Alternative Parametric Boundary Reconstruction Method for Biomedical Imaging

Determining the outline or boundary contour of a two-dimensional object, or the surface of a three-dimensional object poses difficulties particularly when there is substantial measurement noise or uncertainty. By adapting the mathematical approach of stochastic function recovery to this task, it is possible to obtain usable estimates for these boundaries, even in the presence of large amounts of noise. The technique is applied to parametric boundary data and has potential applications in biomedical imaging. It should be considered as one of several techniques to improve the visualization of images

Iterative Solvers for Large, Dense Matrices

Stochastic Interpolation (SI) uses a continuous, centrally symmetric probability distribution function to interpolate a given set of data points, and splits the interpolation operator into a discrete deconvolution followed by a discrete convolution of the data. The method is particularly effective for large data sets, as it does not suffer from the problem of oversampling, where too many data points cause the interpolating function to oscillate wildly. Rather, the interpolation improves every time more data points are added. The method relies on the inversion of relatively large, dense matrices to solve Annx = b for x. Based on the probability distribution function chosen, the matrix Ann may have specific properties that make the problem of solving for x tractable. The iterative Shulz Jones Mayer (SJM) method relies on an initial guess, which is iterated to approximate A�1 nn . We present initial guesses that are guaranteed to converge quadratically for several classes of matrices, including diagonally and tri-diagonally dominant matrices and the structured matrices we encounter in the implementation of SI. We improve the method, creating the Polynomial Shulz Jones Mayer method, and take advantage of the more efficient matrix operations possible for Toeplitz matrices. We calculate error bounds and use those to improve the method’s accuracy, resulting in a method requiring O(nlogn) operations that returns x with double precision. The use of SI and PSJM is illustrated in interpolating functions and images in grey scale and color

The novel stochastic Bernstein method of functional approximation

The stochastic Bernstein method (not to be confused with the Bernstein polynomials) is a novel and significantly improved non-polynomial global method of signal processing that is proving very useful for interpolating and for approximating data. It arose as an obvious extension of the work of Bernstein (it preserves some of the remarkable properties of the Bernstein polynomials). However, this extension means that stochastic interpolation takes on its own properties and additionally can replace the error function by other functions such as the arctangent. The method has a free parameter a known as its diffusivity that can be easily optimized with adaptivity and can interpolate or approximate non-uniformly distributed input data - something that is very awkward to set up with other methods. Adaptivity can also reverse engineer the non-uniformly distributed input data that best recovers a function. This short paper provides an introduction to the new mathematical method that should find wide application in many areas of science and engineering. © 2006 IEEE