P. Mohana Shankar

A general statistical model for ultrasonic backscattering from tissues

The backscattered ultrasonic echo from tissue can be described in terms of Rayleigh distribution or K distribution. Even though both generalized K distribution and homodyned K distribution can account for some of the scattering conditions that exist in tissues, the analytical complexity involved with these distributions is significant. A much simpler generalized model based on the Nakagami distribution is proposed here. This model can describe the statistics of the envelope of the backscattered echo from an ensemble of scatterers with varying number densities, varying cross sections, and the presence or absence of regularly spaced scatterers. Computer simulations and experiments on tissue-mimicking phantoms have been undertaken to test the validity of the model. Results clearly show the versatility of the Nakagami distribution and its parameter to model the backscattered envelope from tissues. It is suggested that Nakagami distribution may be a good model for use in tissue characterization because of its simple analytical nature and ability to encompass different scattering conditions.The backscattered ultrasonic echo from tissue can be described in terms of Rayleigh distribution or K distribution. Even though both generalized K distribution and homodyned K distribution can account for some of the scattering conditions that exist in tissues, the analytical complexity involved with these distributions is significant. A much simpler generalized model based on the Nakagami distribution is proposed here. This model can describe the statistics of the envelope of the backscattered echo from an ensemble of scatterers with varying number densities, varying cross sections, and the presence or absence of regularly spaced scatterers. Computer simulations and experiments on tissue-mimicking phantoms have been undertaken to test the validity of the model. Results clearly show the versatility of the Nakagami distribution and its parameter to model the backscattered envelope from tissues. It is suggested that Nakagami distribution may be a good model for use in tissue characterization because of its simple analytical nature and ability to encompass different scattering conditions.

Bubble size measurements using the nonlinear mixing of two frequencies

A double frequency method of measuring bubble size which has better spatial resolution than previous techniques, is described. Bubbles are insonified simultaneously by a swept, relatively low, ‘‘pumping’’ frequency near their resonance and a high ‘‘imaging’’ frequency. Theoretical expressions are derived for the sound scattered by a bubble under the influence of these two frequencies. Existence of sharp resonance behavior in the scattered sound at the sum frequency is shown both theoretically and experimentally.

Ultrasound speckle analysis based on the K distribution

The departure of speckle magnitude from Rayleigh statistics was applied to examine insonated phantoms with both low and high concentrations of scatterers. A mathematical model, the K distribution of Jakeman, was used to characterize non-Rayleigh statistics. This model contains a parameter, alpha, which characterizes the clustering of the scattering sites in a medium. It is shown from phantom experiments that alpha is linearly proportional to the log-scaled scatterer concentration in a range from about 1 to 30 scatterers per sample volume.

Formation and measurement of tapers in optical fibers

A simple fabrication method is described which is capable of producing physically repeatable biconical tapers in optical fibers. A resistively heated platinum wire furnace provides the heat with standard fiber‐optic positioning equipment to secure the fiber. The method does not require computer control and has the advantages of low cost, fast sample turn‐around time, and the ability to concatenate multiple tapers on a single fiber. Experimental results demonstrating repeatable taper formation and the wavelength filtering property of the tapers is shown. Issues of varying the taper shape are addressed and measurement methods of the physical and optical characteristics are presented. Limited control of the taper symmetry is discussed, and recommendations for enhancement at the cost of simplicity are included.

Effect of modal noise on single mode fiber optic networks

Abstract Modal noise may arise in single mode fiber optic networks from the random coupling of the fundamental model LP01 and the higher order mode LP11. The higher mode LP11 is generated in the fiber optic link from imperfect connectors. A model for the modal noise generated in a single mode fiber optic network consisting of a number of connectors or splices is presented. The effect of modal noise on the link is discussed and expression for signal to noise ratio has been obtained. Suggestions for the reduction of modal noise are given.

STATISTICS OF BOUNDARIES IN ULTRASONIC B-SCAN IMAGES

The existence of edges and boundaries in regions of interest (ROIs) in B-scan images alters the statistics of the backscattered echo from the ROI. Boundaries are the result of at least two different types of scattering scenarios in tissue, and the Nakagami model, which is being used extensively in ultrasound, is unlikely to fit the statistics of the backscattered echo under these conditions. Furthermore, there are very few other statistical models exist that describe the statistics of the backscattered echo from regions containing boundaries. In this work, the gamma mixture density and the recently proposed McKay density are explored as two viable models to fill this void. Justifications of these models are presented along with methods for estimating their parameters. Random number simulations and studies on tissue-mimicking phantoms indicate that the McKay and gamma mixture densities are the best for the modeling of the backscattered echo intensity when boundaries are present in the regions of interest.

Concepts of Probability and Statistics

In this chapter, we examined some theoretical aspects of probability density functions and distributions encountered in the study of fading and shadowing in wireless channels. We started with the basic definition of probability, then discussed density functions and properties relating to the analysis of fading and shadowing. We also examined the transformations of random variables in conjunction with relationships of different types of random variables. This is important in the study of diversity and modeling of specific statistical behavior of the wireless channels. The density functions of some of the functions of two or more random variables were derived. We examined order statistics, placing emphasis on the density functions of interest in diversity analysis. Concepts of stochastic processes and their properties were outlined. Similarly, the characteristics of noise were delineated within the context of signal detection. In addition, this study included an exploration of ways of expressing some of the densities in more compact forms using the hypergeometric functions and MeijerG functions.

Modeling of Fading and Shadowing

Various models to describe the statistical fluctuations in wireless channels are presented. These models range from the simple Rayleigh ones, to cascaded ones, and to complex models such as those based on κ – μ, η – μ, and Nakagami-N-gamma distributions. They are compared in terms of their properties, error rates and outage probabilities. The updated sections provide detailed analysis of the McKay and mixture models for fading and shadowing. The mixture density is expanded to see how it can replace some of the existing statistical models of fading and shadowing. For the updated sections, detailed Matlab scripts are provided (with complete annotation) to supplement the theoretical approach. A section on random number generation relevant to fading and shadowing modeling is also given to complete the pedagogic description of modeling. The properties of Meijer G functions are described in detail in the Appendix.

Modal redistribution and power loss in on-fibre devices

The mode coupling and optical power loss taking place in on-fibre devices are investigated. These devices are fabricated by removing the cladding of a multimode fibre and replacing it with a new material such as electro-optic or magneto-optic. Under an application of an external field the index of this modified cladding increases, leading to coupling among guided modes and coupling to radiation modes, resulting in optical power loss. The coupling among guided modes and the power loss are calculated as a function of the change in the index under the field and the length of the active region for the case when HE1m modes are excited individually or in a group. Results indicate that on-fibre devices may be used as low-loss devices for applications involving sensors and delay lines.

Pedagogy of solutions to a set of linear equations using a Matlab workbook

A Matlab workbook capable of generating complete solutions to a set of linear equations is presented. While standard routines in Matlab and Maple provide solutions to a set of equations, additional work is needed to interpret and relate them to the concepts covered in an undergraduate course in Linear Algebra. One also runs into difficulties when the system of equations is inconsistent. The workbook, started as a simple demo for students, evolved into its present form where the solutions to a set of linear equations are displayed on a single page with explanations, annotations, and equations that match the work by the instructor in class. Regardless of whether the system is consistent or inconsistent with general or unique solutions, the solution set is presented with information such as the rank of a matrix, pivots, null space, free and basic variables, etc., providing the pedagogy. The solutions are compared to those obtained using standard packages from Matlab and Maple demonstrating their shortcomings and limits. In contrast, the results of the workbook provide complete instructive aspects of solutions to a set of linear equations enhancing the educational experience of students.