A Minimalist Approach to Type-Agnostic Detection of Quadrics in Point Clouds

This paper proposes a segmentation-free, automatic and efficient procedure to detect general geometric quadric forms in point clouds, where clutter and occlusions are inevitable. Our everyday world is dominated by man-made objects which are designed using 3D primitives (such as planes, cones, spheres, cylinders, etc.). These objects are also omnipresent in industrial environments. This gives rise to the possibility of abstracting 3D scenes through primitives, thereby positions these geometric forms as an integral part of perception and high level 3D scene understanding. As opposed to state-of-the-art, where a tailored algorithm treats each primitive type separately, we propose to encapsulate all types in a single robust detection procedure. At the center of our approach lies a closed form 3D quadric fit, operating in both primal & dual spaces and requiring as low as 4 oriented-points. Around this fit, we design a novel, local null-space voting strategy to reduce the 4-point case to 3. Voting is coupled with the famous RANSAC and makes our algorithm orders of magnitude faster than its conventional counterparts. This is the first method capable of performing a generic cross-type multi-object primitive detection in difficult scenes. Results on synthetic and real datasets support the validity of our method.Comment: Accepted for publication at CVPR 201

Kernel Based Algebraic Curve Fitting

An algebraic curve is defined as the zero set of a multivariate polynomial. We consider the problem of fitting an algebraic curve to a set of vectors given an additional set of vectors labelled as interior or exterior to the curve. The problem of fitting a linear curve in this way is shown to lend itself to a support vector representation, allowing non-linear curves and high dimensional surfaces to be estimated using kernel functions. The approach is attractive due to the stability of solutions obtained, the range of functional forms made possible (including polynomials), and the potential for applying well understood regularisation operators from the theory of Support Vector Machines

Algebraic Curve Fitting Support Vector Machines

An algebraic curve is defined as the zero set of a multivariate polynomial. We consider the problem of fitting an algebraic curve to a set of vectors given an additional set of vectors labelled as interior or exterior to the curve. The problem of fitting a linear curve in this way is shown to lend itself to a support vector representation, allowing non-linear curves and high dimensional surfaces to be estimated using kernel functions. The approach is attractive due to the stability of solutions obtained, the range of functional forms made ossible (including polynomials), and the potential for applying well understood regularisation operators from the theory of Support Vector Machines

Vector field reconstruction from sparse samples with applications

Abstract. We present a novel algorithm for 2D vector field reconstruction from sparse set of points–vectors pairs. Our approach subdivides the domain adaptively in order to make local piecewise polynomial approximations for the field. It uses partition of unity to blend those local approximations together, generating a global approximation for the field. The flexibility of this scheme allows handling data from very different sources. In particular, this work presents important applications of the proposed method to velocity and acceleration fields ’ analysis, in particular for fluid dynamics visualization

Generic Primitive Detection in Point Clouds Using Novel Minimal Quadric Fits

We present a novel and effective method for detecting 3D primitives in cluttered, unorganized point clouds, without axillary segmentation or type specification. We consider the quadric surfaces for encapsulating the basic building blocks of our environments - planes, spheres, ellipsoids, cones or cylinders, in a unified fashion. Moreover, quadrics allow us to model higher degree of freedom shapes, such as hyperboloids or paraboloids that could be used in non-rigid settings. We begin by contributing two novel quadric fits targeting 3D point sets that are endowed with tangent space information. Based upon the idea of aligning the quadric gradients with the surface normals, our first formulation is exact and requires as low as four oriented points. The second fit approximates the first, and reduces the computational effort. We theoretically analyze these fits with rigor, and give algebraic and geometric arguments. Next, by re-parameterizing the solution, we devise a new local Hough voting scheme on the null-space coefficients that is combined with RANSAC, reducing the complexity from $O(N^4)$ to $O(N^3)$ (three points). To the best of our knowledge, this is the first method capable of performing a generic cross-type multi-object primitive detection in difficult scenes without segmentation. Our extensive qualitative and quantitative results show that our method is efficient and flexible, as well as being accurate.Comment: Submitted to IEEE Transactions on Pattern Analysis and Machine Intelligence (T-PAMI). arXiv admin note: substantial text overlap with arXiv:1803.0719

Doctor of Philosophy

dissertationScene labeling is the problem of assigning an object label to each pixel of a given image. It is the primary step towards image understanding and unifies object recognition and image segmentation in a single framework. A perfect scene labeling framework detects and densely labels every region and every object that exists in an image. This task is of substantial importance in a wide range of applications in computer vision. Contextual information plays an important role in scene labeling frameworks. A contextual model utilizes the relationships among the objects in a scene to facilitate object detection and image segmentation. Using contextual information in an effective way is one of the main questions that should be answered in any scene labeling framework. In this dissertation, we develop two scene labeling frameworks that rely heavily on contextual information to improve the performance over state-of-the-art methods. The first model, called the multiclass multiscale contextual model (MCMS), uses contextual information from multiple objects and at different scales for learning discriminative models in a supervised setting. The MCMS model incorporates crossobject and interobject information into one probabilistic framework, and thus is able to capture geometrical relationships and dependencies among multiple objects in addition to local information from each single object present in an image. The second model, called the contextual hierarchical model (CHM), learns contextual information in a hierarchy for scene labeling. At each level of the hierarchy, a classifier is trained based on downsampled input images and outputs of previous levels. The CHM then incorporates the resulting multiresolution contextual information into a classifier to segment the input image at original resolution. This training strategy allows for optimization of a joint posterior probability at multiple resolutions through the hierarchy. We demonstrate the performance of CHM on different challenging tasks such as outdoor scene labeling and edge detection in natural images and membrane detection in electron microscopy images. We also introduce two novel classification methods. WNS-AdaBoost speeds up the training of AdaBoost by providing a compact representation of a training set. Disjunctive normal random forest (DNRF) is an ensemble method that is able to learn complex decision boundaries and achieves low generalization error by optimizing a single objective function for each weak classifier in the ensemble. Finally, a segmentation framework is introduced that exploits both shape information and regional statistics to segment irregularly shaped intracellular structures such as mitochondria in electron microscopy images

Algebraic Curves That Work Better

An algebraic curve is defined as the zero set of a polynomial in two variables. Algebraic curves are practical for modeling shapes much more complicated than conics or superquadrics. The main drawback in representing shapes by algebraic curves has been the lack of repeatability in fitting algebraic curves to data. A regularized fast linear fitting method based on ridge regression and restricting the representation to well behaved subsets of polynomials is proposed, and its properties are investigated. The fitting algorithm is of sufficient stability for very fast position-invariant shape recognition, position estimation, and shape tracking, based on new invariants and representations, and is appropriate to open as well as closed curves of unorganized data. Among appropriate applications are shape-based indexing into image databases. 1 Introduction Algebraic 2D curves (and 3D surfaces) are extremely powerful for shape recognition and singlecomputation pose estimation because of their ..