Point clouds are unordered sets of coordinates in 3D with no functional relation imposed on them. The Rigid Transformation Universal Manifold Embedding (RTUME) is a mapping of volumetric or surface measurements on a 3D object to matrices, such that when two observations on the same object are related by a rigid transformation, this relation is preserved between their corresponding RTUME matrices, thus providing linear and robust solution to the registration and detection problems. To make the RTUME framework of 3D object detection and registration applicable for processing point cloud observations, there is a need to define a function that assigns each point in the cloud with a value (feature vector), invariant to the action of the transformation group. Since existing feature extraction functions do not achieve the desired level of invariance to rigid transformations, to the variability of sampling patterns, and to model mismatches, we present a novel approach for designing dense feature extraction functions, compatible with the requirements of the RTUME framework. One possible implementation of the approach is to adapt existing feature extracting functions, whether learned or analytic, designed for the estimation of point correspondences, to the RTUME framework. The novel feature-extracting function design employs integration over $SO(3)$ to marginalize the pose dependency of extracted features, followed by projecting features between point clouds using nearest neighbor projection to overcome other sources of model mismatch. In addition, the non-linear functions that define the RTUME mapping are optimized using an MLP model, trained to minimize the RTUME registration errors. The overall RTUME registration performance is evaluated using standard registration benchmarks, and is shown to outperform existing SOTA methods.

We consider the problems of estimating the underlying transformation and the detection of 3-D objects undergoing rigid transformations. It has been shown that the Rigid Transformation Universal Manifold Embedding (RTUME) provides a mapping from the set of all possible observations on some object to a transformation covariant matrix representation, such that its column space is invariant to the geometric transformation. In this paper, we re-derive and adapt the RTUME for the case where the observations are in the form of meshed surfaces. We prove that by evaluating the integrals that define the RTUME operator as surface integrals on the mesh representation of the observed surface, the invariance and covariance properties of the RTUME matrix representation hold, similarly to the case of point cloud observations. It is shown that the RTUME matrix representation can be efficiently evaluated using a barycentric coordinate representation of the observed surface mesh representation. The proposed Mesh-RTUME is shown to outperform the RTUME representation, evaluated from the point cloud representation of the surface, both in transformation estimation and in keypoint detection.

We consider the problem of classifying 3-D objects undergoing rigid transformations. It has been shown that the rigid transformation universal manifold embedding (RTUME) provides a mapping from the orbit of observations on some object to a single low-dimensional linear subspace of Euclidean space. This linear subspace is invariant to the geometric transformations. In the classification problem the RTUME subspace extracted from an experimental observation is tested against a set of subspaces representing the different object manifolds, in search for the nearest class. We elaborate on the design problem of the RTUME operator in the case where the point cloud sampled from the object is sparse, noisy, and non-uniformly sampled. By introducing metric learning and negative-mining techniques into the framework of Grassmannian dimensionality reduction for universal manifold embedding, we improve classification performance for these challenging sampling conditions.

We present a novel method for online background modeling for static video cameras - Dynamic Spatial Predicted Background (DSPB). Our unique method employs a small subset of image pixels to predict the whole scene by exploiting pixel correlations (distant and close). DSPB acts as a hybrid model combining successful elements taken from two major approaches: local-adaptive that propose to fit a distribution pixelwise, and global-linear that reconstruct the background by finding a lowrank version of the scene. To our knowledge, this is the first attempt to combine these approaches in a unified system. DSPB models the scene as a superposition of illumination effects and predicts each pixel's value by a linear estimator comprised of only 5 pixels of the scene and can initialize the background starting from the 5th frame. By doing so, we keep the computational load low, allowing our method to be used in many real-time applications using simple hardware. The suggested prediction model of scene appearance is novel, and the scheme is very accurate and efficient computationally. We show the method merits on an application for video FG-BG separation, and how some of the main existing approaches may be challenged and how their drawbacks are less dominant in our model. Experimental results validate our findings, by computation speed and mean F-measure values on several public datasets. We also examine how results may improve by analyzing each video individually according to its content. DSPB can be successfully incorporated in other image processing tasks like change detection, video compression and video inpainting.

We present a closed form solution to the problem of registration and detection of dense 3-D point clouds undergoing unknown rigid deformations. The solution is obtained by adapting the general framework of the universal manifold embedding (UME) to the case where the deformations the object may undergo are rigid. The UME nonlinearly maps functions (e.g., images, 3D models) related by geometric transformations of coordinates to the same linear subspace of some Euclidean space. Therefore registration, matching and classification are solved as linear problems in a lower dimensional space. In this paper we extend the UME framework to the special case where it is a-priori known that the geometric transformations are rigid (e.g. pose change of a 3-D rigid object). We further demonstrate the applicability of the methodology for the registration of 3-D point clouds. In the case where point correspondences are unknown, the majority of existing methods for registering 3-D point clouds are based on iteratively finding a transformation which minimizes some distance between the object and a model. The method proposed in this paper is notably different as registration is performed using a closed form solution that employs the UME low dimensional representation of the shapes to be registered.

Video foreground-background separation is considered as a basic step for many computer vision applications. Common approaches excel in handling background variances while trying to keep the computational load low. We propose a novel method that models the scene as a superposition of illumination effects while predicting each pixel's value with a linear estimator comprised by a few other pixels of the scene. By doing so, we are able to achieve real-time performance using minimal hardware, which is a crucial consideration for embedding such a system on surveillance cameras. Experimental results on two common datasets show our method's potential by comparing it to state-of-the-art methods.

Consider a set of deformable objects undergoing geometric and radiometric transformations. As a result of the action of these transformations, the set of different realizations of each object is generally a manifold in the space of observations. Assuming the geometric deformations an object undergoes, belong to some finite dimensional family, it has been shown that the universal manifold embedding (UME) provides 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 distinct linear subspace of an Euclidean space. In this paper, we generalize this framework to the case where the observed object undergoes both an affine geometric transformation, and a monotonic radiometric transformation, and present a novel framework for the detection and recognition of the deformable objects. Applying to each of the observations an operator that makes it invariant to monotonic amplitude transformations, but is geometry-covariant with the affine transformation, the set of all possible observations on that object is mapped by the UME into a single linear subspace-invariant with respect to both the geometric and radiometric transformations. The embedding of the space of observations is independent of the specific observed object; hence it is universal. The invariant representation of the object is the basis of a matched manifold detection and tracking framework of objects that undergo complex geometric and radiometric deformations: the observed surface is tessellated into a set of tiles such that the deformation of each one is well approximated by an affine geometric transformation and a monotonic transformation of the measured intensities. Since each tile is mapped by the radiometry invariant UME to a distinct linear subspace, the detection and tracking problems are solved by evaluating distances between linear subspaces. Classification in this context becomes a problem of determining which labeled subspace in a Grassmannian is closest to a subspace in the same Grassmannian, where the latter has been generated by radiometry invariant UME from an unlabeled observation.

We present a novel framework for detection, tracking and recognition of deformable objects undergoing geometric and radiometric transformations. Assuming the geometric deformations an object undergoes, belong to some finite dimensional family, it has been shown that the universal manifold embedding (UME) provides 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 distinct linear subspace. In this paper we generalize this framework to the case where the observed object undergoes both an affine geometric transformation, and a monotonic radiometric transformation. Applying to each of the observations an operator that makes it invariant to monotonic amplitude transformations, but is geometry-covariant with the affine transformation, the set of all possible observations on that object is mapped by the UME into a distinct linear subspace - invariant with respect to both the geometric and radiometric transformations. This invariant representation of the object is the basis of a matched manifold detection and tracking framework of objects that undergo complex geometric and radiometric deformations: The observed surface is tessellated into a set of tiles such that the deformation of each one is well approximated by an affine geometric transformation and a monotonic transformation of the measured intensities. Since each tile is mapped by the radiometry invariant UME to a distinct linear subspace, the detection and tracking problems are solved by evaluating distances between linear subspaces.

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.

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.

The homography between pairs of images is typically computed from the correspondence of key features such as points, lines, conics and other geometric entities, each contributing information to fix the required 8 degrees of freedom (DOF). In the past years there have been attempts to use conics as correspondence features as a minimum of two correspondence pairs are required. However, the resulting methods either place restrictions on the problem (such as pure camera rotation, known calibration) or result in iterative non-linear solutions or in over-parameterized linear problems. Throughout this paper we propose a simple, direct linear transformation (DLT) like solution to the problem of homography estimation using local affine frames, without any restrictions on the physical model. We also provide approximated statistical analysis of the proposed algorithm and a comparison with the performance of the DLT algorithm. In addition, this method can be easily adapted to similar problems which employ a DLT-like algorithm.

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.

The problem of jointly estimating the number as well as the parameters of 2-D sinusoidal signals, observed in the presence of an additive colored noise field, is considered. We begin by establishing the strong consistency of the nonlinear least squares estimator of the parameters of 2-D sinusoids, when the number of sinusoidal signals assumed in the field is incorrect. Based on these results, we prove the strong consistency of a new family of model order selection rules.

We present a novel method for graph based image representation and matching. The image is represented as a graph of affine invariant regions. A local affine invariant coordinate system is used to describe the geometry at the location of each graph vertex. Once an image is encoded as a graph, a graph matching process can be initiated against the graph of any other image. Each pair of matched features induces an affine approximation of the deformation. The similarity of the affine approximations between different matchings is the foundation of a graph matching algorithm presented here, as it greatly reduces the ambiguity in matching. The algorithm is much faster than state of the art feature matching methods and produces similar results in the presence of highly complex scenes that contain background clutter, occlusion, viewpoint changes, and elastic geometric deformations.

We present a novel method to model and estimate elastic geometric deformations of an observed object, whether they are caused by the object's own dynamic behavior, or by the dynamic behavior of the imaging device, or both. A procedure for estimating the space of possible deformations the object may undergo based only on a set of observations is derived. This information is then employed to derive a linear estimator of the elastic deformation in any given observation. Application to change detection is presented.

We introduce a method for geometric deformation estimation of a known object, where the deformation belongs to a known family of deformations. Assume we have a set of observations (for example, images) of different objects, each undergoing different geometric deformation, yet all the deformations belong to the same family of deformations, Q. As a result of the action of Q, the set of different realizations of each object is generally a manifold in the space of observations. The manifolds of the different objects are strongly related. In this paper we obtain explicit estimations for the geometric deformations on the different manifolds, in several specific scenarios. We show that in some specific cases where the set of deformations, Q, 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 linear subspace with the same dimension as that of the manifold. This mapping which we call universal manifold embedding enables the estimation of geometric deformations using classical linear theory. 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 provide two examples of this embedding: for the case of elastic deformations of one-dimensional signals, and for the case of affine deformations of two-dimensional signals. We finally demonstrate the applicability of the solution to the problem of pose estimation in a laboratory setting.

This paper considers the problem of registering two observations of the same object, where the observations differ due to a combined effect of an affine geometric transformation and nonuniform illumination changes. The problem of deriving new representations of the observations that are both invariant to geometric transformations and linear in the illumination model is analyzed. In this framework, we present a novel method for linear estimation of illumination changes in an affine invariant manner, thus, decoupling the original problem into two simpler ones. The computational complexity of the method is low as it requires no more than solving a linear set of equations. The prior step of illumination estimation is shown to improve the accuracy of state-of-the-art registration techniques by a factor of two.

The general problem of defining and determining the sample distribution in the case of continuous-parameter random fields is addressed. Defining a distribution in the case of deterministic functions is straightforward, based on measures of sublevel sets. However, the fields we consider are the sum of a deterministic component (nonrandom multidimensional function) and an i.i.d. random field; an attempt to extend the same notion to the stochastic case immediately raises some fundamental difficulties. We show that by “uniformly sampling” such random fields the difficulties may be avoided and a sample distribution may be compatibly defined and determined. Not surprisingly, the obtained result resembles the known fact that the probability distribution of the sum of two independent random variables is the convolution of their distributions. Finally, we apply the results to derive a solution to the problem of deformation estimation of one- and multidimensional signals in the presence of measurement noise.

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.

We present a novel image transform called scale manipulation (SMT). The transform can be used for object pose estimation and registration of affine transformed images in the presence of non homogenous illumination changes. The transform calculates affine invariant features of objects in a global manner and avoids using any sort of edge detection. The computational load of the method is relatively low since it is linear in the data size. In this paper we introduce the transform and demonstrate its applications for pose estimation in the presence of non uniform varying illumination.

We consider the problem of jointly estimating the affine transformations of multiple objects from a single noisy observation, where each object is undergoing a different affine transformation. The derived algorithm is employed directly, such that prior segmentation of the observation into the distinct objects is avoided. Explicit expressions recovering the parameters of the transformation of each object are derived, assuming only knowledge of the template of each object. In the absence of noise the solution is exact and is not affected by the magnitudes of the deformations. In the presence of noise, the parameters are obtained using a linear least squares solution.

We consider the problem of planar object registration on binary images where the aligning transformation is restricted to the group of affine transformations. Previous approaches usually require established correspondences or the solution of nonlinear optimization problems. Herein we show that it is possible to formulate the problem as the solution of a system of up to third order polynomial equations. These equations are constructed in a simple way using some basic geometric information of binary images. It does not need established correspondences nor the solution of complex optimization problems. The resulting algorithm is fast and provides a direct solution regardless of the magnitude of transformation.

We present a new approach to the general problem of template-based segmentation, detection, and registration. This joint problem is highly nonlinear and high dimensional, due to the large space of possible geometric transformations between a given template and its observed signature. Hence, any attempt to directly solve it inevitably leads to a high dimensional, nonlinear, nonconvex optimization procedure. We propose a novel parametric solution to this problem, by showing that it can be equivalently represented by a low dimensional model, that is linear in the deformation parameters, and biased by the unknown observation background. Classical linear methods are then employed to estimate the deformation parameters, providing an explicit solution for the joint segmentation and registration problem.

We address the problem of object registration when the observation differs from the object both geometrically and radio-metrically. The geometric deformations being considered are affine. The radiometric deformations are due to the a-priori lack of knowledge regarding the locations and intensities of the light sources. Hence, to solve the registration problem, a joint solution for the radiometric and the geometric deformations must be offered. A direct approach for solving the joint registration problem as an optimization problem leads to a high-dimensional non-convex search problem. In this paper, we treat the images as vector valued measurements, such that each element of the vector provides the intensity at a specific spectral (color) band. By applying a set of operators, derived in the paper, to the vector valued data the original high-dimensional search problem is replaced by an equivalent problem, expressed in terms of two systems of linear equations. Their solution provides an exact solution to the joint problem.

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

We consider the problem of jointly estimating the number as well as the parameters of two-dimensional sinusoidal signals, observed in the presence of an additive colored noise field. In this framework we consider the problem of least squares estimation of the parameters of 2-D sinusoidal signals observed in the presence of an additive noise field, when the assumed number of sinusoids is incorrect. In the case where the number of sinusoidal signals is under-estimated we show the almost sure convergence of the least squares estimates to the parameters of the dominant sinusoids. In the case where the number of sinusoidal signals is over-estimated, the estimated parameter vector obtained by the least squares estimator contains a sub-vector that converges almost surely to the correct parameters of the sinusoids. Based on these results, we prove the strong consistency of a large family of model order selection rules

We consider the problem of least squares estimation of the parameters of 2D exponential signals observed in the presence of an additive noise field, when the assumed number of exponentials is incorrect. We consider both the case where the number of exponential signals is under-estimated, and the case where the number of exponential signals is over-estimated. In the case where the number of exponential signals is under-estimated we prove the almost sure convergence of the least squares estimates to the parameters of the dominant exponentials. In the case where the number of exponential signals is over-estimated, the estimated parameter vector obtained by the least squares estimator contains a sub-vector that converges almost surely to the correct parameters of the exponentials.

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.

We consider the problem of estimating jointly the number and the parameters of two-dimensional sinusoidal signals, observed in the presence of an additive white Gaussian noise field. We prove the strong consistency of a large family of model order selection rules, which includes the MAP based rule as a special case.

We consider the problem of least squares estimation of the parameters of two-dimensional (2-D) exponential signals observed in the presence of an additive noise field, when the assumed number of exponentials is incorrect. We consider both the case where the number of exponential signals is underestimated, and the case where the number of exponential signals is overestimated. In the case where the number of exponential signals is underestimated, we prove the almost sure convergence of the least squares estimates (LSE) to the parameters of the dominant exponentials. In the case where the number of exponential signals is overestimated, the estimated parameter vector obtained by the least squares estimator contains a subvector that converges almost surely to the correct parameters of the exponentials.

We consider the general problem of object recognition, based on a set of known templates. 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 affine deformation is expressed using a set on non-linear operators, by a set of linear equations. This system of linear equations is then solved for the transformation parameters.

We introduce Gaussian mixture models of 'structure' and colour features in order to classify coloured textures in images, with a view to the retrieval of textured colour images from databases. Classifications are performed separately using structure and colour and then combined using a confidence criterion. We apply the models to the VisTex database and to the classification of man-made and natural areas in aerial images. We compare these models with others in the literature, and show an overall improvement in performance.

We develop parametric modeling and estimation methods for STAP data based on the results of the 2-D Wold-like decomposition. It is shown that the same parametric model that results from the 2-D Wold-like orthogonal decomposition naturally arises as the physical model in the problem of space-time processing of airborne radar data. This correspondence is exploited to derive a computationally efficient parametric partially adaptive detection algorithm. We prove that it is sufficient to estimate only the spectral support parameters of each interference component in order to obtain a projection matrix onto the subspace orthogonal to the interference subspace. The proposed partially adaptive parametric processing algorithm employs this property. The proposed parametric interference mitigation procedure can be applied when only the information in a single range gate is available, thus achieving high performance gain when the data in the different range gates cannot be assumed stationary.

This paper considers the problem of estimating the parameters of real-valued two-dimensional (2-D) sinusoidal signals observed in colored noise. This problem is a special case of the general problem of estimating the parameters of a real-valued homogeneous random field with mixed spectral distribution from a single observed realization of it. The large sample properties of the least squares (LS) estimator of the parameters of the sinusoidal components are derived, making no assumptions on the type of the probability distribution of the observed field. It is shown that if the disturbance field satisfies a combination of conditions comprised of a strong mixing condition and a condition on the order of its uniformly bounded moments, the normalized estimation error of the LS estimator is consistent asymptotically normal with zero mean and a normalized asymptotic covariance matrix for which a simple expression is derived. It is further shown that the LS estimator is asymptotically unbiased. The normalized asymptotic covariance matrix is block diagonal where each block corresponds to the parameters of a different sinusoidal component. Assuming further that the colored noise field is Gaussian, the LS estimator of the sinusoidal components is shown to be asymptotically efficient.

This paper presents a parametric solution to the problem of estimating the orientation in space of a planar textured surface, from a single, noisy, observed image of it. The coordinate transformation from surface to image coordinates, due to the perspective projection, transforms each homogeneous sinusoidal component of the surface texture into a sinusoid whose frequency is a function of location. The functional dependence of the sinusoid phase in location is uniquely determined by the tilt and slant angles of the surface. Using the phase differencing algorithm we fit a polynomial phase model to a sinusoidal component of the observed texture. Assuming the estimated polynomial coefficients are the coefficients of a Taylor series expansion of the phase, we establish a linear recursive relation between the model parameters and the unknown slant and tilt. A linear least squares solution of the resulting system provides the slant and tilt estimates. To improve accuracy, an iterative refinement procedure is applied in a small neighborhood of these estimates. The performance of the proposed algorithms is evaluated by applying them to images of different planar surfaces, and by comparing their statistical performance with the Cramer-Rao bound. The combined two-stage algorithm is shown to produce estimates that are close to the bound.

This paper presents a computationally and statistically efficient parametric solution to the problem of estimating the orientation in space of a planar textured surface from a single, noisy, observed image of it. The coordinate transformation from surface to image coordinates, due to the perspective projection, transforms each homogeneous sinusoidal component of the surface texture into a sinusoid whose frequency is a function of location. The functional dependence of the sinusoid phase in location is uniquely determined by the tilt and slant angles of the surface. From the physical model of the perspective projection, we derive the Cramer-Rao lower bound on the error variance of estimating the tilt and slant of the observed surface in the presence of observation noise. It is shown in this paper that the phase of each of the sinusoids can be expressed as a linear function of some variables that are related to the surface tilt and slant angles. Using the phase differencing algorithm, we fit a polynomial phase model to a sinusoidal component of the observed texture. Substituting in the derived linear relation, the unknown phase with the one estimated using the phase differencing algorithm, we obtain a closed-form, analytic, and computationally efficient solution to the problem of estimating the tilt and slant angles. The algorithm performance is shown to be close to the Cramer-Rao bound, even for low signal-to-noise ratios, at computational complexity which is considerably lower than that of any existing algorithm.

This paper presents a method for indexing and retrieval of multimedia data through texture segmentation, using the Wold decomposition. The texture field is assumed to be a realisation of a regular homogeneous random field. On the basis of a 2-D Wold-like decomposition, the field is represented as the sum of a purely indeterministic component, a harmonic component and a countable number of evanescent fields. A new rigorous distance measure between textures is derived, using Wold parameters. Adopting the MRF framework, we construct a segmentation procedure using the Wold parameters.

This paper presents a parametric solution to the problem of estimating the orientation in space of a planar textured surface, from a single observed image of it. The coordinate transformation from surface to image coordinates, due to the perspective projection, transforms each homogeneous sinusoidal component of the surface texture into a sinusoid whose frequency is a function of location. Using the phase differencing algorithm we fit a polynomial phase model to a sinusoidal component of the observed texture. Assuming the estimated polynomial coefficients are the coefficients of a Taylor series expansion of the phase, we establish a linear recursive relation between the model parameters and the unknown slant and tilt. A linear least squares solution of the resulting system provides the slant and tilt estimates. To improve accuracy, an iterative refinement procedure is applied in a small neighborhood of these estimates. The combined two-stage algorithm is shown to produce estimates that are close to the Cramer-Rao bound, at a computational complexity which is considerably lower than that of any existing algorithm.

This paper presents a method for indexing and retrieval of multimedia data. The proposed indexing and retrieval strategy is based on the usage of textural information contained in the data imagery components as the indexing keys. On the basis of a 2-D Wold-like decomposition, the texture field is represented as the sum of purely indeterministic, harmonic, and evanescent fields. A new rigorous distance measure between textures which employs their estimated parametric models, is developed. This distance measure is then applied to retrieve multimedia records that contain images with textured segments which are similar to those in a given image. Evaluation of this distance measure is computationally efficient, and hence highly suitable for data base retrieval applications.

This paper considers the problem of estimating the parameters of two-dimensional moving average random fields. We first address the problem of expressing the covariance matrix of a moving average random field, in terms of the model parameters. Assuming the random field is Gaussian, we derive a closed form expression for the Cramer-Rao lower bound on the error variance in jointly estimating the model parameters. A computationally efficient algorithm for estimating the parameters of the moving average model is developed. The algorithm initially fits a two-dimensional autoregressive model to the observed field, then uses the estimated parameters to compute the moving average model.

This paper considers the achievable accuracy in jointly estimating the parameters of a real-valued two-dimensional (2-D) homogeneous random field with mixed spectral distribution, from a single observed realization of it. On the basis of a 2-D Wold-like decomposition, the field is represented as a sum of mutually orthogonal components of three types: purely indeterministic, harmonic, and evanescent. An exact form of the Cramer-Rao lower bound on the error variance in jointly estimating the parameters of the different components is derived. It is shown that the estimation of the harmonic component is decoupled from that of the purely indeterministic and the evanescent components. Moreover, the bound on the parameters of the purely indeterministic and the evanescent components is independent of the harmonic component. Numerical evaluation of the bounds provides some insight into the effects of various parameters on the achievable estimation accuracy.

We consider non-homogeneous 2-D signals which can be represented by a constant modulus polynomial-phase model. In previous papers we developed a computationally efficient estimation algorithm for the parameters of this model, and a novel phase unwrapping method which is based on this estimation algorithm. In this paper we analyze the performance of the algorithm and derive expressions for the mean squared error of the estimated coefficients. Assuming high signal to noise ratio (SNR), we show that the estimates are unbiased, and derive a rule for optimal selection of the algorithm parameters. The theoretical results are verified by Monte-Carlo simulations for selected examples. Finally, we present an approximate error analysis of the estimates for an arbitrary SNR. This analysis is carried out for a specific set of the algorithm parameters, which is selected based on the high SNR analysis.

We consider nonhomogeneous 2-D signals that can be represented by a constant modulus polynomial-phase model. A novel 2-D phase differencing operator is introduced and used to develop a computationally efficient estimation algorithm for the parameters of this model. The operation of the algorithm is illustrated using an example.

We consider the adaptive restoration of inhomogeneous textured images, where the individual regions are modeled using a Wold-like decomposition. A generalized Wiener filter is developed to accommodate mixed spectra, and unsupervised restoration is achieved by using the expectation-maximization (EM) algorithm to estimate the degradation parameters. This algorithm yields superior results when compared with supervised Wiener filtering using autoregressive (AR) image models.

A novel approach for coding textured images is presented. The texture field is assumed to be a realization of a regular homogeneous random field, which can have a mixed spectral distribution. On the basis of a two-dimensional (2-D) Wold-like decomposition, the field is represented as a sum of a purely indeterministic, harmonic, and countable number of evanescent fields. We present an algorithm for estimating and coding the texture model parameters, and show that the suggested algorithm yields high-quality reconstructions at low bit rates. The model and the resulting coding algorithm are seen to be applicable to a wide variety of texture types found in natural images.

This paper presents a maximum-likelihood solution to the general problem of fitting a parametric model to observations from a single realization of a two-dimensional (2-D) homogeneous random field with mixed spectral distribution. On the basis of a 2-D Wold-like decomposition, the field is represented as a sum of mutually orthogonal components of three types: purely indeterministic, harmonic, and evanescent. The suggested algorithm involves a two-stage procedure. In the first stage, we obtain a suboptimal initial estimate for the parameters of the spectral support of the evanescent and harmonic components. In the second stage, we refine these initial estimates by iterative maximization of the conditional likelihood of the observed data, which is expressed as a function of only the parameters of the spectral supports of the evanescent and harmonic components. The solution for the unknown spectral supports of the harmonic and evanescent components reduces the problem of solving for the other unknown parameters of the field to a linear least squares. The Cramer-Rao lower bound on the accuracy of jointly estimating the parameters of the different components is derived, and it is shown that the bounds on the purely indeterministic and deterministic components are decoupled. Numerical evaluation of the bounds provides some insight into the effects of various parameters on the achievable estimation accuracy. The performance of the maximum-likelihood algorithm is illustrated by Monte Carlo simulations and is compared with the Cramer-Rao bound.

Any regular stationary random processes can be represented as the sum of a purely indeterministic process and a deterministic one. This paper considers the achievable accuracy in the joint estimation of the parameters of these two components, from a single observed realization of the process. An exact form of the Cramer-Rao bound (CRB) is derived, as well as a conditional CRB. The relationships between these bounds, and their relations to the previously derived asymptotic bound, are explored by analysis and numerical examples.

We consider non-homogeneous 2-D signals which can be represented by a constant modulus polynomial-phase model. We develop a computationally efficient estimation algorithm for the parameters of this model. The algorithm is based on a phase differencing operator which is introduced in this paper. The basic properties of the operator are derived and used to develop the estimation algorithm.

Abstruct-We present a solution to the problem of modeling, parameter estimation, and synthesis of natural textures.The texture field is assumed to be a realization of a regular homogeneous random field, which can have a mixed spectral distribution.On the basis of a 2-D Wold-like decomposition, the field is represented as a sum of a purely indeterministic component, a harmonic component, and a countable number of evanescent fields.We present a maximum-likelihood solution to the joint parameter estimation problem of these components from a single observed realization of the texture field.The proposed solution is a twostage algorithm.In the first stage, we obtain an estimate for the number of harmonic and evanescent components in the field, and a suboptimal initial estiqate for the parameters of their spectral supports.In the second stage, we refine these initial estimates by iterative maximization of the likelihood function of the observed data.By introducing appropriate parameter transformations the highly nonlinear least-squares problem that results from the maximization of the likelihood function, is transformed into a separable least-squares problem.In this new problem, the solution for the unknown spectral supports of the harmonic and evanescent components reduces the problem of solving for the transformed parameters of the field to linear least squares.Solution of the transformation equations then provides a complete solution of the field-model parameter estimation problem.The Wold-based model and the resulting analysis and synthesis algorithms are seen applicable to a wide variety of texture types found in natural images.The support or shape of the analyzed texture patch may be arbitrary.Our model is very efficient in terms of the number of parameters required to represent and faithfully reconstruct the original texture.

A novel approach for coding textured images is presented. The texture field is assumed to be a realization of a regular homogeneous random field, which can have a mixed spectral distribution. On the basis of a 2D Wold-like decomposition, the field is represented as a sum of purely indeterministic, harmonic, and a countable number of evanescent fields. We present an algorithm for estimating and coding the texture model parameters, and show that the suggested algorithm yields high quality reconstructions at low bit rates. The model and the resulting coding algorithm are seen to be applicable to a wide variety of texture types found in natural images.

We consider the adaptive restoration of inhomogeneous textured images degraded by linear blur and additive white Gaussian noise. The method consists of segmenting the image into individual homogeneous textures and restoring each texture separately. The individual textures are assumed to be realizations of 2-D Wold-decomposition based regular, homogeneous random fields which may possess deterministic components. The conventional Wiener filter assumes that the spectral distribution of the signal is absolutely continuous and, therefore, cannot be directly used to restore the individual textures. A generalized Wiener filter accommodates the unified texture model and is shown to yield minimum mean-squared error estimates for fields with discontinuous spectral distributions. Texture discrimination is performed by obtaining maximum a posteriori estimates for the label field using simulated annealing. The performance of our segmentation algorithm is investigated in the presence of noise.

A maximum-likelihood (ML) estimation procedure for the parameters of homogeneous random fields is presented. On the basis of a 2D Wold-like decomposition, the random field is decomposed into a sum of mutually orthogonal components of three types: purely indeterministic, harmonic, and generalized evanescent. Because, in general, a homogeneous random field is characterized by a mixed spectral distribution, parameter estimation techniques which are solely based on spectral density estimators are not adequate. An algorithm which results in a complete parameter estimation for the purely indeterministic and harmonic components of the homogeneous random field and in estimates for the evanescent component frequencies is presented. The new ML method produces highly accurate estimates of these parameters.>

A new model for the spatial singularities component of the texture field and parameter estimation and synthesis procedures are presented. The spatial singularities field includes the local-structural components of the texture. The field is modeled by a 2-D compound Poisson-cliques model, after testing the fitness of the proposed model to the detected singularities field. According to this model the spatial singularities field is a 2-D point process where each of the events is a clique. The test of fit is based on a discrete version of the Kolmogorov-Smirnov test. In the analysis procedure the intensity of the 2-D point process is estimated, as well as the empirical distribution function of the clique occurrences. These parameters are then used to synthesize the singularities field.>

Imposing a total-order on a two-dimensional discrete homogeneous random field induces an orthogonal decomposition of the random field into two components: a purely indeterministic field and a deterministic one. The purely indeterministic component is shown to have a two-dimensional white-innovations driven MA representation. The two-dimensional deterministic random field can be perfectly predicted from the field's past samples. This field is further orthogonally decomposed into a purely deterministic field that represents the remote past of the field and can thus be perfectly predicted given enough arbitrarily located data samples, and an evanescent component. The evanescent component can be further decomposed into a remote columnwise past component and a column-to-column renewal field.>

A 2-D AR (autoregressive), finite-support, half-plane, causal model for homogeneous random fields is developed and applied to the analysis and synthesis of homogeneous random textures. The conditions under which the finite, discontinuous-support, 2-D Levinson type algorithm can be applied to solve the 2-D normal equations are presented. In the texture analysis case, these conditions are met by first removing all periodic components and subsequently applying a 2-D preemphasis filter. These steps also help reduce the required model order. It is shown that the resulting model is very efficient in terms of both the number of parameters required to achieve a good reconstructed texture (which is usually indistinguishable from the original) and good correlation match.>

A highly nonrestrictive model and procedure for texture analysis and synthesis are presented. The texture is modeled by three independent components: global structural, local structural and stochastic. The texture's global (directional and periodic) components are identified, parameterized and reconstructed through spectral analysis. Local structural components are extracted by parameterizing the line singularities. The stochastic component is modeled, analyzed and synthesized by a 2D half-plane finite-dimension autoregressive model. The model is very efficient in terms of the number of parameters required to represent and reconstruct a given texture faithfully.>