## A review of statistical approaches to level set segmentation: Integrating color, texture, motion and shape (2007)

### Cached

### Download Links

Venue: | International Journal of Computer Vision |

Citations: | 92 - 4 self |

### BibTeX

@ARTICLE{Cremers07areview,

author = {Daniel Cremers and Mikael Rousson and Rachid Deriche},

title = {A review of statistical approaches to level set segmentation: Integrating color, texture, motion and shape},

journal = {International Journal of Computer Vision},

year = {2007},

volume = {72},

pages = {215}

}

### OpenURL

### Abstract

Abstract. Since their introduction as a means of front propagation and their first application to edge-based segmentation in the early 90’s, level set methods have become increasingly popular as a general framework for image segmentation. In this paper, we present a survey of a specific class of region-based level set segmentation methods and clarify how they can all be derived from a common statistical framework. Region-based segmentation schemes aim at partitioning the image domain by progressively fitting statistical models to the intensity, color, texture or motion in each of a set of regions. In contrast to edge-based schemes such as the classical Snakes, region-based methods tend to be less sensitive to noise. For typical images, the respective cost functionals tend to have less local minima which makes them particularly well-suited for local optimization methods such as the level set method. We detail a general statistical formulation for level set segmentation. Subsequently, we clarify how the integration of various low level criteria leads to a set of cost functionals and point out relations between the different segmentation schemes. In experimental results, we demonstrate how the level set function is driven to partition the image plane into domains of coherent color, texture, dynamic texture or motion. Moreover, the Bayesian formulation allows to introduce prior shape knowledge into the level set method. We briefly review a number of advances in this domain.

### Citations

4027 |
Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images
- Geman, Geman
- 1984
(Show Context)
Citation Context ...h which have a minimal cost. Optimization algorithms for these problems include greedy approaches such as the Iterated Conditional Modes (ICM) [2] and continuation methods such as Simulated Annealing =-=[35]-=- or Graduated Non-convexity [5]. Specific classes of graph cut approaches gained in popularity with the re-discovery of efficient global optimization methods, which are based on concepts of dynamic pr... |

3254 | Snakes: Active contour models
- Kass, Witkin, et al.
- 1988
(Show Context)
Citation Context ...e direction of the negative energy gradient using appropriate partial differential equations (pdes). Such pde-based segmentation methods became popular with the seminal paper on Snakes by Kass et al. =-=[44]-=-. In this paper, the contour is implemented by an explicit (parametric) curve C : [0, 1] → Ω which is evolved by locally minimizing the functional � E(C) = − |∇I(C)| 2 � ds + ν1 |Cs| 2 � ds + ν2 |Css|... |

2759 | Normalized Cuts and Image Segmentation
- Shi, Malik
(Show Context)
Citation Context ...ecific classes of graph cut approaches gained in popularity with the re-discovery of efficient global optimization methods, which are based on concepts of dynamic programming [6], on spectral methods =-=[82, 56]-=- or on semidefinite programming techniques [45]. In spatially continuous approaches, the segmentation of the image plane Ω ⊂ IR 2 is considered as a problem of infinite-dimensional optimization. Using... |

1477 | Fast approximate energy minimization via graph cuts
- Boykov, Veksler, et al.
- 2001
(Show Context)
Citation Context ...ted Non-convexity [5]. Specific classes of graph cut approaches gained in popularity with the re-discovery of efficient global optimization methods, which are based on concepts of dynamic programming =-=[6]-=-, on spectral methods [82, 56] or on semidefinite programming techniques [45]. In spatially continuous approaches, the segmentation of the image plane Ω ⊂ IR 2 is considered as a problem of infinite-d... |

1140 | Geodesic active contours
- Caselles, Kimmel, et al.
- 1997
(Show Context)
Citation Context ...lightly different evolution equations for φ(x, t). The first applications of this level set formalism for the purpose of image segmentation were proposed in [10, 58, 57]. Indepdently, Caselles et al. =-=[11]-=- and Kichenassamy et al. [46] proposed a level set formulation IJCV.tex; 19/09/2005; 19:10; p.5s6 for the Snake energy (1) given by: ∂φ ∂t � = |∇φ|div g(I) ∇φ � � � ∇φ = g(I)|∇φ|div + ∇g(I)·∇φ, (7) �∇... |

991 |
Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems
- Mumford, Shah
- 1989
(Show Context)
Citation Context ...n the field of image processing include spatially discrete formulations such as those of Geman and Geman [35] and Besag [2], and spatially continuous formulations such as the ones of Mumford and Shah =-=[63]-=- and Zhu and Yuille [100]. The probabilistic formulation of the segmentation problem presented in the following extends the statistical approaches pioneered in [49, 100, IJCV.tex; 19/09/2005; 19:10; p... |

989 | On the statistical analysis of dirty pictures
- Besag
- 1986
(Show Context)
Citation Context ... and the aim of segmentation is to find cuts of this graph which have a minimal cost. Optimization algorithms for these problems include greedy approaches such as the Iterated Conditional Modes (ICM) =-=[2]-=- and continuation methods such as Simulated Annealing [35] or Graduated Non-convexity [5]. Specific classes of graph cut approaches gained in popularity with the re-discovery of efficient global optim... |

964 | Fronts Propagating with Curvature Dependent Speed: Algorithms Based on HamiltonJacobi Formulations
- Osher, Sethian
- 1988
(Show Context)
Citation Context ... the contour, but only the parameterization. IJCV.tex; 19/09/2005; 19:10; p.4sThere are various methods to evolve implicitly represented contours. The most popular among these is the level set method =-=[29, 30, 65]-=-, in which a contour is propagated by evolving a time-dependent embedding function φ(x, t) according to an appropriate partial differential equation. In the following, we will briefly sketch two alter... |

858 | Active Contours Without Edges
- Chan, Vese
(Show Context)
Citation Context ...ne needs to extend the right-hand side of (5) to the image domain away from the contour. Alternatively to the above derivation, one can obtain a level set equation from a variational formulation (cf. =-=[98, 12]-=-): Rather than deriving an appropriate partial differential equation for φ which implements the contour evolution equation (2), one can embed a variational principle E(C) defined on the space of conto... |

758 | Visual Reconstruction
- BLAKE, ZISSERMAN
- 1987
(Show Context)
Citation Context ...imization algorithms for these problems include greedy approaches such as the Iterated Conditional Modes (ICM) [2] and continuation methods such as Simulated Annealing [35] or Graduated Non-convexity =-=[5]-=-. Specific classes of graph cut approaches gained in popularity with the re-discovery of efficient global optimization methods, which are based on concepts of dynamic programming [6], on spectral meth... |

660 | Region competition: Unifying snakes, region growing, and bayes/mdl for multi-band image segmentation
- Zhu, Yuille
- 1996
(Show Context)
Citation Context ...cessing include spatially discrete formulations such as those of Geman and Geman [35] and Besag [2], and spatially continuous formulations such as the ones of Mumford and Shah [63] and Zhu and Yuille =-=[100]-=-. The probabilistic formulation of the segmentation problem presented in the following extends the statistical approaches pioneered in [49, 100, IJCV.tex; 19/09/2005; 19:10; p.6s66, 87]. In particular... |

504 | Scale-space theory in computer vision
- Lindeberg
- 1994
(Show Context)
Citation Context ...only give a scalar value reflecting the probability of an edge but it also includes the texture orientation. All these properties make this matrix a good descriptor for textures. The structure tensor =-=[34, 4, 70, 55, 36]-=- is given by the matrix of partial derivatives smoothed by a Gaussian kernel Kσ with standard deviation σ: Jσ = Kσ ∗ (∇I∇I ⊤ ) = � Kσ ∗ I 2 x1 Kσ ∗ Ix1 Ix2 Kσ ∗ Ix1 Ix2 Kσ ∗ I 2 x2 15 � . (25) For col... |

496 |
Multiresolution Approximation and Wavelet Orthonormal Bases of L2
- Mallat, G
- 1989
(Show Context)
Citation Context ...lations. More sophisticated features are conceivable as well [53]. 4.2.1. The nonlinear structure tensor as texture feature While texture analysis can rely on texture samples to learn accurate models =-=[38, 26, 59, 83, 99]-=-, unsupervised image segmentation should learn these parameters on-line. Since high-order texture models introduce too many unknown parameter to be estimated in an unsupervised approach, more compact ... |

462 | Shiftable Multi-scale Transforms
- Simoncelli, Freeman, et al.
- 1992
(Show Context)
Citation Context ...lations. More sophisticated features are conceivable as well [53]. 4.2.1. The nonlinear structure tensor as texture feature While texture analysis can rely on texture samples to learn accurate models =-=[38, 26, 59, 83, 99]-=-, unsupervised image segmentation should learn these parameters on-line. Since high-order texture models introduce too many unknown parameter to be estimated in an unsupervised approach, more compact ... |

349 |
Representing and recognizing the visual appearance of materials using threedimensional textons
- Leung, Malik
(Show Context)
Citation Context ...ed by local correlations of pixel values. In the following, we will review a set of basic features which allow to capture these local correlations. More sophisticated features are conceivable as well =-=[53]-=-. 4.2.1. The nonlinear structure tensor as texture feature While texture analysis can rely on texture samples to learn accurate models [38, 26, 59, 83, 99], unsupervised image segmentation should lear... |

349 | A multiphase level set framework for image segmentation using the mumford and shah model
- VESE, CHAN
- 2002
(Show Context)
Citation Context ...egant formulation which only requires log 2(n) level set functions to model n regions. Each of the n regions is characterized by the various level set functions being either positive or negative. See =-=[93]-=- for details. 3.4. Scalar, Vector and Tensor-valued images 3.4.1. Scalar images Let us consider a scalar image made up of two regions, the intensities of which are drawn from a Gaussian distribution: ... |

333 | Statistical shape influence in geodesic active contours
- Leventon, Grimson, et al.
(Show Context)
Citation Context ...l briefly list some of the key contributions in the field of shape priors for level set segmentation. The first application of shape priors for level set segmentation was developed by Leventon et al. =-=[54]-=- who propose to perform principal component analysis on a set of signed distance function embedding a set of sample shapes. The distance functions are sampled on a regular grid to obtain a vector repr... |

332 |
Signal Processing for Computer Vision
- Granlund, Knutsson
- 1995
(Show Context)
Citation Context ...only give a scalar value reflecting the probability of an edge but it also includes the texture orientation. All these properties make this matrix a good descriptor for textures. The structure tensor =-=[34, 4, 70, 55, 36]-=- is given by the matrix of partial derivatives smoothed by a Gaussian kernel Kσ with standard deviation σ: Jσ = Kσ ∗ (∇I∇I ⊤ ) = � Kσ ∗ I 2 x1 Kσ ∗ Ix1 Ix2 Kσ ∗ Ix1 Ix2 Kσ ∗ I 2 x2 15 � . (25) For col... |

331 | High accuracy optical flow estimation based on a theory for warping
- Brox, Bruhn, et al.
- 2004
(Show Context)
Citation Context ..., we will drop the time coordinate in the notation of the velocity field. 12 To allow for variation of the global illumination, one can alternatively assume constancy of higher-order derivatives (cf. =-=[7]-=-). IJCV.tex; 19/09/2005; 19:10; p.22sFor the sake of segmentation, we will assume that the velocity in each of a set of regions can be modeled by a parametric motion of the form 23 v(x) = S(x) · q, (3... |

315 | Contour and texture analysis for image segmentation
- Malik, Belongie, et al.
- 2001
(Show Context)
Citation Context ...ecific classes of graph cut approaches gained in popularity with the re-discovery of efficient global optimization methods, which are based on concepts of dynamic programming [6], on spectral methods =-=[82, 56]-=- or on semidefinite programming techniques [45]. In spatially continuous approaches, the segmentation of the image plane Ω ⊂ IR 2 is considered as a problem of infinite-dimensional optimization. Using... |

295 | S.: Editable dynamic textures
- Doretto, Soatto
- 2002
(Show Context)
Citation Context ... group regions of similar spatio-temporal statistics. The first work addressing this problem was proposed in [32] where the authors made use of recent developments in the modeling of dynamic textures =-=[31]-=-. Due to the scope of this paper, we will merely review the key ideas. Dynamic textures are models of temporally varying textures which assume the image sequence to be generated by a second-order stat... |

293 |
Markov random field texture models
- Cross, Jain
- 1983
(Show Context)
Citation Context ...lations. More sophisticated features are conceivable as well [53]. 4.2.1. The nonlinear structure tensor as texture feature While texture analysis can rely on texture samples to learn accurate models =-=[38, 26, 59, 83, 99]-=-, unsupervised image segmentation should learn these parameters on-line. Since high-order texture models introduce too many unknown parameter to be estimated in an unsupervised approach, more compact ... |

292 |
Finite-element methods for active contour models and balloons for 2-D and 3-D images
- Cohen, Cohen
- 1993
(Show Context)
Citation Context ...ed by local maxima of the image gradient. To overcome these local minima and to drive the contour toward the boundaries of objects of interest, researchers have introduced an additional balloon force =-=[16]-=- which leads to either a shrinking or an expansion of contours. Unfortunately this requires prior knowledge about whether the object of interest is inside or outside the initial contour. Moreover, the... |

265 | A variational level set approach to multiphase motion
- Zhao, Chan, et al.
- 1996
(Show Context)
Citation Context ...ne needs to extend the right-hand side of (5) to the image domain away from the contour. Alternatively to the above derivation, one can obtain a level set equation from a variational formulation (cf. =-=[98, 12]-=-): Rather than deriving an appropriate partial differential equation for φ which implements the contour evolution equation (2), one can embed a variational principle E(C) defined on the space of conto... |

243 | Geodesic active regions and level set methods for supervised texture segmentation, Int
- Paragios, Deriche
- 2002
(Show Context)
Citation Context ...the scene in terms of piecewise parametric motion. 10 The present formulation was proposed in [17, 23] with an earlier (explicit contour) formulation in [21]. Related approaches were also proposed in =-=[60, 67]-=-. The central idea is that we do not precompute local motion vectors. Instead we jointly estimate the segmentation and the motion models for each of a set of regions by minimizing the proposed functio... |

239 |
A fast operator for detection and precise location of distinct points, corners, and centers of circular features
- Forstner, Gulch
- 1987
(Show Context)
Citation Context ...f the structure tensor (also called second order moment matrix) which yields three different feature channels per scale. It has mainly been used to determine the intrinsic dimensionality of images in =-=[3, 34]-=- by providing a continuous measure to detect critical points like edges or corners. Yet, the structure tensor does not only give a scalar value reflecting the probability of an edge but it also includ... |

237 | Random walks for image segmentation - Grady - 2006 |

234 | Constructing simple stable descriptions for image partitioning
- Leclerc
- 1989
(Show Context)
Citation Context ...:10; p.6s66, 87]. In particular, this extension allows the probabilistic framework to be applied to segmentation criteria such as texture and motion, which will be detailed in subsequent sections. In =-=[49]-=-, a segmentation functional is obtained from a Minimum Description Length (MDL) criterion. The link with the Mumford-Shah functional and the equivalence to Bayesian maximum a posteriori (MAP) estimati... |

212 | Gradient flows and geometric active contour models
- Kichenassamy, Kumar, et al.
- 1995
(Show Context)
Citation Context ...quations for φ(x, t). The first applications of this level set formalism for the purpose of image segmentation were proposed in [10, 58, 57]. Indepdently, Caselles et al. [11] and Kichenassamy et al. =-=[46]-=- proposed a level set formulation IJCV.tex; 19/09/2005; 19:10; p.5s6 for the Snake energy (1) given by: ∂φ ∂t � = |∇φ|div g(I) ∇φ � � � ∇φ = g(I)|∇φ|div + ∇g(I)·∇φ, (7) �∇φ| �∇φ| where the gradient |∇... |

206 |
A geometric model for active contours in image processing
- Caselles, Catte, et al.
- 1993
(Show Context)
Citation Context ...g on the chosen embedding, one can obtain slightly different evolution equations for φ(x, t). The first applications of this level set formalism for the purpose of image segmentation were proposed in =-=[10, 58, 57]-=-. Indepdently, Caselles et al. [11] and Kichenassamy et al. [46] proposed a level set formulation IJCV.tex; 19/09/2005; 19:10; p.5s6 for the Snake energy (1) given by: ∂φ ∂t � = |∇φ|div g(I) ∇φ � � � ... |

204 | Filters, random fields and maximum entropy (frame
- Zhu, Wu, et al.
- 1998
(Show Context)
Citation Context |

200 | Topologically Adaptable Snakes
- McInerney, Terzopoulos
- 1995
(Show Context)
Citation Context ... separate textures which 2 It should be pointed out that based on various heuristics, one can successfully incorporate regriding mechanisms and topological changes into explicit representations – cf. =-=[62, 48, 28, 25]-=-. 3 IJCV.tex; 19/09/2005; 19:10; p.3s4 have identical spatial characteristics but differ in their temporal dynamics. In Section 7 we detail how to integrate motion information as a criterion for segme... |

182 |
Beitrag zur theorie des ferromagnetismus, (Contribution to the Theory of Ferromagnetism)”, Zeitschrift für Physik
- Ising
(Show Context)
Citation Context ...on 3.1. Image Segmentation as Bayesian Inference Statistical approaches to image segmentation have a long tradition, they can be traced back to models of magnetism in physics, such as the Ising model =-=[41]-=-, pioneering works in the field of image processing include spatially discrete formulations such as those of Geman and Geman [35] and Besag [2], and spatially continuous formulations such as the ones ... |

180 | Shape priors for level set representations
- Rousson, Paragios
- 2002
(Show Context)
Citation Context ...tudied in [33]. Chen et al. [14] proposed to impose shape information on the zero crossing (rather than on the level set function). Rousson et al. proposed variational integrations of the shape prior =-=[78, 79]-=- based on the assumption of a Gaussian distribution. The use of nonparametric density estimation to model larger classes of level set based shape distributions was developed in [20, 74]. This approach... |

177 |
Multidimensional orientation estimation with applications to texture analysis and optical flow
- Bigun, Granlund, et al.
- 1991
(Show Context)
Citation Context ... these parameters on-line. Since high-order texture models introduce too many unknown parameter to be estimated in an unsupervised approach, more compact features are usually favored. Bigün et al. in =-=[4]-=- addressed this problem with the introduction of the structure tensor (also called second order moment matrix) which yields three different feature channels per scale. It has mainly been used to deter... |

166 | A Riemannian framework for tensor computing
- Pennec, Fillard, et al.
- 2006
(Show Context)
Citation Context ...distance between two members of this family is given by � � � DG(J1, J2) = �1 2� log 2 2 (λi), (24) where λi denote the eigenvalues of the matrix J −1/2 1 J2J −1/2 1 . The same metric was proposed in =-=[68]-=- from a different viewpoint. It verifies the basic properties of a distance (positivity, symmetry and triangle inequality) and it is invariant to inversions: D(J1, J2) = D(J −1 −1 1 , J2 ). The above ... |

146 | Optimal orientation detection of linear symmetry
- Bigün, Granlund
- 1987
(Show Context)
Citation Context ...f the structure tensor (also called second order moment matrix) which yields three different feature channels per scale. It has mainly been used to determine the intrinsic dimensionality of images in =-=[3, 34]-=- by providing a continuous measure to detect critical points like edges or corners. Yet, the structure tensor does not only give a scalar value reflecting the probability of an edge but it also includ... |

146 | Mixture models for optical flow computation
- Jepson, Black
- 1993
(Show Context)
Citation Context ...mallest eigenvalue of Ti. It is normalized, such that the third component is 1. Similar eigenvalue problems arise in motion estimation due to normalization with respect to the velocity magnitude (cf. =-=[4, 43]-=-). Conversely, for fixed motion models qi, a gradient descent on the energy (38) for the boundary C results an evolution equation – cf. (18) – of the form: where ∂φ ∂t ei = − log P (∇3I | qi ; x) = � ... |

141 | Curve evolution implementation of the mumford-shah functional for image segmentation, denoising, interpolation, and magnification - TSAI, YEZZI, et al. - 2001 |

139 |
Hands: A Pattern Theoretic Study of Biological Shapes
- Grenander, Chow, et al.
- 1991
(Show Context)
Citation Context ...ty, the Bayesian formulation of the image segmentation problem allows to introduce higher-level prior knowledge about the shape of expected objects. This idea was pioneered by Grenander and coworkers =-=[37]-=-. In the following, we will briefly list some of the key contributions in the field of shape priors for level set segmentation. The first application of shape priors for level set segmentation was dev... |

130 | A shape-based approach to the segmentation of medical imagery using level sets
- Tsai, Jr, et al.
(Show Context)
Citation Context ...a regular grid to obtain a vector representation. A term is added to the contour evolution equation to drive the embedding function to the most likely shape of the estimated distribution. Tsai et al. =-=[86, 88]-=- proposed a very efficient implementation of shape-driven level set segmentation by directly optimizing in the linear subspace spanned by the principal components. A detailed analysis of various shape... |

116 |
A note on the gradient of a multi-image
- Zenzo
- 1986
(Show Context)
Citation Context ...ion σ: Jσ = Kσ ∗ (∇I∇I ⊤ ) = � Kσ ∗ I 2 x1 Kσ ∗ Ix1 Ix2 Kσ ∗ Ix1 Ix2 Kσ ∗ I 2 x2 15 � . (25) For color images, all channels can be taken into account by summing the tensors of the individual channels =-=[97]-=-. Despite its good properties for texture discrimination, the structure tensor is invariant to intensity changes. In order to segment images with and without texture, a feature vector including the sq... |

112 | A Statistical Approach to Snakes for Bimodal and Trimodal Imagery
- Yezzi, Tsai, et al.
- 1999
(Show Context)
Citation Context ...a function δ is implemented by a smooth approximation – cf. [12]. 3.3. Multiphase Level Set Formulation Several authors have proposed level set formulations which can handle a larger number of phases =-=[98, 96, 66, 8]-=-. These methods use a separate level set function for each region. This clearly increases the computational complexity. Moreover, numerical implementations are somewhat involved since the formation of... |

106 | Diffusion Snakes: Introducing Statistical Shape Knowledge into the Mumford-Shah Functional
- Cremers, Tischhäuser, et al.
- 2002
(Show Context)
Citation Context ... separate textures which 2 It should be pointed out that based on various heuristics, one can successfully incorporate regriding mechanisms and topological changes into explicit representations – cf. =-=[62, 48, 28, 25]-=-. 3 IJCV.tex; 19/09/2005; 19:10; p.3s4 have identical spatial characteristics but differ in their temporal dynamics. In Section 7 we detail how to integrate motion information as a criterion for segme... |

89 | Approximations of shape metrics and application to shape warping and empirical shape statistics
- Charpiat, Faugeras, et al.
- 2005
(Show Context)
Citation Context ...ectly optimizing in the linear subspace spanned by the principal components. A detailed analysis of various shape distances and statistical shape analysis in the level set formulation can be found in =-=[13]-=-. Figure 15 shows the effect of variation along the first principal component on the embedding function and the implicitly represented contour. The use of principal component analysis to model level s... |

88 | Kernel density estimation and intrinsic alignment for shape priors in level set segmentation
- Cremers, Osher, et al.
- 2006
(Show Context)
Citation Context ...he shape prior [78, 79] based on the assumption of a Gaussian distribution. The use of nonparametric density estimation to model larger classes of level set based shape distributions was developed in =-=[20, 74]-=-. This approach allows to model distributions of shape which are not Gaussian – such as the various views of a 3D object [19] or the silhouettes of a walking person [20]. Moreover, in the limit of lar... |

85 | Dynamical statistical shape priors for level set based tracking
- Cremers
- 2006
(Show Context)
Citation Context ...n and to induce a recognition-driven segmentation through the competition of shape priors was developed in [24]. Dynamical statistical shape priors for implicit shape representations were proposed in =-=[18]-=-. The latter approach takes into account that in the context of image sequence segmentation, the probability of a contour IJCV.tex; 19/09/2005; 19:10; p.28sFigure 16. Sample segmentations using statis... |

80 |
Geodesic Active Regions: A New Paradigm to Deal with Frame Partition Problems in Computer Vision
- Paragios, Deriche
- 2002
(Show Context)
Citation Context ...ined from a Minimum Description Length (MDL) criterion. The link with the Mumford-Shah functional and the equivalence to Bayesian maximum a posteriori (MAP) estimation is provided in [100]. Following =-=[66]-=-, an optimal partition P(Ω) of the image plane Ω (i.e. a partition of the image plane into pairwise disjoint regions) can be computed by maximizing the a posteriori probability p(P(Ω) | I) for a given... |

74 | Image Segmentation using Active Contours: Calculus of Variations or Shape Gradients
- Aubert, Barlaud, et al.
(Show Context)
Citation Context ...ations (pdes). Such pde-based segmentation methods became popular with the seminal paper on Snakes by Kass et al. [44]. In this paper, the contour is implemented by an explicit (parametric) curve C : =-=[0, 1]-=- → Ω which is evolved by locally minimizing the functional � E(C) = − |∇I(C)| 2 � ds + ν1 |Cs| 2 � ds + ν2 |Css| 2 ds, (1) where Cs and Css denote the first and second derivative with respect to the c... |

69 |
Deformable meshes with automated topology changes for coarse-to-fine 3D surface extraction
- Lachaud, Montanvert
- 1999
(Show Context)
Citation Context ... separate textures which 2 It should be pointed out that based on various heuristics, one can successfully incorporate regriding mechanisms and topological changes into explicit representations – cf. =-=[62, 48, 28, 25]-=-. 3 IJCV.tex; 19/09/2005; 19:10; p.3s4 have identical spatial characteristics but differ in their temporal dynamics. In Section 7 we detail how to integrate motion information as a criterion for segme... |