Marked regularity models

A physically based, probabilistic model for ultrasonic images incorporating shape, microstructure, and system characteristics

Recent successes with Bayesian methods for analysis of shape in medical images have motivated our development of a shape-based data likelihood for ultrasound, the foundation of which is a computationally feasible, pixel-based, probabilistic image model. Previous probabilistic models for ultrasound generally assume an analytic form, e.g., Rayleigh, Rician, K, Generalized K, etc., then attempt to fit data to the model. Assumptions and intensive computation inherent in such an approach make it unsuitable for our purposes. In the pursuit of a new model, we have described previously a physical model for image formation that incorporates the imaging system characteristics, the surface shape, and the surface microstructure. That physical model was validated via a visual comparison of simulated and actual images of a cadaveric vertebra. In this work, a random phasor sum representation of the physical model provides the basis for a probabilistic form. In contrast to existing probabilistic models, we compute the amplitude mean and variance directly from the physical model. These statistics can be displayed as images to characterize the tissue, but, more importantly, they permit the subsequent assignment of a suitable density function to each pixel for the purposes of constructing a data likelihood. The order of these steps, i.e., first computing the statistics and then assigning a density function, permits the inclusion of the local surface shape, the surface microstructure, and the system characteristics at every image pixel without violating the physical model. Currently, the value of the SNR0, the ratio of the mean to the standard deviation, is used to estimate whether a pixel is Rayleigh- or non-Rayleigh-distributed. This assessment forms the basis for a data likelihood constructed as a product of Rayleigh and Gaussian density functions describing the individual image pixels.

A discrete-scatterer model for ultrasonic images of rough surfaces

Automated analysis of ultrasonic images could be greatly improved with model-based Bayesian methods for image analysis. Such an approach would require an accurate probabilistic image model representing ultrasonic images in terms of the gross shape of underlying anatomical structure. Existing probabilistic models for ultrasonic image data do not adequately incorporate structure shape or system characteristics; thus, a substantially new approach is warranted. Toward that goal, we have developed models for the imaging system and rough surface with the following objectives: (1) accuracy in representation of basic image characteristics such as the texture and intensity, (2) a minimum of computational requirements, and (3) a form that is naturally extendable to an appropriate probabilistic image model. The imaging system was modeled as a linear system with a separable three-dimensional point-spread function with an envelope of Gaussian curves in each dimension. The rough surface was modeled as a collection of discrete scatterers placed on the continuum and parametrized by a surface roughness and scatterer concentration. Models were evaluated by a visual comparison of actual and simulated images of a cadaveric lumbar vertebra. The gross shape of the vertebral surface was estimated from computed tomography images of the vertebra, and simulated images were generated using the models and the gross surface shape. Actual images were registered with the surface and simulated images to within 2 mm. The similarity of the actual and simulated images was quite remarkable considering the simplicity of the models. Differences between the images were less than those between two simulated images separated by 0.4 mm or one-fifth the registration error. Further assessment of the models would require a statistical approach not yet available. The models do, however, provide the basis for the development of a computationally tractable probabilistic image model for image analysis. Such a model will provide the means for a statistical evaluation of the system and surface models.

Generation of non-Rayleigh speckle distributions using marked regularity models

Fully developed speckle patterns observed in coherent imagery are characterized by a Rayleigh-distributed envelope amplitude. Non-Rayleigh distributions are observed in many cases, such as when the number of scatterers in a resolution cell is small or scatterers are organized with some periodicity. Distributions resulting from the assumption of random scatterer phase (random walk models) have been used to describe the speckle amplitude in these cases, leading to K, Rician, and homodyned-K amplitude distributions. An alternative is to incorporate nonrandom phase implicitly by adopting models that directly describe the spatial placement of point scatterers. We examine the consequences of assuming that scattering is described in one dimension by a stationary renewal process in which the arrival times are the locations of ideal point scatterers, the interscatterer distances are drawn from a gamma distribution, and the scatterer amplitudes are allowed to be correlated in space. This model has been called the marked regularity model because variations of the model parameters can generate spatial distributions ranging from clustered to random to nearly periodic. We will demonstrate that all of the non-Rayleigh distributions generated by the previous random phase models can also be generated by the marked regularity model, and we show under what conditions the different distributions will result. We also demonstrate that the regularity model is inherently capable of describing certain sparse scattering conditions. Therefore, the model can represent many cases and provide an intuitively pleasing description of the spatial placement of the scatterers.