Cross-Survey Image Transformation: Enhancing SDSS and DECaLS Images to Near-HSC Quality for Advanced Astronomical Analysis

The Sloan Digital Sky Survey (SDSS) ¹¹1https://www.sdss.org/ is a large-scale astronomical survey project that began in 2000 (York et al., 2000). It has produced the most detailed three-dimensional map of the universe to date, covering approximately one-third of the sky. SDSS provides multi-band deep images ($u$, $g$, $r$, $i$, $z$ bands) and spectral data for over 3 million celestial objects, significantly advancing our understanding of galaxies, dark matter, and large-scale structures in the universe (Gunn et al., 2006; Abazajian et al., 2009).

The SDSS project employs a 2.5-meter telescope located at Apache Point Observatory in New Mexico, USA, which is well-suited for extensive deep-sky observations. However, the telescope’s imaging depth and resolution (typically ranging from 1.5 to 2 arcseconds) are relatively low compared to subsequent high-resolution surveys, limiting its capability to capture fine details. The charge-coupled device (CCD) sensors used in the SDSS camera have a pixel resolution of 0.396 arcseconds per pixel (in the $r$ band). Despite processing techniques such as noise reduction and background light correction, these data may still struggle to resolve the intricate structures of galaxies.

While SDSS data is extensive and provides a wealth of astronomical information, its limitations in spatial resolution and depth hinder detailed structural studies compared to higher-resolution projects like DECaLS and HSC.

The Dark Energy Camera Legacy Survey (DECaLS) ²²2https://www.legacysurvey.org/decamls/ employs the Dark Energy Camera (DECam; Flaugher et al. 2015) to observe cosmic phenomena such as dark matter, dark energy, and galaxy evolution. DECam is mounted on the 4-meter Blanco telescope in Chile and features a wide field of view of 3.2 square degrees, with a pixel resolution of 0.262 arcseconds per pixel. Compared to SDSS, DECaLS significantly enhances image resolution, achieving full widths at half maximum (FWHM) of $1.29^{\prime\prime}$, $1.18^{\prime\prime}$, and $1.11^{\prime\prime}$ in the $g$, $r$, and $z$ bands, respectively. Overall, the image resolution ranges from 0.6 to 1 arcsecond, enabling the detection of finer structures in celestial objects (DePoy et al., 2008).

The DECaLS survey plan provides approximately two-thirds of the optical imaging coverage for the ongoing Dark Energy Spectroscopic Instrument (DESI; Dey et al. 2019), focusing primarily on the northern Galactic cap (declination $\leq 32$ degrees) and the southern Galactic cap (declination $\leq 34$ degrees). Although the coverage area is extensive, there remains potential for further expansion compared to SDSS. In this study, we use the most up-to-date data release (DR15).

The Hyper Suprime-Cam Subaru Strategic Survey (HSC-SSP) ³³3https://hsc-release.mtk.nao.ac.jp/doc/ is another significant astronomical project focused on studying topics such as galaxy evolution, gravitational lensing, supernovae, and galactic structure (Miyazaki et al., 2012, 2018). HSC is mounted on the 8.2-meter Subaru telescope in Japan and features a field of view with a diameter of 1.5 degrees, as well as a pixel resolution of 0.168 arcseconds per pixel . The FWHM in the $i$ band is 0.6 arcseconds, enabling it to resolve the structures of spiral galaxies and other celestial details.

The HSC-SSP survey is divided into three components: wide, deep, and ultra-deep fields, which overlap with those of the SDSS (Ahumada et al., 2020), the Baryon Oscillation Spectroscopic Survey (BOSS; Dawson et al. 2012), the Galaxy and Mass Assembly survey (GAMA; Driver et al. 2011) and the Visible Multi-Object Spectrograph Very Large Telescope Deep Survey (VVDS; Le Fèvre et al. 2005). The third data release (PDR3), published in 2021, covers approximately 670 square degrees across five filters ($g,r,i,z,y$) and achieves a depth of 26 magnitude ($5\sigma$). While the image quality and resolution of HSC surpass those of SDSS and DECaLS, its coverage area remains relatively small.

The main objective of this study is to utilize advanced deep learning methods to convert low-quality images from SDSS and DECaLS into high-quality images comparable to those from HSC. To achieve this, we need to construct a large sample that includes images from these survey projects. Our sample is derived from galaxies in the Galaxy Zoo DECaLS project, all of which are spectroscopic targets from SDSS. Consequently, the sample primarily includes galaxies with brightness greater than $m_{r}=17.77$ and covers redshifts of $z=0.15$ or lower. Additionally, these galaxies are required to have a Petrosian radius (as specified in the PETROTHETA4 column of the NASA–Sloan Atlas v1.0.0) of at least 3 arcseconds (Walmsley et al., 2022).

We first cross-matched the overlapping regions of these three observational projects and randomly selected 15,000 galaxies from these areas as training samples. This selection ensures that the model we build can learn a diverse range of features, including different types and morphologies of galaxies. Furthermore, we selected an additional 2,000 galaxies from these overlapping regions as test samples to evaluate the model’s generalization capabilities and the quality of the generated images. All images of these galaxies were obtained from the Legacy Survey website ⁴⁴4https://www.legacysurvey.org/.

The Legacy Survey website serves as a comprehensive platform for an astronomical project designed to showcase and provide access to observational data produced by DESI Legacy imaging survey. This website consolidates image data from various astronomical projects, including SDSS, DECaLS, HSC, and the Wide-field Infrared Survey Explorer (WISE). Users can access astronomical images and databases of celestial objects generated by the Legacy Survey, allowing them to browse imaging and photometric information for billions of celestial bodies. Additionally, the website provides various interactive tools to help users acquire and analyze this data, which is significant for research on dark energy, galaxy and star evolution, and the exploration of large-scale structures in the universe.

We used the JPEG cropping tool ⁵⁵5https://www.legacysurvey.org/viewer/ provided by the Legacy Survey website to download the corresponding images from SDSS, DECaLS, and HSC based on the right ascension and declination coordinates of the sample galaxies. To ensure consistency in the spatial coverage of images for the same galaxy, we utilized the algorithms provided by the website to adjust the pixel scale of all images to match that of HSC, which has a pixel scale of 0.168 arcseconds per pixel.

The Legacy Survey website also offers various band combinations for color synthesis of the images. After visual inspection, we selected the $g$, $r$, and $i$ bands for the synthesis of SDSS and HSC images, while for DECaLS images, we chose the $g$, $r$, and $z$ bands. This selection ensures that the final generated images maintain visually consistent colors, making images of galaxies from different data sources appear more harmonious and facilitating subsequent scientific analysis and comparison.

Due to hardware resource limitations, we constrained the image size to $128\times 128$ pixels. This size is sufficient to capture enough background sky while preserving the key structures of most galaxies, ensuring that the model can process efficiently under limited resource conditions. It is important to note that our model does not have a strict dependency on the size of the input images; larger or smaller sized images are also applicable. With appropriate adjustments, the model can maintain relatively stable performance across images of different sizes.

Figure 1 presents three examples of sample galaxies, displaying the galaxy images observed in HSC, DECaLS, and SDSS from bottom to top. It is clear that with increased exposure depth and resolution, the structures of the galaxies become more pronounced, and their sizes appear more extended. In images with shallower exposure depths, such as those from the SDSS project, fine structures in the outer regions of the galaxies are difficult to discern. However, with the greater exposure depth observed in the HSC project, these structures become clearer, revealing richer and more distinct internal details of the galaxies.

Our final sample consists of a training set made up of 15,000 groups of galaxies, while the test set consists of 2,000 groups of galaxies, with each group containing the corresponding three images from SDSS, DECaLS, and HSC. All images are adjusted to a resolution of 0.168 arcseconds per pixel and sized at $128\times 128$ pixels, with pixel values ranging from [0, 255], representing the intensity of the colors in the images.

Pix2pix is an image translation model based on conditional GAN, proposed by Isola et al. (2017) at the 2017 CVPR (IEEE Conference on Computer Vision and Pattern Recognition). It is widely used for tasks that involve transforming one image style into another, including converting low-resolution images to high-resolution images, transforming sketches into realistic images, and changing daytime scenes to nighttime scenes (Chen & Hays, 2018; Chao et al., 2019; Liu et al., 2020; Henry et al., 2021; Liu & Xu, 2022; Kumar et al., 2024).

The core concept of pix2pix is to generate images that meet specific conditions using a conditional GAN. The model learns the mapping relationship between paired input and target images, enabling it to produce output images corresponding to the input images. Its network architecture, as illustrated in Figure 2, consists primarily of two components: a generator and a discriminator.

The generator employs a U-Net architecture (Ronneberger et al., 2015; Isola et al., 2017), which is a convolutional neural network (CNN) comprising an encoder and a decoder, specifically designed to transform input images into target images. In our specific task, the encoder is designed to consist of six modules, each composed of a convolutional layer, a batch normalization layer, and a Leaky ReLU activation function. This configuration progressively extracts low-level features while downsampling the images to capture essential information.

The decoder is similarly constructed with six modules, where each module includes a transposed convolution layer, a batch normalization layer, dropout layers (applied to the first three modules), and a Leaky ReLU activation function. This design gradually restores the spatial resolution of the images. In the U-Net architecture, the convolutional layers in the encoder are connected to the corresponding layers in the decoder through skip connections. This design effectively combines low-dimensional features with high-resolution features, preserving fine details and resulting in generated images that are more refined and realistic.

The discriminator employs a conditional PatchGAN architecture (Demir & Unal, 2018). A key feature of the PatchGAN is its ability to divide input images into multiple patches and evaluate their authenticity through local judgments of each patch. This approach not only emphasizes the consistency of local features but also enables the model to focus on details within each patch, effectively enhancing the quality of detail in the generated images.

In our specific task, the discriminator consists of five convolutional layers, each followed by a batch normalization layer and a Leaky ReLU activation function, ultimately producing a patch with the shape of (batchsize, 14, 14, 1). This patch is then averaged along the width and height dimensions to obtain a scalar value, which serves as the final output of the discriminator. The discriminator receives two sets of image pairs as input: one set contains the images outputted by the generator along with their corresponding input images, while the other set includes real/target images and their respective input images, with each set of image pairs concatenated along the channel dimension. The primary task of the discriminator is to learn to differentiate between the authenticity of these two sets of image pairs.

During the training process, the classical pix2pix model optimizes image transformation results primarily through an adversarial learning mechanism between the generator and the discriminator. The model total cost function is divided into generator loss and discriminator loss, with the generator loss comprising both adversarial loss and structural loss.

The adversarial loss aims to maximize the probability that the discriminator recognizes the generated image as a real image. It is calculated using a well-known loss function, the Binary Cross Entropy (BCE), represented by the following formula:

where $x_{\text{low}}$ represents the original low-quality input image, $x_{\text{enhanced}}$ refers to the enhanced image generated by the generator, $y_{\text{true}}$ is a vector of ones of the size of the batch, $G$ denotes the generator and $D$ denotes the discriminator. In TensorFlow 2, the BCE can be implemented using $tf.keras.losses.BinaryCrossentropy()$.

Structural loss is quantified using $L1$ loss to measure the difference between the generated image and the target image, ensuring that the generated image retains detailed information. The calculation formula is as follows:

where, $x_{\text{high}}$ refers to the high-quality target image and $L1$ is also known as the Mean Absolute Error (MAE).

Integrating the above adversarial loss and structural loss, the total loss for the classical pix2pix generator can be expressed as follows:

where $\lambda$ is a hyperparameter that balances the adversarial loss and the structural loss; its value is typically tuned according to the specific task at hand.

The discriminator loss evaluates the discriminator’s ability to distinguish between the target image and the generated image, with the calculation formula expressed as follows:

where $y_{\text{false}}$ is a vector of zeros of the size of the batch.

In each iteration of the training process, the generator first generates an image and calculates its loss. It then updates its weights using backpropagation with the goal of minimizing the total loss of the generator. Meanwhile, the discriminator’s loss is based on its evaluation of both real and generated images, aimed at optimizing its ability to distinguish between them. The discriminator updates its weights accordingly to reduce its loss.

Through this alternating update mechanism, the generator and discriminator mutually reinforce each other throughout the training process, continuously enhancing their performance. As training progresses, the generator increasingly produces high-quality images with a strong sense of realism, thereby ensuring the model’s effectiveness and stability in image-to-image translation tasks.

The classic pix2pix model is based on conditional GAN. However, training GANs is often unstable, leading to fluctuations in the quality of the generated images and potentially resulting in mode collapse (Thanh-Tung & Tran, 2020). Mode collapse refers to a phenomenon where the generator exploits the weaknesses of the discriminator by optimizing along specific paths in the parameter space that can achieve high scores from the discriminator, yet fail to produce meaningful or diverse outputs. As a result, the generator begins to output only a limited range of results, failing to capture the rich features of the training data adequately. To address these issues, we employ WGAN-GP (Gulrajani et al., 2017; Chen, 2021) to improve the loss function structure of the classic pix2pix model and apply it to the specific task at hand . We refer to the modified model as Pix2WGAN.

Specifically, unlike the classical pix2pix model, which utilizes Jensen-Shannon (JS) divergence for binary classification in the discriminator, Pix2WGAN is trained by minimizing the Wasserstein distance between the distributions of real and generated data (Adler & Lunz, 2018). The Wasserstein distance, also known as the “Earth Mover’s” distance, quantifies the amount of “work” required to transform one probability distribution into another and serves as an effective method for measuring the distance between two distributions. A key advantage of this approach is its ability to provide a smoother and more continuous optimization curve, thereby mitigating the gradient vanishing problem that can arise with JS divergence in certain scenarios.

Additionally, WGAN-GP incorporates a Gradient Penalty technique to ensure that the gradients of the discriminator remain within an appropriate range across all input samples, thus satisfying the 1-Lipschitz condition. This method further enhances the model’s stability, enabling WGAN-GP to effectively generate high-quality samples across a wider range of application scenarios.

In our Pix2WGAN, the adversarial loss function of the generator has been modified to:

The structural loss term remains and is calculated using $L1$ loss as before. Therefore, the total loss function for the generator is:

where $\lambda$ is the hyperparameter that balances the adversarial loss and structural loss; in our study, it is set to a value of 100.

For the discriminator, the loss function has been modified to consist of two parts. The first part employs an averaging strategy similar to that of the classical pix2pix model, but its purpose is to measure the difference between real predictions and fake predictions:

The second part is the gradient penalty term, which is used to enforce gradient penalties. This term is defined by first setting $x_{\text{interp}}$ as the interpolation between the target image and the generated image, as follows:

where $\alpha\in(0,1)$ is a random number. The gradient penalty is then defined as:

Where $E$ denotes the mathematical expectation. These two terms are combined to define the loss of the discriminator, as follows:

where $w_{p}$ is the weight for the gradient penalty, which is set to 10 in our study.

After improving the loss function of the classical pix2pix model, we developed a new Pix2WGAN hybrid model that exhibits substantial stability during the training process. In the next section, we will present the remarkable performance of this model in the task of transforming SDSS and DECaLS images into HSC-level images.

Data preprocessing is a critical step in ensuring the model’s performance. Before feeding the images into the model, we normalized the pixel values of the input images. In the downloaded raw images, all pixel values are restricted to the range of [0, 255]. To accelerate the model’s training speed and better retain the details in the images, we further scaled the pixel values of each image to the range of [-1, 1]. Specifically, let the original data be denoted as $x$; we calculate the scaled data $x^{*}$ using the following formula:

In this formula, 127.5 serves as the midpoint of the [0, 255] range. This scaling technique centers the pixel values while maintaining relative contrast, thus aiding in accelerating the model’s convergence and improving the quality of the generated images.

In image-to-image transformation, a smaller difference between images typically simplifies the generator’s task, leading to generated images that more closely match the target images. This reduced disparity helps the model learn the mapping between input and target images more effectively. Given that the difference between DECaLS and HSC images is smaller than that between SDSS and HSC images, this subsection will begin by applying the Pix2WGAN model for the transformation from DECaLS to HSC.

We implemented the Pix2WGAN model using the Keras and TensorFlow 2 libraries (Abadi et al., 2016), and conducted training and testing on a computer equipped with an NVIDIA RTX 3090 GPU. During training, we set the batch size to 32. Both the generator and discriminator utilized the ADAM optimizer (Kingma, 2014), configured with parameters $\beta_{1}=0.5$ and $\beta_{2}=0.999$. The initial learning rate was set to 0.0002 and was reduced to $1/1.00004$ of its previous value after each epoch. To improve the stability and effectiveness of the training process, we adopted a strategy of updating the generator once every three iterations of training the discriminator. This approach helps maintain a balance between the model’s generative and discriminative capabilities, effectively enhancing the transformation from low-quality to high-quality images. The training was concluded after 500 epochs.

To qualitatively evaluate the performance of the Pix2WGAN model and the effects of image transformation, we visually examined the generated images and selected six representative galaxies from the testing set for presentation. These galaxies were chosen based on their diverse morphological features and varying complexities, enabling us to assess the model’s ability to capture different structures and details effectively.

For clarity, we organized the images of these selected galaxies into two figures: Figure 3 and Figure 4. Each figure displays three galaxies, illustrating their images before and after transformation alongside the target images. The first row presents the original DECaLS images, the second row showcases the model-transformed pseudo-HSC images, and the third row features the corresponding actual HSC images.

Through these images, we can clearly observe the quality and detail of the pseudo-HSC images generated from various DECaLS galaxy images after the Pix2WGAN transformation. This observation validates the effectiveness of the transformation method and its ability to enhance the features and details of celestial objects. By comparing the galaxy images in Figures 3 and 4, we note that the pseudo-HSC images produced by the Pix2WGAN model (second row) exhibit clearer structural features and richer details than the original DECaLS images (first row).

For example, in the first galaxy (leftmost column in both figures), the Pix2WGAN model nearly perfectly reconstructs the spiral structure with relatively smooth lines. Additionally, the disk structure of the second galaxy in Figure 4 (middle column) is more pronounced. Furthermore, the tidal tail of the second galaxy in Figure 3 (middle column) and the point-like structure of the third galaxy in Figure 4 (rightmost column) are also clearly reproduced by the model, whereas these features are nearly invisible in the original DECaLS images. This clearly demonstrates Pix2WGAN’s exceptional performance in image transformation.

Furthermore, through systematic comparisons of model-reconstructed pseudo-HSC images and actual HSC observational images, we found that the Pix2WGAN model can generate images that outperform actual results in specific cases, particularly when the observational HSC images exhibit artifacts, noise, or overexposure. To illustrate this, we selected three galaxies from the test samples that display noticeable issues, as shown in Figure 5. In the bottom row of the figure, the first galaxy shows streak-like artifacts, the second exhibits prominent noise, and the third is partially overexposed. After applying the Pix2WGAN model to transform the DECaLS images, the reconstructed pseudo-HSC images effectively removed artifacts, reduced noise, and corrected overexposure.

On the other hand, we also found that when the input images contain poor-quality data (such as artifacts, significant noise, and blurriness), the pseudo-HSC images generated after transformation by our Pix2WGAN model can effectively alleviate these issues. Figure 6 presents examples in this regard, where these galaxies exhibit clear problems in the DECaLS images: the leftmost galaxy shows noticeable streak-like artifacts, the middle galaxy has significant noise interference, and the rightmost galaxy’s image appears blurry. We observed that after the model transformation, the issues of all three galaxies showed significant improvement. Specifically, the streak-like artifacts in the leftmost DECaLS galaxy almost disappeared after transformation, the noise in the middle galaxy was greatly reduced, and the blurriness in the rightmost galaxy’s image was also alleviated to some extent.

In fact, a careful analysis of Figures 3 and 4 also shows that the Pix2WGAN model effectively reduces background noise during the generation of pseudo-HSC images. These results further highlight the potential of the Pix2WGAN model in addressing issues related to image artifacts, noise, overexposure, and blurriness. Particularly when the quality of observational images is limited or flawed, the model demonstrates the ability to provide effective image enhancement and repair capabilities.

However, it is important to note that the model-predicted pseudo-HSC images (second row) shown in Figures 3 and 4 exhibit similarities in structure and detail to the actual HSC observational images (third row); however, they still fall short in overall clarity and detail representation. This suggests that while deep learning models can significantly enhance the quality of the original images, they cannot completely replace the need for hardware upgrades and improvements. Therefore, the most effective strategy for improving image quality is to combine advanced deep learning techniques with high-performance hardware.

In this subsection, we explore the image transformation from SDSS to HSC. Compared to the transformation from DECaLS to HSC, there is a significant difference in the quality of the observed images produced by the SDSS and HSC survey projects. For example, the pixel resolution of SDSS images is 0.396 arcseconds per pixel, while HSC images have a resolution of 0.168 arcseconds per pixel, indicating that HSC images possess approximately 2.357 times the resolution advantage. This disparity is notably greater than the roughly 1.511 times difference between DECaLS and HSC, which makes the direct application of the Pix2WGAN model for transforming SDSS images to HSC images less than optimal.

To achieve better results in image transformation, we further enhanced the architecture of the hybrid model Pix2WGAN, allowing it to utilize intermediate images as a bridge. These intermediate images can be either DECaLS images or degraded HSC images (e.g., reducing the resolution of HSC images by approximately half). In this study, we have chosen DECaLS images as the bridging images, enabling us to simultaneously incorporate images from SDSS, DECaLS, and HSC for cascade training. We refer to this modified model as the Cascade Pix2WGAN model. The Cascade architecture of the model is illustrated in Figure 7, and the detailed training process is described as follows:

First, the SDSS images are input into the Cascade Pix2WGAN model, which generates pseudo-DECaLS images using the first generator, $G_{1}$. Next, the pseudo-DECaLS images generated by $G_{1}$ are paired with the original SDSS images to form one set, while another set consists of real DECaLS images along with the original SDSS images. Both sets of images are then input into the first discriminator, $D_{1}$, whose task is to measure the difference between the real and fake images in these sets.

Subsequently, the pseudo-DECaLS images generated by $G_{1}$ are fed into the second generator, $G_{2}$, to produce pseudo-HSC images. The pseudo-HSC images are paired with their corresponding pseudo-DECaLS images to form one set, while another set consists of the corresponding real HSC images and pseudo-DECaLS images. These two sets of images are then input into the second discriminator, $D_{2}$, which learns to measure the difference between the real and fake images in this context.

All the generators and discriminators in the Cascade Pix2WGAN model share the same structures and parameter settings as described in Section 4.2. During each training epoch, the generators $G_{1}$ and $G_{2}$, along with the discriminators $D_{1}$ and $D_{2}$, are optimized simultaneously. After multiple iterations, the final trained model can directly transform the original SDSS images into pseudo-HSC images using generators $G_{1}$ and $G_{2}$, without the need for DECaLS images as an intermediate bridge.

To facilitate comparison with the previously mentioned results, we selected six galaxies from the test set that are the same as those shown in Figures 3 and 4. The transformation effects of the cascade model are displayed in Figures 8 and 9. Each figure showcases the transformation visual effects of the three galaxies: the top row displays the original SDSS images, the middle row presents the pseudo-HSC images generated by the Cascade Pix2WGAN model, and the bottom row shows the corresponding real HSC images.

From Figures 8 and 9, it is evident that our Cascade Pix2WGAN model has achieved remarkable results. Many structural features that are not visible to the naked eye in SDSS—such as spiral arms, disks, tidal tails, bridges, and fine structures near the centers of galaxies—are clearly presented. The pseudo-HSC images generated by the model closely resemble the actual observed HSC images in terms of galaxy structure, morphology, and color. Compared to the original SDSS images, the transformed images exhibit significant improvements in resolution and detail recognition capability. We believe that these transformed SDSS images will serve as reliable data sources for future research in galaxy morphology classification and structural feature recognition. Additionally, as noted in the previous section, the generated pseudo-HSC images effectively reduce background noise, thereby improving the signal-to-noise ratio of the galaxy images.

Furthermore, we compared the middle row panels of Figures 8 and 9 with those in Figures 3 and 4. The results indicate that the pseudo-HSC images generated directly from SDSS images using the Cascade Pix2WGAN model perform slightly worse than those generated from DECaLS images using the Pix2WGAN model. This observation suggests that while the model’s performance is important, the quality of the initial input images is even more critical. We will explore this point in more detail in our subsequent quantitative comparisons.

In sections 4.2 and 4.3, we conducted a visual inspection of the generated pseudo-HSC images. The results indicated that the Pix2WGAN model significantly enhances the quality of SDSS and DECaLS images, and the generated pseudo-HSC images exhibit a high degree of similarity to the real HSC images. To further validate the model’s performance, this subsection introduces several quantitative evaluation metrics to assess the differences between the generated pseudo-HSC images and the real HSC images. These metrics include Mean Squared Error (MSE), Peak Signal-to-Noise Ratio (PSNR), and Structural Similarity Index (SSIM; Sheikh et al. 2004), as well as two perceptual metrics: Learned Perceptual Image Patch Similarity (LPIPS; Zhang et al. 2018a) and Fréchet Inception Distance (FID; Heusel et al. 2017).

MSE is a commonly used image similarity metric that measures the degree of difference between two images. This algorithm calculates the squared differences between corresponding pixels of the two images and then takes the average of these values to obtain a similarity score. A smaller MSE value indicates greater similarity between the two images, with a value of 0 representing that they are identical. For two multi-channel images of size $M\times N\times C$, the calculation formula is:

where $I$ is the predicted image generated by the model and $K$ is the real image.

PSNR is also an important metric widely used to evaluate the performance of various image processing algorithms. Its value is derived from the MSE, establishing an interdependent relationship between the two metrics. PSNR is typically expressed in decibels (dB), where a higher value indicates greater similarity between the generated image and the real image, as well as better image quality. The calculation formula for PSNR is as follows:

where $MAX$ represents the maximum possible pixel value in the image; in this paper, the maximum pixel value is 255.

Unlike MSE and PSNR, which evaluate the quality of generated images based on pixel differences, SSIM measures the similarity between the real image and the generated image by comparing their brightness, contrast, and structural information. In contrast to MSE and PSNR, SSIM places greater emphasis on structural similarity rather than on pixel-by-pixel brightness differences, making it a more accurate reflection of human visual perception of image similarity. The SSIM value ranges from [-1, 1], with higher values indicating greater similarity between the two images. Its calculation involves comparing multiple features, including mean, variance, covariance, and two constants used for stability, as shown in the following formula:

In the formula, $x$ and $y$ represent the generated image and the real image, respectively, while $\mu_{x}$, $\mu_{y}$, $\sigma_{x}^{2}$, $\sigma_{y}^{2}$, and $\sigma_{xy}$ represent the mean, variance, and covariance of the generated image $x$ and the real image $y$. $C_{1}=(K_{1}L)^{2}$ and $C_{2}=(K_{2}L)^{2}$, where $L$ is the dynamic range of pixel values, and $K1$ and $K2$ are small constants used to ensure stability in calculations. In this study, $L=255$, $K_{1}=0.01$, and $K_{2}=0.03$.

In addition, to better assess the differences between generated images and real images from the perspective of the human visual system, we introduce two commonly used metrics in deep learning: LPIPS and FID. These metrics evaluate image differences based on features extracted by deep learning models, enabling a more effective capture of perceptual similarity. Smaller values of LPIPS and FID indicate higher perceptual similarity between the two images. For detailed definitions and calculation methods of LPIPS and FID, please refer to the studies by Zhang et al. (2018a) and Szegedy et al. (2015). Compared to traditional image quality assessment methods, these metrics provide a more comprehensive analysis and complement the evaluation of generated image quality.

Using these evaluation metrics, we quantitatively assessed the similarity between the model-generated pseudo-HSC images and the real HSC observational images, as well as the similarity between the pre-transformed images and the real HSC images. The results for the test set are presented in Table 1. In the table, the superscript “1” after the term “pseudo-HSC” indicates that these images were generated from DECaLS images using the Pix2WGAN model, while the superscript “2” denotes that the images were generated from SDSS images using the Cascaded Pix2WGAN model.

From Table 1, it is evident that the similarity between the pseudo-HSC images generated by the Cascaded Pix2WGAN model and the real HSC images is significantly higher across all metrics compared to the similarity between the original SDSS images and the real HSC images. Specifically, the SSIM measure increased significantly from 0.43 to 0.71, PSNR improved from 17.59 dB to 23.19 dB, and MSE decreased from 5.24 to 3.25. At the same time, the LPIPS and FID metrics also showed substantial improvement, with LPIPS decreasing from 0.19 to 0.11 and FID dropping from 179.59 to 105.16.

Similarly, the similarity between the pseudo-HSC images generated by the Pix2WGAN model and the real HSC images is significantly higher on most metrics compared to the similarity between the original DECaLS images and the real HSC images. Specifically, the SSIM metric improved from 0.56 to 0.72, PSNR increased from 19.95 dB to 24.21 dB, and MSE decreased from 3.58 to 2.99. This indicates that the Pix2WGAN model excels in reconstructing the details and optical quality of the images.

However, it is worth noting that the LPIPS and FID metrics showed limited change, with LPIPS rising slightly from 0.06 to 0.09 and FID increasing from 78.91 to 98.46. This may be because both LPIPS and FID rely on pre-trained deep learning models (such as VGG and Inception) to capture image features and assess similarity between images, and the performance of these models largely depends on the datasets on which they were trained. Generally, the pre-trained models for LPIPS and FID are trained on the ImageNet dataset, which encompasses a wide variety of objects and scenes but does not include astronomical images. Therefore, when evaluating galaxy images, these models may not effectively capture the unique structures, features, and visual information specific to galaxy images, leading to small differences in LPIPS and FID that do not accurately reflect the similarity or differences between galaxy images.

Nonetheless, the overall evaluation still indicates that the Pix2WGAN model effectively enhances image quality, particularly in terms of preserving structural integrity and visual information.

Furthermore, the results in Table 1 indicate that the pseudo-HSC images generated from DECaLS images using the Pix2WGAN model outperform those generated from SDSS images with the Cascaded Pix2WGAN model across all evaluation metrics. This suggests that while optimizing the performance of the model itself is crucial for the generated results, the quality of the original input images should not be overlooked. High-quality input images provide the model with richer details and more accurate information, significantly enhancing the transformation results. This is also visually evident in the comparisons between Figures 3, 4, and Figures 8, 9: the pseudo-HSC images generated from DECaLS images show superior performance in detail restoration, signal-to-noise ratio, and clarity compared to the results generated from SDSS images.