This paper is concerned with an inverse random source scattering problem in an inhomogeneous background medium. The wave propagation is modeled by the stochastic Helmholtz equation with the source driven by additive white noise. The goal is to reconstruct the statistical properties of the random source such as the mean and variance from the boundary measurement of the radiated random wave field at multiple frequencies. Both the direct and inverse problems are considered. We show that the direct problem has a unique mild solution by a constructive proof. For the inverse problem, we derive Fredholm integral equations, which connect the boundary measurement of the radiated wave field with the unknown source function. A regularized block Kaczmarz method is developed to solve the ill-posed integral equations. Numerical experiments are included to demonstrate the effectiveness of the proposed method.

Membranous nephropathy is one of the most prevalent conditions responsible for nephrotic syndrome in adults. It is clinically nonspecific and mainly diagnosed by kidney biopsy pathology, with three prevalent techniques: light microscopy, electron microscopy, and immunofluorescence microscopy. Manual observation of glomeruli one by one under the microscope is very time-consuming, and there are certain differences in the observation results between physicians. This study makes use of whole-slide images scanned by a light microscope as well as immunofluorescence images to classify patients with membranous nephropathy. The framework mainly includes a glomerular segmentation module, a confidence coefficient extraction module, and a multi-modal fusion module. This framework first identifies and segments the glomerulus from whole-slide images and immunofluorescence images, and then a glomerular classifier is trained to extract the features of each glomerulus. The results are then combined to produce the final diagnosis. The results of the experiments show that the F1-score of image classification results obtained by combining two kinds of features, which can reach 97.32%, is higher than those obtained by using only light-microscopy-observed images or immunofluorescent images, which reach 92.76% and 93.20%, respectively. Experiments demonstrate that considering both WSIs and immunofluorescence images is effective in improving the diagnosis of membranous nephropathy.