Helmholtz-Hodge Decomposition of vector fields on 2-manifolds

posterA Morse-like Decomposition ? - Morse-Smale decomposition for gradient (of scalar) fields is an interesting way of decomposing the domain into regions of unidirectional flow (from a source to a sink ). - But works for gradient fields, which are conservative (irrotational), only. - Can such a decomposition and analysis be extended to generic (consisting rotational component) vector fields ? - Can we extract the rotational component out from generic vector fields ? Feature Identification ? - Analysis on the decomposed components of fields is simpler. eg Identification of critical points in the potentials of the two components is easy. Topological Consistency ? - Is there any relation between the topology of the components and the topology of the original field ? Limitation - So far, HH Decomposition exists only for piece-wise constant vector fields. Such a decomposition for piece-wise linear fields is desirable

Topology Based Flow Analysis and Superposition Effects

Using topology for feature analysis in flow fields faces several problems. First of all, not all features can be detected using topology based methods. Second, while in flow feature analysis the user is interested in a quantification of feature parameters like position, size, shape, radial velocity and other parameters of feature models, many of these parameters can not be determined using topology based methods alone. Additionally, in some applications it is advantageous to regard the vector field as a superposition of several, possibly simple, features. As topology based methods are quite sensitive to superposition effects, their precision and usability is limited in these cases. In this paper, topology based analysis and visualization of flow fields is estimated and compared to other feature based approaches demonstrating these problems

Visualizing High-Order Symmetric Tensor Field Structure with Differential Operators

The challenge of tensor field visualization is to provide simple and comprehensible representations of data which vary both directionally and spatially. We explore the use of differential operators to extract features from tensor fields. These features can be used to generate skeleton representations of the data that accurately characterize the global field structure. Previously, vector field operators such as gradient, divergence, and curl have previously been used to visualize of flow fields. In this paper, we use generalizations of these operators to locate and classify tensor field degenerate points and to partition the field into regions of homogeneous behavior. We describe the implementation of our feature extraction and demonstrate our new techniques on synthetic data sets of order 2, 3 and 4

Estimation of Vector Fields in Unconstrained and Inequality Constrained Variational Problems for Segmentation and Registration

Vector fields arise in many problems of computer vision, particularly in non-rigid registration. In this paper, we develop coupled partial differential equations (PDEs) to estimate vector fields that define the deformation between objects, and the contour or surface that defines the segmentation of the objects as well. We also explore the utility of inequality constraints applied to variational problems in vision such as estimation of deformation fields in non-rigid registration and tracking. To solve inequality constrained vector field estimation problems, we apply tools from the Kuhn-Tucker theorem in optimization theory. Our technique differs from recently popular joint segmentation and registration algorithms, particularly in its coupled set of PDEs derived from the same set of energy terms for registration and segmentation. We present both the theory and results that demonstrate our approach

The natural Helmholtz-Hodge decomposition for open-boundary flow analysis

pre-printThe Helmholtz-Hodge decomposition (HHD), which describes a flow as the sum of an incompressible, an irrotational, and a harmonic flow, is a fundamental tool for simulation and analysis. Unfortunately, for bounded domains, the HHD is not uniquely defined, traditionally, boundary conditions are imposed to obtain a unique solution. However, in general, the boundary conditions used during the simulation may not be known known, or the simulation may use open boundary conditions. In these cases, the flow imposed by traditional boundary conditions may not be compatible with the given data, which leads to sometimes drastic artifacts and distortions in all three components, hence producing unphysical results. This paper proposes the natural HHD, which is defined by separating the flow into internal and external components. Using a completely data-driven approach, the proposed technique obtains uniqueness without assuming boundary conditions a priori. As a result, it enables a reliable and artifact-free analysis for flows with open boundaries or unknown boundary conditions. Furthermore, our approach computes the HHD on a point-wise basis in contrast to the existing global techniques, and thus supports computing inexpensive local approximations for any subset of the domain. Finally, the technique is easy to implement for a variety of spatial discretizations and interpolated fields in both two and three dimensions

Integration of operator-validated contours in deformable image registration for dose accumulation in radiotherapy

BACKGROUND AND PURPOSE: Deformable image registration (DIR) is a core element of adaptive radiotherapy workflows, integrating daily contour propagation and/or dose accumulation in their design. Propagated contours are usually manually validated and may be edited, thereby locally invalidating the registration result. This means the registration cannot be used for dose accumulation. In this study we proposed and evaluated a novel multi-modal DIR algorithm that incorporated contour information to guide the registration. This integrates operator-validated contours with the estimated deformation vector field and warped dose. MATERIALS AND METHODS: The proposed algorithm consisted of both a normalized gradient field-based data-fidelity term on the images and an optical flow data-fidelity term on the contours. The Helmholtz-Hodge decomposition was incorporated to ensure anatomically plausible deformations. The algorithm was validated for same- and cross-contrast Magnetic Resonance (MR) image registrations, Computed Tomography (CT) registrations, and CT-to-MR registrations for different anatomies, all based on challenging clinical situations. The contour-correspondence, anatomical fidelity, registration error, and dose warping error were evaluated. RESULTS: The proposed contour-guided algorithm considerably and significantly increased contour overlap, decreasing the mean distance to agreement by a factor of 1.3 to 13.7, compared to the best algorithm without contour-guidance. Importantly, the registration error and dose warping error decreased significantly, by a factor of 1.2 to 2.0. CONCLUSIONS: Our contour-guided algorithm ensured that the deformation vector field and warped quantitative information were consistent with the operator-validated contours. This provides a feasible semi-automatic strategy for spatially correct warping of quantitative information even in difficult and artefacted cases

Topological visualization of tensor fields using a generalized Helmholtz decomposition

Analysis and visualization of fluid flow datasets has become increasing important with the development of computer graphics. Even though many direct visualization methods have been applied in the tensor fields, those methods may result in much visual clutter. The Helmholtz decomposition has been widely used to analyze and visualize the vector fields, and it is also a useful application in the topological analysis of vector fields. However, there has been no previous work employing the Helmholtz decomposition of tensor fields. We present a method for computing the Helmholtz decomposition of tensor fields of arbitrary order and demonstrate its application. The Helmholtz decomposition can split a tensor field into divergence-free and curl-free parts. The curl-free part is irrotational, and it is useful to isolate the local maxima and minima of divergence (foci of sources and sinks) in the tensor field without interference from curl-based features. And divergence-free part is solenoidal, and it is useful to isolate centers of vortices in the tensor field. Topological visualization using this decomposition can classify critical points of two-dimensional tensor fields and critical lines of 3D tensor fields. Compared with several other methods, this approach is not dependent on computing eigenvectors, tensor invariants, or hyperstreamlines, but it can be computed by solving a sparse linear system of equations based on finite difference approximation operators. Our approach is an indirect visualization method, unlike the direct visualization which may result in the visual clutter. The topological analysis approach also generates a single separating contour to roughly partition the tensor field into irrotational and solenoidal regions. Our approach will make use of the 2nd order and the 4th order tensor fields. This approach can provide a concise representation of the global structure of the field, and provide intuitive and useful information about the structure of tensor fields. However, this method does not extract the exact locations of critical points and lines

Computation of Localized Flow for Steady and Unsteady Vector Fields and its Applications

We present, extend, and apply a method to extract the contribution of a subregion of a data set to the global flow. To isolate this contribution, we decompose the flow in the subregion into a potential flow that is induced by the original flow on the boundary and a localized flow. The localized flow is obtained by subtracting the potential flow from the original flow. Since the potential flow is free of both divergence and rotation, the localized flow retains the original features and captures the region-specific flow that contains the local contribution of the considered subdomain to the global flow. In the remainder of the paper, we describe an implementation on unstructured grids in both two and three dimensions for steady and unsteady flow fields. We discuss the application of some widely used feature extraction methods on the localized flow and describe applications like reverse-flow detection using the potential flow. Finally, we show that our algorithm is robust and scalable by applying it to various flow data sets and giving performance figures