Rami R. Hagege

Parametric Estimation of Affine Transformations: An Exact Linear Solution

We consider the problem of estimating the geometric deformation of an object, with respect to some reference observation on it. Existing solutions, set in the standard coordinate system imposed by the measurement system, lead to high-dimensional, non-convex optimization problems. We propose a novel framework that employs a set of non-linear functionals to replace this originally high dimensional problem by an equivalent problem that is linear in the unknown transformation parameters. The proposed solution includes the case where the deformation relating the observed signature of the object and the reference template is composed both of the geometric deformation due to the affine transformation of the coordinate system and a constant amplitude gain. The proposed solution is unique and exact and is applicable to any affine transformation regardless of its magnitude.

Estimation of multidimensional homeomorphisms for object recognition in noisy environments

In this paper, we consider the general problem of object recognition based on a set of known templates, where the available observations are noisy. While the set of templates is known, the tremendous set of possible transformations and deformations between the template and the observed signature, makes any detection and recognition problem ill-defined unless this variability is taken into account. We propose a method that reduces the high dimensional problem of evaluating the orbit created by applying the set of all possible transformations in the group to a template, into a problem of analyzing a function in a low dimensional Euclidian space. In this setting, the problem of estimating the parametric model of the deformation is transformed using a set on nonlinear operators into a set of equations which is solved by a linear least squares solution in the low dimensional Euclidian space. For the case where the signal to noise ratio is high, and the nonlinear operators are polynomial compositions, a maximum-likelihood estimator is derived, as well.

Linear Estimation of Sequences of Multi-Dimensional Affine Transformations

We consider the general framework of planar object registration and tracking. Given a sequence of observations on an object, subject to an unknown sequence of affine transformations of it, our goal is to estimate the deformation that transforms some pre-chosen representation of this object (template) into the current sequence of observations. We propose a method that employs a set of non-linear operators to replace the original high dimensional and non-linear problem by an equivalent linear problem, expressed in terms of the unknown affine transformation parameters. We investigate two modelling and estimation solutions: the first, estimates the affine transformation relating any two consecutive observations, followed by a least squares fit of a global model to the estimated sequence of instantaneous deformations. The second, is a global solution that fits a time-dependent affine model to the entire set of observed data

Parametric estimation of multi-dimensional affine transformations: an exact linear solution [image recognition applications]

We consider the general framework of planar object recognition based on a set of known templates. Given an observation on one of the known objects, subject to an unknown affine transformation of it, our goal is to estimate the deformation that transforms some pre-chosen representation of this object (template) into the current observation. The direct approach for estimating the transformation is to apply each of the deformations in the affine group to the template to search for the deformed template that matches the observation. We propose a method that employs a set of non-linear operators to replace this high-D problem by an equivalent linear problem, expressed in terms of the unknown affine transformation parameters. This solution is further extended to include the case where the deformation relating the observed signature of the object and the template is composed both of the geometric deformation due to the affine transformation of the coordinate system and a constant illumination change. The proposed solution is unique and exact and is applicable to any affine transformation regardless of the magnitude of the deformation.

2D-3D Pose Estimation of Heterogeneous Objects Using a Region Based Approach

Recently, region based methods for estimating the 3D pose of an object from a 2D image have gained increasing popularity. They do not require prior knowledge of the object’s texture, making them particularity attractive when the object’s texture is unknown a priori. Region based methods estimate the 3D pose of an object by finding the pose which maximizes the image segmentation in to foreground and background regions. Typically the foreground and background regions are described using global appearance models, and an energy function measuring their fit quality is optimized with respect to the pose parameters. Applying a region based approach on standard 2D-3D pose estimation databases shows its performance is strongly dependent on the scene complexity. In simple scenes, where the statistical properties of the foreground and background do not spatially vary, it performs well. However, in more complex scenes, where the statistical properties of the foreground or background vary, the performance strongly degrades. The global appearance models used to segment the image do not sufficiently capture the spatial variation. Inspired by ideas from local active contours, we propose a framework for simultaneous image segmentation and pose estimation using multiple local appearance models. The local appearance models are capable of capturing spatial variation in statistical properties, where global appearance models are limited. We derive an energy function, measuring the image segmentation, using multiple local regions and optimize it with respect to the pose parameters. Our experiments show a substantially higher probability of estimating the correct pose for heterogeneous objects, whereas for homogeneous objects there is minor improvement.

Decoupled Linear Estimation of Affine Geometric Deformations and Nonlinear Intensity Transformations of Images

We consider the problem of registering two observations on an arbitrary object, where the two are related by a geometric affine transformation of their coordinate systems, and by a nonlinear mapping of their intensities. More generally, the framework is that of jointly estimating the geometric and radiometric deformations relating two observations on the same object. We show that the original high-dimensional, nonlinear, and nonconvex search problem of simultaneously recovering the geometric and radiometric deformations can be represented by an equivalent sequence of two linear systems. A solution of this sequence yields an exact, explicit, and efficient solution to the joint estimation problem.

Detection and recognition of deformable objects using structured dimensionality reduction

We present a novel framework for detection and recognition of deformable objects undergoing geometric deformations. Assuming the geometric deformations belong to some finite dimensional family, it is shown that there exists a set of nonlinear operators that universally maps each of the different manifolds, where each manifold is generated by the set all of possible appearances of a single object, into a unique linear subspace. In this paper we concentrate on the case where the deformations are affine. Thus, all affine deformations of some object are mapped by the above universal manifold embedding into the same linear subspace, while any affine deformation of some other object is mapped by the above universal manifold embedding into a different subspace. It is therefore shown that the highly nonlinear problems of detection and recognition of deformable objects can be formulated in terms of evaluating distances between linear subspaces. The performance of the proposed detection and recognition solutions is evaluated in various settings.

Linear Estimation of Time-Warped Signals

We introduce a novel methodology for estimating the time-axis deformation between two observations on a time-warped signal. Since the problem of estimating the warping function is nonlinear, existing methods iteratively minimize some metric between the observation and a hypothesized deformed template. Assuming the family of possible deformations the signal may undergo admits a finite-dimensional representation, we show that there is a nonlinear mapping from the space of observations to a low-dimensional linear space, such that in this space the problem of estimating the parametric model of the warping function is solved by a linear system of equations. We call the family of estimators derived based on this representation, linear warping estimators (LWE). The new representation of the problem enables an analytic analysis of the behavior of the solution in the presence of model mismatches, which is prohibitive when iterative methods are employed. The ability to achieve this major simplification both in the solution and in analyzing its performance results from the representation of the problem in a new coordinate system which is natural to the properties of the problem, instead of representing it in the standard coordinate system imposed by the sampling mechanism. The proposed solution is unique and exact, as it provides a closed-form expression for evaluating each of the parameters of the warping model using only measurements of the amplitude information of the observed and reference signals. The solution is applicable to any elastic warping regardless of its magnitude. We analyze the behavior of the LWE in the presence of noise and obtain a minimum variance unbiased estimator for the model parameters, by finding an optimal set of nonlinear operators for mapping the original problem into a low-dimensional linear space.

Parametric estimation of multi-dimensional affine transformations in the presence of noise: a linear solution

We consider the general framework of planar object registration and recognition based on a set of known templates. While the set of templates is known, the tremendous set of possible affine transformations that may relate the template and the observed signature, makes any detection and recognition problem ill-defined unless this variability is taken into account. Given a noisy observation on one of the known objects, subject to an unknown affine transformation of it, our goal is to estimate the deformation that transforms some pre-chosen representation of this object (template) into the current observation. We propose a method that employs a set of nonlinear operators to replace the original high dimensional and non-linear problem by an equivalent linear least-squares problem, expressed in terms of the unknown affine transformation parameters. The proposed solution is unique and is applicable to any affine transformation regardless of the magnitude of the deformation

Universal Manifold Embedding for Geometrically Deformed Functions

Assume we have a set of observations (for example, images) of different objects, each undergoing a different geometric deformation, yet all the deformations belong to the same family. As a result of the action of these deformations, the set of different observations on each object is generally a manifold in the ambient space of observations. In this paper we show that in those cases where the set of deformations admits a finite-dimensional representation, there is a mapping from the space of observations to a low-dimensional linear space. The manifold corresponding to each object is mapped to a distinct linear subspace of Euclidean space. The dimension of the subspace is the same as that of the manifold. This mapping, which we call universal manifold embedding, enables the estimation of geometric deformations using the classical linear theory. The universal manifold embedding further enables the representation of the object classification and detection problems in a linear subspace matching framework. The embedding of the space of observations depends on the deformation model, and is independent of the specific observed object; hence, it is universal. We study two cases of this embedding: that of elastic deformations of 1-D signals, and the case of affine deformations of n-dimensional signals.