🍩 Database of Original & Non-Theoretical Uses of Topology

(found 2 matches in 0.000877s)
  1. Generalized Penalty for Circular Coordinate Representation (2020)

    Hengrui Luo, Alice Patania, Jisu Kim, Mikael Vejdemo-Johansson
    Abstract Topological Data Analysis (TDA) provides novel approaches that allow us to analyze the geometrical shapes and topological structures of a dataset. As one important application, TDA can be used for data visualization and dimension reduction. We follow the framework of circular coordinate representation, which allows us to perform dimension reduction and visualization for high-dimensional datasets on a torus using persistent cohomology. In this paper, we propose a method to adapt the circular coordinate framework to take into account sparsity in high-dimensional applications. We use a generalized penalty function instead of an \$L_\2\\$ penalty in the traditional circular coordinate algorithm. We provide simulation experiments and real data analysis to support our claim that circular coordinates with generalized penalty will accommodate the sparsity in high-dimensional datasets under different sampling schemes while preserving the topological structures.
  2. Combining Geometric and Topological Information in Image Segmentation (2019)

    Hengrui Luo, Justin Strait
    Abstract A fundamental problem in computer vision is image segmentation, where the goal is to delineate the boundary of an object in the image. The focus of this work is on the segmentation of grayscale images and its purpose is two-fold. First, we conduct an in-depth study comparing active contour and topology-based methods in a statistical framework, two popular approaches for boundary detection of 2-dimensional images. Certain properties of the image dataset may favor one method over the other, both from an interpretability perspective as well as through evaluation of performance measures. Second, we propose the use of topological knowledge to assist an active contour method, which can potentially incorporate prior shape information. The latter is known to be extremely sensitive to algorithm initialization, and thus, we use a topological model to provide an automatic initialization. In addition, our proposed model can handle objects in images with more complex topological structures, including objects with holes and multiple objects within one image. We demonstrate this on artificially-constructed image datasets from computer vision, as well as real medical image data.