MathVisionTools: Medical Image Analysis Prototyping

This paper describes a new international collaborative initiative for the assembly of a high level medical image analysis toolkit, based on Mathematica, to do efficient design and development of advanced computer vision algorithms for computer-aided diagnosis. Mathematica has come to a point that the combined symbolic and numeric power makes it a versatile and efficient framework for prototyping of complex algorithms.This is a first announcement, and call for interest and participation

Rapid prototyping of biomedical image analysis applications with Mathematica

We report on 2.5 years successful use of high level rapid prototyping in education and research of biomedical image analysis, using Mathematica

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)

Invertible orientation scores as an application of generalized wavelet theory

Inspired by the visual system of many mammals, we consider the construction of—and reconstruction from—an orientation score of an image, via a wavelet transform corresponding to the left-regular representation of the Euclidean motion group in (R2) and oriented wavelet f ¿ (R2). Because this representation is reducible, the general wavelet reconstruction theorem does not apply. By means of reproducing kernel theory, we formulate a new and more general wavelet theory, which is applied to our specific case. As a result we can quantify the well-posedness of the reconstruction given the wavelet f and deal with the question of which oriented wavelet f is practically desirable in the sense that it both allows a stable reconstruction and a proper detection of local elongated structures. This enables image enhancement by means of left-invariant operators on orientation scores

Image processing via shift-twist invariant operations on orientation bundle functions

Inspired by our own visual system we consider the construction of-and reconstruction from- an orientation bundle function (OBF) Uf : R2 ¿ S1 ¿C as a local orientation score of an image, f : R2 ¿ R, via a wavelet transform W¿ corresponding to a representation of the Euclidean motion group onto L2(R2) and oriented wavelet ¿ ¿L2(R2). This wavelet transform is a unitary mapping with stable inverse, which allows us to directly relate each operation ¿ on OBF’s to an operation ¿ on images in a robust manner. We examine the geometry of the domain of an OBF and show that the only sensible operations on OBF’s are non-linear and shift-twist invariant. As an example we consider all linear 2nd order shift-twist invariant evolution equations on OBF’s corresponding to stochastic processes on the Euclidean motion group in order to construct nonlinear shift-twist invariant operations on OBF’s. Given two such stochastic processes we derive the probability density that particles of the different processes collide. As an application we detect elongated structures in images and automatically close the gaps between them

GPU-based ray-casting of spherical functions applied to high angular resolution diffusion imaging

Any sufficiently smooth, positive, real-valued function $psi: S^2 rightarrow realset^+$ on a sphere $S^2$ can be expanded by a Laplace expansion into a sum of spherical harmonics. Given the Laplace expansion coefficients, we provide a CPU and GPU-based algorithm that renders the radial graph of $psi$ in a fast and efficient way by ray-casting the glyph of $psi$ in the fragment shader of a GPU. The proposed rendering algorithm has proven highly useful in the visualization of high angular resolution diffusion imaging (HARDI) data. Our implementation of the rendering algorithm can display simultaneously thousands of glyphs depicting the local diffusivity of water. The rendering is fast enough to allow for interactive manipulation of large HARDI data sets

Fast and sleek glyph rendering for interactive HARDI data exploration

High angular resolution diffusion imaging (HARDI) is an emerging magnetic resonance imaging (MRI) technique that overcomes some decisive limitations of its predecessor diffusion tensor imaging (DTI). HARDI can resolve locally more than one direction in the diffusion pattern of water molecules and thereby opens up the opportunity to display and track crossing fibers. Showing the local structure of the reconstructed, angular probability profiles in a fast, detailed, and interactive way can improve the quality of the research in this area and help to move it into clinical application. In this paper we present a novel approach for HARDI glyph visualization or, more generally, for the visualization of any function that resides on a sphere and that can be expressed by a Laplace series. Our GPU-accelerated glyph rendering improves the performance of the traditional way of HARDI glyph visualization as well as the visual quality of the reconstructed data, thus offering interactive HARDI data exploration of the local structure of the white brain matter in-vivo. In this paper we exploit the capabilities of modern GPUs to overcome the large, processor-intensive and memory-consuming data visualization