On the non-local geometry of turbulence

A multi-scale methodology for the study of the non-local geometry of eddy structures in turbulence is developed. Starting from a given three-dimensional field, this consists of three main steps: extraction, characterization and classification of structures. The extraction step is done in two stages. First, a multi-scale decomposition based on the curvelet transform is applied to the full three-dimensional field, resulting in a finite set of component three-dimensional fields, one per scale. Second, by iso-contouring each component field at one or more iso-contour levels, a set of closed iso-surfaces is obtained that represents the structures at that scale. The characterization stage is based on the joint probability density function (p.d.f.), in terms of area coverage on each individual iso-surface, of two differential-geometry properties, the shape index and curvedness, plus the stretching parameter, a dimensionless global invariant of the surface. Taken together, this defines the geometrical signature of the iso-surface. The classification step is based on the construction of a finite set of parameters, obtained from algebraic functions of moments of the joint p.d.f. of each structure, that specify its location as a point in a multi-dimensional ‘feature space’. At each scale the set of points in feature space represents all structures at that scale, for the specified iso-contour value. This then allows the application, to the set, of clustering techniques that search for groups of structures with a common geometry. Results are presented of a first application of this technique to a passive scalar field obtained from 5123 direct numerical simulation of scalar mixing by forced, isotropic turbulence (Reλ = 265). These show transition, with decreasing scale, from blob-like structures in the larger scales to blob- and tube-like structures with small or moderate stretching in the inertial range of scales, and then toward tube and, predominantly, sheet-like structures with high level of stretching in the dissipation range of scales. Implications of these results for the dynamical behaviour of passive scalar stirring and mixing by turbulence are discussed

Shape Spectrum Based View Grouping and Matching of 3D Free-Form Objects

We address the problem of constructing view aspects of 3D free-form objects for efficient matching during recognition. We introduce a novel view representation based on `shape spectrum' features, and propose a general and powerful technique for organizing multiple views of objects of complex shape and geometry into compact and homogeneous clusters. Our view grouping technique obviates the need for surface segmentation and edge detection. Experiments on 6,400 synthetically generated views of 20 free-form objects and 100 real range images of 10 sculpted objects demonstrate the good performance of our shape spectrum based model view selection technique. Key words: Free-form objects, sculpted surfaces, 3D object representation, cosmos, shape spectrum, clustering, view matching. 1 Introduction A 3D rigid object can give rise to arbitrarily many different 2D appearances (views). For an object with free-form or sculpted surfaces, only a part of a surface will typically be visible from a sin..

Digital Library Services for Three-Dimensional Models

With the growth in computing, storage and networking infrastructure, it is becoming increasingly feasible for multimedia professionals—such as graphic designers in commercial, manufacturing, scientific and entertainment areas—to work with 3D digital models of the objects with which they deal in their domain. Unfortunately most of these models exist in individual repositories, and are not accessible to geographically distributed professionals who are in need of them. Building an efficient digital library system presents a number of challenges. In particular, the following issues need to be addressed: (1) What is the best way of representing 3D models in a digital library, so that the searches can be done faster? (2) How to compress and deliver the 3D models to reduce the storage and bandwidth requirements? (3) How can we represent the user\u27s view on similarity between two objects? (4) What search types can be used to enhance the usability of the digital library and how can we implement these searches, what are the trade-offs? In this research, we have developed a digital library architecture for 3D models that addresses the above issues as well as other technical issues. We have developed a prototype for our 3D digital library (3DLIB) that supports compressed storage, along with retrieval of 3D models. The prototype also supports search and discovery services that are targeted for 3-D models. The key to 3DLIB is a representation of a 3D model that is based on “surface signatures”. This representation captures the shape information of any free-form surface and encodes it into a set of 2D images. We have developed a shape similarity search technique that uses the signature images to compare 3D models. One advantage of the proposed technique is that it works in the compressed domain, thus it eliminates the need for uncompressing in content-based search. Moreover, we have developed an efficient discovery service consisting of a multi-level hierarchical browsing service that enables users to navigate large sets of 3D models. To implement this targeted browsing (find an object that is similar to a given object in a large collection through browsing) we abstract a large set of 3D models to a small set of representative models (key models). The abstraction is based on shape similarity and uses specially tailored clustering techniques. The browsing service applies clustering recursively to limit the number of key models that a user views at any time. We have evaluated the performance of our digital library services using the Princeton Shape Benchmark (PSB) and it shows significantly better precision and recall, as compared to other approaches

Solid Shape Processing in Area V4

Coding transformations in ventral visual cortex convert image information into knowledge about object structure and identity. Neurons in area V4, an intermediate ventral pathway stage in the macaque brain, are known to represent 2D contour, binocular disparity, and 3D orientation. In the experiments outlined in this thesis, I address fundamental questions about the relationship between 2D image signals and solid shape perception in the ventral visual stream. In chapter 1, I demonstrate that a substantial fraction of V4 neurons are more responsive to solid 3D than to planar 2D shape. I constructed simple solid shape forms around medial axis skeletons having randomized connectivity, orientation, curvature, and surface structure, and I leveraged realistic shading cues to render them as solid objects. Using micro-electrode recording in awake, fixating monkeys, I analyzed the tuning of V4 neurons to these solid shapes and their planar silhouettes and found that most V4 neurons (~ 70%) prefer solid shape fragments. I then parameterized stimulus shapes in terms of local geometric measures of contours, surfaces, and medial axes, producing a point cloud description of each stimulus in a multi-dimensional geometric space. Analysis of this space suggests that solid shape models more generally account for V4 responses, while slices through these models might explain responses to planar shapes. In chapter 2, I describe two-photon imaging experiments in V4 in anesthetized animals using OGB to measure neuronal responses. Again, a substantial fraction of neurons responded more strongly to solid shapes than to their planar silhouettes. I observed strong local clustering of solid-/planar-preferring neurons in separate cortical patches, and neighboring patches were most responsive to congruent solid and planar shapes. This suggests that derivation of solid shape from image information is a constraint on micro-organization in area V4. Previous studies using gratings, complex planar shapes, and naturalistic images fail to address whether shape processing is inherently 3D, or whether planar and solid shape are processed in parallel. By rigorously studying V4 with solid shape stimuli, we can postulate general principles of coding transformations in the ventral visual pathway