Joseph M. Francos

A unified texture model based on a 2-D Wold-like decomposition

A unified texture model that is applicable to a wide variety of texture types found in natural images is presented. This model leads to the derivation of texture analysis and synthesis algorithms designed to estimate the texture parameters and to reconstruct the original texture field from these parameters. The texture field is assumed to be a realization of a regular homogeneous random field, which is characterized in general by a mixed spectral distribution. The texture field is orthogonally decomposed into a purely indeterministic component and a deterministic component. The deterministic component is further orthogonally decomposed into a harmonic component, and a generalized-evanescent component. Both analytical and experimental results show that the deterministic components should be parameterized separately from the purely indeterministic component. The model is very efficient in terms of the number of parameters required to faithfully represent textures. Reconstructed textures are practically indistinguishable from the originals. >

Estimation of amplitude and phase parameters of multicomponent signals

This paper considers the problem of estimating signals consisting of one or more components of the form a(t)e/sup j/spl phi/(t/), where the amplitude and phase functions are represented by a linear parametric model. The Cramer-Rao bound (CRB) on the accuracy of estimating the phase and amplitude parameters is derived. By analyzing the CRB for the single-component case, if is shown that the estimation of the amplitude and the phase are decoupled. Numerical evaluation of the CRB provides further insight into the dependence of estimation accuracy on signal-to-noise ratio (SNR) and the frequency separation of the signal components. A maximum likelihood algorithm for estimating the phase and amplitude parameters is also presented. Its performance is illustrated by Monte-Carlo simulations, and its statistical efficiency is verified. >

Model based phase unwrapping of 2-D signals

A parametric model and a corresponding parameter estimation algorithm for unwrapping 2-D phase functions are presented. The proposed algorithm performs global analysis of the observed signal. Since this analysis is based on parametric model fitting, the proposed phase unwrapping algorithm has low sensitivity to phase aliasing due to low sampling rates and noise, as well as to local errors. In its first step, the algorithm fits a 2-D polynomial model to the observed phase. The estimated phase is then. Used as a reference information that directs the actual phase unwrapping process. The phase of each sample of the observed field is unwrapped by increasing (decreasing) it by the multiple of 2/spl pi/, which is the nearest to the difference between the principle value of the phase and the estimated phase value at this coordinate. In practical applications, the entire phase function cannot be approximated by a single 2-D polynomial model. Hence, the observed field is segmented, and each segment is fit with its own model. Once the phase model of the observed field has been estimated, we can repeat the model-based unwrapping procedure described earlier for the case of a single segment and a single model field.

Bounds for estimation of multicomponent signals with random amplitude and deterministic phase

We study a class of nonstationary multicomponent signals, where each component has the form a(t) exp j/spl phi/(t), where a(t) is a random amplitude function, and /spl phi/(t) is a deterministic phase function. The amplitude function consists of a stationary Gaussian process and a time varying mean. The phase and the amplitude mean are characterized by a linear parametric model, while the covariance of the amplitude function is parameterized in some general manner. This model encompasses signals that are commonly used in communications, radar, sonar, and other engineering systems. We derive the Cramer-Rao bound (CRB) for the estimates of the amplitude and phase parameters, and of functions of these parameters, such as the instantaneous frequencies of the signal components. >

Maximum likelihood parameter estimation of textures using a Wold-decomposition based model

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 two-stage algorithm. In the first stage, we obtain an estimate for the number of harmonic and evanescent components in the field, and a suboptimal initial estimate 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. 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 a 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 applicable to a wide variety of texture types found in natural images.

An estimation algorithm for 2-D polynomial phase signals

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.

Bounds for estimation of complex exponentials in unknown colored noise

We consider the problem of estimating the parameters of complex exponentials in the presence of complex additive Gaussian noise with unknown covariance. Bounds are derived for the accuracy of jointly estimating the parameters of the exponentials and the noise. We first present an exact Cramer-Rao bound (CRB) for this problem and specialize it for the cases of circular Gaussian processes and autoregressive processes. We also derive an approximate expression for the CRB, which is related to the conditional likelihood function. Numerical evaluation of these bounds provides some insights on the effect of various signal and noise parameters on the achievable estimation accuracy. >

The two-dimensional Wold decomposition for segmentation and indexing in image libraries

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.

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.