Scalable Bayesian Optimization of Composite Functions for Image-Based Inverse Problems in Materials Characterization

# Scalable Bayesian Optimization of Composite Functions for Image-Based Inverse Problems in Materials Characterization
Source: [https://arxiv.org/html/2609.02126](https://arxiv.org/html/2609.02126)
Dasol Yoon1,2Poompol Buathong311footnotemark:1Chia\-Hao Lee1Yujia Zhang3 David A\. Muller1,422footnotemark:2Peter I\. Frazier522footnotemark:2 1School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA2Department of Materials Science and Engineering, Cornell University, Ithaca, NY 14853, USA3Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA4Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, NY 14853, USA5School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USAdy327@cornell\.edupb482@cornell\.educhia\-hao\.lee@cornell\.eduyz685@cornell\.edudavid\.a\.muller@cornell\.edupf98@cornell\.edu††thanks:These authors contributed equally to this work\.††thanks:Corresponding authors:dy327@cornell\.edu,david\.a\.muller@cornell\.edu,pf98@cornell\.edu

###### Abstract

Estimating physical parameters from scientific images is a common inverse problem in materials characterization that often relies on expensive physics\-based simulations\. In electron microscopy, specimen thickness and crystal mistilt are critical parameters that govern how electrons scatter through the sample, and therefore the accuracy of any atomic\-scale structure recovered from it\. They are commonly inferred by matching experimental position\-averaged convergent\-beam electron diffraction \(PACBED\) patterns to simulated ones, but grid searches scale poorly and neural\-network methods require extensive pretraining that may not transfer to new conditions\. Here, we propose scalable Bayesian optimization of composite functions \(SBOCF\), a simulation\-efficient method that exploits the known composite structure of the image\-matching objective and the intermediate information contained in simulated images\. By representing PACBED images with patch\-level summaries and two correction terms, SBOCF preserves the original pixel\-wise objective while reducing the number of modeled outputs from24,64924\{,\}649to1111\. Under a budget of 50 simulator evaluations, SBOCF outperformed standard Bayesian optimization with expected improvement on synthetic SrTiO3benchmarks with thick and thin specimens, reducing the median final SSE by up to290×290\\timesin the thick\-sample case\. On experimental data, SBOCF produced parameter estimates consistent with previously reported values without task\-specific pretraining\. For a simulated mistilted specimen, using the SBOCF estimates in a downstream ptychographic reconstruction recovered sharp atoms that were otherwise blurred\. These results establish SBOCF as a promising approach for inverse problems involving expensive simulators and high\-dimensional structured outputs\.

## 1Introduction

Estimating unknown parameters by matching experimental measurements to physics\-based simulations is a common inverse problem across scientific fields, where the cost of each simulation can limit characterization throughput\. Electron microscopy is a representative instance: a beam of high\-energy electrons is transmitted through a thin specimen, scattering off the atoms inside, and a detector records the electrons that emerge\. Because the scattering depends on how the atoms are arranged, this signal encodes the atomic structure, but recovering the atomic positions requires a computational reconstruction\. Such structural analysis depends on two quantities rarely known beforehand: the specimen*thickness*, which sets how many times electrons scatter; and the crystal*mistilt*, the angular deviation between the beam and the crystallographic zone axis, the direction along which the atoms line up into columns\. Exact zone\-axis alignment is difficult in practice, so a residual mistilt of a few milliradians is typical, and errors in either quantity propagate into the inferred atomic positions\.

Both parameters can be estimated from the same measurement used for structural analysis\. In four\-dimensional scanning transmission electron microscopy \(4D\-STEM\), a focused electron probe is scanned across the specimen and a two\-dimensional diffraction pattern is recorded at each position, giving a four\-dimensional dataset\. Averaging these patterns over probe positions spanning at least one unit cell yields the position\-averaged convergent\-beam electron diffraction \(PACBED\) pattern, a single image whose fringe structure and symmetry depend sensitively on thickness and tilt\[[1](https://arxiv.org/html/2609.02126#bib.bib1)\]\. The mapping from a PACBED pattern back to the parameters that produced it is not available in closed form, but the forward direction is: for candidate values, a multislice simulation\[[2](https://arxiv.org/html/2609.02126#bib.bib2)\]propagates the electron wave through the specimen slice by slice and predicts the resulting pattern\. Estimating thickness and mistilt is therefore an inverse problem solved by forward simulation: we seek the parameters whose simulated pattern best matches the measured one, and each simulation is expensive\. These estimates are valuable on their own and also benefit downstream methods: multislice electron ptychography \(MEP\)\[[3](https://arxiv.org/html/2609.02126#bib.bib3)\]recovers an atomic\-resolution, depth\-resolved image from the same 4D\-STEM dataset, but degrades when thickness and mistilt are inaccurate\[[4](https://arxiv.org/html/2609.02126#bib.bib4)\]; estimating them beforehand removes them from the reconstruction’s variables\.

Existing PACBED\-based approaches face a trade\-off between cost and generality\. Conventional methods perform a grid search, exhaustively simulating patterns over a dense grid of thickness and tilt values and comparing them via least\-squares metrics\[[5](https://arxiv.org/html/2609.02126#bib.bib5)\]\. This scales poorly as the parameter space grows\. More recent methods train convolutional neural networks to estimate parameters rapidly\[[6](https://arxiv.org/html/2609.02126#bib.bib6),[7](https://arxiv.org/html/2609.02126#bib.bib7),[8](https://arxiv.org/html/2609.02126#bib.bib8)\], but they are tied to specific material systems and imaging conditions, requiring large simulated libraries and retraining for new settings\.

To address this challenge, we introduce*scalable Bayesian optimization of composite functions*\(SBOCF\), a data\-efficient method that requires neither exhaustive search nor a task\-specific training dataset\. SBOCF builds on Bayesian optimization \(BO\)\[[9](https://arxiv.org/html/2609.02126#bib.bib9),[10](https://arxiv.org/html/2609.02126#bib.bib10)\], which models the final image\-discrepancy objective using a surrogate to guide the optimization and reduce the number of costly simulations, and Bayesian optimization of composite functions \(BOCF\)\[[11](https://arxiv.org/html/2609.02126#bib.bib11)\], which models the individual pixel intensities and exploits the known composite structure through which they are combined to form the objective\. SBOCF instead partitions each image into patches and expresses the original image\-discrepancy objective in a new composite form defined over lower\-dimensional patch summaries and correction terms\. This reduces the number of modeled outputs by more than 2,000\-fold, retaining BOCF’s ability to exploit the objective’s composite structure while making it practical for high\-resolution PACBED images\. We validate SBOCF on synthetic and experimental SrTiO3PACBED benchmarks, where it yields more accurate parameter estimates than standard BO and random sampling without a large task\-specific training set\. We also demonstrate that SBOCF estimates improve downstream MEP quality\. More broadly, SBOCF offers a promising approach for inverse problems involving expensive simulators and high\-dimensional structured outputs\.

Fig\.[1](https://arxiv.org/html/2609.02126#S1.F1)illustrates the complete workflow, from experimental 4D\-STEM acquisition to PACBED\-based specimen\-parameter estimation using BO\-based methods\.

![Refer to caption](https://arxiv.org/html/2609.02126v1/figures/main-fig1-workflow.png)Figure 1:Conceptual workflow for PACBED\-based specimen\-parameter estimation prior to MEP\. \(a\) A 4D\-STEM dataset is acquired from a specimen with unknown thickness and mistilt\. \(b\) Conventionally these parameters are estimated jointly with the other reconstruction variables, and \(c\) uncorrected mistilt degrades the result\. \(d\) In the proposed workflow they are estimated first, by comparing the experimental PACBED pattern with multislice simulations: conventional methods evaluate a predefined grid, whereas BO\-based methods adaptively select candidates from previous results\. \(e\) The improved reconstruction expected after correction\.
## 2PACBED Parameter Estimation with Bayesian Optimization

### 2\.1Problem formulation

We consider a target PACBED image withM×MM\\times Mpixels\. Let𝐲true∈ℝM×M\\mathbf\{y\}^\{\\mathrm\{true\}\}\\in\\mathbb\{R\}^\{M\\times M\}denote the experimental PACBED image, whereyk,ℓtruey^\{\\mathrm\{true\}\}\_\{k,\\ell\}is the intensity at pixel\(k,ℓ\)\(k,\\ell\)\. Let𝒳⊆ℝd\\mathcal\{X\}\\subseteq\\mathbb\{R\}^\{d\}denote the specimen parameter space\. In this work, we considerd=3d=3parameters: specimen thickness and the tilt components along thexx\- andyy\-directions\.

For a given parameter vector𝐱∈𝒳\\mathbf\{x\}\\in\\mathcal\{X\}, a multislice simulation generates a PACBED image𝐲⁡\(𝐱\)∈ℝM×M\\mathbf\{y\}\(\\mathbf\{x\}\)\\in\\mathbb\{R\}^\{M\\times M\}, whereyk,ℓ​\(𝐱\)y\_\{k,\\ell\}\(\\mathbf\{x\}\)denotes the simulated intensity at pixel\(k,ℓ\)\(k,\\ell\)\. We measure the discrepancy between the simulated and experimental PACBED patterns using the pixel\-wise sum of squared errors \(SSE\),

f⁡\(𝐱\):=∑k=1M∑ℓ=1M\(yk,ℓ​\(𝐱\)−yk,ℓtrue\)2\.f\(\\mathbf\{x\}\):=\\sum\_\{k=1\}^\{M\}\\sum\_\{\\ell=1\}^\{M\}\\left\(y\_\{k,\\ell\}\(\\mathbf\{x\}\)\-y^\{\\mathrm\{true\}\}\_\{k,\\ell\}\\right\)^\{2\}\.\(1\)The parameter estimation problem is then

𝐱∗∈arg⁡min𝐱∈𝒳⁡f⁡\(𝐱\)\.\\mathbf\{x\}^\{\*\}\\in\\arg\\min\_\{\\mathbf\{x\}\\in\\mathcal\{X\}\}f\(\\mathbf\{x\}\)\.\(2\)
Fig\.[2](https://arxiv.org/html/2609.02126#S2.F2)\(a\) illustrates how the simulated pattern𝐲⁡\(𝐱\)\\mathbf\{y\}\(\\mathbf\{x\}\)varies over the thickness and tilt ranges considered in this work\.

![Refer to caption](https://arxiv.org/html/2609.02126v1/figures/fig2_20260826.png)Figure 2:\(a\) PACBED patterns simulated for SrTiO3across thickness \(columns\) and tilt along two orthogonal axes, tiltX \(rows\) and tiltY \(offset diagonally\)\. Thickness reorganizes the fringe structure non\-monotonically, while tilt along either axis breaks the pattern’s symmetry in a direction\-dependent way\. \(b\) Pixel\-wise SSEf⁡\(𝐱\)f\(\\mathbf\{x\}\)versus patch\-level SSEfP​\(𝐱\)f^\{P\}\(\\mathbf\{x\}\)for simulated PACBED images; their systematic relationship motivates using the patch\-level SSE with correction terms\.
### 2\.2Bayesian optimization

Bayesian optimization \(BO\) is a sequential approach for optimizing expensive black\-box functions, applied to problems such as hyperparameter tuning\[[12](https://arxiv.org/html/2609.02126#bib.bib12)\], materials design\[[13](https://arxiv.org/html/2609.02126#bib.bib13)\], and experimental optimization\[[14](https://arxiv.org/html/2609.02126#bib.bib14)\]\. In electron microscopy it has been used to select reconstruction parameters for electron ptychography\[[15](https://arxiv.org/html/2609.02126#bib.bib15)\]and to align the microscope\[[16](https://arxiv.org/html/2609.02126#bib.bib16)\]\. Both target the instrument or the reconstruction; we instead use BO to estimate specimen thickness and mistilt by matching simulated and experimental PACBED patterns, then supply the estimates to a subsequent reconstruction\.

Standard BO places a Gaussian process \(GP\) prior on the scalar objectivef⁡\(𝐱\)f\(\\mathbf\{x\}\)\. Given observations𝒟n=\{\(𝐱i,f⁡\(𝐱i\)\)\}i=1n,\\mathcal\{D\}\_\{n\}=\\left\\\{\\left\(\\mathbf\{x\}\_\{i\},f\(\\mathbf\{x\}\_\{i\}\)\\right\)\\right\\\}\_\{i=1\}^\{n\},the GP posterior yields the predictive distribution

f\(𝐱\)∣𝒟n∼𝒩\(μn\(𝐱\),σn2\(𝐱\)\),𝐱∈𝒳,f\(\\mathbf\{x\}\)\\mid\\mathcal\{D\}\_\{n\}\\sim\\mathcal\{N\}\\\!\\left\(\\mu\_\{n\}\(\\mathbf\{x\}\),\\sigma\_\{n\}^\{2\}\(\\mathbf\{x\}\)\\right\),\\qquad\\mathbf\{x\}\\in\\mathcal\{X\},whereμn​\(𝐱\)\\mu\_\{n\}\(\\mathbf\{x\}\)andσn2​\(𝐱\)\\sigma\_\{n\}^\{2\}\(\\mathbf\{x\}\)are the posterior mean and variance, respectively; see Appendix[A](https://arxiv.org/html/2609.02126#A1)\. An acquisition function uses this distribution to balance exploitation of inputs with promising predicted objective values and exploration of inputs with high posterior uncertainty\.

We primarily use the expected improvement \(EI\) acquisition function\[[17](https://arxiv.org/html/2609.02126#bib.bib17)\],

αnEI​\(𝐱\)=𝔼n​\[max⁡\{fnmin−f⁡\(𝐱\),0\}\],\\alpha\_\{n\}^\{\\mathrm\{EI\}\}\(\\mathbf\{x\}\)=\\mathbb\{E\}\_\{n\}\\left\[\\max\\left\\\{f\_\{n\}^\{\\min\}\-f\(\\mathbf\{x\}\),0\\right\\\}\\right\],\(3\)wherefnmin=mini=1,…,n⁡f⁡\(𝐱i\)f\_\{n\}^\{\\min\}=\\min\_\{i=1,\\ldots,n\}f\(\\mathbf\{x\}\_\{i\}\)is the best objective value observed so far\. Under the Gaussian predictive distribution off⁡\(𝐱\)f\(\\mathbf\{x\}\), EI admits a closed\-form expression given in Appendix[B](https://arxiv.org/html/2609.02126#A2)\. At each iteration, BO selects

𝐱n\+1∈arg⁡max𝐱∈𝒳​αnEI​\(𝐱\)\.\\mathbf\{x\}\_\{n\+1\}\\in\\arg\\max\_\{\\mathbf\{x\}\\in\\mathcal\{X\}\}\\alpha\_\{n\}^\{\\mathrm\{EI\}\}\(\\mathbf\{x\}\)\.\(4\)A PACBED pattern is then simulated at𝐱n\+1\\mathbf\{x\}\_\{n\+1\}, its SSEf⁡\(𝐱n\+1\)f\(\\mathbf\{x\}\_\{n\+1\}\)is evaluated, and the resulting observation is added to𝒟n\\mathcal\{D\}\_\{n\}to form𝒟n\+1\\mathcal\{D\}\_\{n\+1\}\. This process is repeated until the evaluation budget is exhausted\. We also consider the knowledge gradient \(KG\)\[[18](https://arxiv.org/html/2609.02126#bib.bib18),[19](https://arxiv.org/html/2609.02126#bib.bib19)\]as an additional BO baseline; its definition is provided in Appendix[C](https://arxiv.org/html/2609.02126#A3)\.

While standard BO modelsf⁡\(𝐱\)f\(\\mathbf\{x\}\)directly as a scalar black\-box objective, the PACBED objective in Eq\. \([1](https://arxiv.org/html/2609.02126#S2.E1)\) has additional known structure: the specimen parameters𝐱\\mathbf\{x\}are first mapped through the simulator to a PACBED image𝐲⁡\(𝐱\)\\mathbf\{y\}\(\\mathbf\{x\}\), and the resulting pixel intensities are then combined through the SSE function\. We present how to exploit this structure to improve BO sampling efficiency in the next section\.

### 2\.3Bayesian optimization of composite functions

The PACBED objective in Eq\. \([1](https://arxiv.org/html/2609.02126#S2.E1)\) has the known composite form

f⁡\(𝐱\)=g⁡\(𝐲⁡\(𝐱\)\),f\(\\mathbf\{x\}\)=g\\\!\\left\(\\mathbf\{y\}\(\\mathbf\{x\}\)\\right\),\(5\)where the known outer functionggcomputes the pixel\-wise SSE:

g⁡\(𝐲\)=∑k=1M∑ℓ=1M\(yk,ℓ−yk,ℓtrue\)2\.g\(\\mathbf\{y\}\)=\\sum\_\{k=1\}^\{M\}\\sum\_\{\\ell=1\}^\{M\}\\left\(y\_\{k,\\ell\}\-y^\{\\mathrm\{true\}\}\_\{k,\\ell\}\\right\)^\{2\}\.
Bayesian optimization of composite functions \(BOCF\)\[[11](https://arxiv.org/html/2609.02126#bib.bib11)\]exploits this structure by modeling the intermediate output𝐲⁡\(𝐱\)\\mathbf\{y\}\(\\mathbf\{x\}\)with a multi\-output GP and propagating its predictive uncertainty throughggto obtain a predictive distribution overf⁡\(𝐱\)f\(\\mathbf\{x\}\)\. This distribution is then used to evaluate EI and select the next candidate\. Because applying the nonlinear functionggto the Gaussian predictive distribution of𝐲⁡\(𝐱\)\\mathbf\{y\}\(\\mathbf\{x\}\)generally produces a non\-Gaussian distribution overf⁡\(𝐱\)f\(\\mathbf\{x\}\), EI in this case does not admit a closed\-form expression and is instead estimated using Monte Carlo sampling\[[11](https://arxiv.org/html/2609.02126#bib.bib11)\]\.

Directly applying standard BOCF to PACBED requires modelingM2M^\{2\}intermediate pixel intensities𝐲⁡\(𝐱\)\\mathbf\{y\}\(\\mathbf\{x\}\), which becomes computationally impractical for typical PACBED images withM≳128M\\gtrsim 128\. This computational challenge motivates our proposed method, presented in the next section, which uses a lower\-dimensional intermediate representation\.

## 3Proposed Method: Scalable Bayesian Optimization of Composite Functions

To address the computational challenge of GP modeling in standard BOCF, we propose*scalable Bayesian optimization of composite functions*\(SBOCF\)\. SBOCF replaces the pixel\-level intermediate output with lower\-dimensional patch\-level representations while retaining the original pixel\-wise SSE as the optimization objective\.

#### Patch\-level representation\.

We partition anM×MM\\times MPACBED image intoPPdisjoint patches𝒫=\{𝒫1,…,𝒫P\}\\mathcal\{P\}=\\left\\\{\\mathcal\{P\}\_\{1\},\\ldots,\\mathcal\{P\}\_\{P\}\\right\\\}\. For each patch𝒫j\\mathcal\{P\}\_\{j\}, we define a scalar summary

hj\(𝐱\)=Aj\(𝐲\(𝐱\)\),j=1,…,P,h\_\{j\}\(\\mathbf\{x\}\)=A\_\{j\}\\\!\\left\(\\mathbf\{y\}\(\\mathbf\{x\}\)\\right\),\\qquad j=1,\\ldots,P,whereAjA\_\{j\}is an aggregation operator applied to the pixels in𝒫j\\mathcal\{P\}\_\{j\}\. In our experiments,AjA\_\{j\}sums the pixel intensities within each patch:

Aj​\(𝐲⁡\(𝐱\)\)=∑\(k,ℓ\)∈𝒫jyk,ℓ​\(𝐱\)\.A\_\{j\}\\\!\\left\(\\mathbf\{y\}\(\\mathbf\{x\}\)\\right\)=\\sum\_\{\(k,\\ell\)\\in\\mathcal\{P\}\_\{j\}\}y\_\{k,\\ell\}\(\\mathbf\{x\}\)\.\(6\)We similarly define the corresponding target summary ashjtrue=Aj​\(𝐲true\)h\_\{j\}^\{\\mathrm\{true\}\}=A\_\{j\}\\\!\\left\(\\mathbf\{y\}^\{\\mathrm\{true\}\}\\right\)\. Collecting the patch summaries gives𝐡⁡\(𝐱\)=\(h1​\(𝐱\),…,hP​\(𝐱\)\)⊤\\mathbf\{h\}\(\\mathbf\{x\}\)=\\left\(h\_\{1\}\(\\mathbf\{x\}\),\\ldots,h\_\{P\}\(\\mathbf\{x\}\)\\right\)^\{\\top\}\.

Using these summaries, we define the patch\-level SSE

fP​\(𝐱\)=gP​\(𝐡⁡\(𝐱\)\)=∑j=1P\(hj​\(𝐱\)−hjtrue\)2\.f^\{P\}\(\\mathbf\{x\}\)=g^\{P\}\\\!\\left\(\\mathbf\{h\}\(\\mathbf\{x\}\)\\right\)=\\sum\_\{j=1\}^\{P\}\\left\(h\_\{j\}\(\\mathbf\{x\}\)\-h\_\{j\}^\{\\mathrm\{true\}\}\\right\)^\{2\}\.This representation reduces the number of intermediate quantities fromM2M^\{2\}pixel intensities toPPpatch summaries\. However, aggregation discards spatial information within each patch, so the minimizers offP​\(𝐱\)f^\{P\}\(\\mathbf\{x\}\)are generally not equal to those of the original pixel\-wise objectivef⁡\(𝐱\)f\(\\mathbf\{x\}\)\.

#### Correction for aggregation error\.

AlthoughfP​\(𝐱\)f^\{P\}\(\\mathbf\{x\}\)does not exactly recoverf⁡\(𝐱\)f\(\\mathbf\{x\}\), the two quantities exhibit a strong systematic relationship across simulated PACBED images, as shown in Fig\.[2](https://arxiv.org/html/2609.02126#S2.F2)\(b\)\. We therefore represent the original objective as

f⁡\(𝐱\)=ϵ⁡\(𝐱\)​fP​\(𝐱\)\+δ⁡\(𝐱\),f\(\\mathbf\{x\}\)=\\epsilon\(\\mathbf\{x\}\)f^\{P\}\(\\mathbf\{x\}\)\+\\delta\(\\mathbf\{x\}\),\(7\)whereϵ⁡\(𝐱\)\\epsilon\(\\mathbf\{x\}\)is an input\-dependent multiplicative correction andδ⁡\(𝐱\)\\delta\(\\mathbf\{x\}\)captures the remaining discrepancy\. This decomposition is not unique\. We therefore constrain the multiplicative correction to a prescribed range and select the decomposition with the smallest absolute residual:

minϵ⁡\(𝐱\),δ⁡\(𝐱\)\\displaystyle\\min\_\{\\epsilon\(\\mathbf\{x\}\),\\,\\delta\(\\mathbf\{x\}\)\}\|δ⁡\(𝐱\)\|\\displaystyle\|\\delta\(\\mathbf\{x\}\)\|\(8\)s\.t\.\\displaystyle\\mathrm\{s\.t\.\}\|log10⁡ϵ⁡\(𝐱\)\|≤c,\\displaystyle\|\\log\_\{10\}\\epsilon\(\\mathbf\{x\}\)\|\\leq c,f⁡\(𝐱\)=ϵ⁡\(𝐱\)​fP​\(𝐱\)\+δ⁡\(𝐱\),\\displaystyle f\(\\mathbf\{x\}\)=\\epsilon\(\\mathbf\{x\}\)f^\{P\}\(\\mathbf\{x\}\)\+\\delta\(\\mathbf\{x\}\),wherec\>0c\>0controls the allowable range of the multiplicative correction\. This formulation assigns the systematic difference betweenfP​\(𝐱\)f^\{P\}\(\\mathbf\{x\}\)andf⁡\(𝐱\)f\(\\mathbf\{x\}\)primarily to the bounded scaling term, whileδ⁡\(𝐱\)\\delta\(\\mathbf\{x\}\)captures the discrepancy that cannot be explained by this scaling\. A closed\-form solution of optimization problem \([8](https://arxiv.org/html/2609.02126#S3.E8)\) is provided in Appendix[D](https://arxiv.org/html/2609.02126#A4)\.

#### Composite surrogate model\.

SBOCF models the reduced intermediate representation

𝐡~​\(𝐱\)=\(h1​\(𝐱\),…,hP​\(𝐱\),ϵ⁡\(𝐱\),δ⁡\(𝐱\)\)⊤\\tilde\{\\mathbf\{h\}\}\(\\mathbf\{x\}\)=\\left\(h\_\{1\}\(\\mathbf\{x\}\),\\ldots,h\_\{P\}\(\\mathbf\{x\}\),\\epsilon\(\\mathbf\{x\}\),\\delta\(\\mathbf\{x\}\)\\right\)^\{\\top\}using independent GP surrogates for itsP\+2P\+2outputs\. It then applies BOCF to this reduced representation rather than to the full pixel\-level PACBED image\. The original pixel\-wise SSE is recovered through the known outer function

f​\(𝐱\)=g~​\(𝐡~​\(𝐱\)\),f\(\\mathbf\{x\}\)=\\tilde\{g\}\\\!\\left\(\\tilde\{\\mathbf\{h\}\}\(\\mathbf\{x\}\)\\right\),\(9\)whereg~\\tilde\{g\}takes the form:

g~​\(𝐡~\)=h~P\+1​∑j=1P\(h~j−hjtrue\)2\+h~P\+2\.\\tilde\{g\}\(\\tilde\{\\mathbf\{h\}\}\)=\\tilde\{h\}\_\{P\+1\}\\sum\_\{j=1\}^\{P\}\\left\(\\tilde\{h\}\_\{j\}\-h\_\{j\}^\{\\mathrm\{true\}\}\\right\)^\{2\}\+\\tilde\{h\}\_\{P\+2\}\.Thus, SBOCF reduces the number of modeled intermediate outputs fromM2M^\{2\}toP\+2P\+2while preserving the original pixel\-wise SSE objective\.

#### Sequential SBOCF procedure\.

The complete SBOCF procedure is summarized in Fig\.[3](https://arxiv.org/html/2609.02126#S3.F3)\. Starting from an initial set of parameter–image pairs, SBOCF constructs the reduced representations, fits independent GP surrogates for theP\+2P\+2outputs, and propagates posterior samples through the known outer function\. It then maximizes EI to select and simulate the next candidate, appends the resulting parameter–image pair to the dataset, and repeats this process until the simulation budget is exhausted\.

Initial designParameter–image pairs𝒟n=\{\(𝐱i,𝐲⁡\(𝐱i\)\)\}i=1n\\mathcal\{D\}\_\{n\}=\\\{\(\\mathbf\{x\}\_\{i\},\\mathbf\{y\}\(\\mathbf\{x\}\_\{i\}\)\)\\\}\_\{i=1\}^\{n\}Build reduced representationsPartition each image and computehj​\(𝐱\)h\_\{j\}\(\\mathbf\{x\}\)Obtainϵ⁡\(𝐱\)\\epsilon\(\\mathbf\{x\}\)andδ⁡\(𝐱\)\\delta\(\\mathbf\{x\}\)using Eqs\. \([6](https://arxiv.org/html/2609.02126#S3.E6)\)–\([8](https://arxiv.org/html/2609.02126#S3.E8)\)Fit the multi\-output GPModel the reduced representation𝐡~​\(𝐱\)∈ℝP\+2\\tilde\{\\mathbf\{h\}\}\(\\mathbf\{x\}\)\\in\\mathbb\{R\}^\{P\+2\}Induce the posterior offfPropagate GP samples through Eq\. \([9](https://arxiv.org/html/2609.02126#S3.E9)\)f​\(𝐱\)=g~​\(𝐡~​\(𝐱\)\)f\(\\mathbf\{x\}\)=\\tilde\{g\}\\\!\\left\(\\tilde\{\\mathbf\{h\}\}\(\\mathbf\{x\}\)\\right\)Select the next candidateMaximize EI in Eq\. \([4](https://arxiv.org/html/2609.02126#S2.E4)\)𝐱n\+1∈arg​max𝐱∈𝒳​αnEI​\(𝐱\)\\mathbf\{x\}\_\{n\+1\}\\in\\underset\{\\mathbf\{x\}\\in\\mathcal\{X\}\}\{\\operatorname\{arg\\,max\}\}\\;\\alpha\_\{n\}^\{\\mathrm\{EI\}\}\(\\mathbf\{x\}\)Run the simulationGenerate the PACBED image𝐲⁡\(𝐱n\+1\)\\mathbf\{y\}\(\\mathbf\{x\}\_\{n\+1\}\)Budget exhausted?Return best𝐱∗\\mathbf\{x\}^\{\*\}Lowest SSEAppend the new pair\(𝐱n\+1,𝐲⁡\(𝐱n\+1\)\)\\bigl\(\\mathbf\{x\}\_\{n\+1\},\\mathbf\{y\}\(\\mathbf\{x\}\_\{n\+1\}\)\\bigr\)to the datasetYesNoRepeatSBOCFsequential optimization loop

Figure 3:Flowchart of the proposed SBOCF algorithm\.
#### Comparison of methods\.

Table[1](https://arxiv.org/html/2609.02126#S3.T1)summarizes the differences among standard BO, BOCF, and SBOCF\. Row \(a\) describes how each method processes the simulated PACBED image before GP modeling: standard BO reduces the image to a scalar SSE, BOCF retains all pixel intensities, and SBOCF partitions the image into patches\. We use the3×33\\times 3square partition shown in the table throughout the main experiments\. Row \(b\) lists the quantities modeled by the GP surrogates, while Row \(c\) gives the corresponding numbers of GP outputs:11for standard BO,M2M^\{2\}for BOCF, andP\+2P\+2for SBOCF\. Finally, Row \(d\) compares their objective representations: standard BO modelsf⁡\(𝐱\)f\(\\mathbf\{x\}\)directly, BOCF uses the full composite formg⁡\(𝐲⁡\(𝐱\)\)g\(\\mathbf\{y\}\(\\mathbf\{x\}\)\), and SBOCF uses the reduced composite formg~​\(𝐡~​\(𝐱\)\)\\tilde\{g\}\(\\tilde\{\\mathbf\{h\}\}\(\\mathbf\{x\}\)\)\.

Table 1:Comparison of standard BO, BOCF, and SBOCF for PACBED parameter estimation\.

## 4Numerical Experiments

In this section, we first present the comparison methods, optimization settings, and evaluation metric, followed by the simulated and experimental PACBED benchmarks used to evaluate SBOCF\.

### 4\.1Comparison methods and experimental setup

We compare SBOCF with standard BO using a single\-output GP with either expected improvement \(BO\(EI\)\) or knowledge gradient \(BO\(KG\)\), as well as with random sampling \(Random\)\. For all GP\-based methods, we use a constant mean function and a Matérn\-5/25/2covariance kernel, with hyperparameters estimated by maximum likelihood estimation \(MLE\)\. Each algorithm is initialized with the same 7\-point Sobol design and run for a total of 50 evaluations\. We repeat each experiment over 20 independent trials with different initial designs\. For SBOCF, we use the3×33\\times 3square partition shown in Row \(a\) of Table[1](https://arxiv.org/html/2609.02126#S3.T1)and set the parameterccin Eq\. \([8](https://arxiv.org/html/2609.02126#S3.E8)\) to 1 for all problems considered\.

We measure optimization performance by the best \(lowest\) pixel\-wise SSE observed by each iteration, averaged over the 20 trials\. We also quantify the performance gain of SBOCF using the ratio of the median baseline metric to the corresponding median SBOCF metric, considering both the final SSE and absolute parameter\-estimation error\. A ratio greater than one indicates an improvement over the baseline\. In addition, we report the average acquisition\-function optimization time per iteration, excluding the PACBED simulation time, which is common to all methods\.

All methods are implemented in Python using*PyTorch*\[[20](https://arxiv.org/html/2609.02126#bib.bib20)\]and*BoTorch*\[[21](https://arxiv.org/html/2609.02126#bib.bib21)\]\. The code is available at[https://github\.com/dasol\-yoon/bott](https://github.com/dasol-yoon/bott)\. Experiments are run on a single computing node with one NVIDIA GeForce RTX 3090 GPU and four CPU cores\. Further optimization details are provided in Appendix[E](https://arxiv.org/html/2609.02126#A5)\.

### 4\.2Benchmarks

#### Simulated SrTiO3PACBED\.

We first evaluate the methods on synthetic SrTiO3PACBED benchmarks, where the target PACBED images are generated by the same simulator used during optimization\. This setting provides a controlled test case in which the ground\-truth specimen parameters are known and, in the noiseless case, the global optimum corresponds to the parameter setting that exactly reproduces the target image\.

PACBED patterns are simulated on\-the\-fly using the abTEM package\[[22](https://arxiv.org/html/2609.02126#bib.bib22)\]during the optimization process, with BO adaptively selecting the simulation variables\. Each evaluation simulates a 4D\-STEM scan over probe positions spanning a unit cell and averages the resulting diffraction patterns to form the PACBED pattern\. We follow the forward simulation settings for SrTiO3considered in\[[6](https://arxiv.org/html/2609.02126#bib.bib6)\]\. Specifically, we use a beam energy of 200 keV, a convergence semi\-angle of 19\.1 mrad, a defocus of 0 nm, a collection angle of 31 mrad, a scan step size of0\.3Å0\.3\\text\{\\,\}\\mathrm\{\\SIUnitSymbolAngstrom\}, and a potential extent of62\.6Å62\.6\\text\{\\,\}\\mathrm\{\\SIUnitSymbolAngstrom\}\. Each PACBED pattern has a resolution of157×157157\\times 157pixels\.

The unknown parameters are the specimen thickness and the beam tilts along thexx\- andyy\-axes, searched over\[5,500\]\[5,500\]Å and\[−10,10\]\[\-10,10\]mrad, respectively\. We consider three simulated benchmarks:Sim380andSim100, thick and thin specimens with ground\-truth thicknesses of 380 and 100 Å and tilts of\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad; andSim200, at 200 Å and\(3,−5\)\(3,\-5\)mrad\. Results forSim100andSim200are deferred to Appendices[F](https://arxiv.org/html/2609.02126#A6)and[G](https://arxiv.org/html/2609.02126#A7)\.

We additionally evaluate robustness to measurement noise by adding varying noise levels to the target images; see Appendix[H](https://arxiv.org/html/2609.02126#A8)\. We also compare the3×33\\times 3square partition with an alternative partition comprising a high\-intensity circular center and four outer quadrants; see Appendix[I](https://arxiv.org/html/2609.02126#A9)\.

#### Experimental SrTiO3PACBED\.

We next evaluate the methods on an experimental SrTiO3PACBED pattern from\[[6](https://arxiv.org/html/2609.02126#bib.bib6)\], using the same simulation setup, optimization variables, and search space as for the simulated benchmarks\. Unlike the simulated targets, the experimental pattern cannot be reproduced exactly by the simulator, and its true specimen parameters are unknown\. We therefore evaluate the parameter estimates using the reference values reported in\[[6](https://arxiv.org/html/2609.02126#bib.bib6)\]\. Specifically, we use a thickness of380380Å and beam tilts of\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad\. We refer to this benchmark asExp380\.

#### Downstream MEP reconstruction\.

We useSim200as a representative test case for downstream MEP reconstruction\. The corresponding 4D\-STEM dataset is generated using the same specimen parameters and microscope settings, and the SBOCF\-estimated thickness and mistilt are then used in the reconstruction\.

## 5Results and discussion

#### Simulated SrTiO3PACBED:

Fig\.[4](https://arxiv.org/html/2609.02126#S5.F4)summarizes the results forSim380; result figures forSim100andSim200are provided in Appendices[F](https://arxiv.org/html/2609.02126#A6)and[G](https://arxiv.org/html/2609.02126#A7), respectively\. In each figure, Panel \(a\) shows the best observed pixel\-wise SSE at each iteration; Panels \(b\)–\(d\) show the final thickness, tilt\-X, and tilt\-Y estimates; and Panel \(e\) compares the average acquisition runtime per iteration with the final SSE\. Both standard BO baselines outperform random sampling, while SBOCF achieves the lowest final SSE and the most accurate and stable parameter estimates\.

ForSim380, SBOCF achieves final\-SSE improvement factors of290×290\\timesand377×377\\timesover BO\(EI\) and BO\(KG\), respectively\. For thickness, tilt\-X, and tilt\-Y, the corresponding improvement factors are\(24×,26×,35×\)\(24\\times,26\\times,35\\times\)relative to BO\(EI\) and\(23×,22×,51×\)\(23\\times,22\\times,51\\times\)relative to BO\(KG\)\. ForSim100, the final\-SSE improvement factors are47×47\\timesand320×320\\times, with corresponding parameter\-error improvements of\(1\.9×,9\.4×,14×\)\(1\.9\\times,9\.4\\times,14\\times\)and\(5\.4×,26×,33×\)\(5\.4\\times,26\\times,33\\times\)relative to BO\(EI\) and BO\(KG\), respectively\. ForSim200, the final\-SSE improvement factors are 37×\\timesand 202×\\times, with corresponding parameter\-error improvements of\(2\.2×,6\.5×,6\.0×\)\(\\text\{2\.2\}\\times,\\text\{6\.5\}\\times,\\text\{6\.0\}\\times\)and\(4\.2×,16×,13×\)\(\\text\{4\.2\}\\times,\\text\{16\}\\times,\\text\{13\}\\times\)\.

SBOCF is moderately more expensive per iteration than BO\(EI\) because it models 11 GP outputs rather than one, but achieves substantially lower final SSE\. It is also faster and more accurate than BO\(KG\), whose fantasy\-model construction and high\-dimensional one\-shot optimization incur greater computational cost\. Thus, SBOCF provides a favorable trade\-off between runtime and parameter\-estimation accuracy\.

Figure 4:Results forSim380with ground\-truth thickness380380Å and tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad\. Panels show \(a\) best observed pixel\-wise SSE, final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y, and \(e\) acquisition runtime versus final SSE\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.
#### Experimental SrTiO3PACBED:

Fig\.[5](https://arxiv.org/html/2609.02126#S5.F5)summarizes theExp380results using the same panel layout as Fig\.[4](https://arxiv.org/html/2609.02126#S5.F4)\. SBOCF achieves the lowest final SSE, with improvement factors of1\.08×1\.08\\timesand1\.13×1\.13\\timesover BO\(EI\) and BO\(KG\), respectively\. These gains are less pronounced than in the synthetic cases, as measurement noise and simulation–experiment mismatch make the experimental problem more challenging\. Nevertheless, SBOCF delivers the strongest overall performance, achieving the lowest final SSE and improving most parameter estimates while remaining competitive on the others\.

For thickness, SBOCF achieves improvement factors of2\.4×2\.4\\timesand3\.0×3\.0\\timesrelative to BO\(EI\) and BO\(KG\), respectively\. The corresponding factors for tilt\-X are0\.90×0\.90\\timesand1\.78×1\.78\\times, while those for tilt\-Y are1\.41×1\.41\\timesand1\.26×1\.26\\times\. The resulting estimates are close to the reference thickness of380380Å and the tilt values reported in\[[6](https://arxiv.org/html/2609.02126#bib.bib6)\]\.

Although SBOCF incurs greater acquisition\-function overhead than BO\(EI\), it achieves a lower final SSE\. SBOCF also achieves a lower final SSE than BO\(KG\) while incurring substantially less acquisition\-function overhead\. These results demonstrate that SBOCF can provide competitive experimental parameter estimates without problem\-specific neural\-network training or a large simulated training set\.

#### Downstream MEP reconstruction:

Fig\.[6](https://arxiv.org/html/2609.02126#S5.F6)illustrates the downstream consequence of accurate parameter estimates\. For a simulatedSrTiO3\\text\{SrTiO\}\{\\vphantom\{\\text\{X\}\}\}\_\{\\vphantom\{\\text\{2\}\}\\smash\[t\]\{\\text\{3\}\}\}^\{\\vphantom\{\\smash\[t\]\{\\text\{2\}\}\}\\hphantom\{\\text\{3\}\}\\text\{\}\}specimen of thickness200200Å with tilts of33and−5\-5mrad, ignoring the tilt elongates the atomic columns—rows of atoms stacked along the beam direction, which appear as bright dots in a projected view—and tilts them with respect to the beam direction \(a\)–\(c\), whereas reconstructing with the SBOCF estimates recovers sharp, vertical columns \(d\)–\(f\)\. Those estimates match the ground truth to within55Å in thickness and0\.10\.1mrad in tilt\.

Figure 5:Results for theExp380experimental SrTiO3PACBED benchmark using the target image from\[[6](https://arxiv.org/html/2609.02126#bib.bib6)\]\. Layout as in Fig\.[4](https://arxiv.org/html/2609.02126#S5.F4), except that dashed lines in Panels \(b\)–\(d\) indicate the reference estimates reported in\[[6](https://arxiv.org/html/2609.02126#bib.bib6)\]\.![Refer to caption](https://arxiv.org/html/2609.02126v1/figures/fig_mep_tilt.png)Figure 6:Effect of specimen mistilt on multislice electron ptychography and its correction using SBOCF estimates, for a simulatedSrTiO3\\text\{SrTiO\}\{\\vphantom\{\\text\{X\}\}\}\_\{\\vphantom\{\\text\{2\}\}\\smash\[t\]\{\\text\{3\}\}\}^\{\\vphantom\{\\smash\[t\]\{\\text\{2\}\}\}\\hphantom\{\\text\{3\}\}\\text\{\}\}specimen of thickness200200Å with tilts of33mrad \(tilt\-X\) and−5\-5mrad \(tilt\-Y\)\. \(a\)–\(c\) Reconstruction of the dataset without accounting for the tilt: \(a\) projected view, \(b\)xx–zzand \(c\)yy–zzdepth sections\. \(d\)–\(f\) The corresponding reconstruction using the SBOCF\-estimated parameters

## 6Conclusion

We introduced scalable Bayesian optimization of composite functions \(SBOCF\) for simulation\-based estimation of specimen thickness and mistilt from PACBED images\. By representing each simulated image with patch\-level outputs and two correction terms, SBOCF exploits image\-level information while preserving the pixel\-wise SSE objective and reducing the number of modeled outputs by a factor of approximately2,0002\{,\}000relative to standard BOCF\.

Under a limited simulation budget, SBOCF converged faster and achieved lower SSE than BO\(EI\), reducing the median final SSE by a factor of approximately290290on the noise\-free syntheticSim380dataset\. On experimental data, SBOCF produced thickness and mistilt estimates consistent with previous neural\-network\-based estimates without requiring a large simulated training dataset\.

Future work will extend SBOCF to additional PACBED\-sensitive parameters, including probe aberrations and atomic\-vibration amplitudes\. More broadly, SBOCF provides a practical approach to inverse problems involving expensive simulations and high\-dimensional structured outputs\.

## Acknowledgments and Disclosure of Funding

This research was supported by the Center for Alkaline Based Energy Solutions \(CABES\), an Energy Frontier Research Center funded by the U\.S\. Department of Energy \(DOE\), Office of Science, Basic Energy Sciences \(BES\), under Award \#DE\-SC0019445\. PB would like to thank the DPST scholarship program from IPST, Ministry of Education, Thailand, for providing financial support\. CHL is supported by the Eric and Wendy Schmidt AI in Science Postdoctoral Fellowship, a program of Schmidt Sciences, LLC\.

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Appendix

## Appendix AGaussian process posterior distribution

Assume that

f⁡\(⋅\)∼𝒢​𝒫​\(μ0​\(⋅\),k0​\(⋅,⋅\)\),f\(\\cdot\)\\sim\\mathcal\{GP\}\(\\mu\_\{0\}\(\\cdot\),k\_\{0\}\(\\cdot,\\cdot\)\),whereμ0​\(⋅\)\\mu\_\{0\}\(\\cdot\)is the prior mean function andk0​\(⋅,⋅\)k\_\{0\}\(\\cdot,\\cdot\)is the prior covariance function\.

Given a dataset𝒟n=\{\(𝐱i,f⁡\(𝐱i\)\)\}i=1n\\mathcal\{D\}\_\{n\}=\\\{\(\\mathbf\{x\}\_\{i\},f\(\\mathbf\{x\}\_\{i\}\)\)\\\}\_\{i=1\}^\{n\}, conditioning on this data set, the posterior distribution off⁡\(𝐱\)f\(\\mathbf\{x\}\)follows\[[23](https://arxiv.org/html/2609.02126#bib.bib23)\]:

f⁡\(𝐱\)\|𝒟n∼𝒩⁡\(μn​\(𝐱\),σn2​\(𝐱\)\),f\(\\mathbf\{x\}\)\|\\mathcal\{D\}\_\{n\}\\sim\\mathcal\{N\}\(\\mu\_\{n\}\(\\mathbf\{x\}\),\\sigma\_\{n\}^\{2\}\(\\mathbf\{x\}\)\),whereμn​\(𝐱\)\\mu\_\{n\}\(\\mathbf\{x\}\)andσn2​\(𝐱\)\\sigma\_\{n\}^\{2\}\(\\mathbf\{x\}\)are posterior mean and posterior variance given by

μn​\(𝐱\)=k0​\(𝐱,𝐗\)T​K​\(𝐗,𝐗\)−1​\(f⁡\(𝐗\)−μ0​\(𝐗\)\)\+μ0​\(𝐱\)\\mu\_\{n\}\(\\mathbf\{x\}\)=k\_\{0\}\(\\mathbf\{x\},\\mathbf\{X\}\)^\{T\}K\(\\mathbf\{X\},\\mathbf\{X\}\)^\{\-1\}\(f\(\\mathbf\{X\}\)\-\\mu\_\{0\}\(\\mathbf\{X\}\)\)\+\\mu\_\{0\}\(\\mathbf\{x\}\)and

σn2​\(𝐱\)=k0​\(𝐱,𝐱\)−k0​\(𝐱,𝐗\)T​K​\(𝐗,𝐗\)−1​k0​\(𝐱,𝐗\)\.\\sigma\_\{n\}^\{2\}\(\\mathbf\{x\}\)=k\_\{0\}\(\\mathbf\{x\},\\mathbf\{x\}\)\-k\_\{0\}\(\\mathbf\{x\},\\mathbf\{X\}\)^\{T\}K\(\\mathbf\{X\},\\mathbf\{X\}\)^\{\-1\}k\_\{0\}\(\\mathbf\{x\},\\mathbf\{X\}\)\.
Here,𝐗=\[𝐱1𝐱2…𝐱n\]\\mathbf\{X\}=\\begin\{bmatrix\}\\mathbf\{x\}\_\{1\}&\\mathbf\{x\}\_\{2\}&\\ldots&\\mathbf\{x\}\_\{n\}\\end\{bmatrix\}is the matrix ofnnevaluated inputs,k0​\(𝐱,𝐗\)T=\[k0​\(𝐱,𝐱1\)k0​\(𝐱,𝐱2\)…k0​\(𝐱,𝐱n\)\]k\_\{0\}\(\\mathbf\{x\},\\mathbf\{X\}\)^\{T\}=\\begin\{bmatrix\}k\_\{0\}\(\\mathbf\{x\},\\mathbf\{x\}\_\{1\}\)&k\_\{0\}\(\\mathbf\{x\},\\mathbf\{x\}\_\{2\}\)&\\ldots&k\_\{0\}\(\\mathbf\{x\},\\mathbf\{x\}\_\{n\}\)\\end\{bmatrix\}is a kernel vector between the queried point𝐱\\mathbf\{x\}and evaluated points𝐱i\\mathbf\{x\}\_\{i\}fori=1,…,ni=1,\\ldots,n,K⁡\(𝐗,𝐗\)K\(\\mathbf\{X\},\\mathbf\{X\}\)is an×nn\\times nkernel matrix betweennnevaluated points whose entries are given byK​\(𝐗,𝐗\)i,j=k0​\(𝐱i,𝐱j\)K\(\\mathbf\{X\},\\mathbf\{X\}\)\_\{i,j\}=k\_\{0\}\(\\mathbf\{x\}\_\{i\},\\mathbf\{x\}\_\{j\}\)\.μ0​\(𝐗\)T=\[μ0​\(𝐱1\)μ0​\(𝐱2\)…μ0​\(𝐱n\)\]\\mu\_\{0\}\(\\mathbf\{X\}\)^\{T\}=\\begin\{bmatrix\}\\mu\_\{0\}\(\\mathbf\{x\}\_\{1\}\)&\\mu\_\{0\}\(\\mathbf\{x\}\_\{2\}\)&\\ldots&\\mu\_\{0\}\(\\mathbf\{x\}\_\{n\}\)\\end\{bmatrix\}is a mean vector andf​\(𝐗\)T=\[f⁡\(𝐱1\)f⁡\(𝐱2\)…f⁡\(𝐱n\)\]f\(\\mathbf\{X\}\)^\{T\}=\\begin\{bmatrix\}f\(\\mathbf\{x\}\_\{1\}\)&f\(\\mathbf\{x\}\_\{2\}\)&\\ldots&f\(\\mathbf\{x\}\_\{n\}\)\\end\{bmatrix\}is an observation vector\.

## Appendix BClosed\-form expected improvement formula

Given the Gaussian posterior distribution

f⁡\(𝐱\)\|𝒟n∼𝒩⁡\(μn​\(𝐱\),σn2​\(𝐱\)\),f\(\\mathbf\{x\}\)\|\\mathcal\{D\}\_\{n\}\\sim\\mathcal\{N\}\(\\mu\_\{n\}\(\\mathbf\{x\}\),\\sigma\_\{n\}^\{2\}\(\\mathbf\{x\}\)\),EI admits the closed\-form formula as follows:

αnEI​\(𝐱\)=\(fnmin−μn​\(𝐱\)\)​Φ​\(fnmin−μn​\(𝐱\)σn​\(𝐱\)\)\+σn​\(𝐱\)​ϕ​\(fnmin−μn​\(𝐱\)σn​\(𝐱\)\),\\alpha^\{\\mathrm\{EI\}\}\_\{n\}\(\\mathbf\{x\}\)=\(f^\{\\min\}\_\{n\}\-\\mu\_\{n\}\(\\mathbf\{x\}\)\)\\Phi\\\!\\left\(\\frac\{f^\{\\min\}\_\{n\}\-\\mu\_\{n\}\(\\mathbf\{x\}\)\}\{\\sigma\_\{n\}\(\\mathbf\{x\}\)\}\\right\)\+\\sigma\_\{n\}\(\\mathbf\{x\}\)\\phi\\\!\\left\(\\frac\{f^\{\\min\}\_\{n\}\-\\mu\_\{n\}\(\\mathbf\{x\}\)\}\{\\sigma\_\{n\}\(\\mathbf\{x\}\)\}\\right\),\(B1\)wherefnmin=mini=1,…,n⁡f⁡\(𝐱i\)f\_\{n\}^\{\\min\}=\\min\_\{i=1,\\ldots,n\}f\(\\mathbf\{x\}\_\{i\}\)is the best objective value observed so far\.Φ⁡\(⋅\)\\Phi\(\\cdot\)andϕ⁡\(⋅\)\\phi\(\\cdot\)denote the standard normal cumulative distribution function \(CDF\) and probability density function \(PDF\), respectively\. Derivation can be found, for example, in\[[24](https://arxiv.org/html/2609.02126#bib.bib24)\]\.

## Appendix CKnowledge gradient formula

Given the Gaussian predictive distribution

f⁡\(𝐱\)\|𝒟n∼𝒩⁡\(μn​\(𝐱\),σn2​\(𝐱\)\),f\(\\mathbf\{x\}\)\|\\mathcal\{D\}\_\{n\}\\sim\\mathcal\{N\}\(\\mu\_\{n\}\(\\mathbf\{x\}\),\\sigma^\{2\}\_\{n\}\(\\mathbf\{x\}\)\),KG is constructed as follows:

αnKG​\(𝐱\)=min𝐱′∈𝒳⁡μn​\(𝐱′\)−𝔼⁡\[min𝐱′∈𝒳⁡μn\+1​\(𝐱′\)\|𝐱n\+1=𝐱\],\\alpha^\{\\mathrm\{KG\}\}\_\{n\}\(\\mathbf\{x\}\)=\\min\_\{\\mathbf\{x\}^\{\\prime\}\\in\\mathcal\{X\}\}\\mu\_\{n\}\(\\mathbf\{x\}^\{\\prime\}\)\-\\mathbb\{E\}\\left\[\\min\_\{\\mathbf\{x\}^\{\\prime\}\\in\\mathcal\{X\}\}\\mu\_\{n\+1\}\(\\mathbf\{x\}^\{\\prime\}\)\\;\\middle\|\\;\\mathbf\{x\}\_\{n\+1\}=\\mathbf\{x\}\\right\],\(C1\)where the first term is the current best inferred value, and the second term represents the expected future best inferred SSE after evaluating at𝐱\\mathbf\{x\}\. Thus,αnKG​\(𝐱\)\\alpha\_\{n\}^\{\\mathrm\{KG\}\}\(\\mathbf\{x\}\)quantifies the expected reduction in the best inferred SSE resulting from the additional evaluation\. Unlike EI, KG does not have an analytical formula and has to be evaluated using Monte Carlo approximation and optimized in a one\-shot style\[[21](https://arxiv.org/html/2609.02126#bib.bib21)\]\.

## Appendix DClosed\-form solution forϵ⁡\(𝐱\)\\epsilon\(\\mathbf\{x\}\)andδ⁡\(𝐱\)\\delta\(\\mathbf\{x\}\)decomposition

For a fixed input𝐱\\mathbf\{x\}, consider the optimization problem

minϵ⁡\(𝐱\),δ⁡\(𝐱\)\\displaystyle\\min\_\{\\epsilon\(\\mathbf\{x\}\),\\,\\delta\(\\mathbf\{x\}\)\}\|δ⁡\(𝐱\)\|\\displaystyle\|\\delta\(\\mathbf\{x\}\)\|\(D1\)s\.t\.\\displaystyle\\text\{s\.t\.\}\|log10⁡ϵ⁡\(𝐱\)\|≤c,\\displaystyle\|\\log\_\{10\}\\epsilon\(\\mathbf\{x\}\)\|\\leq c,f⁡\(𝐱\)=ϵ⁡\(𝐱\)​f𝒫​\(𝐱\)\+δ⁡\(𝐱\),\\displaystyle f\(\\mathbf\{x\}\)=\\epsilon\(\\mathbf\{x\}\)f^\{\\mathcal\{P\}\}\(\\mathbf\{x\}\)\+\\delta\(\\mathbf\{x\}\),wherec\>0c\>0is a user\-specified constant\.

We derive the closed\-form solution below\.

From the constraint

\|log10⁡ϵ⁡\(𝐱\)\|≤c,\|\\log\_\{10\}\\epsilon\(\\mathbf\{x\}\)\|\\leq c,we obtain

10−c≤ϵ⁡\(𝐱\)≤10c\.10^\{\-c\}\\leq\\epsilon\(\\mathbf\{x\}\)\\leq 10^\{c\}\.Using the equality constraint in \([D1](https://arxiv.org/html/2609.02126#A4.E1)\), we can eliminateδ⁡\(𝐱\)\\delta\(\\mathbf\{x\}\):

δ⁡\(𝐱\)=f⁡\(𝐱\)−ϵ⁡\(𝐱\)​f𝒫​\(𝐱\)\.\\delta\(\\mathbf\{x\}\)=f\(\\mathbf\{x\}\)\-\\epsilon\(\\mathbf\{x\}\)f^\{\\mathcal\{P\}\}\(\\mathbf\{x\}\)\.Substituting this into the objective gives the equivalent one\-variable problem

minϵ⁡\(𝐱\)∈\[10−c,10c\]⁡\|f⁡\(𝐱\)−ϵ⁡\(𝐱\)​f𝒫​\(𝐱\)\|\.\\min\_\{\\epsilon\(\\mathbf\{x\}\)\\in\[10^\{\-c\},\\,10^\{c\}\]\}\\left\|f\(\\mathbf\{x\}\)\-\\epsilon\(\\mathbf\{x\}\)f^\{\\mathcal\{P\}\}\(\\mathbf\{x\}\)\\right\|\.\(D2\)
We first consider the casef𝒫​\(𝐱\)\>0f^\{\\mathcal\{P\}\}\(\\mathbf\{x\}\)\>0\. In this case, minimizing

\|f⁡\(𝐱\)−ϵ⁡\(𝐱\)​f𝒫​\(𝐱\)\|\|f\(\\mathbf\{x\}\)\-\\epsilon\(\\mathbf\{x\}\)f^\{\\mathcal\{P\}\}\(\\mathbf\{x\}\)\|is equivalent to choosingϵ⁡\(𝐱\)\\epsilon\(\\mathbf\{x\}\)as close as possible to the unconstrained minimizer

ϵ⁡\(𝐱\)=f⁡\(𝐱\)f𝒫​\(𝐱\)\.\\epsilon\(\\mathbf\{x\}\)=\\frac\{f\(\\mathbf\{x\}\)\}\{f^\{\\mathcal\{P\}\}\(\\mathbf\{x\}\)\}\.Therefore, the constrained optimum is obtained by projectingf⁡\(𝐱\)f𝒫​\(𝐱\)\\frac\{f\(\\mathbf\{x\}\)\}\{f^\{\\mathcal\{P\}\}\(\\mathbf\{x\}\)\}onto the interval\[10−c,10c\]\[10^\{\-c\},\\,10^\{c\}\], i\.e\.,

ϵ∗​\(𝐱\)=min⁡\{10c,max⁡\{10−c,f⁡\(𝐱\)f𝒫​\(𝐱\)\}\}\.\\epsilon^\{\*\}\(\\mathbf\{x\}\)=\\min\\left\\\{10^\{c\},\\max\\left\\\{10^\{\-c\},\\frac\{f\(\\mathbf\{x\}\)\}\{f^\{\\mathcal\{P\}\}\(\\mathbf\{x\}\)\}\\right\\\}\\right\\\}\.The corresponding optimalδ⁡\(𝐱\)\\delta\(\\mathbf\{x\}\)is then recovered from the equality constraint:

δ∗​\(𝐱\)=f⁡\(𝐱\)−ϵ∗​\(𝐱\)​f𝒫​\(𝐱\)\.\\delta^\{\*\}\(\\mathbf\{x\}\)=f\(\\mathbf\{x\}\)\-\\epsilon^\{\*\}\(\\mathbf\{x\}\)f^\{\\mathcal\{P\}\}\(\\mathbf\{x\}\)\.
Next, consider the casef𝒫​\(𝐱\)=0f^\{\\mathcal\{P\}\}\(\\mathbf\{x\}\)=0\. Then the equality constraint reduces to

f⁡\(𝐱\)=δ⁡\(𝐱\),f\(\\mathbf\{x\}\)=\\delta\(\\mathbf\{x\}\),soδ⁡\(𝐱\)\\delta\(\\mathbf\{x\}\)is fixed and the objective no longer depends onϵ⁡\(𝐱\)\\epsilon\(\\mathbf\{x\}\)\. In this case, anyϵ⁡\(𝐱\)\\epsilon\(\\mathbf\{x\}\)satisfying10−c≤ϵ⁡\(𝐱\)≤10c10^\{\-c\}\\leq\\epsilon\(\\mathbf\{x\}\)\\leq 10^\{c\}is optimal\. For convenience, we set

ϵ⁡\(𝐱\)=1,δ⁡\(𝐱\)=f⁡\(𝐱\)\.\\epsilon\(\\mathbf\{x\}\)=1,\\qquad\\delta\(\\mathbf\{x\}\)=f\(\\mathbf\{x\}\)\.

## Appendix EOptimization details

In this section, we provide optimization details for each method\. All Bayesian optimization methods are implemented inBoTorch\[[21](https://arxiv.org/html/2609.02126#bib.bib21)\]\.

For the EI acquisition function, we use the built\-in analytical implementation of log Expected Improvement \(logEI\), which has been shown to be substantially more numerically stable than the standard EI formulation\[[25](https://arxiv.org/html/2609.02126#bib.bib25)\]\. The acquisition function is optimized using the default multi\-start optimization procedure inBoTorchwith 20 restarts and 100 raw samples\.

For the KG acquisition function, we follow the one\-shot KG implementation inBoTorch\. In particular, Eq\. \([C1](https://arxiv.org/html/2609.02126#A3.E1)\) is approximated using 16 fantasy samples and optimized through a deterministic sample\-average approximation enabled by the reparameterization trick and automatic differentiation\. The resulting optimization problem is solved using 20 restarts and 100 raw samples\.

For SBOCF, we follow the implementation strategy from prior work on Bayesian optimization of composite functions\[[11](https://arxiv.org/html/2609.02126#bib.bib11)\]\. Since the acquisition function does not admit a closed\-form expression, we use Monte Carlo approximation with 512 samples drawn from a Sobol quasi\-Monte Carlo \(QMC\) sampler together with the reparameterization trick and automatic differentiation to enable gradient\-based optimization\. The acquisition function is optimized with 50 restarts and 100 raw samples\. Since the method involves compositions of Gaussian process models, the resulting acquisition landscape is substantially more complex, making the use of additional optimization effort reasonable\.

## Appendix FFigure forSim100

Fig\.[F1](https://arxiv.org/html/2609.02126#A6.F1)shows theSim100results referred to in Section[5](https://arxiv.org/html/2609.02126#S5); the corresponding improvement factors are reported there\.

Figure F1:Results forSim100with ground\-truth thickness100100Å and tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad\. Layout and conventions as in Fig\.[4](https://arxiv.org/html/2609.02126#S5.F4)\.
## Appendix GFigure forSim200

Fig\.[G1](https://arxiv.org/html/2609.02126#A7.F1)shows theSim200results referred to in Section[5](https://arxiv.org/html/2609.02126#S5); the corresponding improvement factors are reported there\. Since we use this problem as a representative case for the downstream ptychography application, we additionally note that SBOCF recovers the ground\-truth parameters to within55Å in thickness and0\.10\.1mrad in tilt\.

![Refer to caption](https://arxiv.org/html/2609.02126v1/figures/performance_200.png)Figure G1:Results forSim200with ground\-truth thickness200200Å and tilts\(3,−5\)\(3,\-5\)mrad\. Layout and conventions as in Fig\.[4](https://arxiv.org/html/2609.02126#S5.F4)\.
## Appendix HAblation study: Simulated test cases with ground\-truth noise

In this section, we consider the simulated benchmark problems introduced in the main paper, namelySim380,Sim100andSim200\. To better reflect realistic experimental conditions where measurements are corrupted by noise and the simulator cannot perfectly reproduce the observed image, we conduct an additional ablation study in which different levels of noise are added to the ground\-truth PACBED images\.

Specifically, we add Poisson noise to the ground\-truth images after normalizing the image intensities and scaling them to simulated photon counts with a specified peak count parameter\. Poisson\-distributed counts are then sampled independently for each pixel before the noisy image is rescaled back to the original intensity range\. For each problem, we consider three noise settings with peak photon count values of 150, 100, and 50, resulting in the benchmark problemsSim380\-P150,Sim380\-P100,Sim380\-P50,Sim100\-P150,Sim100\-P100,Sim100\-P50,Sim200\-P150,Sim200\-P100, andSim200\-P50\. Smaller peak photon count values correspond to noisier observations due to larger relative Poisson fluctuations\.

The original ground\-truth PACBED patterns and their noisy counterparts under different Poisson noise levels are shown in Fig\.[H1](https://arxiv.org/html/2609.02126#A8.F1)forSim380\(380 Å\), Fig\.[H2](https://arxiv.org/html/2609.02126#A8.F2)forSim100\(100 Å\), and Fig\.[H3](https://arxiv.org/html/2609.02126#A8.F3)forSim200\(200 Å\)\. Each figure presents the original noise\-free ground truth image together with three noisy versions generated using peak photon count levels of P150, P100, and P50\.

![Refer to caption](https://arxiv.org/html/2609.02126v1/si-fig1-compare-sim380-noise.png)Figure H1:Comparison between the original ground truth PACBED pattern and noisy observations for theSim380benchmark problem with a sample thickness of 380 Å\. From left to right, the figure shows the original noise\-free image and noisy versions generated using peak photon counts of P150, P100, and P50\. Lower peak photon counts correspond to higher levels of Poisson noise\.![Refer to caption](https://arxiv.org/html/2609.02126v1/si-fig2-compare-sim100-noise.png)Figure H2:Comparison between the original ground truth PACBED pattern and noisy observations for theSim100benchmark problem with a sample thickness of 100 Å\. From left to right, the figure shows the original noise\-free image and noisy versions generated using peak photon counts of P150, P100, and P50\. Lower peak photon counts correspond to higher levels of Poisson noise\.![Refer to caption](https://arxiv.org/html/2609.02126v1/si-fig2-1-compare-sim200-noise.png)Figure H3:Comparison between the original ground truth PACBED pattern and noisy observations for theSim200benchmark problem with a sample thickness of 200 Å\. From left to right, the figure shows the original noise\-free image and noisy versions generated using peak photon counts of P150, P100, and P50\. Lower peak photon counts correspond to higher levels of Poisson noise\.Figs\.[H4](https://arxiv.org/html/2609.02126#A8.F4)–[H6](https://arxiv.org/html/2609.02126#A8.F6),[H7](https://arxiv.org/html/2609.02126#A8.F7)–[H9](https://arxiv.org/html/2609.02126#A8.F9), and[H10](https://arxiv.org/html/2609.02126#A8.F10)–[H12](https://arxiv.org/html/2609.02126#A8.F12)summarize the noisySim380,Sim100, andSim200results, respectively, for peak photon counts of 150, 100, and 50\. Each figure shows \(a\) the best observed pixel\-wise SSE, \(b\)–\(d\) the final thickness, tilt\-X, and tilt\-Y estimates, and \(e\) acquisition runtime versus final SSE\. Dashed lines in panels \(b\)–\(d\) indicate the ground\-truth parameter values\.

Overall, we observe that the benefits of leveraging composite structure become less pronounced as the observation noise increases\. In particular, under the highest noise setting, the performance gap between SBOCF and standard BO methods narrows substantially, especially for the thinner\-sampleSim100problems\. This behavior is expected since heavy pixel\-level noise partially obscures the underlying composite structure exploited by SBOCF\. Nevertheless, SBOCF remains consistently competitive across all noise levels and continues to provide accurate parameter estimates in many settings\.

We additionally observe that the thinner\-sample cases are substantially more sensitive to noise than the thicker\-sample cases\. In theSim380experiments, SBOCF continues to provide clear improvements in both optimization performance and parameter estimation consistency even under noisy conditions\. In contrast, for the thinner\-sampleSim100andSim200problems, the optimization quality and parameter estimation consistency degrade more noticeably as the noise level increases\. This observation is consistent with a prior PACBED study\[[6](https://arxiv.org/html/2609.02126#bib.bib6)\], which suggests that PACBED patterns contain more reliable structural information for samples thicker than 200 Å, leading to more stable parameter recovery\.

Figure H4:Results forSim380\-P150with ground\-truth thickness380380Å, tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad, and Poisson noise with a peak photon count of150150\. Panels show \(a\) best observed pixel\-wise SSE, final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y, and \(e\) acquisition runtime versus final SSE\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure H5:Results forSim380\-P100with ground\-truth thickness380380Å, tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad, and Poisson noise with a peak photon count of100100\. Panels show \(a\) best observed pixel\-wise SSE, final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y, and \(e\) acquisition runtime versus final SSE\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure H6:Results forSim380\-P50with ground\-truth thickness380380Å, tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad, and Poisson noise with a peak photon count of5050\. Panels show \(a\) best observed pixel\-wise SSE, final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y, and \(e\) acquisition runtime versus final SSE\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure H7:Results forSim100\-P150with ground\-truth thickness100100Å, tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad, and Poisson noise with a peak photon count of150150\. Panels show \(a\) best observed pixel\-wise SSE, final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y, and \(e\) acquisition runtime versus final SSE\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure H8:Results forSim100\-P100with ground\-truth thickness100100Å, tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad, and Poisson noise with a peak photon count of100100\. Panels show \(a\) best observed pixel\-wise SSE, final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y, and \(e\) acquisition runtime versus final SSE\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure H9:Results forSim100\-P50with ground\-truth thickness100100Å, tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad, and Poisson noise with a peak photon count of5050\. Panels show \(a\) best observed pixel\-wise SSE, final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y, and \(e\) acquisition runtime versus final SSE\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure H10:Results forSim200\-P150with ground\-truth thickness200200Å, tilts\(3,−5\)\(3,\-5\)mrad, and Poisson noise with a peak photon count of150150\. Panels show \(a\) best observed pixel\-wise SSE, final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y, and \(e\) acquisition runtime versus final SSE\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure H11:Results forSim200\-P100with ground\-truth thickness200200Å, tilts\(3,−5\)\(3,\-5\)mrad, and Poisson noise with a peak photon count of100100\. Panels show \(a\) best observed pixel\-wise SSE, final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y, and \(e\) acquisition runtime versus final SSE\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure H12:Results forSim200\-P50with ground\-truth thickness200200Å, tilts\(3,−5\)\(3,\-5\)mrad, and Poisson noise with a peak photon count of5050\. Panels show \(a\) best observed pixel\-wise SSE, final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y, and \(e\) acquisition runtime versus final SSE\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.
## Appendix IAblation study: Alternative partition pattern

In this section, we conduct an additional ablation study to investigate the effect of the image partitioning pattern used in SBOCF\. In the main experiments, we use thesquarepattern shown in Panel \(b\) of Fig\.[2](https://arxiv.org/html/2609.02126#S2.F2), where the PACBED image is divided into a regular grid of square patches\. Here, we compare this choice with an alternativedomainpattern motivated by the intensity structure of PACBED images\.

Thedomainpattern partitions the PACBED image into five regions: one circular region containing the high\-intensity pixels near the center of the image, and four outer quadrant regions containing lower\-intensity pixels\. This pattern is illustrated in Fig\.[I1](https://arxiv.org/html/2609.02126#A9.F1)\. We refer to this partition as thedomainpattern because it separates the image into broad intensity\-based regions rather than regular spatial patches\.

![Refer to caption](https://arxiv.org/html/2609.02126v1/figures/si-fig9-domain-pattern.png)Figure I1:Illustration of thedomainPACBED segmentation pattern used in the ablation study\. The image is partitioned into five regions: one central circular region containing high\-intensity pixels and four surrounding quadrant regions containing lower\-intensity pixels\.We perform this comparison on all three synthetic problems,Sim380,Sim100, andSim200, under both noiseless and noisy settings\. For the noisy cases, we consider peak photon countsP150,P100, andP50\. Figs\.[I2](https://arxiv.org/html/2609.02126#A9.F2)–[I5](https://arxiv.org/html/2609.02126#A9.F5),[I6](https://arxiv.org/html/2609.02126#A9.F6)–[I9](https://arxiv.org/html/2609.02126#A9.F9), and[I10](https://arxiv.org/html/2609.02126#A9.F10)–[I13](https://arxiv.org/html/2609.02126#A9.F13)show the results forSim380,Sim100, andSim200, respectively\. In each figure, Panel \(a\) shows the best observed pixel\-wise SSE, while Panels \(b\)–\(d\) show the final thickness, tilt\-X, and tilt\-Y estimates\.

Overall, thesquarepattern performs better than, or at least comparably to, thedomainpattern across the tested settings\. The advantage of thesquarepattern is especially clear forSim380, where it achieves faster reduction in pixel\-SSE and produces more stable final parameter estimates under both noiseless and noisy observations\. For thinner problems, e\.g\.Sim100andSim200, the difference is smaller in some settings, but thesquarepattern still generally gives comparable or lower final pixel\-SSE and more stable parameter estimates\.

One possible explanation is that thedomainpattern groups pixels primarily according to broad high\- and low\-intensity regions\. Although this choice is physically motivated, it may be too coarse for this parameter estimation problem and may discard spatial information that is useful for distinguishing thickness and tilt effects in PACBED images\. In contrast, thesquarepattern preserves more local spatial information by dividing the image into smaller regular patches\. These results suggest that the choice of partition pattern can affect the performance of SBOCF, and designing problem\-adaptive partition patterns is worth further investigation\.

Figure I2:Comparison of SBOCF using thesquareanddomainpartition patterns on the noiselessSim380problem with ground\-truth thickness380380Å and tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad\. Panels show \(a\) best observed pixel\-wise SSE and final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure I3:Comparison of SBOCF using thesquareanddomainpartition patterns on theSim380\-P150problem with ground\-truth thickness380380Å, tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad and Poisson noise with a peak photon count of 150\. Panels show \(a\) best observed pixel\-wise SSE and final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure I4:Comparison of SBOCF using thesquareanddomainpartition patterns on theSim380\-P100problem with ground\-truth thickness380380Å, tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad and Poisson noise with a peak photon count of 100\. Panels show \(a\) best observed pixel\-wise SSE and final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure I5:Comparison of SBOCF using thesquareanddomainpartition patterns on theSim380\-P50problem with ground\-truth thickness380380Å, tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad and Poisson noise with a peak photon count of 50\. Panels show \(a\) best observed pixel\-wise SSE and final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure I6:Comparison of SBOCF using thesquareanddomainpartition patterns on the noiselessSim100problem with ground\-truth thickness100100Å and tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad\. Panels show \(a\) best observed pixel\-wise SSE and final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure I7:Comparison of SBOCF using thesquareanddomainpartition patterns on theSim100\-P150problem with ground\-truth thickness100100Å, tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad and Poisson noise with a peak photon count of 150\. Panels show \(a\) best observed pixel\-wise SSE and final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure I8:Comparison of SBOCF using thesquareanddomainpartition patterns on theSim100\-P100problem with ground\-truth thickness100100Å, tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad and Poisson noise with a peak photon count of 100\. Panels show \(a\) best observed pixel\-wise SSE and final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure I9:Comparison of SBOCF using thesquareanddomainpartition patterns on theSim100\-P50problem with ground\-truth thickness100100Å, tilts\(1\.5,−1\.5\)\(1\.5,\-1\.5\)mrad and Poisson noise with a peak photon count of 50\. Panels show \(a\) best observed pixel\-wise SSE and final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure I10:Comparison of SBOCF using thesquareanddomainpartition patterns on the noiselessSim200problem with ground\-truth thickness200200Å and tilts\(3,−5\)\(3,\-5\)mrad\. Panels show \(a\) best observed pixel\-wise SSE and final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure I11:Comparison of SBOCF using thesquareanddomainpartition patterns on theSim200\-P150problem with ground\-truth thickness200200Å, tilts\(3,−5\)\(3,\-5\)mrad and Poisson noise with a peak photon count of 150\. Panels show \(a\) best observed pixel\-wise SSE and final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure I12:Comparison of SBOCF using thesquareanddomainpartition patterns on theSim200\-P100problem with ground\-truth thickness200200Å, tilts\(3,−5\)\(3,\-5\)mrad and Poisson noise with a peak photon count of 100\. Panels show \(a\) best observed pixel\-wise SSE and final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.Figure I13:Comparison of SBOCF using thesquareanddomainpartition patterns on theSim200\-P50problem with ground\-truth thickness200200Å, tilts\(3,−5\)\(3,\-5\)mrad and Poisson noise with a peak photon count of 50\. Panels show \(a\) best observed pixel\-wise SSE and final estimates of \(b\) thickness, \(c\) tilt\-X, and \(d\) tilt\-Y\. Solid lines in Panel \(a\) show the mean across 20 trials, and the shaded regions indicate standard error\. Dashed lines in Panels \(b\)–\(d\) indicate the ground truth\.