A Survey on Non-rigid 3D Shape Analysis <s> INTRODUCTION <s> Statistical analysis of longitudinal imaging data is crucial for understanding normal anatomical development as well as disease progression. This fundamental task is challenging due to the difficulty in modeling longitudinal changes, such as growth, and comparing changes across different populations. We propose a new approach for analyzing shape variability over time, and for quantifying spatiotemporal population differences. Our approach estimates 4D anatomical growth models for a reference population (an average model) and for individuals in different groups. We define a reference 4D space for our analysis as the average population model and measure shape variability through diffeomorphisms that map the reference to the individuals. Conducting our analysis on this 4D space enables straightforward statistical analysis of deformations as they are parameterized by momenta vectors that are located at homologous locations in space and time. We evaluate our method on a synthetic shape database and clinical data from a study that seeks to quantify brain growth differences in infants at risk for autism. <s> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> INTRODUCTION <s> We present an algorithm for recognition and reconstruction of scanned 3D indoor scenes. 3D indoor reconstruction is particularly challenging due to object interferences, occlusions and overlapping which yield incomplete yet very complex scene arrangements. Since it is hard to assemble scanned segments into complete models, traditional methods for object recognition and reconstruction would be inefficient. We present a search-classify approach which interleaves segmentation and classification in an iterative manner. Using a robust classifier we traverse the scene and gradually propagate classification information. We reinforce classification by a template fitting step which yields a scene reconstruction. We deform-to-fit templates to classified objects to resolve classification ambiguities. The resulting reconstruction is an approximation which captures the general scene arrangement. Our results demonstrate successful classification and reconstruction of cluttered indoor scenes, captured in just few minutes. <s> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> INTRODUCTION <s> Shape morphometry of subcortical surfaces plays an important role in analyzing normal developmental changes as well as in quantifying disease-related effects in the human brain. We present a geometric approach for joint registration, deformation, and statistical analysis of shapes of subcortical surfaces. Here, subcortical surfaces are mathematically represented by vector-fields, termed square-root normal vector fields (SRNFs), on the spherical domain. The SRNF representation allows modeling of the action of the re-parameterization group to by isometries under the standard Hilbert norm, and thus enables an elastic shape analysis. This elastic analysis results in optimal deformations between shapes and helps to quantify shape differences using geodesic lengths. Importantly, the joint registration of observed shapes in a shape class removes nuisance variability due to mis-registration in the shape data and results in parsimonious statistical shape models with improved inferences for characterizing population-based subcortical structural variability. We demonstrate the ideas for shape matching and statistical analysis of hippocampal shapes for (N=120) subjects from the Alzheimer's disease Neuroimaging Initiative (ADNI) dataset. Specifically, we present results for assessing group differences both by using global descriptors such as principal components and local descriptors such as deformations induced by the tangent vectors from the mean shapes to the individuals. <s> BIB003
S HAPE is an important physical property of natural and manmade objects that characterizes their external appearances. Understanding differences between shapes and modeling the variability within and across shape classes are fundamental problems and building blocks to many applications, ranging from computer vision and computer graphics to biology and medicine. In medicine, studying shapes of 3D anatomical structures and modeling their growth patterns are of particular interest since many diseases can be linked to alterations of these shapes  , BIB001 , BIB003 . In computer graphics, 3D modeling using low level primitives (e.g. polygons) is a tedious and timeconsuming process. With the growing availability of 3D models via online repositories, recent research efforts have shifted their focus towards data-driven techniques for generating 3D shape variations from existing ones. Data-driven techniques are also gaining momentum in the field of 3D reconstruction and modeling from images, range scans and noisy point clouds BIB002 , which are ill-posed problems. By leveraging shared information across multiple objects, datadriven techniques aim at discovering patterns, which can be used as geometric, structural, and semantic priors for driving the 3D reconstruction and modeling processes. • Hamid Laga is with the School of Engineering and IT, Murdoch University, Australia and with the Phenomics and Bioinformatics Research Center, University of South Australia. E-mail: H.Laga@murdoch.edu.au. Manuscript received April 19, 2005 ; revised December 27, 2012. When dealing with numbers, statistics is the science for studying and modeling variabilities. Geometric shapes are fundamentally different. Research in this area, driven by the computer vision, medical imaging, computer graphics and statistics communities, seeks to develop mathematical tools for manipulating shapes in the same way we manipulate numbers. This includes: • Comparing the shapes of two objects, i.e. saying whether two objects have similar shape or not, and more importantly, quantifying and localizing the similarities and differences. • Computing summary statistics and prototypes, such as the mean shape, modes of variations and higher order statistics, of a collection of 3D models. • Mathematical modeling of shape variations, using, for example, probability distributions as generative models. These models form priors for random sampling and for statistical inferences, and • Exploring the shape variations for synthesizing valid shapes and performing interpolations, extrapolations, statistical inferences, and regressions. Implementing these ideas, for 3D models, requires solving several challenges, each one has been the subject of important research and contributions. The first one is about the representation of shape. When selecting any representation, it is important to ensure that two different shapes cannot have the same representation and that a given representation can always