Optimal surface parameterization using inverse curvature map

Planar Visualization of Treelike Structures

We present a novel method to create planar visualizations of treelike structures (e.g., blood vessels and airway trees) where the shape of the object is well preserved, allowing for easy recognition by users familiar with the structures. Based on the extracted skeleton within the treelike object, a radial planar embedding is first obtained such that there are no self-intersections of the skeleton which would have resulted in occlusions in the final view. An optimization procedure which adjusts the angular positions of the skeleton nodes is then used to reconstruct the shape as closely as possible to the original, according to a specified view plane, which thus preserves the global geometric context of the object. Using this shape recovered embedded skeleton, the object surface is then flattened to the plane without occlusions using harmonic mapping. The boundary of the mesh is adjusted during the flattening step to account for regions where the mesh is stretched over concavities. This parameterized surface can then be used either as a map for guidance during endoluminal navigation or directly for interrogation and decision making. Depth cues are provided with a grayscale border to aid in shape understanding. Examples are presented using bronchial trees, cranial and lower limb blood vessels, and upper aorta datasets, and the results are evaluated quantitatively and with a user study.

Colon flattening using heat diffusion Riemannian metric

We propose a new colon flattening algorithm that is efficient, shape-preserving, and robust to topological noise. Unlike previous approaches, which require a mandatory topological denoising to remove fake handles, our algorithm directly flattens the colon surface without any denoising. In our method, we replace the original Euclidean metric of the colon surface with a heat diffusion metric that is insensitive to topological noise. Using this heat diffusion metric, we then solve a Laplacian equation followed by an integration step to compute the final flattening. We demonstrate that our method is shape-preserving and the shape of the polyps are well preserved. The flattened colon also provides an efficient way to enhance the navigation and inspection in virtual colonoscopy. We further show how the existing colon registration pipeline is made more robust by using our colon flattening. We have tested our method on several colon wall surfaces and the experimental results demonstrate the robustness and the efficiency of our method.

Interactive visibility retargeting in VR using conformal visualization

In Virtual Reality, immersive systems such as the CAVE provide an important tool for the collaborative exploration of large 3D data. Unlike head-mounted displays, these systems are often only partially immersive due to space, access, or cost constraints. The resulting loss of visual information becomes a major obstacle for critical tasks that need to utilize the users' entire field of vision. We have developed a conformal visualization technique that establishes a conformal mapping between the full 360° field of view and the display geometry of a given visualization system. The mapping is provably angle-preserving and has the desirable property of preserving shapes locally, which is important for identifying shape-based features in the visual data. We apply the conformal visualization to both forward and backward rendering pipelines in a variety of retargeting scenarios, including CAVEs and angled arrangements of flat panel displays. In contrast to image-based retargeting approaches, our technique constructs accurate stereoscopic images that are free of resampling artifacts. Our user study shows that on the visual polyp detection task in Immersive Virtual Colonoscopy, conformal visualization leads to improved sensitivity at comparable examination times against the traditional rendering approach. We also develop a novel user interface based on the interactive recreation of the conformal mapping and the real-time regeneration of the view direction correspondence.

Automated construction of low-resolution, texture-mapped, class-optimal meshes

In this paper, we present a framework for the groupwise processing of a set of meshes in dense correspondence. Such sets arise when modeling 3D shape variation or tracking surface motion over time. We extend a number of mesh processing tools to operate in a groupwise manner. Specifically, we present a geodesic-based surface flattening and spectral clustering algorithm which estimates a single class-optimal flattening. We also show how to modify an iterative edge collapse algorithm to perform groupwise simplification while retaining the correspondence of the data. Finally, we show how to compute class-optimal texture coordinates for the simplified meshes. We present alternative algorithms for topologically symmetric data which yield a symmetric flattening and low-resolution mesh topology. We present flattening, simplification, and texture mapping results on three different data sets and show that our approach allows the construction of low-resolution 3D morphable models.

Context preserving maps of tubular structures

When visualizing tubular 3D structures, external representations are often used for guidance and display, and such views in 2D can often contain occlusions. Virtual dissection methods have been proposed where the entire 3D structure can be mapped to the 2D plane, though these will lose context by straightening curved sections. We present a new method of creating maps of 3D tubular structures that yield a succinct view while preserving the overall geometric structure. Given a dominant view plane for the structure, its curve skeleton is first projected to a 2D skeleton. This 2D skeleton is adjusted to account for distortions in length, modified to remove intersections, and optimized to preserve the shape of the original 3D skeleton. Based on this shaped 2D skeleton, a boundary for the map of the object is obtained based on a slicing path through the structure and the radius around the skeleton. The sliced structure is conformally mapped to a rectangle and then deformed via harmonic mapping to match the boundary placement. This flattened map preserves the general geometric context of a 3D object in a 2D display, and rendering of this flattened map can be accomplished using volumetric ray casting. We have evaluated our method on real datasets of human colon models.

Conformal Visualization for Partially-Immersive Platforms

Current immersive VR systems such as the CAVE provide an effective platform for the immersive exploration of large 3D data. A major limitation is that in most cases at least one display surface is missing due to space, access or cost constraints. This partially-immersive visualization results in a substantial loss of visual information that may be acceptable for some applications, however it becomes a major obstacle for critical tasks, such as the analysis of medical data. We propose a conformal deformation rendering pipeline for the visualization of datasets on partially-immersive platforms. The angle-preserving conformal mapping approach is used to map the 360°3D view volume to arbitrary display configurations. It has the desirable property of preserving shapes under distortion, which is important for identifying features, especially in medical data. The conformal mapping is used for rasterization, realtime raytracing and volume rendering of the datasets. Since the technique is applied during the rendering, we can construct stereoscopic images from the data, which is usually not true for image-based distortion approaches. We demonstrate the stereo conformal mapping rendering pipeline in the partially-immersive 5-wall Immersive Cabin (IC) for virtual colonoscopy and architectural review.

Almost isometric mesh parameterization through abstract domains

In this paper, we propose a robust, automatic technique to build a global hi-quality parameterization of a two-manifold triangular mesh. An adaptively chosen 2D domain of the parameterization is built as part of the process. The produced parameterization exhibits very low isometric distortion, because it is globally optimized to preserve both areas and angles. The domain is a collection of equilateral triangular 2D regions enriched with explicit adjacency relationships (it is abstract in the sense that no 3D embedding is necessary). It is tailored to minimize isometric distortion, resulting in excellent parameterization qualities, even when meshes with complex shape and topology are mapped into domains composed of a small number of large continuous regions. Moreover, this domain is, in turn, remapped into a collection of 2D square regions, unlocking many advantages found in quad-based domains (e.g., ease of packing). The technique is tested on a variety of cases, including challenging ones, and compares very favorably with known approaches. An open-source implementation is made available.

Computing Teichmüller shape space

Shape indexing, classification, and retrieval are fundamental problems in computer graphics. This work introduces a novel method for surface indexing and classification based on Teichmuller theory. The Teichmuller space for surfaces with the same topology is a finite dimensional manifold, where each point represents a conformal equivalence class, a curve represents a deformation process from one class to the other. We apply Teichmuller space coordinates as shape descriptors, which are succinct, discriminating and intrinsic; invariant under the rigid motions and scalings, insensitive to resolutions. Furthermore, the method has solid theoretic foundation, and the computation of Teichmuller coordinates is practical, stable and efficient. This work focuses on the surfaces with negative Euler numbers, which have a unique conformal Riemannian metric with -1 Gaussian curvature. The coordinates which we will compute are the lengths of a special set of geodesics under this special metric. The metric can be obtained by the curvature flow algorithm, the geodesics can be calculated using algebraic topological method. We tested our method extensively for indexing and comparison of about one hundred of surfaces with various topologies, geometries and resolutions. The experimental results show the efficacy and efficiency of the length coordinate of the Teichmuller space.