Quantitative ultrasound (QUS) techniques using radiofrequency (RF) backscattered signals have been used for tissue characterization of numerous organ systems. One approach is to use the magnitude and frequency dependence of backscatter echoes to quantify tissue structures. Another approach is to use first-order statistical properties of the echo envelope as a signature of the tissue microstructure. We propose a unification of these QUS concepts. For this purpose, a mixture of homodyned K-distributions is introduced to model the echo envelope, together with an estimation method and a physical interpretation of its parameters based on the echo signal spectrum. In particular, the total, coherent and diffuse signal powers related to the proposed mixture model are expressed explicitly in terms of the structure factor previously studied to describe the backscatter coefficient (BSC). Then, this approach is illustrated in the context of red blood cell (RBC) aggregation. It is experimentally shown that the total, coherent and diffuse signal powers are determined by a structural parameter of the spectral Structure Factor Size and Attenuation Estimator. A two-way repeated measures ANOVA test showed that attenuation (p-value of 0.077) and attenuation compensation (p-value of 0.527) had no significant effect on the diffuse to total power ratio. These results constitute a further step in understanding the physical meaning of first-order statistics of ultrasound images and their relations to QUS techniques. The proposed unifying concepts should be applicable to other biological tissues than blood considering that the structure factor can theoretically model any spatial distribution of scatterers.

The Horn--Schunck (HS) method, which amounts to the Jacobi iterative scheme in the interior of the image, was one of the first optical flow algorithms. In this paper, we prove the convergence of the HS method whenever the problem is well-posed. Our result is shown in the framework of a generalization of the HS method in dimension $n\geq1$, with a broad definition of the discrete Laplacian. In this context, the condition for the convergence is that the intensity gradients not all be contained in the same hyperplane. Two other works ([A. Mitiche and A. Mansouri, IEEE Trans. Image Process., 13 (2004), pp. 848--852] and [Y. Kameda, A. Imiya, and N. Ohnishi, A convergence proof for the Horn-Schunck optical-flow computation scheme using neighborhood decomposition, in Combinatorial Image Analysis, Springer, Berlin, 2008, pp. 262--273]) claimed to solve this problem in the case $n=2$, but it appears that both of these proofs are erroneous. Moreover, we explain why some standard results about the convergence of the Jacobi method do not apply for the HS problem, unless $n=1$. It is also shown that the convergence of the HS scheme implies the convergence of the Gauss--Seidel and successive overrelaxation schemes for the HS problem.

The homodyned K-distribution appears naturally in the context of random walks and provides a useful model for the distribution of the received intensity in a wide range of non-Gaussian scattering configurations, including medical ultrasonics. An estimation method for the homodyned K-distribution based on the first moment of the intensity and two log-moments (XU method), namely the $X$- and $U$-statistics previously studied in the special case of the K-distribution, is proposed as an alternative to a method based on the first three moments of the intensity (MI method) or the amplitude (MA method), and a method based on the signal-to-noise ratio (SNR), the skewness, and the kurtosis of two fractional orders of the amplitude (labeled RSK method). Properties of the $X$- and $U$-statistics for the homodyned K-distribution are proved, except for one conjecture. Using these properties, an algorithm based on the bisection method for monotonous functions was developed. The algorithm has a geometric rate of convergence. Various tests were performed to study the behavior of the estimators. It was shown with simulated data samples that the estimations of the parameters $1/\alpha$ and $1/(\kappa+1)$ of the homodyned K-distribution are preferable to the direct estimations of the clustering parameter $\alpha$ and the structure parameter $\kappa$ (with respective relative root mean squared errors (RMSEs) of $0.63$ and $0.13$, as opposed to $1.04$ and $4.37$, when $N=1000$). Tests on simulated ultrasound images with only diffuse scatterers (up to $10$ per resolution cell) indicated that the XU estimator is overall more reliable than the other three estimators for the estimation of $1/\alpha$, with relative RMSEs of 0.79 (MI), 0.61 (MA), 0.53 (XU), and 0.67 (RSK). For the parameter $1/(\kappa+1)$, the relative RMSEs were equal to 0.074 (MI), 0.075 (MA), 0.069 (XU), and 0.100 (RSK). In the case of a large number of scatterers (11 to 20 per resolution cell), the relative RMSEs of $1/\alpha$ were equal to 1.43 (MI), 1.27 (MA), 1.25 (XU), and 1.33 (RSK), and the relative RMSEs of $1/(\kappa+1)$ were equal to 0.14 (MI), 0.16 (MA), 0.17 (XU), and 0.20 (RSK). The four methods were also tested on simulated ultrasound images with a variable density of periodic scatterers to test images with a coherent component. The addition of noise on ultrasound images was also studied. Results showed that the XU estimator was better overall than the other three. Finally, on the simulated ultrasound images, the average computation times per image were equal to 6.0 ms (MI), 8.0 ms (MA), 6.8 ms (XU), and 500 ms (RSK). Thus, a fast, reliable, and novel algorithm for the estimation of the homodyned K-distribution was proposed.

The goal of this work is to perform a segmentation of the intimamedia thickness (IMT) of carotid arteries in view of computing various dynamical properties of that tissue, such as the elasticity distribution (elastogram). The echogenicity of a region of interest comprising the intima-media layers, the lumen, and the adventitia in an ultrasonic B-mode image is modeled by a mixture of three Nakagami distributions. In a first step, we compute the maximum a posteriori estimator of the proposed model, using the expectation maximization (EM) algorithm. We then compute the optimal segmentation based on the estimated distributions as well as a statistical prior for disease-free IMT using a variant of the exploration/selection (ES) algorithm. Convergence of the ES algorithm to the optimal solution is assured asymptotically and is independent of the initial solution. In particular, our method is well suited to a semi-automatic context that requires minimal manual initialization. Tests of the proposed method on 30 sequences of ultrasonic B-mode images of presumably disease-free control subjects are reported. They suggest that the semi-automatic segmentations obtained by the proposed method are within the variability of the manual segmentations of two experts.

In this paper, we present a new model for deformations of shapes. A pseudo-likelihood is based on the statistical distribution of the gradient vector field of the gray level. The prior distribution is based on the Probabilistic Principal Component Analysis (PPCA). We also propose a new model based on mixtures of PPCA that is useful in the case of greater variability in the shape. A criterion of global or local object specificity based on a preliminary color segmentation of the image, is included into the model. The localization of a shape in an image is then viewed as minimizing the corresponding Gibbs field. We use the Exploration/Selection (E/S) stochastic algorithm in order to find the optimal deformation. This yields a new unsupervised statistical method for localization of shapes. In order to estimate the statistical parameters for the gradient vector field of the gray level, we use an Iterative Conditional Estimation (ICE) procedure. The color segmentation of the image can be computed with an Exploration/Selection/Estimation (ESE) procedure.

In this paper, we present a Hidden Markov Random Field (HMRF) data-fusion model. The proposed model is applied to the segmentation of natural images based on the fusion of colors and textons into Julesz ensembles. The corresponding Exploration/ Selection/Estimation (ESE) procedure for the estimation of the parameters is presented. This method achieves the estimation of the parameters of the Gaussian kernels, the mixture proportions, the region labels, the number of regions, and the Markov hyper-parameter. Meanwhile, we present a new proof of the asymptotic convergence of the ESE procedure, based on original finite time bounds for the rate of convergence.

We propose a new stochastic algorithm for computing useful Bayesian estimators of hidden Markov random field (HMRF) models that we call exploration/selection/estimation (ESE) procedure. The algorithm is based on an optimization algorithm of O. François, called the exploration/selection (E/S) algorithm. The novelty consists of using the a posteriori distribution of the HMRF, as exploration distribution in the E/S algorithm. The ESE procedure computes the estimation of the likelihood parameters and the optimal number of region classes, according to global constraints, as well as the segmentation of the image. In our formulation, the total number of region classes is fixed, but classes are allowed or disallowed dynamically. This framework replaces the mechanism of the split-and-merge of regions that can be used in the context of image segmentation. The procedure is applied to the estimation of a HMRF color model for images, whose likelihood is based on multivariate distributions, with each component following a Beta distribution. Meanwhile, a method for computing the maximum likelihood estimators of Beta distributions is presented. Experimental results performed on 100 natural images are reported. We also include a proof of convergence of the E/S algorithm in the case of nonsymmetric exploration graphs.

In this paper, we present a robust method for estimating the model parameters in a mixture of probabilistic principal component analyzers. This method is based on the stochastic version of the expectation maximization (SEM) algorithm. Parameters of this mixture model are herein used to constrain the 3D reconstruction problem of scoliotic rib cage from a pair of planar and conventional calibrated radiographic images (postero-anterior with normal incidence (I/sub PA/) and lateral (I/sub LAT/)). More precisely, the proposed PPCA mixture model is herein robustly exploited for dimensionality reduction and to get a set of probabilistic prior models associated to each detected class of pathological deformations observed on a representative training scoliotic rib cage population. By using an appropriate likelihood and for each considered class-conditional prior model, the proposed 3D reconstruction is stated as an energy function minimization problem, which is solved with a stochastic optimization algorithm. The optimal 3D reconstruction then corresponds to the class of deformation and parameters leading to the minimal energy. This 3D method of reconstruction has been successfully tested on several biplanar radiographic images, yielding very promising results.

In this paper, we describe a statistical model for the gradient vector field of the gray level in images validated by different experiments. Moreover, we present a global constrained Markov model for contours in images that uses this statistical model for the likelihood. Our model is amenable to an Iterative Conditional Estimation (ICE) procedure for the estimation of the parameters; our model also allows segmentation by means of the Simulated Annealing (SA) algorithm, the Iterated Conditional Modes (ICM) algorithm, or the Modes of Posterior Marginals (MPM) Monte Carlo (MC) algorithm. This yields an original unsupervised statistical method for edge-detection, with three variants. The estimation and the segmentation procedures have been tested on a total of 160 images. Those tests indicate that the model and its estimation are valid for applications that require an energy term based on the log-likelihood ratio. Besides edge-detection, our model can be used for semiautomatic extraction of contours, localization of shapes, non-photo-realistic rendering; more generally, it might be useful in various problems that require a statistical likelihood for contours.

In this paper, we describe a new hidden Markov random field model, which we call hierarchical multidata model, and which is based on a triplet of random fields (two hidden random fields and one observed field) in order to capture interscale and within-scale dependencies between various scales of resolution of wavelet-based texture features. We present a variation of the iterated conditional modes (ICM) algorithm for the segmentation, and an adaptation of the iterative conditional estimation (ICE) procedure for the estimation of the statistical parameters of the model. Results of tests performed on 75 mosaics of Brodatz textures are reported.

In this paper, we describe an unsupervised segmentation method for contours which proves quite adapted for the images obtained by electronic acquisition. We present two statistical models for the norm of the gradient of the gray level at the pixels of an Image, one for contour points and one for points outside contours. We also describe a Markov model with constraint which incorporates those two statistical distributions as likelihood together with a simple a priori model. Our model is suitable for an iterative conditional estimation (ICE) procedure for the estimation of the parameters and an iterated conditional modes (ICM) algorithm, or simulated annealing, for the segmentation. A preliminary step proceeds to the segmentation of the image into sub-regions and uses a Markov model without constraint based on the gray level distribution on the image.