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

Faugeras9972228的研究人员和他们的同事研究了位于任意余维光滑流形上的点集的自动聚类问题。他们使用测地线蛇形方法，其中他们建立了一个偏微分方程(PDE)，该方程在由数据点创建的某个势场中驱动流形的演化，使得PDE的解收敛到非常接近数据点(理想情况下包含数据点)的流形。这个流形的“形状”(包括它的共维)表征了这组点。他们研究了任意余维流形按水平集方法的计算效率实现，因为它们具有健壮性和灵活性。这些实现被用于研究三维空间中曲线的欧几里得运动和仿射-曲率运动。调查人员正在追查两份申请。第一个应用是检测磁共振血管成像(MRA)图像中的血管。在MRA图像中，血管表现为明亮而嘈杂的曲线样图案，可能有间隙。重要的是独立于噪声检测这些模式，并弥合差距。作为第一近似值，血管的检测可以被认为是三维情况的一个例子，即协维二维。第二个应用是高维空间中特征点的聚类，着眼于两个具体的问题，一个是簇的“形状”的描述，另一个是基于内容的自动图像检索系统的设计。给定的图案，例如图像，可以由某些维空间中的特征向量来表示。同一类中的图案通常会分成若干簇，这些簇可能会有一些非常复杂的“形状”。研究人员和他们的同事认为，他们的方法可以提供从样本特征向量集合中自动获取形状描述的方法，从而方便识别算法的设计和提高识别算法的性能。一个目标应用是设计基于内容的自动化图像检索系统。研究人员和他们的同事研究了将可能的高维空间中的一组点自动拟合到一组光滑流形上的问题。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。他们目前正在寻求这些技术的两种应用。第一个是磁共振血管成像(MRA)图像中血管的检测。在MRA图像中，血管表现为明亮而嘈杂的曲线样图案，可能有间隙。重要的是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一簇点的“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。与普通市民特别相关的一个目标应用是设计基于内容的自动化图像检索系统。在这样的系统中，每个图像可以由某个高维(10到20)特征空间中的一个点来表示，并且同一类中的图像，例如汽车图像，聚集在一些流形上，例如“汽车流形”。该流形的形状描述(其比类中对象的真实形状更简单)然后可用于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流形更接近相应的流形。

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.