Quantitative and Qualitative Measures for the Evaluation of the Superquadric Models

In this paper we discuss the evaluation criteria for superquadric models recovered from the range data. We present arguments to support our belief that both quantitative and qualitative measures are required in order to evaluate a superquadric fit. The concept of superquadric contraction and dilation is introduced and used to derive a novel interpretation of the modified superquadric inside-outside function in terms of contraction/expansion factor. The same concept also gives a close initial guess for the numerical procedure computing the minimum Euclidean distance of a point from a superquadric model. The minimum Euclidean distance map is introduced as a qualitative criterion for interpretation of fit. View-dependent qualitative measures like the contour-difference map and the z-distance map are shown to be essential for the complete evaluation of the models. Analytical solution and techniques for the contour generator on superquadric models are presented. Finally, examples of real objects are given to generate the measures

Superquadric representation of scenes from multi-view range data

Object representation denotes representing three-dimensional (3D) real-world objects with known graphic or mathematic primitives recognizable to computers. This research has numerous applications for object-related tasks in areas including computer vision, computer graphics, reverse engineering, etc. Superquadrics, as volumetric and parametric models, have been selected to be the representation primitives throughout this research. Superquadrics are able to represent a large family of solid shapes by a single equation with only a few parameters. This dissertation addresses superquadric representation of multi-part objects and multiobject scenes. Two issues motivate this research. First, superquadric representation of multipart objects or multi-object scenes has been an unsolved problem due to the complex geometry of objects. Second, superquadrics recovered from single-view range data tend to have low confidence and accuracy due to partially scanned object surfaces caused by inherent occlusions. To address these two problems, this dissertation proposes a multi-view superquadric representation algorithm. By incorporating both part decomposition and multi-view range data, the proposed algorithm is able to not only represent multi-part objects or multi-object scenes, but also achieve high confidence and accuracy of recovered superquadrics. The multi-view superquadric representation algorithm consists of (i) initial superquadric model recovery from single-view range data, (ii) pairwise view registration based on recovered superquadric models, (iii) view integration, (iv) part decomposition, and (v) final superquadric fitting for each decomposed part. Within the multi-view superquadric representation framework, this dissertation proposes a 3D part decomposition algorithm to automatically decompose multi-part objects or multiobject scenes into their constituent single parts consistent with human visual perception. Superquadrics can then be recovered for each decomposed single-part object. The proposed part decomposition algorithm is based on curvature analysis, and includes (i) Gaussian curvature estimation, (ii) boundary labeling, (iii) part growing and labeling, and (iv) post-processing. In addition, this dissertation proposes an extended view registration algorithm based on superquadrics. The proposed view registration algorithm is able to handle deformable superquadrics as well as 3D unstructured data sets. For superquadric fitting, two objective functions primarily used in the literature have been comprehensively investigated with respect to noise, viewpoints, sample resolutions, etc. The objective function proved to have better performance has been used throughout this dissertation. In summary, the three algorithms (contributions) proposed in this dissertation are generic and flexible in the sense of handling triangle meshes, which are standard surface primitives in computer vision and graphics. For each proposed algorithm, the dissertation presents both theory and experimental results. The results demonstrate the efficiency of the algorithms using both synthetic and real range data of a large variety of objects and scenes. In addition, the experimental results include comparisons with previous methods from the literature. Finally, the dissertation concludes with a summary of the contributions to the state of the art in superquadric representation, and presents possible future extensions to this research

Part Description and Segmentation Using Contour, Surface and Volumetric Primitives

The problem of part definition, description, and decomposition is central to the shape recognition systems. The Ultimate goal of segmenting range images into meaningful parts and objects has proved to be very difficult to realize, mainly due to the isolation of the segmentation problem from the issue of representation. We propose a paradigm for part description and segmentation by integration of contour, surface, and volumetric primitives. Unlike previous approaches, we have used geometric properties derived from both boundary-based (surface contours and occluding contours), and primitive-based (quadric patches and superquadric models) representations to define and recover part-whole relationships, without a priori knowledge about the objects or object domain. The object shape is described at three levels of complexity, each contributing to the overall shape. Our approach can be summarized as answering the following question : Given that we have all three different modules for extracting volume, surface and boundary properties, how should they be invoked, evaluated and integrated? Volume and boundary fitting, and surface description are performed in parallel to incorporate the best of the coarse to fine and fine to coarse segmentation strategy. The process involves feedback between the segmentor (the Control Module) and individual shape description modules. The control module evaluates the intermediate descriptions and formulates hypotheses about parts. Hypotheses are further tested by the segmentor and the descriptors. The descriptions thus obtained are independent of position, orientation, scale, domain and domain properties, and are based purely on geometric considerations. They are extremely useful for the high level domain dependent symbolic reasoning processes, which need not deal with tremendous amount of data, but only with a rich description of data in terms of primitives recovered at various levels of complexity

Physics-Based Modeling of Nonrigid Objects for Vision and Graphics (Dissertation)

This thesis develops a physics-based framework for 3D shape and nonrigid motion modeling for computer vision and computer graphics. In computer vision it addresses the problems of complex 3D shape representation, shape reconstruction, quantitative model extraction from biomedical data for analysis and visualization, shape estimation, and motion tracking. In computer graphics it demonstrates the generative power of our framework to synthesize constrained shapes, nonrigid object motions and object interactions for the purposes of computer animation. Our framework is based on the use of a new class of dynamically deformable primitives which allow the combination of global and local deformations. It incorporates physical constraints to compose articulated models from deformable primitives and provides force-based techniques for fitting such models to sparse, noise-corrupted 2D and 3D visual data. The framework leads to shape and nonrigid motion estimators that exploit dynamically deformable models to track moving 3D objects from time-varying observations. We develop models with global deformation parameters which represent the salient shape features of natural parts, and local deformation parameters which capture shape details. In the context of computer graphics, these models represent the physics-based marriage of the parameterized and free-form modeling paradigms. An important benefit of their global/local descriptive power in the context of computer vision is that it can potentially satisfy the often conflicting requirements of shape reconstruction and shape recognition. The Lagrange equations of motion that govern our models, augmented by constraints, make them responsive to externally applied forces derived from input data or applied by the user. This system of differential equations is discretized using finite element methods and simulated through time using standard numerical techniques. We employ these equations to formulate a shape and nonrigid motion estimator. The estimator is a continuous extended Kalman filter that recursively transforms the discrepancy between the sensory data and the estimated model state into generalized forces. These adjust the translational, rotational, and deformational degrees of freedom such that the model evolves in a consistent fashion with the noisy data. We demonstrate the interactive time performance of our techniques in a series of experiments in computer vision, graphics, and visualization

Knowledge-Based Deformable Surface Model with Application to Segmentation of Brain Structures in MRI

We have developed a knowledge-based deformable surface for segmentation of medical images. This work has been done in the context of segmentation of hippocampus from brain MRI, due to its challenge and clinical importance. The model has a polyhedral discrete structure and is initialized automatically by analyzing brain MRI sliced by slice, and finding few landmark features at each slice using an expert system. The expert system decides on the presence of the hippocampus and its general location in each slice. The landmarks found are connected together by a triangulation method, to generate a closed initial surface. The surface deforms under defined internal and external force terms thereafter, to generate an accurate and reproducible boundary for the hippocampus. The anterior and posterior (AP) limits of the hippocampus is estimated by automatic analysis of the location of brain stem, and some of the features extracted in the initialization process. These data are combined together with a priori knowledge using Bayes method to estimate a probability density function (pdf) for the length of the structure in sagittal direction. The hippocampus AP limits are found by optimizing this pdf. The model is tested on real clinical data and the results show very good model performance.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85930/1/Fessler166.pd

The mitral valve computational anatomy and geometry analysis

We present a novel methodology to characterize and quantify the Mitral Valve (MV) geometry and physical attributes in a multi-resolution framework. A multi-scale decomposition was implemented to model the MV geometry by using superquadric shape primitives and spectral reconstruction of the finer-scale geometric details. Superquadrics provide a basis to normalize the size and approximate a basic model of the MV geometry. The point-wise difference between the original geometry and the superquadric model denotes the finer-scale geometric details, which can be modeled as a scalar attribute for the MV model development. The additive decomposition of the basic MV geometry from geometric details (attributes) allows recovering the actual geometry by superposition of the superquadric approximation and the finer-details model. We implemented a lasso optimization algorithm to perform spectral analysis and develop the Fourier reconstruction of the geometric details. The spectral modeling enabled us to resample the geometric details or use spectral filters in order to adjust the spatial resolution in the model reconstruction. It also provides the basis to control the level of detail in the final model reconstruction by applying low-pass filters in the frequency domain. The higher-order attributes such as internal fiber architecture can be integrated with the geometric models using the same framework. We applied our pipeline to create models of three ovine MVs based on computed-tomography 3D images with micrometer resolution. We were able to quantify the MV leaflet geometry, reconstruct models with custom level of geometric details, and develop medial representation of the MV leaflet structure. The results show that our methodology for geometry analysis provides a basis for assessing patient-specific geometries and facilitates developing population-averaged models. Ultimately, this approach allows building personalized image-based computational models for medical device design and surgical treatment simulations.Mechanical Engineerin

Surface and Volumetric Segmentation of Complex 3-D Objects Using Parametric Shape Models

The problem of part definition, description, and decomposition is central to the shape recognition systems. In this dissertation, we develop an integrated framework for segmenting dense range data of complex 3-D scenes into their constituent parts in terms of surface and volumetric primitives. Unlike previous approaches, we use geometric properties derived from surface, as well as volumetric models, to recover structured descriptions of complex objects without a priori domain knowledge or stored models. To recover shape descriptions, we use bi-quadric models for surface representation and superquadric models for object-centered volumetric representation. The surface segmentation uses a novel approach of searching for the best piecewise description of the image in terms of bi-quadric (z = f(x,y)) models. It is used to generate the region adjacency graphs, to localize surface discontinuities, and to derive global shape properties of the surfaces. A superquadric model is recovered for the entire data set and residuals are computed to evaluate the fit. The goodness-of-fit value based on the inside-outside function, and the mean-squared distance of data from the model provide quantitative evaluation of the model. The qualitative evaluation criteria check the local consistency of the model in the form of residual maps of overestimated and underestimated data regions. The control structure invokes the models in a systematic manner, evaluates the intermediate descriptions, and integrates them to achieve final segmentation. Superquadric and bi-quadric models are recovered in parallel to incorporate the best of the coarse-to-fine and fine-to-coarse segmentation strategies. The model evaluation criteria determine the dimensionality of the scene, and decide whether to terminate the procedure, or selectively refine the segmentation by following a global-to-local part segmentation approach. The control module generates hypotheses about superquadric models at clusters of underestimated data and performs controlled extrapolation of the part-model by shrinking the global model. As the global model shrinks and the local models grow, they are evaluated and tested for termination or further segmentation. We present results on real range images of scenes of varying complexity, including objects with occluding parts, and scenes where surface segmentation is not sufficient to guide the volumetric segmentation. We analyze the issue of segmentation of complex scenes thoroughly by studying the effect of missing data on volumetric model recovery, generating object-centered descriptions, and presenting a complete set of criteria for the evaluation of the superquadric models. We conclude by discussing the applications of our approach in data reduction, 3-D object recognition, geometric modeling, automatic model generation. object manipulation, and active vision

Multi-Level Shape Representation Using Global Deformations and Locally Adaptive Finite Elements

We present a model-based method for the multi-level shape, pose estimation and abstraction of an object’s surface from range data. The surface shape is estimated based on the parameters of a superquadric that is subjected to global deformations (tapering and bending) and a varying number of levels of local deformations. Local deformations are implemented using locally adaptive finite elements whose shape functions are piecewise cubic functions with C1 continuity. The surface pose is estimated based on the model\u27s translational and rotational degrees of freedom. The algorithm first does a coarse fit, solving for a first approximation to the translation, rotation and global deformation parameters and then does several passes of mesh refinement, by locally subdividing triangles based on the distance between the given datapoints and the model. The adaptive finite element algorithm ensures that during subdivision the desirable finite element mesh generation properties of conformity, non-degeneracy and smoothness are maintained. Each pass of the algorithm uses physics-based modeling techniques to iteratively adjust the global and local parameters of the model in response to forces that are computed from approximation errors between the model and the data. We present results demonstrating the multi-level shape representation for both sparse and dense range data