after distortioncorrection of the image with hexagonal pattern.Distortion correction was first carried out using an energy slice close to the valence band maximum of theWSe band structure mapping data, because its sharp features, the K , K and Γ points [32, 33], allow relatively7 ncorrected Non-iterativeprojective Iterativeprojective Thin platespline IterativeTPS10 S y mm e t r y sc o r e s ( a . u . ) Measures of outcome
CSMARMSRS
Non-iterative projective Iterative projective a b
Thin plate spline ce gi fj k m
Iterative TPS dhl
Figure 2: Comparison of the non-iterative and the iterative approaches of symmetrization using an energy slicefrom the WSe band structure mapping measurement. (a)-(d) show the reference (black cross) and target (redcross) points in the uncorrected image used for each method shown in the title. (e)-(h) show the deformed grid (ordeformation field) obtained from the corresponding symmetrization transform. (i)-(l) show the corrected imagesafter applying the transform to all pixels in each case. An additional rotation is applied afterwards to align asymmetry axis with an image axis for presentation. The dashed blue and red circles in (c) and (i)-(l), respectively,guide the eye for symmetry evaluation. (m) The continuous symmetry measure (CSM), the area retainmentmeasure (ARM) and the symmetry recovery measure (SRS) computed for the uncorrected and corrected imagesfor quantitative comparison (see values in Table 1). 8 b cd e fg h i It e r a t i v e p r o j e c t i v e It e r a t i v e t h i n p l a t e s p li ne U n c o rr e c t ed H i ghLo w I n t en s i t y Figure 3: Application of the symmetrization transforms obtained from the iterative approaches shown in Fig. 2 toimage slices at other photoelectron energies in the lower valence bands of WSe from the same measured dataset.(a)-(c) show the uncorrected images, (d)-(f) show the corrected images using the projective transform and (g)-(i)using the thin plate spline, both parametrized from iterative estimation. The circular masks are used in (d)-(i) toremove the extraneous intensity features near the aperture edge.precise landmark localization. The outcomes using both the non-iterative and the iterative methods are shownin Fig. 2a-l, including the energy slice before and after the correction, the landmarks and the deformation fieldretrieved from the nonrigid registration.The symmetry scores calculated for the outcomes are presented in Table 1 and visualized in Fig. 2m. In bothdistortion correction using projective and TPS transforms, the iterative approach shows quantitative improve-ments from its non-iterative counterpart by all metrics, which demonstrates the effectiveness of the constraintsin enhancing the level of rotational symmetry while preserving the overall shape of the region defined by thelandmarks. The center point was not used in the parameter estimation for the projective transform in orderto avoid overconstraining the optimization, because of its relatively fewer number of parameters. The order ofmagnitude differences between the outcomes using the projective and the TPS transforms is due to the differencein the degrees of freedom these transforms possess.Next, we applied the same symmetrization transform for the hexagon pattern to other energy slices withdistorted symmetry from the same volumetric measurement (see examples in Fig. 3). Even though some of theimages have less obvious symmetry-equivalent landmarks, the transformed images show well-restored symmetry. Inother measurements without strictly point-like symmetry equivalents, one may use manually selected approximatelandmarks as inputs to the algorithm to solve for the symmetrization transform. The procedures described in theMethods section were implemented in the Python package symmetrize [20] and tested on a number of experimental9atasets. For specific alternative use cases, the package may be extended by adding user-defined cost functions. We have formulated image symmetrization in terms of nonrigid image registration guided by symmetry constraintsand have proposed an iterative framework to compensate for the localization errors of point landmarks in thereference image. The framework incorporates the knowledge of the pattern’s symmetry to construct the imageregistration target and to formulate the geometric constraints for iterative optimization. Using distorted electronicband structure mapping data as a test case, we compared the iterative and non-iterative approaches for estimatingthe symmetrization transform for a two-dimensional image with distorted discrete rotation-symmetric elements. Bythe use of scalar metrics to quantify the symmetry and the area change, we show that the symmetry constraints inthe iterative approach allow further reduction of symmetry distortion while better preserving the area of the pattern,thereby outperforming its non-iterative counterpart. In addition to multidimensional photoemission spectroscopy,our approach should find convenient adoption and adaptation in other experimental methods in which distortioncorrection is necessary to recover the symmetry in the measured data to improve their analyses. The open sourcecodebase enables further development and reuse by a broader community. Finally, the symmetrization procedureis an important step for the development of a data processing pipeline to extract symmetry-related quantities andphysical insights from multidimensional photoemission measurements.
Acknowledgements
This project has received funding from the Max Planck Society and from the European Research Council (ERC)under the European Union’s Horizon 2020 research and innovation program (Grant Agreements No. ERC-2015-CoG-682843). We thank S. Dong, Y. W. Windsor, M. Dendzik, M. Puppin and C. W. Nicholson for contributionsin the instrument construction and S. Dong for performing the measurement. We thank M. Wolf for his support,P. Hofmann for helpful discussion, S. Schülke and G. Schnapka for providing the computing infrastructure. L.Rettig acknowledges funding from the DFG in the Emmy Noether program under grant number RE 3977/1.
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