Sampling and Visualizing Creases with Scale-Space Particles

Abstract

Particle systems have gained importance as a methodology for sampling implicit surfaces and segmented objects to improve mesh generation and shape analysis. We propose that particle systems have a significantly more general role in sampling structure from unsegmented data. We describe a particle system that computes samplings of crease features (i.e. ridges and valleys, as lines or surfaces) that effectively represent many anatomical structures in scanned medical data. Because structure naturally exists at a range of sizes relative to the image resolution, computer vision has developed the theory of scale-space, which considers an n-D image as an (n + 1)-D stack of images at different blurring levels. Our scale-space particles move through continuous four-dimensional scale-space according to spatial constraints imposed by the crease features, a particle-image energy that draws particles towards scales of maximal feature strength, and an inter-particle energy that controls sampling density in space and scale. To make scale-space practical for large three-dimensional data, we present a spline-based interpolation across scale from a small number of pre-computed blurrings at optimally selected scales. The configuration of the particle system is visualized with tensor glyphs that display information about the local Hessian of the image, and the scale of the particle. We use scale-space particles to sample the complex three-dimensional branching structure of airways in lung CT, and the major white matter structures in brain DTI.

Superquadric Glyphs for Symmetric Second-Order Tensors

“…One edge definition is zero-crossing on the second directional derivative along the gradient direction, f = n T Hn. This edge surface is sampled by a particle system [33] in Fig. 9(c), showing the Hessians at the edge locations, and revealing close similarities with the geometry tensors on the isosurface in Fig.…”

“…Faced with indefinite tensors, a frequent strategy is to map them to positive-definite tensors prior to visualization [34,22,21,52,33]. Even when bijective mappings are used (so mathematically, no information is lost), such mappings still visually obscure the difference between positive and negative eigenvalues, which is a fundamental qualitative aspect in various applications.…”

“…Such high-level tensor visualization frequently requires complex algorithms. More basic techniques, including glyphs, play an important part in their debugging and validation [33,49].…”

“…Alternatively, some authors enforce positivedefiniteness for visualization purposes, for example by taking the exponential of eigenvalues [34], by shifting them by the amount of the most negative eigenvalue [52], or other application-specific mappings [22,21,33]. An entirely different approach is to define symmetric tensor glyphs via algebraic equations, such as the quadratic surface of the tensor, known as the Cauchy stress quadric in geomechanics [18] and as the Dupin indicatrix in differential geometry [17], or the Reynolds glyph [18].…”

Pulmonary lobe segmentation based on ridge surface sampling and shape model fitting

“…We build on this work and automatically identify fissure locations by fitting a shape model to fissure locations detected by means of a particle system for ridge surface sampling previously described by Ref. 19. Other fissure identification and enhancement schemes have been proposed, namely.…”