On growth and formlets: Sparse multi-scale coding of planar shape

We propose a sparse representation of 2D planar shape through the composition of warping functions, termed formlets, localized in scale and space. Each formlet subjects the 2D space in which the shape is embedded to a localized isotropic radial deformation. By constraining these localized warping transformations to be diffeomorphisms, the topology of shape is preserved, and the set of simple closed curves is closed under any sequence of these warpings. A generative model based on a composition of formlets applied to an embryonic shape, e.g., an ellipse, has the advantage of synthesizing only those shapes that could correspond to the boundaries of physical objects. To compute the set of formlets that represent a given boundary, we demonstrate a greedy coarse-to-fine formlet pursuit algorithm that serves as a non-commutative generalization of matching pursuit for sparse approximations. We evaluate our method by pursuing partially occluded shapes, comparing performance against a contour-based sparse shape coding framework.     

Statistical shape analysis: clustering, learning, and testing

Using a differential-geometric treatment of planar shapes, we present tools for: 1) hierarchical clustering of imaged objects according to the shapes of their boundaries, 2) learning of probability models for clusters of shapes, and 3) testing of newly observed shapes under competing probability models. Clustering at any level of hierarchy is performed using a minimum variance type criterion and a Markov process. Statistical means of clusters provide shapes to be clustered at the next higher level, thus building a hierarchy of shapes. Using finite-dimensional approximations of spaces tangent to the shape space at sample means, we (implicitly) impose probability models on the shape space, and results are illustrated via random sampling and classification (hypothesis testing). Together, hierarchical clustering and hypothesis testing provide an efficient framework for shape retrieval. Examples are presented using shapes and images from ETH, Surrey, and AMCOM databases.     

Blended deformable models

This paper develops a new class of parameterized models based on the linear interpolation of two parameterized shapes along their main axes, using a blending function. This blending function specifies the relative contribution of each component shape on the resulting blended shape. The resulting blended shape can have aspects of each of the component shapes. Using a small number of additional parameters, blending extends the coverage of shape primitives while also providing abstraction of shape. In particular, it offers the ability to construct shapes whose genus can change. Blended models are incorporated into a physics-based shape estimation framework which uses dynamic deformable models. Finally, we present experiments involving the extraction of complex shapes from range data including examples of dynamic genus change     

Partial Matching of Nonrigid Shapes by Learning Piecewise Smooth Functions

Learning functions defined on non-flat domains, such as outer surfaces of non-rigid shapes, is a central task in computer vision and geometry processing. Recent studies have explored the use of neural fields to represent functions like light reflections in volumetric domains and textures on curved surfaces by operating in the embedding space. Here, we choose a different line of thought and introduce a novel formulation of partial shape matching by learning a piecewise smooth function on a surface. Our method begins with pairing sparse landmarks defined on a full shape and its part, using feature similarity. Next, a neural representation is optimized to fit these landmarks, efficiently interpolating between the matched features that act as anchors. This process results in a function that accurately captures the partiality. Unlike previous methods, the proposed neural model of functions is intrinsically defined on the given curved surface, rather than the classical embedding Euclidean space. This representation is shown to be particularly well-suited for representing piecewise smooth functions. We further extend the proposed framework to the more challenging part-to-part setting, where both shapes exhibit missing parts. Comprehensive experiments highlight that the proposed method effectively addresses partiality in shape matching and significantly outperforms leading state-of-the-art methods in challenging benchmarks. Code is available at https://github.com/davidgip74/Learning-Partiality-with-Implicit-Intrinsic-Functions     

Projectively invariant decomposition and recognition of planar shapes

An algorithm is presented for computing a decomposition of planar shapes into convex subparts represented. by ellipses. The method is invariant to projective transformations of the shape, and thus the conic primitives can be used for matching and definition of invariants in the same way as points and lines. The method works for arbitrary planar shapes admitting at least four distinct tangents and it is based on finding ellipses with four points of contact to the given shape. The cross ratio computed from the four points on the ellipse can then be used as a projectively invariant index. It is demonstrated that a given shape has a unique parameter-free decomposition into a finite set of ellipses with unit cross ratio. For a given shape, each pair of ellipses can be used to compute two independent projective invariants. The set of invariants computed for each ellipse pair can be used as indexes to a hash table from which model hypothesis can be generated Examples of shape decomposition and recognition are given for synthetic shapes and shapes extracted from grey level images of real objects using edge detection.     

String Methods for Stochastic Image and Shape Matching

Matching of images and analysis of shape differences is traditionally pursued by energy minimization of paths of deformations acting to match the shape objects. In the large deformation diffeomorphic metric mapping (LDDMM) framework, iterative gradient descents on the matching functional lead to matching algorithms informally known as Beg algorithms. When stochasticity is introduced to model stochastic variability of shapes and to provide more realistic models of observed shape data, the corresponding matching problem can be solved with a stochastic Beg algorithm, similar to the finite-temperature string method used in rare event sampling. In this paper, we apply a stochastic model compatible with the geometry of the LDDMM framework to obtain a stochastic model of images and we derive the stochastic version of the Beg algorithm which we compare with the string method and an expectation-maximization optimization of posterior likelihoods. The algorithm and its use for statistical inference is tested on stochastic LDDMM landmarks and images.     

Intrinsic Bayesian Active Contours for Extraction of Object Boundaries in Images

We present a framework for incorporating prior information about high-probability shapes in the process of contour extraction and object recognition in images. Here one studies shapes as elements of an infinite-dimensional, non-linear quotient space, and statistics of shapes are defined and computed intrinsically using differential geometry of this shape space. Prior models on shapes are constructed using probability distributions on tangent bundles of shape spaces. Similar to the past work on active contours, where curves are driven by vector fields based on image gradients and roughness penalties, we incorporate the prior shape knowledge in the form of vector fields on curves. Through experimental results, we demonstrate the use of prior shape models in the estimation of object boundaries, and their success in handling partial obscuration and missing data. Furthermore, we describe the use of this framework in shape-based object recognition or classification.     

Elastic shapes models for improving segmentation of object boundaries in synthetic aperture sonar images

We present a variational framework for naturally incorporating prior shape knowledge in guidance of active contours for boundary extraction in images. This framework is especially suitable for images collected outside the visible spectrum, where boundary estimation is difficult due to low contrast, low resolution, and presence of noise and clutter. Accordingly, we illustrate this approach using the segmentation of various objects in synthetic aperture sonar (SAS) images of underwater terrains. We use elastic shape analysis of planar curves in which the shapes are considered as elements of a quotient space of an infinite dimensional, non-linear Riemannian manifold. Using geodesic paths under the elastic Riemannian metric, one computes sample mean and covariances of training shapes in each classes and derives statistical models for capturing class-specific shape variability. These models are then used as shape priors in a variational setting to solve for Bayesian estimation of desired contours as follows. In traditional active contour models curves are driven towards minimum of an energy composed of image and smoothing terms. We introduce an additional shape term based on shape models of relevant shape classes. The minimization of this total energy, using iterated gradient-based updates of curves, leads to an improved segmentation of object boundaries. This is demonstrated using a number of shape classes in two large SAS image datasets.     

Modeling the machine layout problem

In this paper, a mathematical model for the machine layout problem is presented. The objective function of the model minimizes the total cost involved in transporting material between each pair of machines. The constraints ensure that: (i) the pickup and drop-off points of a machine fall within the boundaries of the machine; (ii) no two machines in the layout overlap; and (iii) the machines are located inside the building. A simulated annealing algorithm for solving the model is also presented. The solution provided by the algorithm indicates the MHS paths, the relative positioning of each machine, the pick-up and drop-off points for each machine, and clearance between machines.     

Data‐Driven Sparse Priors of 3D Shapes

We present a sparse optimization framework for extracting sparse shape priors from a collection of 3D models. Shape priors are defined as point‐set neighborhoods sampled from shape surfaces which convey important information encompassing normals and local shape characterization. A 3D shape model can be considered to be formed with a set of 3D local shape priors, while most of them are likely to have similar geometry. Our key observation is that the local priors extracted from a family of 3D shapes lie in a very low‐dimensional manifold. Consequently, a compact and informative subset of priors can be learned to efficiently encode all shapes of the same family. A comprehensive library of local shape priors is first built with the given collection of 3D models of the same family. We then formulate a global, sparse optimization problem which enforces selecting representative priors while minimizing the reconstruction error. To solve the optimization problem, we design an efficient solver based on the Augmented Lagrangian Multipliers method (ALM). Extensive experiments exhibit the power of our data‐driven sparse priors in elegantly solving several high‐level shape analysis applications and geometry processing tasks, such as shape retrieval, style analysis and symmetry detection.     

Recursive scan-matching SLAM

This paper presents Scan-SLAM, a new generalization of simultaneous localization and mapping (SLAM). SLAM implementations based on extended Kalman filter (EKF) data fusion have traditionally relied on simple geometric models for defining landmarks. This limits EKF-SLAM to environments suited to such models and tends to discard much potentially useful data. The approach presented in this paper is a marriage of EKF-SLAM and scan correlation. Landmarks are no longer defined by analytical models; instead they are defined by templates composed of raw sensed data. These templates can be augmented as more data become available so that the landmark definition improves with time. A new generic observation model is derived that is generated by scan correlation, and this permits stochastic location estimation for landmarks with arbitrary shape within the Kalman filter framework. The statistical advantages of an EKF representation are augmented with the general applicability of scan matching. Scan matching also serves to enhance data association reliability by providing a shape metric for landmark disambiguation. Experimental results in an outdoor environment are presented which validate the algorithm.     

Local shape blending using coherent weighted regions

We present a novel local shape blending method that maps a sparse configuration of facial markers captured from an actor onto target models. The advantage of local shape blending methods is that, given a small set of key shapes for each local region, their combination can generate various facial expressions. However, they have the common problem that they use the pre-determined (fixed) regions and compute the combination of local key shapes for each region independently of each other. So, they have a risk of breaking natural correlations between the regions. We present a stochastic method of computing the regions and the blending weight vectors simultaneously. To do so, we formulate local shape blending as a problem of finding an optimal distribution of blending weight vectors of all control points in MAP?CMRF framework.     

Shape transformation by boundary representation interpolation: A recursive approach to establishing face correspondences

The issues relating to the shape transformation problem are discussed and a new algorithm is presented for computing the transformation of one shape into another. In this algorithm, the boundary definitions of the two initial shapes are used and a mapping is established between the vertices and edges of the respective objects. New vertices and edges are introduced into the object definitions when necessary to establish a one-to-one vertex correspondence and to match connectivity relationships between vertices. These can then be used to do a vertex-to-vertex interpolation that maintains valid polyhedral topologies for all of the intermediate shapes. The algorithm establishes a mapping between areas of the object such that adjacency relationships are preserved. These areas are recursively subdivided so that adjacency relationships of subareas are also preserved. During subdivision, vertices and edges are added to the boundaries of subareas so that a one-to-one mapping is established between them. Subdivision continues until each subarea consists of a single face. The algorithm presented works for objects that are topologically equivalent to spheres and can easily be extended to other pairs of objects as long as they are topologically equivalent to each other.     

A sparse collocation method for solving time-dependent HJB equations using multivariate B-splines

This paper presents a sparse collocation method for solving the time-dependent Hamilton–Jacobi–Bellman (HJB) equation associated with the continuous-time optimal control problem on a fixed, finite time-horizon with integral cost functional. Through casting the problem in a recursive framework using the value-iteration procedure, the value functions of every iteration step is approximated with a time-varying multivariate simplex B-spline on a certain state domain of interest. In the collocation scheme, the time-dependent coefficients of the spline function are further approximated with ordinary univariate B-splines to yield a discretization for the value function fully in terms of piece-wise polynomials. The B-spline coefficients are determined by solving a sequence of highly sparse quadratic programming problems. The proposed algorithm is demonstrated on a pair of benchmark example problems. Simulation results indicate that the method can yield increasingly more accurate approximations of the value function by refinement of the triangulation.     

Statistical Model of Shape Moments with Active Contour Evolution for Shape Detection and Segmentation

This paper describes a novel method for shape representation and robust image segmentation. The proposed method combines two well known methodologies, namely, statistical shape models and active contours implemented in level set framework. The shape detection is achieved by maximizing a posterior function that consists of a prior shape probability model and image likelihood function conditioned on shapes. The statistical shape model is built as a result of a learning process based on nonparametric probability estimation in a PCA reduced feature space formed by the Legendre moments of training silhouette images. A greedy strategy is applied to optimize the proposed cost function by iteratively evolving an implicit active contour in the image space and subsequent constrained optimization of the evolved shape in the reduced shape feature space. Experimental results presented in the paper demonstrate that the proposed method, contrary to many other active contour segmentation methods, is highly resilient to severe random and structural noise that could be present in the data.     

Recognition of 3D planar objects in canonical frames

An algorithm for recognizing 3D planar objects by their boundaries is presented. Extreme points on a shape are extracted for constructing canonical frames, under which signatures are then generated for determining the similarity between shapes. The method is efficient and yields a high recognition rate.