3D time series analysis of cell shape using Laplacian approaches

Background: Fundamental cellular processes such as cell movement, division or food uptake critically depend on cells being able to change shape. Fast acquisition of three-dimensional image time series has now become possible, but we lack efficient tools for analysing shape deformations in order to understand the real three-dimensional nature of shape changes. Results: We present a framework for 3D+time cell shape analysis. The main contribution is three-fold: First, we develop a fast, automatic random walker method for cell segmentation. Second, a novel topology fixing method is proposed to fix segmented binary volumes without spherical topology. Third, we show that algorithms used for each individual step of the analysis pipeline (cell segmentation, topology fixing, spherical parameterization, and shape representation) are closely related to the Laplacian operator. The framework is applied to the shape analysis of neutrophil cells. Conclusions: The method we propose for cell segmentation is faster than the traditional random walker method or the level set method, and performs better on 3D time-series of neutrophil cells, which are comparatively noisy as stacks have to be acquired fast enough to account for cell motion. Our method for topology fixing outperforms the tools provided by SPHARM-MAT and SPHARM-PDM in terms of their successful fixing rates. The different tasks in the presented pipeline for 3D+time shape analysis of cells can be solved using Laplacian approaches, opening the possibility of eventually combining individual steps in order to speed up computations

Learning SO(3) Equivariant Representations with Spherical CNNs

We address the problem of 3D rotation equivariance in convolutional neural networks. 3D rotations have been a challenging nuisance in 3D classification tasks requiring higher capacity and extended data augmentation in order to tackle it. We model 3D data with multi-valued spherical functions and we propose a novel spherical convolutional network that implements exact convolutions on the sphere by realizing them in the spherical harmonic domain. Resulting filters have local symmetry and are localized by enforcing smooth spectra. We apply a novel pooling on the spectral domain and our operations are independent of the underlying spherical resolution throughout the network. We show that networks with much lower capacity and without requiring data augmentation can exhibit performance comparable to the state of the art in standard retrieval and classification benchmarks.Comment: Camera-ready. Accepted to ECCV'18 as oral presentatio

Geometric modeling and optimization over regular domains for graphics and visual computing

The effective construction of parametric representation of complicated geometric objects can facilitate many design, analysis, and simulation tasks in Computer-Aided Design (CAD), Computer-Aided Manufacturing (CAM), and Computer-Aided Engineering (CAE). Given a 3D shape, the procedure of finding such a parametric representation upon a canonical domain is called geometric parameterization. Regular geometric regions, such as polycubes and spheres, are desirable domains for parameterization. Parametric representations defined upon regular geometric domains have many desirable mathematical properties and can facilitate or simplify various surface/solid modeling and processing computation. This dissertation studies the construction of parameterization on regular geometric domains and explores their applications in shape modeling and computer-aided design. Specifically, we studies (1) the surface parameterization on the spherical domain for closed genus-zero surfaces; (2) the surface parameterization on the polycube domain for general closed surfaces; and (3) the volumetric parameterization for 3D-manifolds embedded in 3D Euclidean space. We propose novel computational models to solve these geometric problems. Our computational models reduce to nonlinear optimizations with various geometric constraints. Hence, we also need to explore effective optimization algorithms. The main contributions of this dissertation are three-folded. (1) We developed an effective progressive spherical parameterization algorithm, with an efficient nonlinear optimization scheme subject to the spherical constraint. Compared with the state-of-the-art spherical mapping algorithms, our algorithm demonstrates the advantages of great efficiency, lower distortion, and guaranteed bijectiveness, and we show its applications in spherical harmonic decomposition and shape analysis. (2) We propose a first topology-preserving polycube domain optimization algorithm that simultaneously optimizes polycube domain together with the parameterization to balance the mapping distortion and domain simplicity. We develop effective nonlinear geometric optimization algorithms dealing with variables with and without derivatives. This polycube parameterization algorithm can benefit the regular quadrilateral mesh generation and cross-surface parameterization. (3) We develop a novel quaternion-based optimization framework for 3D frame field construction and volumetric parameterization computation. We demonstrate our constructed 3D frame field has better smoothness, compared with state-of-the-art algorithms, and is effective in guiding low-distortion volumetric parameterization and high-quality hexahedral mesh generation

Non-rigid registration of 2-D/3-D dynamic data with feature alignment

In this work, we are computing the matching between 2D manifolds and 3D manifolds with temporal constraints, that is we are computing the matching among a time sequence of 2D/3D manifolds. It is solved by mapping all the manifolds to a common domain, then build their matching by composing the forward mapping and the inverse mapping. At first, we solve the matching problem between 2D manifolds with temporal constraints by using mesh-based registration method. We propose a surface parameterization method to compute the mapping between the 2D manifold and the common 2D planar domain. We can compute the matching among the time sequence of deforming geometry data through this common domain. Compared with previous work, our method is independent of the quality of mesh elements and more efficient for the time sequence data. Then we develop a global intensity-based registration method to solve the matching problem between 3D manifolds with temporal constraints. Our method is based on a 4D(3D+T) free-from B-spline deformation model which has both spatial and temporal smoothness. Compared with previous 4D image registration techniques, our method avoids some local minimum. Thus it can be solved faster and achieve better accuracy of landmark point predication. We demonstrate the efficiency of these works on the real applications. The first one is applied to the dynamic face registering and texture mapping. The second one is applied to lung tumor motion tracking in the medical image analysis. In our future work, we are developing more efficient mesh-based 4D registration method. It can be applied to tumor motion estimation and tracking, which can be used to calculate the read dose delivered to the lung and surrounding tissues. Thus this can support the online treatment of lung cancer radiotherapy