\begin{filecontents}{references.bib}


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\end{filecontents}

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\begin{document}

\section*{Supplementary Information}
\subsection*{Supplementary Note1: Photon Budget Estimation}

\noindent To provide a quantitative benchmark for comparing photon budgets discussed in this work, we estimated the minimum number of incident photons required for HHG-based XUV ghost imaging, HHG-based coherent diffraction imaging (CDI), and confocal soft X-ray scanning transmission microscopy (STXM) under identical imaging conditions. The estimation assumes an image size of $80 \times 80$ pixels and a target SNR of 5. According to the Rose criterion, this corresponds to 25 detected photons per pixel. The photon detection efficiency of the detector was assumed to be 0.95, and the sample transmission was assumed to be 0.1 for all cases.

\noindent For XUV GI, the optical path between the source and detector contains only an XUV mask and the sample. When using a single-pixel (bucket) detector, at least 6,400 measurements are required to reconstruct an $80 \times 80$ image. Assuming a transmission of 0.15 for the XUV mask, as shown in Fig.~1(b), the minimum required number of incident photons is estimated to be $1.12 \times 10^{7}$.

\noindent In the case of HHG-based CDI, additional oversampling of the diffraction pattern is required to ensure stable phase retrieval. An oversampling factor of 2 was assumed. The optical path between the HHG source and the detector includes a monochromator with a harmonic selection efficiency of 0.001 and two reflective mirrors with reflectivities of 0.35 each. Based on these parameters, the minimum number of incident photons required for CDI is estimated to be approximately $2.75 \times 10^{10}$ \cite{dinh2015coherent}.

\noindent For STXM, image acquisition is performed by sequentially scanning each spatial position. Between the source and detector, a focusing zone plate is used with a photon efficiency of 0.08. An order-sorting aperture (OSA) for first-order transmission is assumed to have a photon efficiency of 1. The detection zone plate has a photon efficiency of approximately $6 \times 10^{-7}$, followed by a pinhole with a photon efficiency of 0.1. Considering the cumulative efficiency of the entire optical system, the minimum required number of incident photons is estimated to be $3.51 \times 10^{14}$ \cite{spath2015confocal}.

\noindent While these comparisons do not represent absolute quantitative benchmarks—as they depend on the specific optical configurations and components of each system—they serve as a valuable reference for evaluating photon requirements across different imaging modalities. Furthermore, although additional processing steps inherent to each technique may influence overall efficiency, such factors were excluded from this simplified estimation to ensure a consistent baseline for comparison.


\newpage
\subsection*{Supplementary Note 2: XUV Ghost Imaging Setup}

\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{s_fig/s1.png}
\caption[Tabletop XUV Ghost Imaging Setup and Simulation Results]{\textbf{Tabletop XUV Ghost Imaging Setup and Simulation Results.} (a) A photograph of the experimental setup, showing the wafer holder, X-Y stage for precise object positioning, and the XUV mask. (b) The target image used for simulations. (c-e) Reconstructed images from ideal simulations using differential Hadamard, negative Hadamard, and positive Hadamard patterns with the compressed sensing algorithm, respectively.}\label{fig:s1}
\end{figure}

\noindent The experimental setup, illustrated in S.Fig.\ref{fig:s1}(a), utilizes a high-precision X-Y stage to ensure accurate alignment between the static XUV mask and the target object. The XUV mask is mounted on a wafer holder driven by two linear translation stages (PI M112), providing a unidirectional repeatability of $\pm$0.125~$\mu$m. Geometrically, the object is positioned 1.5~cm from the XUV mask and 2.5~cm from the XUV detector. Despite the high precision of the translation stages, slight misalignments can still occur due to mechanical clearances within the holder and the screw-based fixation method, both of which can hinder perfectly horizontal movement. In CGI, the reconstruction algorithm ideally assumes that the numerical patterns perfectly match the physical illumination on the object. However, even a single-pixel misalignment between the structured illumination pattern and the object leads to a critical discrepancy, significantly degrading the final image quality. As demonstrated in the simulations shown in S.Fig.\ref{fig:s1}(c--e), such mismatches result in significant artifacts and a marked degradation of image fidelity.

\newpage
\subsection*{Supplementary Note 3: Reconstruction Algorithms, Pattern Strategies, and Simulation Performance}

\noindent GI reconstructs an object $G(x,y)$ by correlating known illumination patterns $I_n(x,y)$ with bucket detector measurements $S_n$. TGI utilizes second-order correlation, expressed as $G(x,y) = \frac{1}{N} \sum_{n=1}^{N} [I_n(x,y) - \langle I \rangle][S_n - \langle S \rangle]$ \cite{oh2023, oh2025}. While early GI relied on random speckle patterns for their versatility, their reconstruction efficiency in TGI is often limited by the lack of strict orthogonality between successive realizations, requiring a large number of measurements to suppress background noise. However, when combined with CS, random illumination patterns can alleviate the need for the large number of measurements typically required in conventional TGI, as sparsity-based reconstruction enables stable image recovery from undersampled datasets \cite{Katz2009}.

\noindent The reconstruction resolution of XUV GI is strongly dependent on both the number of illumination patterns and their spatial feature size. To enhance illumination utilization and SNR within these resolution constraints, structured deterministic bases such as Hadamard and Fourier patterns can be employed. Hadamard patterns form a strictly orthogonal binary basis, allowing efficient differential measurements that suppress background noise and significantly reduce the number of realizations required for stable reconstruction \cite{Zhang2017, Hahamovich2021, Zhang2017Fourier, Wang2021}. Fourier-based GI, on the other hand, encodes spatial frequency information by projecting sinusoidal illumination patterns. Using phase-shifting schemes, such as the 4-step Fourier method, enables reconstruction through inverse Fourier transformation, thereby providing direct access to both amplitude and phase information \cite{Zhang2017Fourier}.

\begin{figure}[h]
\centering
\includegraphics[width=15cm, height=7cm]{s_fig/simu_gi.png} 
\caption{\textbf{Numerical Simulation of Ghost Imaging with Various Illumination Patterns.} (a) Target object used in the simulation. (b,c) Reconstructions using random patterns with (b) TGI and (c) CS. (d) Reconstruction using a 4-step Fourier pattern. (e) Reconstruction using differential Hadamard patterns with inverse Hadamard transform. (f,g) Reconstructions using (f) negative and (g) positive Hadamard patterns combined with CS.}
\label{fig_simu_gi}
\end{figure}

\noindent To quantitatively evaluate the reconstruction performance of different illumination strategies, we performed numerical GI simulations using multiple pattern types and reconstruction algorithms. Due to the limited number of illumination patterns that can be practically implemented on the physical XUV mask, the complexity of the target objects was necessarily restricted. Consequently, structurally simple geometries, such as the cross-shaped target shown in S.Fig.\ref{fig_simu_gi} a), were selected to validate the reconstruction performance under these sampling constraints. These simulations reveal a strong dependence of reconstruction quality on the selected illumination and reconstruction approach.

\noindent As shown in S.Fig.\ref{fig_simu_gi}b), TGI reconstruction using random patterns exhibits substantial noise, making the target structure barely distinguishable. Incorporating CS with random patterns (S.Fig.~\ref{fig_simu_gi}c) slightly improves reconstruction fidelity but still produces blurred structural features. The binary 4-step Fourier method (S.Fig.\ref{fig_simu_gi}d) provides clearer reconstruction compared with random illumination; however, discretization of the sinusoidal patterns introduces residual artifacts. In contrast, Hadamard-based illumination consistently demonstrates superior performance. Differential Hadamard reconstruction using an inverse transform (S.Fig.~\ref{fig_simu_gi}e) produces high-contrast and noise-suppressed images. Furthermore, both positive and negative Hadamard patterns combined with CS (S.Figs.~\ref{fig_simu_gi}f and \ref{fig_simu_gi}g) achieve nearly artifact-free reconstructions.

\noindent These results demonstrate that structured orthogonal illumination, particularly Hadamard-based strategies, provides significant advantages in photon-limited XUV GI by improving reconstruction stability, reducing measurement requirements, and enabling high-fidelity structural recovery.


\newpage
\subsection*{Supplementary Note 4: Quantitative Analysis of Image Quality}

\begin{figure}[h]
\centering
\includegraphics[width=0.8\linewidth]{s_fig/s2.png}
\caption[Quantitative Analysis of Image Fidelity]{\textbf{Quantitative Analysis of Image Fidelity.} Correlation matrix of SSIM values for reconstructed cross-shaped targets. The matrix compares six reconstructed reference images and their noisy counterparts. The red box shows the average SSIM between a reference image and its noisy version, while the blue box shows the average SSIM when comparing a different reference image. The green box represents the average SSIM between a reference images and a noise-added version.}
\label{fig:s2}
\end{figure}

\noindent The correlation matrix in S.Fig.\ref{fig:s2} quantitatively validates the structural integrity and robustness of our reconstruction method. The six reference images (Ref.Img) were acquired directly using the XUV detector to serve as ground truth. The upper section of the figure highlights two benchmark results derived from average SSIM values, which establish the metric's baseline performance and sensitivity. First, a high structural consistency ($\text{SSIM} = 0.79$), indicated by the light blue box, was achieved by comparing the XUV-acquired object image with a reference from an alternative XUV source. Second, a low similarity score ($\text{SSIM} = 0.08$), indicated by the green box, was obtained by comparing the original XUV camera image with its noise-added version. This significantly low value of 0.08 underscores the high specificity of the SSIM metric. It confirms that even when a noise-contaminated image retains clearly discernible structural features, minor perturbations can lead to a substantial drop in the SSIM score. This result demonstrates that the metric is exceptionally sensitive to localized structural discrepancies that may otherwise seem negligible to the naked eye.

\noindent Following the analysis in S.Fig.\ref{fig:s2}, we established a practical scale to help interpret what the SSIM values mean within our XUVGI system:

\begin{itemize}[leftmargin=1.5em]
    \item $0.6 < \mathrm{SSIM} \leq 1.0$ (\textbf{High Fidelity}): The two images are virtually indistinguishable, showing nearly perfect structural agreement.
    \item $0.1 < \mathrm{SSIM} \leq 0.6$ (\textbf{Structural Consistency}): Minor noise or artifacts are present, but the primary structural features remain well-preserved.
    \item $0.05 < \mathrm{SSIM} \leq 0.1$ (\textbf{Moderate Similarity}): Basic features are discernible, though significantly obscured by noise.
    \item $0 < \mathrm{SSIM} \leq 0.05$ (\textbf{Structural Failure}): The two images show little to no correlation due to overwhelming noise or significant misalignment.
\end{itemize}

\newpage
\subsection*{Supplementary Note 5: XUV Ghost Imaging with Diverse Objects}

\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{s_fig/s3.png}
\caption[XUV Ghost Imaging of a Window-shaped Object]{\textbf{XUV Ghost Imaging of a Window-shaped Object.} 
\textbf{Top row:} Microscopic target image (200~$\mu$m scale bar) followed by reconstructions using Diff. Hada, Neg. Hada, and Pos. Hada patterns with CS. 
\textbf{Bottom row:} Comparison of post-processing stages, showing the original target image followed by reconstructions using Ideal pattern, Position-calibrated ideal pattern, Real pattern, Position-calibrated real pattern, and final Histogram-equalized real pattern.}\label{fig:S3}
\end{figure}

\noindent To further validate the versatility of our tabletop XUV ghost imaging system, we performed experiments on a window-shaped target featuring a four-square array geometry. Unlike the single cross-shaped object, this target serves as a benchmark for the system's ability to resolve multiple spatially separated features.

\noindent As shown in the top row of S.Fig.\ref{fig:S3}, all Hadamard-based reconstruction methods successfully recovered the fundamental window geometry. While the images from differential, negative, and positive Hadamard patterns exhibit pixelation and blurred boundaries, they clearly resolve the four distinct square openings. These results demonstrate the fundamental robustness of our approach in imaging multi-featured structures.

\noindent The bottom row of S.Fig.\ref{fig:S3} underscores the critical impact of the data processing pipeline on reconstruction fidelity. When using an ideal numerical pattern, the reconstruction yields a notably low SSIM of 0.0340. However, applying position calibration to the ideal pattern increases the SSIM to 0.0687. A more significant improvement is observed when utilizing the "Real pattern" captured directly by the XUV camera. The integration of position calibration with the real pattern further pushes the SSIM to 0.0876. Interestingly, while the subsequent application of histogram equalization visually enhances contrast, it results in a slight decrease in the structural score (SSIM = 0.0822).

\begin{figure}[h]
\centering
\includegraphics[width=0.8\linewidth]{s_fig/s4.png}
\caption[High-Resolution XUV Ghost Imaging of a Small Cross-Shaped Object]{\textbf{Imaging Resolution Limits with a 100~$\mu$m Cross-shaped Object.} (a) Microscopic image of the target (100~$\mu$m scale bar). Reconstructions using (b) differential Hadamard, (c) negative, (d) positive Hadamard, and (e) 4-step Fourier patterns. (f) Final reconstruction using a position-calibrated and histogram-equalized real pattern.}
\label{fig:s4}
\end{figure}

\noindent To investigate the resolution limits of our tabletop setup, we performed experiments on a smaller cross-shaped object with a width of 100~$\mu$m—half the size of the 200~$\mu$m targets discussed previously. As illustrated in S.Fig.\ref{fig:s4}(b--e), the initial reconstructions using Hadamard and Fourier patterns fail to recover any recognizable features. Even with the full data processing pipeline, the final reconstructed image in S.Fig.\ref{fig:s4}(f) lacks sharp boundaries and appears significantly blurred compared to the larger targets in S.Fig.\ref{fig:s4}. This performance degradation confirms that reconstruction fidelity is fundamentally governed by the ratio between the illumination pattern's feature size and the object's dimensions. In XUV ghost imaging, as the target size decreases, the spatial sampling provided by the mask must be proportionally refined.

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