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## Kernel Density Estimation and Intrinsic Alignment for Knowledge-driven Segmentation: Teaching Level Sets to Walk (2004)

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Venue: | International Journal of Computer Vision |

Citations: | 113 - 16 self |

### Citations

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Citation Context ...dic process. – Kernel density estimators were shown to converge to the true distribution in the limit of infinite (independent and identically distributed) training samples (Devroye and Györfi, 1985; =-=Silverman, 1992-=-). In the context of shape representations, this implies that our approach is capable of accurately representing arbitrarily complex shape deformations. – By not imposing a linear subspace, we circumv... |

1407 | Geodesic active contours
- Caselles, Kimmel, et al.
- 1997
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Citation Context ...vel set method [16, 10] has become a popular framework for image segmentation. The level set framework has been applied to segment images based on numerous low-level criteria such as edge consistency =-=[13, 2, 11]-=-, intensity homogeneity [3, 22], texture information [17, 1] and motion information [6]. More recently, it was proposed to integrate prior knowledge about the shape of expected objects into the level ... |

1273 | Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems
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(Show Context)
Citation Context ...which makes it well-suited for the segmentation of multiple or multiply-connected objects. In the present paper, we use a level set formulation of the piecewise constant Mumford-Shah functional, c.f. =-=[15, 22, 3]-=-. In particular, a two-phase segmentation of an image I : Ω → R can be generated by minimizing the functional [3]: � Ecv(φ) = (I −u+) 2 � Hφ(x)dx + (I −u−) 2� 1−Hφ(x) � � dx + ν |∇Hφ|dx, (2) Ω Ω with ... |

1180 |
On estimation of a probability density function and mode
- Parzen
- 1962
(Show Context)
Citation Context ..., 20, 8, 9, 4]. Building up on these developments, we propose in this paper two contributions. Firstly, we introduce a statistical shape prior which is based on the classical kernel density estimator =-=[19, 18]-=- extended to the level set domain. In contrast to existing approaches of shape priors in level set segmentation, this prior allows to well approximate arbitrary distributions of shapes. Secondly, we p... |

1179 | Active contours without edges
- Chan, Vese
(Show Context)
Citation Context ...a popular framework for image segmentation. The level set framework has been applied to segment images based on numerous low-level criteria such as edge consistency [13, 2, 11], intensity homogeneity =-=[3, 22]-=-, texture information [17, 1] and motion information [6]. More recently, it was proposed to integrate prior knowledge about the shape of expected objects into the level set framework [12, 21, 5, 20, 8... |

1165 | Fronts Propagating with Curvature Dependent Speed. Algorithms Based on Hamilton-Jacobi Formulations
- Osher, Sethian
- 1988
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Citation Context ...as been devoted to imitating such an integration of prior knowledge into machinevision problems, in particular in the context of image segmentation. Among variational approaches, the level set method =-=[16, 10]-=- has become a popular framework for image segmentation. The level set framework has been applied to segment images based on numerous low-level criteria such as edge consistency [13, 2, 11], intensity ... |

1065 |
Level Set Methods and Dynamic Implicit Surfaces
- Osher, Fedkiw
(Show Context)
Citation Context ...in the analysis of random shapes by Fréchet (1961) and in the school of mathematical morphology founded by Matheron (1975). Osher and Sethian introduced the level set method (Osher and Sethian, 1988; =-=Osher and Fedkiw, 2002-=-; Osher and Paragios, 2003) as a means of propagating contours (independent of parameterization) by evolving associated embedding functions via partial differential equations. For a precursor containi... |

798 | Shape modeling with front propagation: A level set approach
- Malladi, Sethian, et al.
- 1995
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Citation Context ...vel set method [16, 10] has become a popular framework for image segmentation. The level set framework has been applied to segment images based on numerous low-level criteria such as edge consistency =-=[13, 2, 11]-=-, intensity homogeneity [3, 22], texture information [17, 1] and motion information [6]. More recently, it was proposed to integrate prior knowledge about the shape of expected objects into the level ... |

770 | Statistical Shape Analysis - Dryden, Mardia - 1998 |

635 | Random sets and integral geometry - Matheron - 1975 |

614 |
Active shape models-their training and application
- Cootes, Taylor, et al.
- 1995
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Citation Context ...ity measures amounts to a problem of density estimation. In the case of explicitly represented boundaries, this has been addressed by modeling the space of familiar shapes by linear subspaces (PCA) ( =-=Cootes et al., 1995-=-) and the related Gaussian distribution (Cremers et al., 2002), by mixture models (Cootes and Taylor, 1999) or nonlinear (multi-modal) representations via simple models in appropriate feature spaces (... |

575 | A level set approach for computing solutions to incompressible two-phase flow - Sussman, Smerka, et al. - 1994 |

502 |
Remarks on some nonparametric estimates of a density function
- Rosenblatt
- 1997
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Citation Context ..., 20, 8, 9, 4]. Building up on these developments, we propose in this paper two contributions. Firstly, we introduce a statistical shape prior which is based on the classical kernel density estimator =-=[19, 18]-=- extended to the level set domain. In contrast to existing approaches of shape priors in level set segmentation, this prior allows to well approximate arbitrary distributions of shapes. Secondly, we p... |

392 | Statistical Shape Influence in Geodesic Active Contours
- Leventon, Grimson, et al.
- 2000
(Show Context)
Citation Context ...mogeneity [3, 22], texture information [17, 1] and motion information [6]. More recently, it was proposed to integrate prior knowledge about the shape of expected objects into the level set framework =-=[12, 21, 5, 20, 8, 9, 4]-=-. Building up on these developments, we propose in this paper two contributions. Firstly, we introduce a statistical shape prior which is based on the classical kernel density estimator [19, 18] exten... |

304 | Geodesic active regions and level set methods for supervised texture segmentation
- Paragios, Deriche
- 2002
(Show Context)
Citation Context ... segmentation. The level set framework has been applied to segment images based on numerous low-level criteria such as edge consistency [13, 2, 11], intensity homogeneity [3, 22], texture information =-=[17, 1]-=- and motion information [6]. More recently, it was proposed to integrate prior knowledge about the shape of expected objects into the level set framework [12, 21, 5, 20, 8, 9, 4]. Building up on these... |

235 | Gradient flows and geometric active contour models
- Kichenassamy, Kumar, et al.
- 1995
(Show Context)
Citation Context ...vel set method [16, 10] has become a popular framework for image segmentation. The level set framework has been applied to segment images based on numerous low-level criteria such as edge consistency =-=[13, 2, 11]-=-, intensity homogeneity [3, 22], texture information [17, 1] and motion information [6]. More recently, it was proposed to integrate prior knowledge about the shape of expected objects into the level ... |

202 | Shape priors for level set representations
- Rousson, Paragios
- 2002
(Show Context)
Citation Context ...mogeneity [3, 22], texture information [17, 1] and motion information [6]. More recently, it was proposed to integrate prior knowledge about the shape of expected objects into the level set framework =-=[12, 21, 5, 20, 8, 9, 4]-=-. Building up on these developments, we propose in this paper two contributions. Firstly, we introduce a statistical shape prior which is based on the classical kernel density estimator [19, 18] exten... |

175 |
Curve evolution implementation of the Mumford-Shah functionalal for image segmentation, denoising, dnterpolation, and magnification
- Tsai, Yezzi, et al.
(Show Context)
Citation Context ...a popular framework for image segmentation. The level set framework has been applied to segment images based on numerous low-level criteria such as edge consistency [13, 2, 11], intensity homogeneity =-=[3, 22]-=-, texture information [17, 1] and motion information [6]. More recently, it was proposed to integrate prior knowledge about the shape of expected objects into the level set framework [12, 21, 5, 20, 8... |

168 | Analysis of planar shapes using geodesic paths on shape spaces
- Klassen, Srivastava, et al.
- 2004
(Show Context)
Citation Context .... (1988) and Gdalyahu and Weinshall (1999)). Yet, factoring out the reparameterization group and identifying an initial point correspondence (when matching shapes) are numerically involved processes (=-=Klassen et al., 2004-=-), especially when generalizing to higher dimensions (surface matching). Moreover, extensions of explicit representations to model multiply-connected objects are not straightforward. Finally, the noti... |

147 | HANDS. A Pattern Theoretical Study of Biological Shapes - Grenander, Chow, et al. - 1991 |

145 |
Nonparametric Density Estimation: the L1 View
- Devroye, Györfi
- 1985
(Show Context)
Citation Context ... from an essentially periodic process. – Kernel density estimators were shown to converge to the true distribution in the limit of infinite (independent and identically distributed) training samples (=-=Devroye and Györfi, 1985-=-; Silverman, 1992). In the context of shape representations, this implies that our approach is capable of accurately representing arbitrarily complex shape deformations. – By not imposing a linear sub... |

144 | Computable elastic distances between shapes - Younes |

130 | Flexible syntactic matching of curves and its application to automatic hierarchical classification of silhouettes - Gdalyahu, Weinshall - 1999 |

129 | Diffusion snakes: Introducing statistical shape knowledge into the Mumford-Shah functional
- Cremers, Tischhäuser, et al.
(Show Context)
Citation Context ...the case of explicitly represented boundaries, this has been addressed by modeling the space of familiar shapes by linear subspaces (PCA) ( Cootes et al., 1995) and the related Gaussian distribution (=-=Cremers et al., 2002-=-), by mixture models (Cootes and Taylor, 1999) or nonlinear (multi-modal) representations via simple models in appropriate feature spaces (Cremers et al., 2003). For level set based shape representati... |

121 | A mixture model for representing shape variation
- Cootes, Taylor
- 1999
(Show Context)
Citation Context ...ies, this has been addressed by modeling the space of familiar shapes by linear subspaces (PCA) ( Cootes et al., 1995) and the related Gaussian distribution (Cremers et al., 2002), by mixture models (=-=Cootes and Taylor, 1999-=-) or nonlinear (multi-modal) representations via simple models in appropriate feature spaces (Cremers et al., 2003). For level set based shape representations, it was suggested (Leventon et al., 2000;... |

121 | Diffeomorphism groups and pattern matching in image analysis - Trouve - 1998 |

120 | Deformotion: Deforming motion, shape average and the joint registration and approximation of structures in images
- Yezzi, Soatto
- 2003
(Show Context)
Citation Context ...of Kernel Density Estimation and Intrinsic Alignment 337 explicit parameters and numerically optimizing the segmentation functional by gradient descent (Chen et al., 2002; Rousson and Paragios, 2002; =-=Yezzi and Soatto, 2003-=-). This iterative optimization not only requires a delicate tuning of associated gradient descent time step sizes (in order to guarantee a stable evolution). It is also not clear in what order and how... |

118 | Determining the similarity of deformable shapes
- Basri, Costa, et al.
- 1998
(Show Context)
Citation Context ...ly based on an explicit representation of shape. In contrast to implicit representations, these allow to easily define correspondence of parts and the notions of contour shrinking and stretching (cf. =-=[1, 27]-=-). Yet, factoring out the reparameterization group and identifying an initial point correspondence are numerically involved processes [33], especially when generalizing to higher dimensions (surface m... |

101 | Dynamical statistical shape priors for level set based tracking - Cremers - 2006 |

94 | Shape statistics in kernel space for variational image segmentation
- Cremers, Kohlberger, et al.
(Show Context)
Citation Context ...shape priors which are commonly based on the assumption of a Gaussian distribution (cf. [12]), the distribution in (5) is a multimodal one (thereby allowing more complex training shapes). We refer to =-=[7]-=- for an alternative multi-modal prior for spline-based shape representations.s4 Daniel Cremers, Stanley J. Osher, and Stefano Soatto 4 Translation Invariance by Intrinsic Alignment By construction the... |

91 | Approximations of shape metrics and application to shape warping and empirical shape statistics - Charpiat, Faugeras, et al. - 2004 |

87 |
On the choice of smoothing parameter for Parzen estimators of probability density functions
- Duin
- 1976
(Show Context)
Citation Context ...l width σ , based on asymptotic expansions such as the parametric method (Deheuvels, 1977), heuristic estimates (Wagner, 1975; Silverman, 1978) or maximum likelihood optimization by cross validation (=-=Duin, 1976-=-; Chow et al., 1983). We refer to Devroye and Györfi (1985); Silverman (1992) for a detailed discussion. For this work, we simply fix σ 2 to be the mean squared nearest-neighbor distance: σ 2 = 1 N N�... |

77 |
A Willsky. Model-based curve evolution technique for image segmentation
- Yezzi, Wells, et al.
- 2001
(Show Context)
Citation Context ...mogeneity [3, 22], texture information [17, 1] and motion information [6]. More recently, it was proposed to integrate prior knowledge about the shape of expected objects into the level set framework =-=[12, 21, 5, 20, 8, 9, 4]-=-. Building up on these developments, we propose in this paper two contributions. Firstly, we introduce a statistical shape prior which is based on the classical kernel density estimator [19, 18] exten... |

70 | R.: Active unsupervised texture segmentation on a diffusion based feature space - Rousson, Brox, et al. - 2003 |

68 | The diffusion of shape - Kendall - 1977 |

50 | Level set based shape prior segmentation
- Chan, Zhu
(Show Context)
Citation Context |

47 | Motion Competition. A variational framework for piecewise parametric motion segmentation - Cremers, Soatto - 2005 |

47 | Particle filtering for geometric active contours and application to tracking deforming objects - Rathi, Vaswani, et al. - 2005 |

46 | S.: A pseudo-distance for shape priors in level set segmentation
- Cremers, Soatto
- 2003
(Show Context)
Citation Context |

44 | Efficient kernel density estimation of shape and intensity priors for level set segmentation - Rousson, Cremers - 2005 |

33 | Multiphase dynamic labeling for variational recognition-driven image segmentation
- Cremers, Sochen, et al.
- 2004
(Show Context)
Citation Context |

32 | A multiphase dynamic labeling model for variational recognition-driven image segmentation
- Cremers, Sochen, et al.
- 2005
(Show Context)
Citation Context ...egions (Cremers, Sochen and Schnörr, 2003; Chan and Zhu, 2003), or to impose multiple competing shape priors so as to simultaneously reconstruct several independent objects in a given image sequence (=-=Cremers et al., 2006-=-). The above approaches allow to improve the level set based segmentation of corrupted images of familiar objects. Yet, existing methods to impose statistical shape information on the evolving embeddi... |

32 | Natural image statistics for natural image segmentation - Heiler, Schnorr |

32 | Implicit active shape models for 3D segmentation - Rousson, Paragios, et al. - 2004 |

31 | Unlevel-sets: Geometry and prior-based segmentation - Riklin-Raviv, Kiryati, et al. - 2004 |

29 | A TV flow based local scale measure for texture discrimination
- Brox, Weickert
- 2004
(Show Context)
Citation Context ... segmentation. The level set framework has been applied to segment images based on numerous low-level criteria such as edge consistency [13, 2, 11], intensity homogeneity [3, 22], texture information =-=[17, 1]-=- and motion information [6]. More recently, it was proposed to integrate prior knowledge about the shape of expected objects into the level set framework [12, 21, 5, 20, 8, 9, 4]. Building up on these... |

28 | C.: Towards recognition-based variational segmentation using shape priors and dynamic labeling
- Cremers, Sochen, et al.
- 2003
(Show Context)
Citation Context ...) and the related Gaussian distribution (Cremers et al., 2002), by mixture models (Cootes and Taylor, 1999) or nonlinear (multi-modal) representations via simple models in appropriate feature spaces (=-=Cremers et al., 2003-=-). For level set based shape representations, it was suggested (Leventon et al., 2000; Tsai et al., 2001; Rousson et al., 2004) to fit a linear sub-space to the samKernel Density Estimation and Intrin... |

27 | A variational framework for image segmentation combining motion estimation and shape regularization
- Cremers
- 2003
(Show Context)
Citation Context ...amework has been applied to segment images based on numerous low-level criteria such as edge consistency [13, 2, 11], intensity homogeneity [3, 22], texture information [17, 1] and motion information =-=[6]-=-. More recently, it was proposed to integrate prior knowledge about the shape of expected objects into the level set framework [12, 21, 5, 20, 8, 9, 4]. Building up on these developments, we propose i... |

27 |
Estimation non paramétrique de la densité par histogrammes généralisés
- Deheuvels
- 1977
(Show Context)
Citation Context ...e spanned by the training data (Rousson and Cremers, 2005). There exist extensive studies on how to optimally choose the kernel width σ , based on asymptotic expansions such as the parametric method (=-=Deheuvels, 1977-=-), heuristic estimates (Wagner, 1975; Silverman, 1978) or maximum likelihood optimization by cross validation (Duin, 1976; Chow et al., 1983). We refer to Devroye and Györfi (1985); Silverman (1992) f... |

21 |
Consistent cross-validated density estimation
- Chow, Geman, et al.
- 1987
(Show Context)
Citation Context ...based on asymptotic expansions such as the parametric method (Deheuvels, 1977), heuristic estimates (Wagner, 1975; Silverman, 1978) or maximum likelihood optimization by cross validation (Duin, 1976; =-=Chow et al., 1983-=-). We refer to Devroye and Györfi (1985); Silverman (1992) for a detailed discussion. For this work, we simply fix σ 2 to be the mean squared nearest-neighbor distance: σ 2 = 1 N N� i=1 min j�=i d2 (H... |

19 |
Nonparametric estimates of probability densities
- Wagner
- 1975
(Show Context)
Citation Context ... and Cremers, 2005). There exist extensive studies on how to optimally choose the kernel width σ , based on asymptotic expansions such as the parametric method (Deheuvels, 1977), heuristic estimates (=-=Wagner, 1975-=-; Silverman, 1978) or maximum likelihood optimization by cross validation (Duin, 1976; Chow et al., 1983). We refer to Devroye and Györfi (1985); Silverman (1992) for a detailed discussion. For this w... |

14 | How to deal with point correspondences and tangential velocities in the level set framework
- Pons, Hermosillo, et al.
- 2003
(Show Context)
Citation Context ...xplicit representations to model multiply-connected objects are not straightforward. Finally, the notion of point-wise correspondence can be introduced into implicit boundary representations as well (=-=Pons et al., 2003-=-). In this work, we therefore adopt the implicit shape representation given by the level set framework. 1.2. Prior Shape Knowledge in Level Set Segmentation Among variational approaches, the level set... |

13 |
A finite element method for the simulation of Raleigh–Taylor instability
- Dervieux, Thomasset
- 1980
(Show Context)
Citation Context ...as been devoted to imitating such an integration of prior knowledge into machinevision problems, in particular in the context of image segmentation. Among variational approaches, the level set method =-=[16, 10]-=- has become a popular framework for image segmentation. The level set framework has been applied to segment images based on numerous low-level criteria such as edge consistency [13, 2, 11], intensity ... |

11 |
Using Shape Priors in Geometric Active Contours in a Variational Framework
- Chen, Tagare, et al.
- 2002
(Show Context)
Citation Context |

9 | Tracking objects with the chan-vese algorithm
- Moelich, Chan
- 2003
(Show Context)
Citation Context ...erating the evolution (3) 100 times for each frame (using the previous result as initialization). For a similar application of the Chan-Vese functional (without statistical shape priors), we refer to =-=[14]-=-. The set of sample frames in Figure 2 clearly demonstrates that this purely image-driven segmentation scheme is not capable of separating the object of interest from the occluding bar and similarly s... |

9 | A priori information in image segmentation: Energy functional based on shape statistical model and image information
- Bresson, Vandergheynst, et al.
- 2003
(Show Context)
Citation Context ...also takes into account shape discrepancies within the second shape. It gives a more informative measure of the shape dissimilarity and therefore allows for more powerful shape priors. Alternatively (=-=Bresson et al., 2003-=-) one can constrain the integration in (4) to the contour C1 represented by φ1 (i.e. to the area where φ = 0): d 2 2 (φ1,φ2) � = φ C1 2 2 dC1 � = φ � 2 2 (x) δ(φ1)|∇φ1| dx. (7) Due to the definition o... |

7 | 2004b, ‘Learning and Bayesian Shape Extraction for Object Recognition - Mio, Srivastava, et al. |

5 |
The Measurement of Biological Shape and Shape Changes, volume 24
- Bookstein
(Show Context)
Citation Context ...rtain transformation groups. Based on the concept of landmarks (associated with a specific parameterization), a statistical analysis of shape deformations has been developed among others by Bookstein =-=[2]-=-, Cootes et al. [11] and Cremers et al. [13]. We refer to the book by Dryden and Mardia [23] for an overview. A mathematical representation of shape which is independent of parameterization was pionee... |

5 | Les courbes aléatoires - Fréchet - 1961 |

1 | volume 2351 of Lect - Vis |