Biomedical Image Analysis: Rapid prototyping with Mathematica

Digital acquisition techniques have caused an explosion in the production of medical images, especially with the advent of multi-slice CT and volume MRI. One third of the financial investments in a modern hospital's equipment are dedicated to imaging. Emerging screening programs add to this flood of data. The capabilities of many recent computer-aided diagnosis (CAD) programs are compelling, and have recently lead to many new CAD companies. This calls for many new algorithms for image analysis and dedicated scientists for the job.Image analysis software libraries abound, but unfortunately are often limited in functionality, are too specific, or need a rather dedicated environment and have a long learning curve. Today's computer vision algorithms are based on solid mathematics, requiring a highly versatile, high level mathematical prototyping environment. We have chosen Mathematica by Wolfram Research Inc., and describe the successful results of the first 2.5 years of its use in the training of biomedical engineers in image analysis

The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D-Euclidean motion group

We provide the solutions of linear, left-invariant, 2nd-order stochastic evolution equations on the 2D-Euclidean motion group. These solutions are given by group-convolution with the corresponding Green’s functions that we derive in explicit form in Fourier space. A particular case coincides with the hitherto unsolved forward Kolmogorov equation of the so-called direction process, the exact solution of which is required in the field of image analysis for modeling the propagation of lines and contours. By approximating the left-invariant base elements of the generators by left-invariant generators of a Heisenberg-type group, we derive simple, analytic approximations of the Green’s functions. We provide the explicit connection and a comparison between these approximations and the exact solutions. Finally, we explain the connection between the exact solutions and previous numerical implementations, which we generalize to cope with all linear, left-invariant, 2nd-order stochastic evolution equations

The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D-Euclidean motion group

We provide the solutions of linear, left-invariant, 2nd-order stochastic evolution equations on the 2D-Euclidean motion group. These solutions are given by group-convolution with the corresponding Green’s functions that we derive in explicit form in Fourier space. A particular case coincides with the hitherto unsolved forward Kolmogorov equation of the so-called direction process, the exact solution of which is required in the field of image analysis for modeling the propagation of lines and contours. By approximating the left-invariant base elements of the generators by left-invariant generators of a Heisenberg-type group, we derive simple, analytic approximations of the Green’s functions. We provide the explicit connection and a comparison between these approximations and the exact solutions. Finally, we explain the connection between the exact solutions and previous numerical implementations, which we generalize to cope with all linear, left-invariant, 2nd-order stochastic evolution equations

Edge Preserving Smoothing with Euclidean Shortening flow

Edge preserving smoothing is a locally adaptive process, where the size of the blurkernel, applied to suppress the noise, is a local function of the edge strength [2, 3]. Blurring is described by a partial differential equation, the diffusion equation. At strong edges, the 'conductivity' of the diffusion is reduced.This was first introduced into the realm of computer vision by Perona and Malik [4] in 1991. It was a huge success, as it also enhanced the remaining edges. But at strong edges, the noise remained, and there was a parameter k that had to be set. Alvarez [5] came up with an elegant solution for both issues, by proposing a new nonlinear image evolution scheme, where the local edge direction was taken into account. This solution is known as Euclidean Shortening Flow.We first give the theory and implementation of Perona & Malik nonlinear diffusion, then we focus on Euclidean shortening flow. The last section discusses an implementation on a noise ultrasound image. This paper is based on Chapter 21 of the book "Front-End Vision & Multi-Scale Image Analysis" (ter Haar Romeny, 2003)