Multi-region active contours with a single level set function

Anastasia Dubrovina Karni, Ron KimmelComputer Science, Technion,

Haifa, 32000, Israelnastyad@cs.technion.ac.il, ron@cs.technion.ac.il

Guy RosmanCSAIL, MIT,

Cambridge, MA, USArosman@csail.mit.edu

Abstract

Segmenting an image into an arbitrary number of coher-ent regions is at the core of image understanding. We pro-pose a novel method for segmenting images into multipleregions using an axiomatic variational approach. The pro-posed approach was shown to obtain accurate segmentationresults for various natural 2D and 3D images, comparableto state-of-the-art image segmentation algorithms.

1. IntroductionImage segmentation constitutes an important part of var-

ious image processing and computer vision tasks, such asobject detection and classification, scene understanding,etc. Various formulations of the segmentation problem havebeen suggested over the past years. These formulations in-clude, among others, axiomatic functionals, which are hardto implement and analyze, and graph-based alternatives,which may impose a non-geometric metric on the problem.Here, we propose a novel method for segmenting an imageinto an arbitrary number of regions using an axiomatic vari-ational approach.

Specifically, we consider the active contours approach,according to which boundaries of different image regionsare modelled by parametric curves, and the segmentationis obtained by minimizing a certain segmentation criterionmodelled as an energy functional. Important advantages ofthe active contours approach include significant flexibilityin the design of the energy functional, as well as its ability todetect accurate region boundaries, with minimal geometricartefacts.

The level set framework is commonly employed to com-pute the optimal contour [9], though the conventional levelset approach allows to perform only binary image segmen-tation. Various methods were developed to alleviate thislimitation. These methods either require managing multi-ple level set functions to denote multiple regions, e.g. [11],or were designed to minimize only a certain type of seg-mentation criteria, while modelling multiple regions using

Figure 1. Multi-region level set function. Left: synthetic imagewith region boundaries shown in red. Right: its correspondinglevel set function.

a single piecewise constant function, e.g. [7].We propose a new level set framework for segmenting

images into multiple regions. In contrast to most existingapproaches, here, multiple region boundaries are modelledby a single non-negative level set function. The zero levelset of this function coincides with these boundaries, as il-lustrated in Figure 1. The level set evolution is efficientlyperformed through the Voronoi Implicit Interface Method(VIIM), recently introduced by Saye and Sethian [10] fornumerical simulations of fluid dynamics. It implicitly dealswith region merging and splitting, naturally handles com-plex topological structures such as multi-point junctions,and produces region boundaries with sub-pixel precision,while avoiding metrication errors.

The proposed method allows to incorporate variousgeneric region appearance models into the minimized en-ergy functional. Specifically, it was applied with the well-known region competition model of Zhu and Yuille [12],the piecewise constant variant of the Mamford-Shah model[8], and the recently suggested pairwise region dissimilar-ity models [1, 6]. We derived their corresponding multi-region active contours evolution rules, and, subsequently,their multi-region level set formulation. The algorithm isnot limited to the segmentation models described above; itcan be easily extended for texture image segmentation, em-ploy various additional edge alignment functionals, or dif-ferent classes of segmentation models, e.g. [5].

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Figure 2. Segmentation of volumetric MRI scan. Top left: singleMRI slice and cross-sections of the initial surfaces (yellow cir-cles). Top right: cross-section of the obtained 3D segmentation.Bottom: white matter exterior boundary, sagittal and axial views.

Further advantages of this new level set framework in-clude the ability to segment an image without prior knowl-edge of image region number or region intensity statistics,as well as good segmentation results produced for variousinitial conditions. In terms of memory consumption, themethod requires additional space in the order of the numberof image pixels. Level set function evolution is efficientlyperformed using the narrow band approach. Thus, the com-putational complexity of the algorithm is determined by thecardinality of the narrow band around the evolving regionboundaries, as well as by the complexity of the segmenta-tion model it is applied with. In terms of computation timesand the segmentation accuracy, the algorithm is comparableto the state-of-art methods, e.g. [2, 3].

Segmentation results obtained with the proposed methodfor both 2D and volumetric images are shown in Figures 2and 3. Further details may be found in [4].

2. ConclusionsTo conclude, we addressed the problem of segmenting

an image into an arbitrary number of regions using a novelmulti-region active contours formulation. The proposedframework treats multiple regions in a uniform manner byutilising the new Voronoi implicit interface method, whileavoiding metrication errors. It can be applied with variousregion and boundary appearance priors, for both 2D andvolumetric image segmentation, for which it produces ac-curate and detailed segmentation results.

3. AcknowledgementThis research was supported by the European Commu-

nity’s FP7-ERC program, grant agreement 267414.

Figure 3. Segmentation results obtained using the proposedmethod, for images from the Berkeley Segmentation Dataset.

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