Abstract. Numerous dimensionality reduction problems in data analysis involve the recovery of lowdimensional models or the learning of manifolds underlying sets of data. Many manifold learning methods require the estimation of the tangent space of the manifold at a point from locally available data samples. Local sampling conditions such as (i) the size of the neighborhood (sampling width) and (ii) the number of samples in the neighborhood (sampling density) affect the performance of learning algorithms. In this work, we propose a theoretical analysis of local sampling conditions for the estimation of the tangent space at a point P lying on a m-dimensional Riemannian manifold S in R n . Assuming a smooth embedding of S in R n , we estimate the tangent space T P S by performing a Principal Component Analysis (PCA) on points sampled from the neighborhood of P on S. Our analysis explicitly takes into account the second order properties of the manifold at P , namely the principal curvatures as well as the higher order terms. We consider a random sampling framework and leverage recent results from random matrix theory to derive conditions on the sampling width and the local sampling density for an accurate estimation of tangent subspaces. We measure the estimation accuracy by the angle between the estimated tangent space T P S and the true tangent space T P S and we give conditions for this angle to be bounded with high probability. In particular, we observe that the local sampling conditions are highly dependent on the correlation between the components in the second-order local approximation of the manifold. We finally provide numerical simulations to validate our theoretical findings.

This paper describes a novel approach to partially reconstruct high-resolution 4D light fields from a stack of differently focused photographs taken with a fixed camera. First, a focus map is calculated from this stack using a simple approach combining gradient detection and region expansion with graph cut. Then, this focus map is converted into a depth map thanks to the calibration of the camera. We proceed after this with the tomographic reconstruction of the epipolar images by backprojecting the focused regions of the scene only. We call it masked back-projection. The angles of back-projection are calculated from the depth map. Thanks to the high angular resolution we achieve by suitably exploiting the image content captured over a large interval of focus distances, we are able to render puzzling perspective shifts although the original photographs were taken from a single fixed camera at a fixed position.

Abstract-Supervised manifold learning methods for data classification map data samples residing in a high-dimensional ambient space to a lower-dimensional domain in a structurepreserving way, while enhancing the separation between different classes in the learned embedding. Most nonlinear supervised manifold learning methods compute the embedding of the manifolds only at the initially available training points, while the generalization of the embedding to novel points, known as the out-of-sample extension problem in manifold learning, becomes especially important in classification applications. In this work, we propose a semi-supervised method for building an interpolation function that provides an out-of-sample extension for general supervised manifold learning algorithms studied in the context of classification. The proposed algorithm computes a radial basis function (RBF) interpolator that minimizes an objective function consisting of the total embedding error of unlabeled test samples, defined as their distance to the embeddings of the manifolds of their own class, as well as a regularization term that controls the smoothness of the interpolation function in a direction-dependent way. The class labels of test data and the interpolation function parameters are estimated jointly with a progressive procedure. Experimental results on face and object images demonstrate the potential of the proposed out-of-sample extension algorithm for the classification of manifold-modeled data sets.

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Abstract. We present a performance analysis for image registration with gradient descent. We consider a typical multiscale registration setting where the global 2-D translation between a pair of images is estimated by smoothing the images and minimizing the distance between them with gradient descent. Our study particularly concentrates on the effect of noise and low-pass filtering on the alignment accuracy. We analyze the well-behavedness of the image distance function by estimating the neighborhood of translations for which it is free of undesired local minima. This is the neighborhood of translations that are correctly computable with a simple gradient descent minimization. We show that the area of this neighborhood increases at least quadratically with the smoothing filter size. We then examine the effect of noise on the alignment accuracy and derive an upper bound for the alignment error in terms of the noise properties and filter size. Our main finding is that the error increases at a rate that is at least linear with respect to the filter size. Therefore, smoothing improves the well-behavedness of the distance function; however, this comes at the cost of amplifying the alignment error in noisy settings. Our results provide a mathematical insight about why hierarchical techniques are effective in image registration, suggesting that the multiscale alignment strategy of these techniques is very suitable from the perspective of the trade-off between the well-behavedness of the objective function and the registration accuracy. To the best of our knowledge, this is the first such study for descent-based image registration.