Projection (set theory)

About: Projection (set theory) is a research topic. Over the lifetime, 47870 publications have been published within this topic receiving 469696 citations.

Visual learning and recognition of 3-D objects from appearance

Abstract: The problem of automatically learning object models for recognition and pose estimation is addressed. In contrast to the traditional approach, the recognition problem is formulated as one of matching appearance rather than shape. The appearance of an object in a two-dimensional image depends on its shape, reflectance properties, pose in the scene, and the illumination conditions. While shape and reflectance are intrinsic properties and constant for a rigid object, pose and illumination vary from scene to scene. A compact representation of object appearance is proposed that is parametrized by pose and illumination. For each object of interest, a large set of images is obtained by automatically varying pose and illumination. This image set is compressed to obtain a low-dimensional subspace, called the eigenspace, in which the object is represented as a manifold. Given an unknown input image, the recognition system projects the image to eigenspace. The object is recognized based on the manifold it lies on. The exact position of the projection on the manifold determines the object's pose in the image. A variety of experiments are conducted using objects with complex appearance characteristics. The performance of the recognition and pose estimation algorithms is studied using over a thousand input images of sample objects. Sensitivity of recognition to the number of eigenspace dimensions and the number of learning samples is analyzed. For the objects used, appearance representation in eigenspaces with less than 20 dimensions produces accurate recognition results with an average pose estimation error of about 1.0 degree. A near real-time recognition system with 20 complex objects in the database has been developed. The paper is concluded with a discussion on various issues related to the proposed learning and recognition methodology.

Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization

Abstract: An iterative algorithm, based on recent work in compressive sensing, is developed for volume image reconstruction from a circular cone-beam scan. The algorithm minimizes the total variation (TV) of the image subject to the constraint that the estimated projection data is within a specified tolerance of the available data and that the values of the volume image are non-negative. The constraints are enforced by the use of projection onto convex sets (POCS) and the TV objective is minimized by steepest descent with an adaptive step-size. The algorithm is referred to as adaptive-steepest-descent-POCS (ASD-POCS). It appears to be robust against cone-beam artifacts, and may be particularly useful when the angular range is limited or when the angular sampling rate is low. The ASD-POCS algorithm is tested with the Defrise disk and jaw computerized phantoms. Some comparisons are performed with the POCS and expectation-maximization (EM) algorithms. Although the algorithm is presented in the context of circular cone-beam image reconstruction, it can also be applied to scanning geometries involving other x-ray source trajectories.

Finite element interpolation of nonsmooth functions satisfying boundary conditions

Abstract: In this paper, we propose a modified Lagrange type interpolation operator to approximate functions in Sobolev spaces by continuous piecewise polynomials. In order to define interpolators for "rough" functions and to preserve piecewise polynomial boundary conditions, the approximated functions are averaged appropriately either on dor (d 1)-simplices to generate nodal values for the interpolation operator. This combination of averaging and interpolation is shown to be a projection, and optimal error estimates are proved for the projection error.