Population-based fitting of medial shape models with correspondence optimization

pre-printA crucial problem in statistical shape analysis is establishing the correspondence of shape features across a population. While many solutions are easy to express using boundary representations, this has been a considerable challenge for medial representations. This paper uses a new 3-D medial model that allows continuous interpolation of the medial manifold and provides a map back and forth between it and the boundary. A measure defined on the medial surface then allows one to write integrals over the boundary and the object interior in medial coordinates, enabling the expression of important object properties in an object-relative coordinate system.We use these integrals to optimize correspondence during model construction, reducing variability due to the model parameterization that could potentially mask true shape change effects. Discrimination and hypothesis testing of populations of shapes are expected to benefit, potentially resulting in improved significance of shape differences between populations even with a smaller sample size

Multiscale medial shape-based analysis of image objects

pre-printMedial representation of a three-dimensional (3-D) object or an ensemble of 3-D objects involves capturing the object interior as a locus of medial atoms, each atom being two vectors of equal length joined at the tail at the medial point. Medial representation has a variety of beneficial properties, among the most important of which are 1) its inherent geometry, provides an object-intrinsic coordinate system and thus provides correspondence between instances of the object in and near the object(s); 2) it captures the object interior and is, thus, very suitable for deformation; and 3) it provides the basis for an intuitive object-based multiscale sequence leading to efficiency of segmentation algorithms and trainability of statistical characterizations with limited training sets. As a result of these properties, medial representation is particularly suitable for the following image analysis tasks; how each operates will be described and will be illustrated by results: 1) segmentation of objects and object complexes via deformable models; 2) segmentation of tubular trees, e.g., of blood vessels, by following height ridges of measures of fit of medial atoms to target images; 3) object-based image registration via medial loci of such blood vessel trees; 4) statistical characterization of shape differences between control and pathological classes of structures. These analysis tasks are made possible by a new form of medial representation called m-reps, which is described

Optimal Separable Algorithms to Compute the Reverse Euclidean Distance Transformation and Discrete Medial Axis in Arbitrary Dimension

In binary images, the distance transformation (DT) and the geometrical skeleton extraction are classic tools for shape analysis. In this paper, we present time optimal algorithms to solve the reverse Euclidean distance transformation and the reversible medial axis extraction problems for $d$-dimensional images. We also present a $d$-dimensional medial axis filtering process that allows us to control the quality of the reconstructed shape

Venation Skeleton-Based Modeling Plant Leaf Wilting

A venation skeleton-driven method for modeling and animating plant leaf wilting is presented. The proposed method includes five principal processes. Firstly, a three-dimensional leaf skeleton is constructed from a leaf image, and the leaf skeleton is further used to generate a detailed mesh for the leaf surface. Then a venation skeleton is generated interactively from the leaf skeleton. Each vein in the venation skeleton consists of a segmented vertices string. Thirdly, each vertex in the leaf mesh is banded to the nearest vertex in the venation skeleton. We then deform the venation skeleton by controlling the movement of each vertex in the venation skeleton by rotating it around a fixed vector. Finally, the leaf mesh is mapped to the deformed venation skeleton, as such the deformation of the mesh follows the deformation of the venation skeleton. The proposed techniques have been applied to simulate plant leaf surface deformation resulted from biological responses of plant wilting

Structuring 3D Medial Skeletons: A Comparative Study

International audienceMedial skeletons provide an effective alternative to boundary or volumetric representations for applications that focus on shape structure. This capability is provided by the skeletal structure, i.e., the curves and surfaces computed from centers of maximally inscribed balls by a process called structuration. Many several structuration methods exist, all having various challenges in terms of delivering a high-quality medial skeleton. This paper provides a first overview of existing structuration methods. We formally define the skeletal structure by giving its theoretical properties, and use these properties to propose quality criteria for structurations. We next review existing structuration methods and compare them using the established criteria. The obtained insights help both practitioners in choosing a suitable structuration method and researchers in further perfecting such methods

Smooth surfaces, umbilics, lines of curvatures, foliations, ridges and the medial axis: a concise overview

The understanding of surfaces embedded in R^3 requires local and global concepts, which are respectively evocative of differential geometry and differential topology. While the local theory has been classical for decades, global objects such as the foliations defined by the lines of curvature, or the medial axis still pose challenging mathematical problems. This duality is also tangible from a practical perspective, since algorithms manipulating sampled smooth surfaces (meshes or point clouds) are more developed in the local than the global category. As an example and assuming this makes sense for the applications encompassed, we are not aware as of today of any algorithm able to report ---under reasonable assumptions--- a topologically correct medial axis or foliation from a sampled surface. As a prerequisite for those interested in the development of algorithms for the manipulation of surfaces, we propose a concise overview of global objects related to curvature properties of a smooth generic surface. Gathering from differential topology and singularity theory sources, our presentation focuses on the geometric intuition rather than the technicalities. We first recall the classification of umbilics, of curvature lines, and describe the corresponding stable foliations. Next, fundamentals of contact and singularity theory are recalled, together with the classification of points induced by the contact of the surface with a sphere. This classification is further used to define ridges and their properties, and to recall the stratification properties of the medial axis. From a theoretical perspective, we expect this survey to ease the access to intricate notions scattered over several sources. From a practical standpoint, we hope it will be helpful for those interested in the manipulation of surfaces without using global parametrizations, and also for those aiming at producing globally coherent approximations of surfaces

Differential topology and geometry of smooth embedded surfaces: selected topics

The understanding of surfaces embedded in E3 requires local and global concepts, which are respectively evocative of differential geometry and differential topology. While the local theory has been classical for decades, global objects such as the foliations defined by the lines of curvature, or the medial axis still pose challenging mathematical problems. This duality is also tangible from a practical perspective, since algorithms manipulating sampled smooth surfaces (meshes or point clouds) are more developed in the local than the global category. As a prerequisite for those interested in the development of algorithms for the manipulation of surfaces, we propose a concise overview of core concepts from differential topology applied to smooth embedded surfaces. We first recall the classification of umbilics, of curvature lines, and describe the corresponding stable foliations. Next, fundamentals of contact and singularity theory are recalled, together with the classification of points induced by the contact of the surface with a sphere. This classification is further used to define ridges and their properties, and to recall the stratification properties of the medial axis. Finally, properties of the medial axis are used to present sufficient conditions ensuring that two embedded surfaces are ambient isotopic. From a theoretical perspective, we expect this survey to ease the access to intricate notions scattered over several sources. From a practical standpoint, we hope it will be useful for those interested in certified approximations of smooth surfaces