Evolution of Co-dimension k>1 Manifolds Embedded in R^n and Applications to Medical Image Analysis

研究者和他们的同事研究了任意协维光滑流形上的点的自动聚类问题。他们使用测地线蛇形方法，在这种方法中，他们建立了一个偏微分方程（PDE），该方程驱动由数据点创建的一些潜在场中的流形的演变，这样PDE的解收敛到非常接近数据点（理想情况下包含它们）的流形。这个流形的“形状”（包括它的协维）表征了点的集合。由于其鲁棒性和灵活性，他们研究了通过水平集方法实现任意协维流形的计算效率。这些实现被用于研究三维空间中曲线的欧几里得和仿射曲率运动。调查人员正在调查两项申请。第一个应用是在磁共振血管造影（MRA）图像中检测血管。在核磁共振成像图像中，血管表现为明亮而嘈杂的曲线状图案，可能有间隙。重要的是要独立于噪声检测这些模式并弥合差距。血管的检测可以考虑，作为第一个近似，作为三维情况的一个实例，协维二。第二个应用是高维空间中特征点的聚类，着眼于两个具体问题，即聚类“形状”的描述和基于内容的自动图像检索系统的设计。一种给定的模式，例如：一个图像，可以用一个维空间的特征向量来表示。同一类中的模式通常属于可能具有一些非常棘手的“形状”的集群。研究人员及其同事认为，他们的方法可以提供从样本特征向量集自动获取形状描述的方法，从而促进识别算法的设计并提高其性能。目标应用程序是基于内容的自动图像检索系统的设计。研究人员和他们的同事研究了一个问题，即在可能的高维空间中自动拟合一组点到一组光滑流形。这一点很重要，因为在许多实际应用中，数据可以表示为高维空间中的点。他们目前正在研究这些技术的两种应用。首先是磁共振血管造影（MRA）图像中血管的检测。在核磁共振成像图像中，血管呈明亮而嘈杂的曲线状，可能有间隙。重要的是要独立于噪声检测这些模式并弥合差距。该检测结果可用于疾病的诊断和手术计划。第二个应用是描述高维空间中点簇的“形状”。研究人员和他们的同事正在研究从一组样本中自动获得这种描述的方法。他们认为这将有助于识别算法的设计并提高其性能。一个与普通市民特别相关的目标应用是基于内容的自动图像检索系统的设计。在这样的系统中，每个图像都可以用高维（10到20）特征空间中的一个点来表示，并且同一类的图像，例如汽车图像，聚集在一些流形上，例如“汽车流形”。这个流形的形状描述（比类中对象的实际形状更简单）可以用来检索数据库中的所有图像，例如汽车，因为它们在高维空间中的表示将比任何其他形式更接近相应的流形。

Faugeras9972228 The investigators and their colleagues study the problem ofautomatically clustering sets of points that lie on smoothmanifolds of arbitrary co-dimension. They use the geodesic snakeapproach in which they set up a Partial Differential Equation(PDE) that drives the evolution of a manifold in some potentialfield created by the data points, such that the solution of thePDE converges to a manifold that is very close to the data points(ideally contains them). The "shape" of this manifold (includingits co-dimension) characterizes the set of points. Theyinvestigate computationally efficient implementations forevolving manifolds of arbitrary co-dimension by level setsmethods, because of their robustness and flexibility. Theseimplementations are being used to study the Euclidean and affinemean-curvature motion of curves in three-dimensional spaces. Twoapplications are being pursued by the investigators. The firstapplication is the detection of blood vessels in MagneticResonance Angiography (MRA) images. In MRA images, blood vesselsappear as bright and noisy curve-like patterns, possibly withgaps. It is important to detect those patterns independently ofthe noise and to bridge the gaps. The detection of blood vesselscan be considered, as a first approximation, as an instance ofthe three-dimensional case, co-dimension two. The secondapplication is the clustering of feature points in higherdimensional spaces with an eye on two specific problems, thedescription of the "shape" of a cluster and the design ofautomated content-based image retrieval systems. A given pattern,e.g. an image, can be represented by a vector of features in somen-dimensional space. Patterns in the same class usually fall intoclusters that may have some very tricky "shapes". Theinvestigators and their collegues believe that their methods canprovide ways of automatically obtaining shape descriptions fromsets of sample feature vectors thereby facilitating the design ofrecognition algorithms and improving their performances. A targetapplication is the design of automated content-based imageretrieval systems. The investigators and their colleagues study the problem ofautomatically fitting a set of points in a possiblyhigh-dimensional space to a set of smooth manifolds. This isimportant because in many practical applications data can berepresented as points in high-dimensional spaces. They arecurrently pursuing two applications of these techniques. Thefirst one is the detection of blood vessels in Magnetic ResonanceAngiography (MRA) images. In MRA images, blood vessels appear asbright and noisy curve-like patterns, possibly with gaps. It isimportant to detect those patterns independently of the noise andto bridge the gaps. The results of this detection can be used inthe diagnosis of diseases and in planning surgery. The secondapplication is the description of the "shape" of a cluster ofpoints in a high-dimensional space. The investigators and theircolleagues are investigating means of automatically obtainingsuch descriptions from a set of samples. They believe that thiswill facilitate the design of recognition algorithms and increasetheir performances. A target application that is particularlyrelevant to the average citizen is the design of automatedcontent-based image retrieval systems. In such systems each imagecan be represented by a point in some high-dimensional (10 to 20)feature space, and images in the same class, e.g. images of cars,cluster on some manifold, e.g a "car-manifold". The descriptionof the shape of this manifold (which is simpler than the realshapes of the objects in the class) can then be used to retrieveall images of, e.g. cars, in the data base because theirrepresentations in the high-dimensional space will be closer tothe corresponding manifold than to any other.