Robust Feature Detection and Local Classification for Surfaces Based on Moment Analysis

The stable local classification of discrete surfaces with respect to features such as edges and corners or concave and convex regions, respectively, is as quite difficult as well as indispensable for many surface processing applications. Usually, the feature detection is done via a local curvature analysis. If concerned with large triangular and irregular grids, e.g., generated via a marching cube algorithm, the detectors are tedious to treat and a robust classification is hard to achieve. Here, a local classification method on surfaces is presented which avoids the evaluation of discretized curvature quantities. Moreover, it provides an indicator for smoothness of a given discrete surface and comes together with a built-in multiscale. The proposed classification tool is based on local zero and first moments on the discrete surface. The corresponding integral quantities are stable to compute and they give less noisy results compared to discrete curvature quantities. The stencil width for the integration of the moments turns out to be the scale parameter. Prospective surface processing applications are the segmentation on surfaces, surface comparison, and matching and surface modeling. Here, a method for feature preserving fairing of surfaces is discussed to underline the applicability of the presented approach.

Feature preserving variational smoothing of terrain data

Journal ArticleIn this paper, we present a novel two-step, variational and feature preserving smoothing method for terrain data. The first step computes the field of 3D normal vectors from the height map and smoothes them by minimizing a robust penalty function of curvature. This penalty function favors piecewise planar surfaces; therefore, it is better suited for processing terrain data then previous methods which operate on intensity images. We formulate the total curvature of a height map as a function of its normals. Then, the gradient descent minimization is implemented with a second-order partial differential equation (PDE) on the field of normals. For the second step, we define another penalty function that measures the mismatch between the the 3D normals of a height map model and the field of smoothed normals from the first step. Then, starting with the original height map as the initialization, we fit a non-parametric terrain model to the smoothed normals minimizing this penalty function. This gradient descent minimization is also implemented with a second-order PDE. We demonstrate the effectiveness of our approach with a ridge/gully detection application

PDEs for tensor image processing

Methods based on partial differential equations (PDEs) belong to those image processing techniques that can be extended in a particularly elegant way to tensor fields. In this survey paper the most important PDEs for discontinuity-preserving denoising of tensor fields are reviewed such that the underlying design principles becomes evident. We consider isotropic and anisotropic diffusion filters and their corresponding variational methods, mean curvature motion, and selfsnakes. These filters preserve positive semidefiniteness of any positive semidefinite initial tensor field. Finally we discuss geodesic active contours for segmenting tensor fields. Experiments are presented that illustrate the behaviour of all these methods

Singular diffusivity facets, shocks and more

There is a class of nonlinear evolution equations with singular diffusivity, so that diffusion effect is nonlocal. A simplest one-dimensional example is a diffusion equation of the form u_t = \delta(u_x)u_{xx} for u = u(x; t), where \delta denotes Dirac s delta function. This lecture is intended to provide an overview of analytic aspects of such equations, as well as various applications. Equations with singular diffusivity are applied to describe several phenomena in the applied sciences, and to provide several devices in technology, especially image processing. A typical example is a gradient flow of the total variation of a function, which arises in image processing, as well as in material science to describe the motion of grain boundaries. In the theory of crystal growth the motion of a crystal surface is often described by an anisotropic curvature flow equation with a driving force term. At low temperature the equation includes a singular diffusivity, since the interfacial energy is not smooth. Another example is a crystalline algorithm to calculate curvature flow equations in the plane numerically, which is formally written as an equation with singular diffusivity. Because of singular diffusivity, the notion of solution is not a priori clear, even for the above one-dimensional example. It turns out that there are two systematic approaches. One is variational, and applies to divergence type equations. However, there are many equations like curvature flow equations which are not exactly of divergence type. Fortu-nately, our approach based on comparision principles turns out to be succesful in several interesting problems. It also asserts that a solution can be considered as a limit of solution of an approximate equation. Since the equation has a strong diffusivity at a particular slope of a solution, a flat portion with this slope is formed. In crystal growth ploblems this flat portion is called a facet. The discontinuity of a solution (called a shock) for a scalar conservation law is also considered as a result of singular diffusivity in the vertical direction

Curvature-based transfer functions for direct volume rendering: methods and applications

Journal ArticleDirect volume rendering of scalar fields uses a transfer function to map locally measured data properties to opacities and colors. The domain of the transfer function is typically the one-dimensional space of scalar data values. This paper advances the use of curvature information in multi-dimensional transfer functions, with a methodology for computing high-quality curvature measurements. The proposed methodology combines an implicit formulation of curvature with convolution-based reconstruction of the field. We give concrete guidelines for implementing the methodology, and illustrate the importance of choosing accurate filters for computing derivatives with convolution. Curvature-based transfer functions are shown to extend the expressivity and utility of volume rendering through contributions in three different application areas: nonphotorealistic volume rendering, surface smoothing via anisotropic diffusion, and visualization of isosurface uncertainty

Curvature-driven PDE methods for matrix-valued images

Matrix-valued data sets arise in a number of applications including diffusion tensor magnetic resonance imaging (DT-MRI) and physical measurements of anisotropic behaviour. Consequently, there arises the need to filter and segment such tensor fields. In order to detect edgelike structures in tensor fields, we first generalise Di Zenzo\u27s concept of a structure tensor for vector-valued images to tensor-valued data. This structure tensor allows us to extend scalar-valued mean curvature motion and self-snakes to the tensor setting. We present both two-dimensional and three-dimensional formulations, and we prove that these filters maintain positive semidefiniteness if the initial matrix data are positive semidefinite. We give an interpretation of tensorial mean curvature motion as a process for which the corresponding curve evolution of each generalised level line is the gradient descent of its total length. Moreover, we propose a geodesic active contour model for segmenting tensor fields and interpret it as a minimiser of a suitable energy functional with a metric induced by the tensor image. Since tensorial active contours incorporate information from all channels, they give a contour representation that is highly robust under noise. Experiments on three-dimensional DT-MRI data and an indefinite tensor field from fluid dynamics show that the proposed methods inherit the essential properties of their scalar-valued counterparts

Automatically extracting cellular structures from images generated via electron microscopy

In this paper, we consider mathematical techniques for locating cellular structures in digital images generated via electron microscopy. We approach this problem in two steps: a pre-processing denoising stage and a segmentation stage. For image denoising, we will limit our discussion to Partial Differential Equation (PDE) based methods, primarily focusing on diffusion and total variation methods. Segmentation will also b

Feature preserving smoothing of 3D surface scans

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, February 2004.Includes bibliographical references (p. 63-70).With the increasing use of geometry scanners to create 3D models, there is a rising need for effective denoising of data captured with these devices. This thesis presents new methods for smoothing scanned data, based on extensions of the bilateral filter to 3D. The bilateral filter is a non-linear, edge-preserving image filter; its extension to 3D leads to an efficient, feature preserving filter for a wide class of surface representations, including points and "polygon soups."by Thouis Raymond Jones.S.M

Edge Aware Anisotropic Diffusion for 3D Scalar Data

Fig. 1: The left half of the figure demonstrates the consistency in smoothing of our method compared to the existing method. The right half of the figure demonstrates the de-noising capabilities of our method. All the images from (a-c) were obtained byrenderingan iso-surface of 153. (a) Diffused with an existing diffusion model proposed by Krissian et al. [20] with k = 40, and100 iterations (b) The original Sheep’s heart data. (c) Diffused with our method with σ = 1 and the same number of iterations. The yellow circle indicates aregionwheretheiso-surfacehasbothhighandmediumrangegradient magnitude, and the blue circle marks a region where the gradient magnitude is much lower. Note the inconsistent smoothing in (a) inside the yellow circle. (d) The tooth data contaminated with Poisson noise (SNR=12.8) (e)Theoriginaltoothdata(f)Diffusedwithourmethod(SNR=25.76) withσ = 1 and 25 iterations. We used the exact same transfer function to render all the images in(d-f). Abstract—Inthispaperwepresentanovelanisotropicdiffusionmodel targeted for 3D scalar field data. Our model preserves material boundaries as well as fine tubular structures while noise is smoothed out. One of the major novelties is the use of the directional second derivative to define material boundaries instead of the gradient magnitude for thresholding. This results in a diffusion model that has much lower sensitivity to the diffusion parameter and smoothes material boundaries consistently compared to gradient magnitude based techniques. We empirically analyze the stability and convergence of the proposed diffusion and demonstrate its de-noising capabilities for both analytic and real data. We also discuss applications in the context of volume rendering