General expressions for Rician density and distribution functions

Abstract

The tracking index: A generalized para meter for a-f3 and a·{:l·[" target trackers. General expressions are derived for the density and distribution functions of the amplitude of a linear matched mter output given a nonfluctuatlng target in arbitrary clutter. These generalized Rieian density and distribution functions are based on the density function of the clutter amplitude at the receiver output, and are easily evaluated numerically. Fur thermore, we present error bounds for the numerical calculation of the density function.We also present several numerical example s.

Noise and Signal Estimation in Magnitude MRI and Rician Distributed Images: A LMMSE Approach

“…This kind of noise may be encountered in many practical situations, such as MR data, speckle [10], communication fading problems [21], and many others. The magnitude signal may be expressed (1) being the magnitude image, the original signal level if no noise is present, and and uncorrelated Gaussian noise variables with zero mean and equal variance . The probability density function (PDF) of such an image is, as indicated, a Rician distribution [1], [22] (2) being the 0th-order modified Bessel function of the first kind and the Heaviside step function.…”

“…The magnitude signal may be expressed (1) being the magnitude image, the original signal level if no noise is present, and and uncorrelated Gaussian noise variables with zero mean and equal variance . The probability density function (PDF) of such an image is, as indicated, a Rician distribution [1], [22] (2) being the 0th-order modified Bessel function of the first kind and the Heaviside step function. Although, generally speaking, the moments of this distribution are difficult to calculate, the even-order (non central) moments are simple polynomials as, for instance, the second-order moment (3) In the image background, where the signal-to-noise ratio is zero due to the lack of water-proton density in the air, the Rician PDF simplifies to a Rayleigh distribution [22] with PDF (4)…”

“…This estimation may be also done using nonsquare weighted windows, such as Gaussian functions. Using the relations from (1) and , the LMMSE estimator may finally be written as (13) with (14) Note that the value must be properly estimated. This task is usually done from a selected region from the background pixels, where the signal is assumed zero.…”

“…If the estimator is simplified to be pointwise, vectors and matrices become scalar values. Then and making use of the model in (1) Finally, the LMMSE estimator is (11) Assuming local ergodicity, the expectation may be replaced by its sample estimator , that can be defined (12) 1 As it is well known, this type of estimators model the parameter to be estimated as a sample of a random variable, the parameters of which have a known relation with the parameters of the observation [27]. with a square neighborhood around the pixel .…”

“…N OISE in magnitude magnetic resonance (MR) images is usually modeled by means of a Rician distribution, due to the existence of zero-mean uncorrelated Gaussian noise with equal variance in both the real and imaginary parts of the complex k-space data [1], [2]. This noise may affect the performance of different postprocessing techniques applied to MR data, such as segmentation, registration or tensor estimation in diffusion tensor MRI (DT-MRI) [3].…”