## 1 Introduction  
The use of light microscopy for minimally invasive high-resolution imaging is fundamental to human understanding of biological systems and processes. In clinical pathology, analyzing high-resolution color pathological images from optical microscopes remains a widely used standard method for diagnosing diseases ranging from cancer to blood-borne infections. To perform quantitative and accelerated pathological image analysis, the whole slide imaging (WSI) [1] system is gradually replacing traditional optical microscopes. Fourier ptychographic microscopy (FPM) [2, 3], which has been popularized in recent years, is considered a high-throughput imaging technology that does not require mechanical scanning and has more advantages over conventional whole slide imaging systems in digital pathology. First of all, FPM does not require mechanical scanning to achieve both high resolution (HR) and a large field of view, and it can simply add LED arrays to other microscopes to set up experimental platforms. Secondly, the digital refocusing process is utilized by FPM to correct sample defocusing after acquisition [2, 4]. Furthermore, the phase information recovered from the FPM provides more localized scattering details of the sample, which enhances the accuracy and reliability of pathological diagnosis [5, 6]. However, the application of FPM in pathology still faces some challenges [6]. On the one hand, the time efficiency of FPM acquisition and reconstruction is relatively poor. FPM usually requires continuous acquisition with red, green, and blue separate illumination, composing hundreds of low-resolution (LR) images for better color reconstruction, leading to slower acquisition and reconstruction speeds. On the other hand, FPM color reconstruction images are usually of poor quality due to coherent artifacts caused by light interference and reconstruction errors. Furthermore, the GlaS dataset in Figure 1 is pathological sections of the differentiation tissue of colon cancer glands stained with hematoxylin and eosin (H&E), and the MoNuSeg dataset is composed of various H&E stained histological images across multiple patients and organs obtained from different hospitals. It can be observed in Figure 1 that the color reconstruction results of the same batch of pathological sections will have significant visual differences due to the influence of factors such as collection personnel, technology, and the external environment. In biological and clinical medicine applications, the color information of the sample after staining with reagents helps the observer quickly locate the regions of interest and interpret the relevant information about tissues and cells. Therefore, we use a virtual staining method for the colorful reconstruction of FPM single-channel grayscale images. Recently, significant advances have been made in the field of deep learning, paving the way for the application of neural networks in a variety of image-processing tasks, including virtual tissue staining. To better obtain the color information of FPM-stained tissue sections, image translation can be applied to convert the recovered monochromatic FPM image to the style of regular incoherent microscopy. The mathematical mapping between input and output [7] can be found accurately by this deep learning technology based on the deep convolutional neural network (CNN), and the coherence artifacts of FPM recovery can also be reduced [8], thus improving the quality of colored FPM pathology images. Since its inception, the unsupervised learning image-to-image translation model CycleGAN [9] has shown its powerful transformation ability between two image domains, which makes it a popular method of virtual staining in the field of biomedical imaging. However, the cyclic consistency [9] assumption dictates that the relationship between the two domains must be bijective [10], which is often not ideal, as it may limit the diversity of images.