Magnetopause Boundary Detection Based on a Deep Image Prior Model Using Simulated Lobster-Eye Soft X-Ray Images

Abstract

This study focuses on the problem of identifying and extracting the magnetopause boundary of the Earth’s magnetosphere using the Soft X-ray Imager (SXI) onboard the Solar Wind Magnetosphere Ionosphere Link Explorer (SMILE) mission. The SXI employs lobster-eye optics to perform panoramic imaging of the magnetosphere based on the Solar Wind Charge Exchange (SWCX) mechanism. However, several factors are expected to hinder future in-orbit observations, including the intrinsically low signal-to-noise ratio (SNR) of soft-X-ray emission, pronounced vignetting, and the non-uniform effective-area distribution of lobster-eye optics. These limitations could severely constrain the accurate interpretation of magnetospheric structures—especially the magnetopause boundary. To address these challenges, a boundary detection approach is developed that combines image calibration with denoising based on deep image prior (DIP). The method begins with calibration procedures to correct for vignetting and effective area variations in the SXI images, thereby restoring the accurate brightness distribution and improving spatial uniformity. Subsequently, a DIP-based denoising technique is introduced, which leverages the structural prior inherent in convolutional neural networks to suppress high-frequency noise without pretraining. This enhances the continuity and recognizability of boundary structures within the image. Experiments use ideal magnetospheric images generated from magnetohydrodynamic (MHD) simulations as reference data. The results demonstrate that the proposed method significantly improves the accuracy of magnetopause boundary identification under medium and high solar wind number density conditions (N = 10–20 cm⁻³). The extracted boundary curves consistently achieve a normalized mean squared error (NMSE) below 0.05 compared to the reference models. Additionally, the DIP-processed images show notable improvements in peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM), indicating enhanced image quality and structural fidelity. This method provides adequate technical support for the precise extraction of magnetopause boundary structures in soft X-ray observations and holds substantial scientific and practical value.

1. Introduction

The interaction between the solar wind and the Earth’s magnetic field generates the magnetosphere, which acts as a plasma shield surrounding the planet. The structure of the magnetosphere is directly influenced by the solar wind, and disturbances within it can trigger geomagnetic storms and significant fluctuations in high-energy charged particles, impacting communication, navigation, and satellite systems [1,2]. Therefore, panoramic observations of the magnetosphere are essential for advancing our understanding of its structure and interactions with solar wind [3,4,5]. Such observations contribute to refining the Earth’s magnetospheric models, improving space weather forecasting accuracy, and enhancing the safety of both terrestrial and space-based activities.

The discovery of Solar Wind Charge Exchange (SWCX) radiation has enabled panoramic imaging of Earth’s magnetosphere [6,7]. In this process, high-charge state heavy ions in the solar wind interact with neutral atoms or molecules in Earth’s outer space. This interaction typically involves ions such as carbon (C), nitrogen (N), and oxygen (O), and neutral components primarily composed of hydrogen (H) atoms escaping from Earth’s vicinity. During the interaction, heavy ions capture electrons and emit soft X-ray photons as they decay from excited states [8,9]. SWCX radiation encodes information about the large-scale structure of the magnetosphere, including features such as bow shock and the shape and position of the magnetopause. Observing these soft X-rays provides detailed insights into the interaction between solar wind and the magnetosphere, supporting a deeper exploration of the dynamic behavior of the solar–terrestrial interaction [10,11,12,13].

The Solar Wind Magnetosphere Ionosphere Link Explorer (SMILE) mission, a joint initiative between the Chinese Academy of Sciences and the European Space Agency, is scheduled for launch in 2026. This mission is designed to capture high-resolution, large-scale images of the Earth’s magnetosphere using an onboard Soft X-ray Imager (SXI) [14,15]. The SXI features an advanced lobster-eye optical system inspired by the structure of lobster compound eyes, providing a wide field of view and high sensitivity for soft X-ray imaging [16]. This unique lobster-eye imaging system, employing specialized micropore optics, efficiently reflects and focuses soft X-rays, thus enabling wide-area imaging with high-resolution detection [17]. Using this technology, the SXI conducts detailed observations within the soft X-ray band (0.2–2 keV) to capture large-scale images of the magnetopause and bow shock, allowing scientists to monitor their positions and dynamic behavior.

Once the SMILE satellite is in orbit, the comparatively small flux of solar wind charge exchange (SWCX) photons—together with instrumental noise, contamination from stellar point sources, and the angular-dependent collection efficiency of the lobster-eye optics—is expected to yield Soft X-ray Imager (SXI) frames with a low signal-to-noise ratio (SNR). This photon-limited condition will hamper the reliable extraction and analysis of magnetospheric structures and thus poses a substantial challenge for future scientific investigations. A tangent fitting approach can restore the magnetospheric top by optimally matching the tangent direction [18]. This paper will provide a unique way of determining the magnetopause boundary by improving image quality using deep convolutional networks.

Using an image denoising and structure enhancement method based on deep image prior (DIP) to improve the recognizability of the magnetopause boundary in lobster-eye soft X-ray imaging images. This is the proposed method for the observational data to be obtained by the soft-X-ray imager (SXI) on the SMILE satellite. It builds an imaging simulation scene based on the magnetospheric image generated by magnetohydrodynamics (MHD) simulation and conducts image processing research for boundary detection [19]. The research primarily concentrates on cases with moderate-to-high solar wind number densities (N = 10, 15, 20 cm⁻³). First, the SXI imaging image is calibrated to correct the vignetting effect and effective area change caused by the optical system, restore the accurate brightness distribution of the image, and improve the spatial consistency of the boundary area. The vignetting effect refers to the phenomenon that when the off-axis angle of the incident light increases, the effective area of the optical system decreases accordingly, resulting in the signal intensity at the edge of the field of view being lower than that in the center [20,21]. Subsequently, the DIP denoising method without pre-training is introduced. With the help of the prior constraints of the neural network structure itself, the high-frequency noise in the image is effectively suppressed, and the magnetopause region’s low-frequency continuity and edge structure performance are enhanced. Experimental results show that this method can significantly improve the fitting accuracy of the magnetopause boundary under various solar wind density variations, providing a reliable image basis for the identification and evolution analysis of the magnetospheric structure.

This paper is organized as follows: Section 2 introduces the imaging principles of the lobster-eye soft X-ray technology and the acquisition process of simulated SXI images, Section 3 presents the processing model for SXI images and the algorithms applied to each component. Section 4 discusses the experimental results, with Section 5 providing the summary of the study.

2. Lobster-Eye Imaging

2.1. Principle of Lobster-Eye Imaging

Lobster-eye imaging is a technique that leverages the unique optical structure of a lobster’s eye for X-ray observation, with its primary imaging component being a spherical micropore optic (MPO) [16]. This MPO comprises millions of cube-shaped micropores, all oriented toward a common center of curvature. X-ray photons reflect off these micropores’ ultra-smooth, metal-coated inner walls, maximizing the reflection efficiency and enabling effective X-ray focusing and imaging. When a point light source is focused through the MPO, a distinctive cross-shaped spot is produced on the CCD sensor located on the focal plane. Photons that underwent an even number of reflections within the MPO walls converged at the center of the cross, whereas those with one or three reflections appeared along the arms of the cross. Figure 1 presents a schematic of the lobster-eye point-source imaging principle and the point-source imaging results.

The imaging characteristics of surface light sources are relatively complex and are primarily influenced by their optical properties and system structure. X-ray photons from different directions enter the MPO at specific angles and converge on the focal plane after reflection. In this process, imaging results are constrained and affected by multiple factors, exhibiting characteristics that differ from point-source imaging.

Vignetting is a prominent feature of lobster-eye optical systems for imaging surface light sources [22]. In this system, because of the curvature of the MPO and the tilt of the micropores, X-ray photons near the edge of the field of view enter at larger incident angles than those in the central region, resulting in more reflections within the micropores and relatively lower reflection efficiency. This effect is particularly significant when handling surface light sources, leading to an uneven brightness distribution on the imaging plane, where the central region is brighter, and the edges gradually darken. The effective area also significantly impacts the imaging results, causing a reduction in the photon count, which further affects the brightness and clarity of the image. In practical imaging, owing to the limited number of soft X-ray photons, pixel binning technology is commonly used to display the results of surface light source imaging, thereby enhancing the signal-to-noise ratio. Pixel values represent the total number of photons received within the pixel range. A schematic diagram of the lobster-eye extended source imaging principle and the extended source imaging results are shown in Figure 2.

2.2. SXI Image Simulation

In space physics, the magnetohydrodynamic (MHD) model is typically used to simulate magnetospheric images [19]. In this study, we utilized MHD simulations to model the process of Solar Wind Charge Exchange (SWCX) radiation, generating panoramic imaging data of solar wind–magnetosphere interactions from the orbital perspective of the SMILE satellite at various time intervals. The MHD images were then input into the SXI image simulation program [23], which simulated the propagation paths of the X-ray photons, tracked the reflection behavior of each photon within the Microchannel Plate (MPO), and recorded the final photon distribution on the CCD sensor, thereby producing the imaging simulation results.

A panoramic simulation of the solar wind–magnetosphere interaction from the orbital perspective of the SMILE satellite was conducted using magnetohydrodynamic (MHD) modeling, with the simulation data provided by the authors of the referenced literature [24]. The three-dimensional X-ray emission images are generated by the PPMLR-MHD hydrodynamic model [19], which describes the magnetohydrodynamics of the solar wind–magnetosphere–ionosphere system in the GSM coordinate system. The MHD solar wind simulation parameters were $V_x=400 \mathrm{km}·{\mathrm{s}}^{-1}$ and $V_y=V_z=0$; the planetary magnetic field parameters were $B_x=B_y=0$ and $B_z=-5 \mathrm{nT}$; the spatial resolution and the range are $-10\le x$, $y$ and $z\le 10R_E$; the solar wind number density is 10, 15, and 20 ${\mathrm{cm}}^{-3}$, respectively, with an integration time of 300 s per image, a field of view of 16° × 27°, and a data point interval of 1°, measured in units of $\mathrm{keV}·{\mathrm{cm}}^{-2}·{\mathrm{s}}^{-1}·{\mathrm{sr}}^{-1}$. The PPMLR-MHD provides the plasma parameters in the Earth–space environment and calculates the X-ray intensity by integrating the X-ray emissivity $P_X$ along a specific line of sight [18]:

$$I_X={\displaystyle \frac{1}{4\pi }}{\displaystyle \int P_{X}dr}={\displaystyle \frac{1}{4\pi }}{\displaystyle \int {\alpha }_X{n}_H{n}_{sw}}\sqrt{u_{sw}^2+u_{th}^2}dr$$

(1)

The SXI simulation parameters were set to a field of view of 16° × 27° and a spatial resolution of $0.1°$. The lobster-eye lens array is eight rows and four columns; the optical focal length of 600 mm; the lens thickness is 2 mm; the microchannel hole side length is 0.04 mm; the microchannel hole depth is 1.2 mm; the reflective material is set to iridium; and the reflective material roughness is 0.5 nm. Poisson noise is not added during the SXI simulation.

Each pixel value can be regarded as a point light source in MHD simulation images. Using the MHD simulation outputs as the input light source for the SXI simulation, we generated lobster-eye soft X-ray photon count imaging results. We selected three magnetospheric MHD simulation results with different solar wind number densities for simulation and obtained the following SXI images for subsequent algorithm processing research. To facilitate distinction, we labeled the MHD images MHD_1, MHD_2, and MHD_3, with their respective SXI images named SXI_1, SXI_2, and SXI_3. The spatial resolution of the images is 15 arcmin. The simulation results are shown in Figure 3.

3. Methodology

3.1. SXI Image Processing Method

Lobster-eye soft X-ray imaging technology can directly detect the radiation generated by the Solar Wind Charge Exchange (SWCX) to achieve imaging of the magnetosphere. However, the number of effective photons detected is limited, resulting in a low image signal-to-noise ratio, affecting the resolution of magnetosphere structural information. This feature makes it difficult for the original images obtained to display the structural features of the magnetosphere clearly. Therefore, these images must be further processed and optimized to enhance the magnetosphere information in the images.

The optical system Introduces vignetting effects and effective area Issues Into the images. In the lobster-eye optical system, the incident X-rays need to undergo one or two total reflections on the hole wall before they can be focused. When the off-axis angle increases, part of the beam is reflected only once or blocked by the hole wall, and the fixed frame and baffles further block the light. The geometric blockage and reflection efficiency together cause the effective collection area at the edge of the field of view to drop rapidly, resulting in a vignetting effect. The vignetting effect causes the image to appear brighter in the center and attenuated in the edge areas. Figure 4 shows the vignetting caused by the optical system. This effect compromises the uniformity of the overall image brightness, necessitating a vignetting correction to eliminate edge darkening and achieve a more consistent brightness distribution across the image. Additionally, an effective area correction is required because of variations in the detector response across different regions, affecting the photon counts’ accuracy. By applying effective area correction, the actual photon count for each pixel can be restored to ensure the reliability of the image data.

Another key issue affecting image quality is the significant discreteness of the pixel values. The pixel values display marked discreteness due to the limited photon count, thereby introducing substantial high-frequency noise. Figure 5 presents a frequency-domain comparison between the SXI and ideal MHD images. In the frequency domain of the SXI image, noise predominates at the corners, whereas the central cross-shaped feature is not prominent. This indicates the dominance of high-frequency signals and the weakness of low-frequency signals [25]. Ideally, the frequency domain map of a magnetosphere image should display a clear central cross feature, be dominated by low-frequency signals, and exhibit minimal noise at the corners. High-frequency noise not only obscures image details but also compromises the representation of weaker magnetospheric structures and can even distort the overall image structure. Therefore, during image processing, it is essential to effectively remove high-frequency noise and enhance pixel continuity to improve the overall clarity of the image.

We developed an optimized model for processing lobster-eye X-ray images to address these issues. A flowchart of the algorithm is shown in Figure 6. The initial step in this model is calibration, targeting the correction of vignetting effects and effective area discrepancies. Vignetting correction removes edge darkening and achieves a more spatially uniform brightness distribution across the image. Effective area correction was applied by adjusting the photon response in different regions according to the response curve of the detector at various photon energy levels, thereby restoring the actual photon count in these areas.

In image denoising, traditional filtering methods (such as Gaussian and median filtering) can suppress noise to some degree but often result in the loss of image details, especially affecting the accuracy of high-brightness X-ray source regions. In this study, we employed a denoising method based on the DIP [26]. This approach leverages the structural priors inherent in deep neural networks to capture an image’s global structure and local features using iterative training to generate a clear denoised image. In X-ray imaging, the DIP can better handle areas with strong light sources without affecting their brightness too much. This method can effectively extract and enhance the magnetosphere’s top boundary features and improve the overall clarity of the image by adaptively suppressing image noise. Compared with traditional denoising techniques, the DIP offers notable advantages in preserving the image authenticity and improving the signal-to-noise ratio.

3.2. SXI Image Calibration

In the imaging process of a soft X-ray imager, vignetting effects and variations in the effective area are key factors that affect the image quality and accuracy of scientific data. Vignetting refers to the phenomenon in which the image gradually darkens from the center towards the edges, caused by the characteristics of the optical system and changes in the incident angle. As the X-ray photons’ incident angle increases, the optical system’s reflection and transmission efficiencies decrease, leading to lower photon counts at the image edges than at the center. To correct this effect, a vignetting correction algorithm must be applied during the data processing stage to restore the true brightness distribution of the image. Figure 7 shows the processing flow of the calibration procedure.

By utilizing photons of varying energies and calculating them based on the optical parameters of the imager, we can determine the vignetting distribution under different energy conditions, as shown in Figure 8. These distributions precisely describe the intensity attenuation characteristics of photons within the imager’s field of view. The intensity variation in photons from the center to the edge of the field of view can be quantified by simulating the transmission paths of photons with different energies through the imager and accounting for the characteristics of the optical system [23].

The vignetting correction was applied using the following method: photons in the image were categorized by their energy, resulting in photon datasets at different energy levels, denoted as $I=[I_1,I_2,\dots ,I_n]$, where $I_n$ represents the images at varying photon energy levels. Similarly, the vignetting data can be represented as $V=[V_1,V_2,\dots ,V_n]$, where $V_n$ denotes the vignetting data at the corresponding energy levels. The vignetting correction was implemented using the following formula:

$$R={\displaystyle \sum _{i=1}^n\frac{I_i}{V_i}},$$

(2)

The effective area is a key technical indicator that describes the imager’s ability to collect photons. The larger the effective area, the higher the image quality and the greater the accuracy of the optical system observations. Correcting the effective area is crucial for ensuring the accuracy of photon counting. A reduction in the effective area decreases the number of detected photons, thereby affecting the precision of the observational results. Therefore, it is necessary to correct the effective area to ensure consistent detection efficiency under varying conditions. As the effective area varies significantly with different photon energies, the SXI images must be processed separately according to the photon energy. We obtained photon effective area data at different energies through simulations [23], as shown in Figure 9.

We used $Ge{o}_i$ to represent the geometric area at energy $i$, $Ef{f}_i$ to represent the effective area at that energy, and $I_i$ to represent the SXI image at that energy. The calculation method is as follows:

$$R={\displaystyle \sum _{i=1}^n\frac{Ge{o}_i}{Ef{f}_i}}{I}_i,$$

(3)

Figure 10 presents the image results after applying vignetting correction and effective area processing. Compared to the uncorrected images, the dark regions at the image edges are significantly restored, resulting in significantly improved brightness uniformity. To more clearly demonstrate this improvement, the third panel of Figure 10 presents a normalized difference image. This image reveals a pronounced increase in brightness at the periphery, while the central region shows only marginal changes, indicating that the edge-darkening correction is effective. Additionally, effective area processing recovers lost photon counts owing to variations in detection efficiency, enhancing the accuracy and consistency of photon counts across the entire field of view.

3.3. Depth Image Prior (DIP) Processing

3.3.1. DIP Principle Introduction

Neural networks can enhance image quality. Machine learning applications in this field primarily rely on large-scale datasets for pretraining, enabling neural networks to learn prior knowledge from images and deliver effective image-processing results. Although this approach performs well, it may present limitations in certain scenarios. For instance, when processing SXI-observed magnetosphere images, the absence of real panoramic images of the magnetosphere implies that training on simulation data alone may yield results that deviate from the actual magnetospheric structures.

In this study, we employed a method based on DIP for image denoising and quality enhancement. DIP leverages the structural characteristics of neural networks to restore images without requiring pretraining, enabling a more accurate representation of the observed magnetospheric features. The core approach is akin to adaptive learning, and the training process is described as follows:

$$\theta *=\arg \underset{\theta }{\min }‖x_0-f(\theta |z)‖,$$

(4)

$$x*=f(\theta *|z),$$

(5)

where $x_0$ represents the degraded image; $x*\in R^{H*W}$ is the final processed output image; $z\in R^{H*W}$ is the random noise input to the network; and $\theta$ is a neural network parameter [26]. $f(\theta |·)$ represents a neural network generator [27]. The neural network served as a regularizer to provide prior knowledge, and image denoising was achieved by searching for optimal network parameters to generate the reconstructed image [28].

In processing the SXI images, we adjusted the above formula by applying a mean filter to the input SXI image $x_0$ to increase the signal-to-noise ratio and accelerate parameter optimization.

$${x^{\prime }}_{x,y}={\displaystyle \frac{1}{n^2}}{\displaystyle \sum _{i=-n/2}^{n/2}{\displaystyle \sum _{j=-n/2}^{n/2}{x}_{0_{(x+i,y+j)}}}},$$

(6)

where $n$ represents the size of the filtering kernel, and x′ is the filtered image. The image is then normalized and scaled to a range of 0–100 to eliminate numerical instability:

$${x^{″}}_{i,j}=\left({\displaystyle \frac{{x^{\prime }}_{i,j}}{{x^{\prime }}_{\max }}}\right)\times 100$$

(7)

We added Gaussian noise to the normalized image $x^{″}$, replacing the input variable z in the formula. In this case, the input variable incorporates prior information from the SXI image, which helps the network converge more quickly.

$$g=x^{″}+\sigma ·N,$$

(8)

$${\theta }^{\prime }=\arg \underset{\theta }{\min }‖x^{″}-f(\theta |g)‖,$$

(9)

In this formula, the elements of the noise tensor N follow a normal distribution with a mean of 0 and a standard deviation of 1. The noise amplitude was controlled by multiplying it by the noise intensity coefficient $\sigma$. Adding noise ensures that the network avoids becoming trapped in the local minima, even with the input SXI image’s prior information. We must determine the appropriate parameter ${\theta }^{\prime }$ to generate the denoised image $f({\theta }^{\prime }|g)$.

DIP demonstrates that during the iterative training process of finding parameters, the Convolutional Neural Network (CNN) has low impedance toward learning signals and high impedance toward learning noise. In other words, when fitting $x_0$ with a CNN, the network prioritizes learning the effective signals in the image. The higher the validity of the information, the higher the learning priority in the CNN. We understand this theory from the perspective of the image frequency domain, where the primary information of an image is located in the low-frequency domain, whereas the noise is located in the high-frequency domain [29]. Figure 11 shows the frequency-domain decomposition of the image. This effect can be explained by the CNN prioritizing the learning of low-frequency signals and the gradual learning of high-frequency signals as the number of iterations increases.

The discreteness between adjacent pixels in the SXI images was high, resulting in high-frequency noise in the images. Ideally, a magnetospheric image should have smoothly changing characteristics, which determines that the effective information resides in the low-frequency domain. Therefore, stopping training once the network has learned the effective low-frequency information is necessary. At this point, the image generated by the network with parameter $\theta$ is denoised. Figure 12 shows the relationship between the number of parameter iterations and the frequency of the image learned by the neural network.

3.3.2. DIP Algorithm Implementation

In this part of the algorithm, we input the calibrated SXI image and apply filtering to remove noise while preserving essential image features. Random noise is then added to the filtered image, which serves as the input to the generator to produce a new image. The generated image was evaluated against the filtered image using the mean square error (MSE). If the error does not meet the predetermined threshold, the generator parameters are updated through backpropagation, and the image is regenerated until the error evaluation passes. Once the evaluation is successful, the final generated image is output as the optimized SXI image. A flowchart of the DIP process is shown in Figure 13. It is important to emphasize that in the DIP, this processing optimization is performed independently for each SXI image. The network weights are initialized from scratch and trained on that image alone. Hence, the “training” phase and the “image-processing” phase are one and the same—the network converges while denoising the current frame, and the resulting output is taken as the restored image. No pretrained or shared model is reused across different images.

The generator used in this study adopted a U-Net network architecture with skip connections and was implemented using Convolutional Neural Networks (CNNs) [30]. This network has an encoder–decoder structure composed of a series of progressive convolutional layers that extract features from the input image. The network uses an input image and performs feature extraction using a combination of multiple convolutional layers and nonlinear activation functions. Figure 14 shows the network structure of the generator.

In this architecture, the encoder section is responsible for progressively downsampling the input data and extracting features through convolution operations while gradually reducing the spatial resolution of the feature maps. Each encoding layer consisted of two convolutional layers and a nonlinear activation layer, and the number of channels increased as the network depth increased. The encoder section contained five downsampling layers with channel numbers 8, 16, 32, 64, and 128. The decoder restores the features extracted by the encoder back to the input spatial resolution through upsampling. Each layer of the decoder corresponds symmetrically to a layer in the encoder, and skip connections are used to pass features from the encoding layers to the corresponding decoding layers. These skip connections effectively alleviate the vanishing gradient problem and prevent information loss in the deep networks. The decoder also consisted of five upsampling layers with the same number of channels as the encoder: 128, 64, 32, 16, and 8. Skip connections span four layers with 8, 8, 8, and 4 channel numbers.

For the activation function, the network uses the Swish function, which is a type of nonlinear activation function [31]. Swish retains the advantages of the ReLU function while also having smooth characteristics, which provides nonzero derivatives across the entire input range, resulting in better performance in optimizing SXI images. After each convolution operation in the network, normalization was applied to prevent overfitting.

The output layer of the network used a 1 × 1 convolution kernel to map the final output of the decoder onto the desired number of output channels. The output layer uses a sigmoid function for normalization, ensuring the output values fall within the range [0, 1].

Regarding the selection of network parameters, to balance the complexity of the network and the computational cost, this study sets appropriate numbers of channels and convolution kernel sizes. It uses the Adam optimizer for parameter updates [32]. The mean squared error (MSE) is used as the loss function during training because it performs well in regression tasks such as image denoising. During the experiments, training was stopped, and the generator’s output was produced once the loss function decreased to a specified threshold. In our experiments, we set the maximum number of epochs to 2000 and the loss threshold to 3.0 × 10⁻⁵, and we used PyTorch 2.3.1 as the deep learning framework [33]. The training was performed on a CPU only, with the system environment being Windows 10 x64 and the virtual environment being Anaconda3 with Python 3.12.4.

4. Results

This section will show the SXI image calibration process results and the final image obtained after the DIP method. The SMILE mission focuses on imaging the large-scale magnetopause boundary in magnetospheric observations. To verify the effectiveness of the algorithm in this paper in magnetopause boundary reconstruction, this paper will analyze the algorithm’s restoration performance at the magnetopause position and evaluate and compare the overall image quality.

In Figure 15(a1–a3) are the simulated SXI images, and Figure 15(b1–b3) present the calibrated SXI images labeled CAL_1, CAL_2, and CAL_3, respectively. It can be observed that vignetting at the edges was eliminated in all three images. Figure 15(c1–c3) represent SXI images processed by mean filtering, using FIL_1, FIL_2, and FIL_3 to distinguish. It can be seen that the outline of the magnetosphere can be seen after filtering, but the image quality is still poor. Figure 15(d1–d3) represent the original MHD images. Figure 15(e1–e3) are the final output SXI images, labeled OUT_1, OUT_2, and OUT_3 after DIP processing. The images processed with deep image priors show significant noise reduction; image quality has been significantly improved, and the key features of the magnetosphere boundary have been clearly presented.

We compared the calibrated SXI images with other reference algorithms, including BM3D, wavelet denoising, and supervised learning [24,34,35]. Figure 15(f1–f3) present the classic BM3D denoising method, with the noise standard deviation parameter set to 0.3, labeled B3D. Figure 15(g1–g3) present wavelet denoising methods, where we use the sym4 wavelet function, and the number of decomposition levels is set to 3, labeled WDD. Figure 15(h1–h3) present supervised learning methods, labeled SPL. We can see that all methods reduce the noise of the image. The images of BM3D and wavelet denoising are smoother than those of supervised learning, but the magnetopause recovery effect is not good. The DIP image presents a cleaner image and a clearer and continuous magnetospheric boundary. The DIP performs better for two main reasons: it is optimized for each SXI frame individually, and its low-frequency structure priority convergence naturally preserves the extended magnetopause morphology.

4.1. Comparison of Magnetopause Boundaries

In order to evaluate the effectiveness of the algorithm in restoring the position of the magnetopause boundary in the SXI image, we compared the original magnetopause boundary in the MHD simulated image with the magnetopause boundary deviation extracted from the SXI image after the DIP processing. We searched for the maximum pixel value in the image row by row. We used cubic spline interpolation to fit the extracted data points to obtain the magnetopause top boundary curve. The cubic spline interpolation method approximates the local data interval by constructing a low-order polynomial. The polynomials of adjacent intervals are kept consistent at the connection through continuity and smoothness constraints, thereby generating a smooth continuous boundary curve. The interpolation expression is as follows:

$$S_i(x)=a_i+b_i(x-x_i)+c_i{(x-x_i)}^2+d_i{(x-x_i)}^3,$$

(10)

where $x_i$ is a known boundary point, $x\in [x_i,x_{i+1}]$, $i=0,1,2,\dots ,n-1$. The cubic polynomial function has four unknown coefficients, a, b, c, d, and the spline interpolation function has $4n$ unknowns. We can obtain a smooth curve with a continuous boundary as the magnetosphere by solving it. We perform magnetopause boundary detection on all MHD images, SXI images, filtering images, and images processed by DIP, and the results are shown in Figure 16.

From the fitting curve, it can be seen that the SXI image has more noise, the row maximum value is scattered, the fitting curve is not smooth, and it is difficult to show the position of the magnetosphere boundary directly and clearly. The filtered image’s fitting curve is close to the MHD fitting curve, but it is still not smooth enough. The OUT image has a smoother fitting curve because the high-frequency noise is eliminated.

We use the normalized mean square error to measure the difference between the magnetopause boundary after the DIP processing and the ideal magnetopause boundary in the MHD image. The normalized mean square error is normalized based on the mean square error (MSE) to eliminate the influence of the data scale. NMSE is usually defined as the ratio of MSE to the variance of the reference curve, expressed as follows:

$$NMSE={\displaystyle \frac{{\displaystyle \sum {(y_i-{\widehat{y}}_i)}^2}}{{\displaystyle \sum {(y_i-\overline{y})}^2}}}$$

(11)

where $y_i$ is the actual value of the reference curve; ${\widehat{y}}_i$ is the predicted value of the fitted or compared curve; and $\overline{y}$ is the mean of the reference curve. NMSE ranges from 0 to 1, and the smaller the value, the higher the similarity of the two curves. NMSE eliminates the influence of different data ranges on the error measurement through normalization, which is particularly useful in evaluating the matching degree of curves with different scales or amplitudes. We use SXI to represent the SXI image fitting curve, FIL to represent the filtered image, OUT to represent the output image fitting curve after the DIP processing, and NMSE to evaluate the difference in the magnetopause. We simulated the MHD images of each solar wind number density to generate 100 SXI images for full-process processing and calculated the NMSE values at each stage, obtaining the following results (

Table 1. NMSE values of MHD image fitting curve and other image curves.

Density (cm−3)	10	15	20
SXI	2.852 ± 1.378	1.941 ± 0.746	1.600 ± 0.612
FIL	0.518 ± 0.210	0.470 ± 0.177	0.452 ± 0.165
OUT	0.025 ± 0.008	0.023 ± 0.008	0.022 ± 0.007
B3D	0.578 ± 0.225	0.476 ± 0.206	0.433 ± 0.186
WDD	0.815 ± 0.253	0.736 ± 0.261	0.686 ± 0.249
SPL	0.921 ± 0.272	0.892 ± 0.279	0.761 ± 0.247

Note: all values in the table are presented as mean ± standard deviation.

).

The normalized mean square error (NMSE) results show that the NMSE of the unprocessed SXI image is significantly higher than that of the processed image. Under the same method, a higher solar wind number density is more likely to obtain a better NMSE value. Under different solar wind conditions, the mean NMSE between the magnetopause boundary processed by the DIP algorithm and the ideal magnetopause boundary obtained by MHD simulation is less than 0.05, and the standard deviation is 0.008, which is much smaller than MB3D, wavelet denoising, and supervised learning methods. The results show that the proposed method can maintain high accuracy under different solar wind intensities, ensuring that the position of the magnetopause boundary is highly consistent. The algorithm can more accurately restore the magnetopause structure in SXI data, providing technical support for subsequent large-scale observations of the magnetopause boundary based on soft X-ray imaging.

4.2. Comparison of Image Quality

To evaluate the image quality of the output images, we use the image peak signal-to-noise ratio (PSNR) and the structural similarity index (SSIM) as evaluation metrics [36]. The PSNR is a key indicator for evaluating image processing quality and is typically used to quantify the degree of difference between the processed and original images. Measured in decibels (dB), higher PSNR values indicate better image quality, signifying a smaller difference between the processed and original images. The formula for calculating PSNR is as follows:

$$PSNR(a,b)=10·{\log }_{10}({\displaystyle \frac{MA{X}^2}{\frac{1}{mn}{{‖a-b‖}^2}_F}})$$

(12)

where $a$ and $b$ represent the two images being compared; MAX represents the maximum pixel value of the image; and ${‖·‖}_F$ denotes the Frobenius norm, which is the square root of the sum of the squares of all elements in the matrix. M and n represent the dimensions of the image. Generally, a PSNR value greater than 40 dB indicates a very high image quality, where the difference from the original image is hardly noticeable to the human eye, with minimal distortion. A PSNR between 30 dB and 40 dB indicates good restoration quality, between 20 dB and 30 dB indicates average image quality, and below 20 dB signifies poor image quality.

The Structural Similarity Index (SSIM) is a metric that evaluates the similarity between two images in structure, brightness, and contrast [37]. Compared to the PSNR, the SSIM focuses more on the structural information of the image. The SSIM value ranges from 0 to 1, with higher values indicating greater image similarity. A value of 1 indicates that the two images are identical. The formula for calculating the SSIM is as follows:

$$SSIM(a,b)={\displaystyle \frac{(2{\mu }_a{\mu }_b+C_1)(2{\sigma }_{ab}+C_2)}{({\mu }_a^2+{\mu }_b^2+C_1)({\sigma }_a^2+{\sigma }_b^2+C_2)}}$$

(13)

where ${\mu }_a$ and ${\mu }_b$ represent the mean values of images $a$ and $b$, respectively, which indicate the brightness information of the two images. ${\sigma }_a^2$ and ${\sigma }_b^2$ represent the variances of the two images used to measure the image contrast. ${\sigma }_{ab}$ denotes the covariance between two images, which measures their structural similarity. $C_1$ and $C_2$ are constants introduced to stabilize the calculation and avoid division by zero. They are typically set as $C_1={(K_{1}L)}^2$ and $C_2={(K_{2}L)}^2$, where $K_1=0.01$ and $K_2=0.03$, and $L$ is the dynamic range of the pixel values.

We calculated the MHD, original SXI, calibrated SXI, filtered SXI, and DIP-processed SXI images at each solar wind number density as well as the peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) indicators of the reference algorithm BM3D, wavelet denoising, and supervised learning. All images were normalized before calculation to ensure effective comparative analysis. We calculated 100 images for each stage and used box plots to represent the data distribution of each stage, as shown in Figure 17.

From the peak signal-to-noise ratio (PSNR) statistics in Figure 17a, it can be seen that the images generated by the deep image prior (DIP) method are always better than all baseline methods. For three solar wind density levels (10, 15, and 20 cm⁻³), the median PSNR of OUT is higher than 30 dB, while the original SXI data remains at around 15 dB. The calibration image (CAL) and other reference algorithms have greatly improved the PSNR compared to the original SXI image but are still not as good as the processing effect of DIP. This shows that the deep image prior method can significantly improve the signal-to-noise ratio of the image while effectively denoising, making the processed image closer to the real MHD simulation image. The SSIM data also show that the images processed by the deep image prior are excellent in terms of structural preservation. Under medium and high solar wind density, the SSIM value of OUT is significantly improved. Under the condition of 20 cm⁻³ number density, the median SSIM of OUT reaches 0.85, which is much higher than BM3D (0.66), WDD (0.54), and SPL (0.41), while the median SSIM of the original SXI image is only 0.038. The experimental results show that the deep image prior method not only performs well in noise suppression but also can effectively retain and enhance the structural information of the image so that the key features of the magnetosphere can be clearly presented.

5. Conclusions

This study addresses the challenges of low signal-to-noise ratio and unclear magnetopause boundaries in lobster-eye soft X-ray imaging by proposing an optimized boundary detection method that integrates image calibration with deep image prior (DIP)-based denoising. The approach first applies vignetting and effective area corrections to enhance brightness uniformity across the image. Subsequently, unsupervised denoising is performed using DIP, which leverages the intrinsic structural priors of neural networks to effectively suppress high-frequency noise while enhancing the visibility and continuity of large-scale boundary features such as magnetopause. Experimental results demonstrate that the proposed method enables accurate extraction of the magnetopause boundary under moderate-to-high solar wind number densities. The extracted boundary contours exhibit high consistency with the reference boundaries derived from magnetohydrodynamic (MHD) simulations, achieving normalized mean squared error (NMSE) values consistently below 0.05—significantly outperforming the unprocessed SXI images and other reference algorithms. In addition, improvements in peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) further confirm the method’s effectiveness in magnetospheric structure identification. Even based on simulated SXI data, the magnetopause boundaries extracted by DIP can still provide direct value to space weather research and satellite mission operations—they can make up for the limitations of in situ detection, provide a real-time snapshot of global magnetospheric dynamics, and provide early warning of magnetopause rapid movement events for geosynchronous and highly elliptical orbit satellites.

Despite its strong performance on images with clear magnetopause contrast and distinct boundary features, the method exhibits limitations in cases with uniform brightness or weak structural cues, such as at low solar wind number density (N = 5 cm³). Future research will focus on enhancing the algorithm’s adaptability to diverse SXI imaging conditions, including incorporating additional physical priors and improvements to the network architecture to further advance boundary detection robustness and reliability in the complex observational environments of the SMILE mission. Ultimately, applying this process to real SXI observations is a critical next step. We plan to acquire the first SXI images during the SMILE satellite’s on-orbit commissioning phase (expected in the second quarter of 2026), run DIP denoising and boundary extraction frame by frame, and compare and verify with CCMC MHD predictions for the same period. At that time, we will focus on addressing challenges such as vignetting drift caused by aging of microporous coatings, residual star artifacts, time-varying X-ray background, and potential lower signal-to-noise ratio. Through the above improvements and on-orbit verification, this method is expected to provide a reliable global magnetopause imaging solution for future space weather monitoring.