Efficient energies and algorithms for parametric snakes - EPFL

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2AbstractParametric active contour models are one of the preferred approaches for image segmentationbecause of their computational efficiency and simplicity. However, they have a few drawbacks whichlimit their performance. In this paper, we identify some of these problems and propose efficient solutionsto get around them.The widely-used gradient magnitude-based energy is parameter dependent; its use will negativelyaffect the parametrization of the curve and consequently its stiffness. Hence, we introduce a new edgebasedenergy that is independent of the parameterization. It is also more robust since it takes intoaccount the gradient direction as well. We express this energy term as a surface integral, thus unifyingit naturally with the region-based schemes. The unified framework enables the user to tune the imageenergy to the application at hand.We show that parametric snakes can guarantee low curvature curves, but only if they are describedin the curvilinear abscissa. Since normal curve evolution do not ensure constant arc-length, we proposea new internal energy term that will force this configuration.The curve evolution can sometimes give rise to closed loops in the contour, which will adverselyinterfere with the optimization algorithm. We propose a curve evolution scheme that prevents thiscondition.Index Termsactive contour, snake, segmentation, curve, splineI. INTRODUCTIONSnakes or active contour models have proven to be very effective tools for image segmentation.An active contour model is essentially a curve that evolves from an initial position towards theboundary of an object in such a way as to minimize some energy functional. The popularityof this semi-automatic approach may be attributed to its ability to aid the segmentation processwith a-priori knowledge and user interaction.Extensive research in this area have resulted in many snake variants [1], [2]; these aredistinguished mainly by the type of curve representation used and the choice of the imageenergy term. The popular curve representation schemes in the snake literature are• Point-based snakes, where the curve is an ordered collection of discrete points (also termedas snaxels) [3]–[5].Submitted to IEEE Trans. Image Processing, January 7, 2004.DRAFT
3• Parametric snakes, where the curve is described continuously in a parametric form usingbasis functions such as B-splines [6]–[9], Fourier exponentials [10], [11] etc.• Geometric snakes, where the planar curve is represented as a level set of an appropriate2-D surface [12]–[16].The point-based approach can be viewed as a special case of parametric curve representationwhere the basis functions are uniform translates of a B-spline of degree zero 1 ; likewise, parametricapproaches using smooth basis functions will tend to the point-based scheme as the numberof basis functions increases. In general, however, representations using smooth basis functionsrequire fewer parameters than point-based approaches and thus result in faster optimizationalgorithms [6], [10], [17]. Moreover, such curve models have inherent regularity and hence donot require extra constraints to ensure smoothness [9], [17].Since both the above mentioned schemes represent the curve explicitly, it is easy to introducea priori shape constraints into the snake framework [10], [18]–[20]. It is also straightforwardto accommodate user interaction; this is often done by allowing the user to specify pointsthrough which the curve should go through [3]. However, these models offer less flexibilityin accounting for topological changes during the curve evolution. One will have to performsome extra bookkeeping to accommodate changes in topology.Geometric approaches offer great flexibility as far as the curve topology is considered; theypresently constitute a very promising research area [14]–[16]. However, they tend to be computationallymore complex since they evolve a surface rather than a curve. Also, since thecurve representation is implicit, it is much more challenging to introduce shape priors into thisframework [21].In this paper, we focus on general parametric snakes due to its computational advantages andsimplicity. We will start by taking a critical look at them, identifying some of their limitations,and propose some improvements to make them more attractive.There are many different image energy terms that are used in practice. Most of the commonlyused approaches fall into two broadly defined categories: (i) edge-based schemes which uselocal image information (typically gradient information) [3], [6], [9], [10], [17], [22], and, (ii)⎧⎨ 1 if |x| < 0.51 A B-spline of degree zero is defined as β 0 (x) =⎩ 0 elseSubmitted to IEEE Trans. Image Processing, January 7, 2004.DRAFT
Page 1: 1Efficient energies and algorithms
Page 5 and 6: 5(6,8); t=2C¡¡¡¡¡¡¡¡¡¡¡
Page 7 and 8: 71) Gradient magnitude energy: Many
Page 9 and 10: 9(a) Initialization(b) Magnitude-ba
Page 11 and 12: 11The use of this energy assumes tw
Page 14 and 15: 14second term reduces to∫ Mprovid
Page 16 and 17: 16V. EXTERNAL CONSTRAINT ENERGY.As
Page 18 and 19: 18computed as the curve integral∫
Page 20 and 21: 20whereh 1 (l, m, n) =h 2 (l) =∫
Page 22: d22d sOCFig. 6.Computation of the e
Page 25 and 26: 25(a)(b)(c)(d)Fig. 8. Segmentation
Page 27 and 28: 27In the second step, we make use o
Page 29 and 30: 29APPENDIX - D: INTEGRAL OF THE TAN
Page 31: 31[32] L.D. Cohen and I. Cohen, “

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