Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification

©2001 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.DOI: 10.1109/83.935033In this work, we first address the problem of simultaneous image segmentation and smoothing by approaching the Mumford–Shah paradigm from a curve evolution perspective. In particular, we let a set of deformable contours define the boundaries between regions in an image where we model the data via piecewise smooth functions and employ a gradient flow to evolve these contours. Each gradient step involves solving an optimal estimation problem for the data within each region, connecting curve evolution and the Mumford–Shah functional with the theory of boundary-value stochastic processes. The resulting active contour model offers a tractable implementation of the original Mumford–Shah model (i.e., without resorting to elliptic approximations which have traditionally been favored for greater ease in implementation) to simultaneously segment and smoothly reconstruct the data within a given image in a coupled manner. Various implementations of this algorithm are introduced to increase its speed of convergence.We also outline a hierarchical implementation of this algorithm to handle important image features such as triple points and other multiple junctions. Next, by generalizing the data fidelity term of the original Mumford– Shah functional to incorporate a spatially varying penalty, we extend our method to problems in which data quality varies across the image and to images in which sets of pixel measurements are missing. This more general model leads us to a novel PDE-based approach for simultaneous image magnification, segmentation, and smoothing, thereby extending the traditional applications of the Mumford–Shah functional which only considers simultaneous segmentation and smoothing

State of the Art of Level Set Methods in Segmentation and Registration of Medical Imaging Modalities

Segmentation of medical images is an important step in various applications such as visualization, quantitative analysis and image-guided surgery. Numerous segmentation methods have been developed in the past two decades for extraction of organ contours on medical images. Low-level segmentation methods, such as pixel-based clustering, region growing, and filter-based edge detection, require additional pre-processing and post-processing as well as considerable amounts of expert intervention or information of the objects of interest. Furthermore the subsequent analysis of segmented objects is hampered by the primitive, pixel or voxel level representations from those region-based segmentation. Deformable models, on the other hand, provide an explicit representation of the boundary and the shape of the object. They combine several desirable features such as inherent connectivity and smoothness, which counteract noise and boundary irregularities, as well as the ability to incorporate knowledge about the object of interest. However, parametric deformable models have two main limitations. First, in situations where the initial model and desired object boundary differ greatly in size and shape, the model must be re-parameterized dynamically to faithfully recover the object boundary. The second limitation is that it has difficulty dealing with topological adaptation such as splitting or merging model parts, a useful property for recovering either multiple objects or objects with unknown topology. This difficulty is caused by the fact that a new parameterization must be constructed whenever topology change occurs, which requires sophisticated schemes. Level set deformable models, also referred to as geometric deformable models, provide an elegant solution to address the primary limitations of parametric deformable models. These methods have drawn a great deal of attention since their introduction in 1988. Advantages of the contour implicit formulation of the deformable model over parametric formulation include: (1) no parameterization of the contour, (2) topological flexibility, (3) good numerical stability, (4) straightforward extension of the 2D formulation to n-D. Recent reviews on the subject include papers from Suri. In this chapter we give a general overview of the level set segmentation methods with emphasize on new frameworks recently introduced in the context of medical imaging problems. We then introduce novel approaches that aim at combining segmentation and registration in a level set formulation. Finally we review a selective set of clinical works with detailed validation of the level set methods for several clinical applications

Learning the dynamics of deformable objects and recursive boundary estimation using curve evolution techniques

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 161-176).The primary objective of this thesis is to develop robust algorithms for the incorporation of statistical information in the problem of estimating object boundaries in image data. We propose two primary algorithms, one which jointly estimates the underlying field and boundary in a static image and another which performs image segmentation across a temporal sequence. Some motivating applications come from the earth sciences and medical imaging. In particular, we examine the problems of oceanic front and sea surface temperature estimation in oceanography, soil boundary and moisture estimation in hydrology, and left ventricle boundary estimation across a cardiac cycle in medical imaging. To accomplish joint estimation in a static image, we introduce a variational technique that incorporates the spatial statistics of the underlying field to segment the boundary and estimate the field on either side of the boundary. For image segmentation across a sequence of frames, we propose a method for learning the dynamics of a deformable boundary that uses these learned dynamics to recursively estimate the boundary in each frame over time. In the recursive estimation algorithm, we extend the traditional particle filtering approach by applying sample-based methods to a complex shape space.(cont.) We find a low-dimensional representation for this shape-shape to make the learning of the dynamics tractable and then incorporate curve evolution into the state estimates to recursively estimate the boundaries. Experimental results are obtained on cardiac magnetic resonance images, sea surface temperature data, and soil moisture maps. Although we focus on these application areas, the underlying mathematical principles posed in the thesis are general enough that they can be applied to other applications as well. We analyze the algorithms on data of differing quality, with both high and low SNR data and also full and sparse observations.by Walter Sun.Ph.D

Information Extraction and Modeling from Remote Sensing Images: Application to the Enhancement of Digital Elevation Models

To deal with high complexity data such as remote sensing images presenting metric resolution over large areas, an innovative, fast and robust image processing system is presented. The modeling of increasing level of information is used to extract, represent and link image features to semantic content. The potential of the proposed techniques is demonstrated with an application to enhance and regularize digital elevation models based on information collected from RS images

Image segmentation with variational active contours

An important branch of computer vision is image segmentation. Image segmentation aims at extracting meaningful objects lying in images either by dividing images into contiguous semantic regions, or by extracting one or more specific objects in images such as medical structures. The image segmentation task is in general very difficult to achieve since natural images are diverse, complex and the way we perceive them vary according to individuals. For more than a decade, a promising mathematical framework, based on variational models and partial differential equations, have been investigated to solve the image segmentation problem. This new approach benefits from well-established mathematical theories that allow people to analyze, understand and extend segmentation methods. Moreover, this framework is defined in a continuous setting which makes the proposed models independent with respect to the grid of digital images. This thesis proposes four new image segmentation models based on variational models and the active contours method. The active contours or snakes model is more and more used in image segmentation because it relies on solid mathematical properties and its numerical implementation uses the efficient level set method to track evolving contours. The first model defined in this dissertation proposes to determine global minimizers of the active contour/snake model. Despite of great theoretic properties, the active contours model suffers from the existence of local minima which makes the initial guess critical to get satisfactory results. We propose to couple the geodesic/geometric active contours model with the total variation functional and the Mumford-Shah functional to determine global minimizers of the snake model. It is interesting to notice that the merging of two well-known and "opposite" models of geodesic/geometric active contours, based on the detection of edges, and active contours without edges provides a global minimum to the image segmentation algorithm. The second model introduces a method that combines at the same time deterministic and statistical concepts. We define a non-parametric and non-supervised image classification model based on information theory and the shape gradient method. We show that this new segmentation model generalizes, in a conceptual way, many existing models based on active contours, statistical and information theoretic concepts such as mutual information. The third model defined in this thesis is a variational model that extracts in images objects of interest which geometric shape is given by the principal components analysis. The main interest of the proposed model is to combine the three families of active contours, based on the detection of edges, the segmentation of homogeneous regions and the integration of geometric shape prior, in order to use simultaneously the advantages of each family. Finally, the last model presents a generalization of the active contours model in scale spaces in order to extract structures at different scales of observation. The mathematical framework which allows us to define an evolution equation for active contours in scale spaces comes from string theory. This theory introduces a mathematical setting to process a manifold such as an active contour embedded in higher dimensional Riemannian spaces such as scale spaces. We thus define the energy functional and the evolution equation of the multiscale active contours model which can evolve in the most well-known scale spaces such as the linear or the curvature scale space

Trends in Mathematical Imaging and Surface Processing

Motivated both by industrial applications and the challenge of new problems, one observes an increasing interest in the field of image and surface processing over the last years. It has become clear that even though the applications areas differ significantly the methodological overlap is enormous. Even if contributions to the field come from almost any discipline in mathematics, a major role is played by partial differential equations and in particular by geometric and variational modeling and by their numerical counterparts. The aim of the workshop was to gather a group of leading experts coming from mathematics, engineering and computer graphics to cover the main developments

A data consistent variational segmentation approach suitable for real-time tomography

Computed Tomography (CT) is an imaging technique that allows to reconstruct volumetric information of the analyzed objects from their projections. The most popula