Statistical Modeling of the 3D Geometry and Topology of Botanical Trees

We propose a framework for statistical modeling of the 3D geometry and topology of botanical trees. We treat botanical trees as points in a tree‐shape space equipped with a proper metric that captures the geometric and the topological differences between trees. Geodesics in the tree‐shape space correspond to the optimal sequence of deformations, i.e. bending, stretching, and topological changes, which align one tree onto another. In this way, the 3D tree modeling and synthesis problem becomes a problem of exploring the tree‐shape space either in a controlled fashion, using statistical regression, or randomly by sampling from probability distributions fitted to populations in the tree‐shape space. We show how to use this framework for (1) computing statistical summaries, e.g. the mean and modes of variations, of a population of botanical trees, (2) synthesizing random instances of botanical trees from probability distributions fitted to a population of botanical trees, and (3) modeling, interactively, 3D botanical trees using a simple sketching interface. The approach is fast and only requires as input 3D botanical tree models with a known upright orientation.     

基于机器学习和几何变换的实时2D/3D脊椎配准

在图像引导的脊柱手术中,实时高效的2D/3D配准是一项重要且具有挑战性的任务.通常的2D/3D配准一般是将三维图像投影到二维平面,然后进行2D/2D的配准.由于投影空间涉及到3个平移以及3个旋转参数,其投影空间的复杂度为O（n6）,使得配准很难兼具高准确性和高实时性.本文提出了一个结合机器学习与几何变换的2D/3D配准方法,首先,使用统计形状模型对目标脊椎进行建模,并构建了一种新的投影方式,使得6个投影参数中的4个可以使用几何的方法计算出来;接下来利用回归学习的方法学习目标脊椎的形状与投影参数之间的关系;最终,结合学到的关系和几何变换完成配准.本方法的两个姿态参数的平均预测误差为0.84°和0.81°,平均目标配准误差（Mean target registration error,mTRE）为0.87mm,平均配准时间为0.9s.实验结果表明本方法具有很好的实时性和准确性.     

A generic scheme for progressive point cloud coding

In this paper, we propose a generic point cloud encoder that provides a unified framework for compressing different attributes of point samples corresponding to 3D objects with arbitrary topology. In the proposed scheme, the coding process is led by an iterative octree cell subdivision of the object space. At each level of subdivision, positions of point samples are approximated by the geometry centers of all tree-front cells while normals and colors are approximated by their statistical average within each of tree-front cells. With this framework, we employ attribute-dependent encoding techniques to exploit different characteristics of various attributes. All of these have led to significant improvement in the rate-distortion (R-D) performance and a computational advantage over the state of the art. Furthermore, given sufficient levels of octree expansion, normal space partitioning and resolution of color quantization, the proposed point cloud encoder can be potentially used for lossless coding of 3D point clouds.     

基于轮廓形变的复杂表面重构

本文提出了一种基于自由形变（Free-Form Deformation, FFD）及外轴投影（External Axes Projection, EAP）的复杂表面重构算法.该算法以目标形状的切片轮廓作为输入数据,此后轮廓被嵌入到高维空间有向距离场中,在此隐式空间中,算法主要分为以下三步：生成计算序列,计算序列由计算单元组成,每一个计算单元包含上下相邻的两个轮廓;根据相邻轮廓间的拓扑关系,进行外轴投影（EAP）,以解决潜在的分支问题;在每个计算单元中,根据轮廓长度决定自由形变方向,并进行自由形变,根据自由形变结果,建立轮廓间顶点的一一对应关系,并以此进行表面重构.该方法具有以下特点：输入轮廓可具有任意拓扑结构;所生成表面与输入轮廓完全贴合,生成表面准确,无自我重叠,拓扑关系不发生改变;算法高度并行,执行效率高.实验结果证明,该算法可以解决复杂表面的重构问题.     

Geodesic distance-weighted shape vector image diffusion

This paper presents a novel and efficient surface matching and visualization framework through the geodesic distance-weighted shape vector image diffusion. Based on conformal geometry, our approach can uniquely map a 3D surface to a canonical rectangular domain and encode the shape characteristics (e.g., mean curvatures and conformal factors) of the surface in the 2D domain to construct a geodesic distance-weighted shape vector image, where the distances between sampling pixels are not uniform but the actual geodesic distances on the manifold. Through the novel geodesic distance-weighted shape vector image diffusion presented in this paper, we can create a multiscale diffusion space, in which the cross-scale extrema can be detected as the robust geometric features for the matching and registration of surfaces. Therefore, statistical analysis and visualization of surface properties across subjects become readily available. The experiments on scanned surface models show that our method is very robust for feature extraction and surface matching even under noise and resolution change. We have also applied the framework on the real 3D human neocortical surfaces, and demonstrated the excellent performance of our approach in statistical analysis and integrated visualization of the multimodality volumetric data over the shape vector image.     

Estimating body shape of dressed humans

The paper presents a method to estimate the detailed 3D body shape of a person even if heavy or loose clothing is worn. The approach is based on a space of human shapes, learned from a large database of registered body scans. Together with this database we use as input a 3D scan or model of the person wearing clothes and apply a fitting method, based on ICP (iterated closest point) registration and Laplacian mesh deformation. The statistical model of human body shapes enforces that the model stays within the space of human shapes. The method therefore allows us to compute the most likely shape and pose of the subject, even if it is heavily occluded or body parts are not visible. Several experiments demonstrate the applicability and accuracy of our approach to recover occluded or missing body parts from 3D laser scans.     

3D anatomical shape atlas construction using mesh quality preserved deformable models

3D anatomical shape atlas construction has been extensively studied in medical image analysis research, owing to its importance in model-based image segmentation, longitudinal studies and populational statistical analysis, etc. Among multiple steps of 3D shape atlas construction, establishing anatomical correspondences across subjects, i.e., surface registration, is probably the most critical but challenging one. Adaptive focus deformable model (AFDM) [1] was proposed to tackle this problem by exploiting cross-scale geometry characteristics of 3D anatomy surfaces. Although the effectiveness of AFDM has been proved in various studies, its performance is highly dependent on the quality of 3D surface meshes, which often degrades along with the iterations of deformable surface registration (the process of correspondence matching). In this paper, we propose a new framework for 3D anatomical shape atlas construction. Our method aims to robustly establish correspondences across different subjects and simultaneously generate high-quality surface meshes without removing shape details. Mathematically, a new energy term is embedded into the original energy function of AFDM to preserve surface mesh qualities during deformable surface matching. More specifically, we employ the Laplacian representation to encode shape details and smoothness constraints. An expectation–maximization style algorithm is designed to optimize multiple energy terms alternatively until convergence. We demonstrate the performance of our method via a set of diverse applications, including a population of sparse cardiac MRI slices with 2D labels, 3D high resolution CT cardiac images and rodent brain MRIs with multiple structures. The constructed shape atlases exhibit good mesh qualities and preserve fine shape details. The constructed shape atlases can further benefit other research topics such as segmentation and statistical analysis.     

Break and Splice: A Statistical Method for Non-Rigid Point Cloud Registration

3D object matching and registration on point clouds are widely used in computer vision. However, most existing point cloud registration methods have limitations in handling non-rigid point sets or topology changes (e.g. connections and separations). As a result, critical characteristics such as large inter-frame motions of the point clouds may not be accurately captured. This paper proposes a statistical algorithm for non-rigid point sets registration, addressing the challenge of handling topology changes without the need to estimate correspondence. The algorithm uses a novel Break and Splice framework to treat the non-rigid registration challenges as a reproduction process and a Dirichlet Process Gaussian Mixture Model (DPGMM) to cluster a pair of point sets. Labels are assigned to the source point set with an iterative classification procedure, and the source is registered to the target with the same labels using the Bayesian Coherent Point Drift (BCPD) method. The results demonstrate that the proposed approach achieves lower registration errors and efficiently registers point sets undergoing topology changes and large inter-frame motions. The proposed approach is evaluated on several data sets using various qualitative and quantitative metrics. The results demonstrate that the Break and Splice framework outperforms state-of-the-art methods, achieving an average error reduction of about 60% and a registration time reduction of about 57.8%.     

Space–time blending

Shape transformation between objects of different topology and positions in space is an open modelling problem. We propose a new approach to solving this problem for two given 2D or 3D shapes. The key steps of the proposed algorithm are: increase dimension by converting two input kD shapes into half‐cylinders in (k+1)D space–time, applying bounded blending with added material to the half‐cylinders, and making cross‐sections for getting intermediate shapes under the transformation. The additional dimension is considered as a time coordinate for making animation. We use the bounded blending set operations in space–time defined using R‐functions and displacement functions with the localized area of influence applied to the functionally defined half‐cylinders. The proposed approach is general enough to handle input shapes with arbitrary topology defined as polygonal objects with holes and disjoint components, set‐theoretic objects, or analytically defined implicit surfaces. The obtained unusual amoeba‐like behaviour of the shape combines metamorphosis with the non‐linear motion. Copyright © 2004 John Wiley & Sons, Ltd.     

CustomCut: On‐demand Extraction of Customized 3D Parts with 2D Sketches

Several applications in shape modeling and exploration require identification and extraction of a 3D shape part matching a 2D sketch. We present CustomCut, an on‐demand part extraction algorithm. Given a sketched query, CustomCut automatically retrieves partially matching shapes from a database, identifies the region optimally matching the query in each shape, and extracts this region to produce a customized part that can be used in various modeling applications. In contrast to earlier work on sketch‐based retrieval of predefined parts, our approach can extract arbitrary parts from input shapes and does not rely on a prior segmentation into semantic components. The method is based on a novel data structure for fast retrieval of partial matches: the randomized compound k‐NN graph built on multi‐view shape projections. We also employ a coarse‐to‐fine strategy to progressively refine part boundaries down to the level of individual faces. Experimental results indicate that our approach provides an intuitive and easy means to extract customized parts from a shape database, and significantly expands the design space for the user. We demonstrate several applications of our method to shape design and exploration.     

Discriminative Sketch‐based 3D Model Retrieval via Robust Shape Matching

We propose a sketch‐based 3D shape retrieval system that is substantially more discriminative and robust than existing systems, especially for complex models. The power of our system comes from a combination of a contour‐based 2D shape representation and a robust sampling‐based shape matching scheme. They are defined over discriminative local features and applicable for partial sketches; robust to noise and distortions in hand drawings; and consistent when strokes are added progressively. Our robust shape matching, however, requires dense sampling and registration and incurs a high computational cost. We thus devise critical acceleration methods to achieve interactive performance: precomputing kNN graphs that record transformations between neighboring contour images and enable fast online shape alignment; pruning sampling and shape registration strategically and hierarchically; and parallelizing shape matching on multi‐core platforms or GPUs. We demonstrate the effectiveness of our system through various experiments, comparisons, and user studies.     

An interactive local flattening operator to support digital investigations on artwork surfaces

Analyzing either high-frequency shape detail or any other 2D fields (scalar or vector) embedded over a 3D geometry is a complex task, since detaching the detail from the overall shape can be tricky. An alternative approach is to move to the 2D space, resolving shape reasoning to easier image processing techniques. In this paper we propose a novel framework for the analysis of 2D information distributed over 3D geometry, based on a locally smooth parametrization technique that allows us to treat local 3D data in terms of image content. The proposed approach has been implemented as a sketch-based system that allows to design with a few gestures a set of (possibly overlapping) parameterizations of rectangular portions of the surface. We demonstrate that, due to the locality of the parametrization, the distortion is under an acceptable threshold, while discontinuities can be avoided since the parametrized geometry is always homeomorphic to a disk. We show the effectiveness of the proposed technique to solve specific Cultural Heritage (CH) tasks: the analysis of chisel marks over the surface of a unfinished sculpture and the local comparison of multiple photographs mapped over the surface of an artwork. For this very difficult task, we believe that our framework and the corresponding tool are the first steps toward a computer-based shape reasoning system, able to support CH scholars with a medium they are more used to.     

A Colour Interpolation Scheme for Topologically Unrestricted Gradient Meshes

Gradient meshes are a 2D vector graphics primitive where colour is interpolated between mesh vertices. The current implementations of gradient meshes are restricted to rectangular mesh topology. Our new interpolation method relaxes this restriction by supporting arbitrary manifold topology of the input gradient mesh. Our method is based on the Catmull‐Clark subdivision scheme, which is well‐known to support arbitrary mesh topology in 3D. We adapt this scheme to support gradient mesh colour interpolation, adding extensions to handle interpolation of colours of the control points, interpolation only inside the given colour space and emulation of gradient constraints seen in related closed‐form solutions. These extensions make subdivision a viable option for interpolating arbitrary‐topology gradient meshes for 2D vector graphics.     

Multilevel active registration for kinect human body scans: from low quality to high quality

Registration of 3D human body has been a challenging research topic for over decades. Most of the traditional human body registration methods require manual assistance, or other auxiliary information such as texture and markers. The majority of these methods are tailored for high-quality scans from expensive scanners. Following the introduction of the low-quality scans from cost-effective devices such as Kinect, the 3D data capturing of human body becomes more convenient and easier. However, due to the inevitable holes, noises and outliers in the low-quality scan, the registration of human body becomes even more challenging. To address this problem, we propose a fully automatic active registration method which deforms a high-resolution template mesh to match the low-quality human body scans. Our registration method operates on two levels of statistical shape models: (1) the first level is a holistic body shape model that defines the basic figure of human; (2) the second level includes a set of shape models for every body part, aiming at capturing more body details. Our fitting procedure follows a coarse-to-fine approach that is robust and efficient. Experiments show that our method is comparable with the state-of-the-art methods for high-quality meshes in terms of accuracy and it outperforms them in the case of low-quality scans where noises, holes and obscure parts are prevalent.     

Non-rigid registration using distance functions

This paper deals with the registration of geometric shapes. Our primary contribution is the use of a simple and robust shape representation (distance functions) for global-to-local alignment. We propose a rigid-invariant variational framework that can deal as well with local non-rigid transformations. To this end, the registration map consists of a linear motion model and a local deformations field, incrementally recovered. In order to demonstrate the performance of the selected representation a simple criterion is considered, the sum of square differences. Empirical validation and promising results were obtained on examples that exhibit large global motion as well as important local deformations and arbitrary topological changes.     

Elastic geodesic paths in shape space of parameterized surfaces

This paper presents a novel Riemannian framework for shape analysis of parameterized surfaces. In particular, it provides efficient algorithms for computing geodesic paths which, in turn, are important for comparing, matching, and deforming surfaces. The novelty of this framework is that geodesics are invariant to the parameterizations of surfaces and other shape-preserving transformations of surfaces. The basic idea is to formulate a space of embedded surfaces (surfaces seen as embeddings of a unit sphere in IR3) and impose a Riemannian metric on it in such a way that the reparameterization group acts on this space by isometries. Under this framework, we solve two optimization problems. One, given any two surfaces at arbitrary rotations and parameterizations, we use a path-straightening approach to find a geodesic path between them under the chosen metric. Second, by modifying a technique presented in [25], we solve for the optimal rotation and parameterization (registration) between surfaces. Their combined solution provides an efficient mechanism for computing geodesic paths in shape spaces of parameterized surfaces. We illustrate these ideas using examples from shape analysis of anatomical structures and other general surfaces.     

Constrained 3D shape reconstruction using a combination of surface fitting and registration

We investigate 3D shape reconstruction from measurement data in the presence of constraints. The constraints may fix the surface type or set geometric relations between parts of an object's surface, such as orthogonality, parallelity and others. It is proposed to use a combination of surface fitting and registration within the geometric optimization framework of squared distance minimization (SDM). In this way, we obtain a quasi-Newton like optimization algorithm, which in each iteration simultaneously registers the data set with a rigid motion to the fitting surface and adapts the shape of the fitting surface. We present examples to show the applicability of our method to constrained 3D shape fitting for reverse engineering of CAD models and to high accuracy fitting with kinematic surfaces, which include surfaces of revolution (reconstructed from fragments of archeological pottery) and spiral surfaces, which are fitted to 3D measurement data of shells. Our optimization algorithm can combine registration of multiple scans of an object and model fitting into a single optimization process which is shown to be superior to the traditional procedure, which first registers the data and then fits a model to it.     

A statistical atlas based approach to automated subject-specific FE modeling

Subject-specific modeling is increasingly important in biomechanics simulation. However, how to automatically create high-quality finite element (FE) mesh and how to automatically impose boundary condition are challenging.This paper presents a statistical atlas based approach for automatic meshing of subject-specific shapes. In our approach, shape variations among a shape population are explicitly modeled and the correspondence between a given subject-specific shape and the statistical atlas is sought within the “legal” shape variations. This approach involves three parts: (1) constructing a statistical atlas from a shape population, including the statistical shape model and the FE model of the mean shape; (2) establishing the correspondence between a given subject shape and the atlas; and (3) deforming the atlas to the subject shape based on the shape correspondence. Numerical results on 2D hands, 3D femur bones and 3D aorta demonstrate the effectiveness of the proposed approach.