2 research outputs found
The Theory of Computational Quasi-conformal Geometry on Point Clouds
Quasi-conformal (QC) theory is an important topic in complex analysis, which
studies geometric patterns of deformations between shapes. Recently,
computational QC geometry has been developed and has made significant
contributions to medical imaging, computer graphics and computer vision.
Existing computational QC theories and algorithms have been built on
triangulation structures. In practical situations, many 3D acquisition
techniques often produce 3D point cloud (PC) data of the object, which does not
contain connectivity information. It calls for a need to develop computational
QC theories on PCs. In this paper, we introduce the concept of computational QC
geometry on PCs. We define PC quasi-conformal (PCQC) maps and their associated
PC Beltrami coefficients (PCBCs). The PCBC is analogous to the Beltrami
differential in the continuous setting. Theoretically, we show that the PCBC
converges to its continuous counterpart as the density of the PC tends to zero.
We also theoretically and numerically validate the ability of PCBCs to measure
local geometric distortions of PC deformations. With these concepts, many
existing QC based algorithms for geometry processing and shape analysis can be
easily extended to PC data
Decomposition of Longitudinal Deformations via Beltrami Descriptors
We present a mathematical model to decompose a longitudinal deformation into
normal and abnormal components. The goal is to detect and extract subtle
quivers from periodic motions in a video sequence. It has important
applications in medical image analysis. To achieve this goal, we consider a
representation of the longitudinal deformation, called the Beltrami descriptor,
based on quasiconformal theories. The Beltrami descriptor is a complex-valued
matrix. Each longitudinal deformation is associated to a Beltrami descriptor
and vice versa. To decompose the longitudinal deformation, we propose to carry
out the low rank and sparse decomposition of the Beltrami descriptor. The low
rank component corresponds to the periodic motions, whereas the sparse part
corresponds to the abnormal motions of a longitudinal deformation. Experiments
have been carried out on both synthetic and real video sequences. Results
demonstrate the efficacy of our proposed model to decompose a longitudinal
deformation into regular and irregular components.Comment: arXiv admin note: text overlap with arXiv:1402.6908 by other author