Affine Cones as Images of Affine Spaces

Varieties covered by affine spaces, uniformly rational varieties and their cones

We endow the set of complements of a fixed subspace of a projective space with the structure of an affine space, and show that certain lines of such an affine space are affine reguli or cones over affine reguli. Moreover, we apply our concepts to the problem of describing dual spreads. We do not assume that the projective space is finite-dimensional or pappian.

The Bundle Axiom and Egglike Subsets of Projective Spaces

In this paper, we propose a novel classification method, called local manifold matching (LMM), for face recognition. LMM has great representational capacity of available prototypes and is based on the local linearity assumption that each data point and its k nearest neighbors from the same class lie on a linear manifold locally embedded in the image space. We present a supervised local manifold learning algorithm for learning all locally linear manifold structures. Then we propose the nearest manifold criterion for the classification in which the query feature point is assigned to the most matching face manifold. Experimental results show that kernel PCA incorporated with the LMM classifier achieves the best face recognition performance.

Line matching plays an important role in vision localization and three-dimensional reconstruction of building structures. The conventional method of line matching is not effective for processing stereo images with wide baselines and large viewing angles. This paper proposes a line matching method in an affine projection space, aiming to solve the problem of change of viewing angles in aerial oblique images. Firstly, monocular image orientation can be performed through geometric structures of buildings. Secondly, according to the pose information of the camera, the affine projection matrix is obtained. The original image can be rectified as a conformal image based on this projection matrix, thereby reducing the difference in the viewing angle between images. Then, line matching is performed on the rectified images to get the matched line pairs. Finally, the inverse affine projection matrix is used to back-project the matched line pairs to the original images. The experimental results of five groups of aerial oblique images show that the matched line segments obtained by the proposed method are basically superior to those of the methods which are directly processed on the original image in terms of quantity, correctness, and efficiency.

Use of uncalibrated images has found many applications such as image synthesis. However, it is not easy to specify the desired position of the new image in projective or affine space. This paper proposes to recover Euclidean structure from uncalibrated images using domain knowledge such as distances and angles. The knowledge we have is usually about an object category, but not very precise for the particular object being considered. The variation (fuzziness) is modeled as a Gaussian variable. Six types of common knowledge are formulated. Once we have an Euclidean description, the task to specify the desired position in Euclidean space becomes trivial. The proposed technique is then applied to synthesis of new facial images. A number of difficulties existing in image synthesis are identified and solved. For example, we propose to use edge points to deal with occlusion.

Dynamic textures are sequences of images of moving scenes that show stationarity properties in time. Eg: waves, flame, fountain, etc. Recent attempts at generating, potentially, infinitely long sequences model the dynamic texture as a Linear Dynamic System. This assumes a linear correlation in the input sequence. Most real world sequences however, exhibit nonlinear correlation between frames. In this paper, we propose a technique of generating dynamic textures using a low dimension model that preserves the non-linear correlation. We use nonlinear dimensionality reduction to create an embedding of the input sequence. Using this embedding, a nonlinear mapping is learnt from the embedded space into the image input space. Any input is represented by a linear combination of nonlinear bases functions centered along the manifold in the embedded space. A spline is used to move along the input manifold in this embedded space as a similar manifold is created for the output. The nonlinear mapping learnt on the input is used to map this new manifold into a sequence in the image space. Output sequences, thus created, contain images never present in the original sequence and are very realistic.KeywordsTextureDynamic TextureImage-based RenderingNon Linear Manifold Learning

A new visual control method for the end-effector of a robot to approach an object is proposed in this paper, which consists of tracking, approaching and decision part. Two uncalibrated cameras are employed. One is for track the object in its image space. Another is for approach the object in the direction of affine line between first camera and the object. The distances between the current and desired positions of the end-effector are estimated according to the end-effector's actual movements based on cross ratio invariance. The approaching part is enabled or disenabled by the decision part according to the result of tracking part. With the combination of tracking and approaching, the end-effector is moved to the target object. Experimental results are presented to verify the effectiveness of the proposed method.

Establishing visual correspondence is one of the most fundamental tasks in many applications of computer vision fields. In this paper we propose a robust image matching to address the affine variation problems between two images taken under different viewpoints. Unlike the conventional approach finding the correspondence with local feature matching on fully affine transformed-images, which provides many outliers with a time consuming scheme, our approach is to find only one global correspondence and then utilizes the local feature matching to estimate the most reliable inliers between two images. In order to estimate a global image correspondence very fast as varying affine transformation in affine space of reference and query images, we employ a Bhattacharyya similarity measure between two images. Furthermore, an adaptive tree with affine transformation model is employed to dramatically reduce the computational complexity. Our approach represents the satisfactory results for severe affine transformed-images while providing a very low computational time. Experimental results show that the proposed affine-invariant image matching is twice faster than the state-of-the-art methods at least, and provides better correspondence performance under viewpoint change conditions.

Gibbs manifolds are images of affine spaces of symmetric matrices under the exponential map. They arise in applications such as optimization, statistics and quantum physics, where they extend the ubiquitous role of toric geometry. The Gibbs variety is the zero locus of all polynomials that vanish on the Gibbs manifold. We compute these polynomials and show that the Gibbs variety is low-dimensional. Our theory is applied to a wide range of scenarios, including matrix pencils and quantum optimal transport.

Surjective Nash maps between semialgebraic sets