Conformal Virtual Colon Flattening Wei Hong Xianfeng Gu Feng Qiu Miao Jin Arie Kaufman Center for Visual Computing (CVC) and Department of Computer Science ∗ Stony Brook University Stony Brook, NY 11794-4400, USA (a) Holomorphic 1-form (b) Conformal virtual colon flattening Figure 1: Conformal Virtual Colon Flattening: (a) illustrates the holomorphic one-form on the colon surface by texture-mapping a checker board image. (b) exhibits the conformal flattening induced by (a). Abstract We present an efficient colon flattening algorithm using conformal structure, which is angle-preserving and minimizes the global dis- tortion. Moreover, our algorithm is general which can handle high genus surfaces. First, the colon wall is segmented and extracted from the CT dataset. The topology noise (i.e., minute handle) is lo- cated and removed automatically. The holomorphic 1-form, a pair of orthogonal vector fields, is then computed on the 3D colon sur- face mesh using the conjugate gradient method. The colon surface is cut along a vertical trajectory traced using the holomorphic 1- form. Consequently, the 3D colon surface is conformal mapped to a 2D rectangle. The flattened 2D mesh is then rendered using a direct volume rendering method accelerated with the GPU. Our algorithm is tested with a number of CT datasets of real patholog- ical cases, and gives consistent results. We demonstrated that the shape of the polyps is well preserved on the flattened colon images, which provides an efficient way to enhance the navigation of a vir- tual colonoscopy system. Keywords: Conformal Mapping, Direct Volume Rendering, Vir- tual Colonoscopy 1 Introduction Virtual colonoscopy uses computed tomographic (CT) images of patient’s abdomen and a virtual fly-through visualization system [Hong et al. 1997] that allows the physician to navigate within a 3D model of the colon searching for polyps, the precursors of cancer. Virtual colonoscopy has been successfully demonstrated to be more convenient and efficient than the real optical colonoscopy. How- ever, because of the length of the colon, inspecting the entire colon wall is time consuming, and prone to errors. Moreover, polyps be- ∗ Email: {weihong|gu|qfeng|mjin|ari}@cs.sunysb.edu hind folds may be hidden, which results in incomplete examina- tions. Virtual dissection is an efficient visualization technique for polyp detection, in which the entire inner surface of the colon is displayed as a single 2D image. The straightforward method [Balogh et al. 2002; Wang and Vannier 1995] starts with uniformly resampling the colonic central path. At each sampling point, a cross section orthogonal to the path is computed. The central path is straight- ened and the cross sections are unfolded and remapped into a new 3D volume. The isosurface is then extracted and rendered. In this method, nearby cross sections may overlap at high curvature re- gions. As a consequence, a polyp might appear twice or be missed completely in the flattened image. Balogh et al. [2002] use an itera- tive method to correct cross sections, using two consecutive ones at a time. Wang et al. [1998; 1999] use electrical field lines generated by a local charged path to generate curved cross sections instead of planar sections. If the complete path is charged, then the cross sections tend to diverge, avoiding overlaps. However, due to the expansive computation of the global charge, the authors only lo- cally charge the path, which cannot guarantee that the curved cross sections do not intersect each other any more. Paik et al. [2000] have used cartographic projections to project the whole solid angle of the camera. This approach samples the solid angle of the camera, and maps it onto a cylinder which is mapped finally to the image. However, this method causes distor- tions in shape. Bartrol´ ı et al. [2001b] have proposed a method to move a camera along the central path of the colon. For each camera position a small cylinder tangent to the path is defined. Rays start- ing at the cylinder axis and being orthogonal to the cylinder surface are traced. The cylinder is then opened and mapped to a 2D im- age. The result is a video where each frame shows the projection of a small part of the inner surface of the colon onto the cylinder. This avoids the appearance of double polyps since intersections can only appear between different frames. However, this approach does not provide a complete overview of the colon. They have presented