On the local behavior of spaces of range image patches
- PDF / 4,553,728 Bytes
- 32 Pages / 439.642 x 666.49 pts Page_size
- 11 Downloads / 170 Views
On the local behavior of spaces of range image patches Jinhong Li1 · Shengxiang Xia2 Received: 10 July 2019 / Revised: 20 June 2020 / Accepted: 16 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We focus on the quantitative and local topological properties of range images. We consider the spaces Mm of m × m high-contrast patches of range images for m=3, 5, 7, 9, 11. Using computational topological tools to analyze range image patches, we detect that M3 (M9 , M11 ) has core subsets with the topology of a circle, M3 , M5 , M7 , M9 and that M11 have some subspaces with the topology of a Klein bottle. We also discover that the largest subspace with the Klein bottle’s topology decreases as the measurements of patches increase, which generalizes the results in the paper of H. Adams and G. Carlsson, and demonstrates properties among optical images and range image patches, which are more similar than those established by Lee et al. Keywords Range images · Topology · Persistent homology · Klein bottle · Barcode
1 Introduction Range images have attracted an increasing amount of interest in recent years for two reasons: First, they can be employed in various industrial applications [13, 17]; second, more practical models of optical images can be built by using a range image [4, 14]. Every range image can be considered to be a very high-dimensional vector of a space P . When we directly study a set of images G ⊆ P , we encounter the high-dimensional problem of G and the sparsity of G in P . One approach is to analyze the state space of local patterns of pixel values, which are modeled by small patches of images. Reducing the dimension of the problem is one advantage of locally analyzing a range image space, and another advantage (suggested by Field [9] and Hateren [22]) is that many global statistical properties of
Shengxiang Xia
[email protected] Jinhong Li [email protected] 1
School of Mathematics and Statistics, Qilu University of Technology, Jinan, 250353, People’s Republic of China
2
School of Science, Shandong Jianzhu University, Jinan, 250101, People’s Republic of China
Multimedia Tools and Applications
the space can be provided by local statistics. The authors of [15] present several interesting observations about the resulting spaces of optical and range images 3 × 3 patches; for example, they discover that most of the optical 3 × 3 high-contrast patches are concentrated around a 2-dimensional loop. In [6], Carlsson et al. analyzed optical patches (studied by Lee et al. [15]) by using computational topological tools; they find that there is a large 2-dimensional subset with the same homology as a Klein bottle. The authors establish a relation between the optical patch space and the 2-variable polynomials and applied it to prove the existence of a subspace of the optical patch space with a topology that resembles that of a Klein bottle. In [2], the authors revealed that 5 × 5 and 7 × 7 range image patches possess the topology of a circle. In this study, we u
Data Loading...
