This paper aims to describe the geometrical structure and explicit expressions of family of finitely parametrized probability densities over smooth manifold $M$. The geometry of family of probability densities on $M$ are inherited from probability densities on Euclidean spaces $\left\{U_\alpha \right\}$ via bundle morphisms, induced by an orientation-preserving diffeomorphisms $\rho_\alpha:U_\alpha \rightarrow M$. Current literature inherits densities on $M$ from tangent spaces via Riemannian exponential map $\exp: T_x M \rightarrow M$; densities on $M$ are defined locally on region where the exponential map is a diffeomorphism. We generalize this approach with an arbitrary orientation-preserving bundle morphism; we show that the dualistic geometry of family of densities on $U_\alpha$ can be inherited to family of densities on $M$. Furthermore, we provide explicit expressions for parametrized probability densities on $\rho_\alpha(U_\alpha) \subset M$. Finally, using the component densities on $\rho_\alpha(U_\alpha)$, we construct parametrized mixture densities on totally bounded subsets of $M$. We provide a description of inherited mixture product dualistic geometry of the family of mixture densities.

Macula fovea detection is a crucial prerequisite towards screening and diagnosing macular diseases. Without early detection and proper treatment, any abnormality involving the macula may lead to blindness. However, with the ophthalmologist shortage and time-consuming artificial evaluation, neither accuracy nor effectiveness of the diagnose process could be guaranteed. In this project, we proposed a deep learning approach on ultra-widefield fundus (UWF) images for macula fovea detection. This study collected 2300 ultra-widefield fundus images from Shenzhen Aier Eye Hospital in China. Methods based on U-shape network (Unet) and Fully Convolutional Networks (FCN) are implemented on 1800 (before amplifying process) training fundus images, 400 (before amplifying process) validation images and 100 test images. Three professional ophthalmologists were invited to mark the fovea. A method from the anatomy perspective is investigated. This approach is derived from the spatial relationship between macula fovea and optic disc center in UWF. A set of parameters of this method is set based on the experience of ophthalmologists and verified to be effective. Results are measured by calculating the Euclidean distance between proposed approaches and the accurate grounded standard, which is detected by Ultra-widefield swept-source optical coherence tomograph (UWF-OCT) approach. Through a comparation of proposed methods, we conclude that, deep learning approach of Unet outperformed other methods on macula fovea detection tasks, by which outcomes obtained are comparable to grounded standard method.