## Mean shift: A robust approach toward feature space analysis (2002)

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Venue: | IEEE Transactions on Pattern Analysis and Machine Intelligence |

Citations: | 1607 - 34 self |

### BibTeX

@ARTICLE{Comaniciu02meanshift:,

author = {Dorin Comaniciu and Peter Meer},

title = {Mean shift: A robust approach toward feature space analysis},

journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},

year = {2002},

volume = {24},

pages = {603--619}

}

### Years of Citing Articles

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### Abstract

Abstract A general nonparametric technique is proposed for the analysis of a complex multimodal featurespace and to delineate arbitrarily shaped clusters in it. The basic computational module of the technique is an old pattern recognition procedure, the mean shift. We prove for discrete data the convergence ofa recursive mean shift procedure to the nearest stationary point of the underlying density function and thus its utility in detecting the modes of the density. The equivalence of the mean shift procedureto the Nadaraya-Watson estimator from kernel regression and the robust M-estimators of location is also established. Algorithms for two low-level vision tasks, discontinuity preserving smoothing andimage segmentation are described as applications. In these algorithms the only user set parameter is the resolution of the analysis, and either gray level or color images are accepted as input. Extensiveexperimental results illustrate their excellent performance.

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Citation Context ... an objective function that expresses the quality of the decomposition (i.e.,the index of cluster validity). The objective function typically compares the inter- versus intra-cluster variability [30],=-=[28]-=- or evaluates the isolation and connectivity of the delineated clusters [43]. . Finally,since in most of the cases the decomposition is task dependent,top-down information provided by the user or by a... |

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Citation Context ...mizes an objective function that expresses the quality of the decomposition (i.e.,the index of cluster validity). The objective function typically compares the inter- versus intra-cluster variability =-=[30]-=-,[28] or evaluates the isolation and connectivity of the delineated clusters [43]. . Finally,since in most of the cases the decomposition is task dependent,top-down information provided by the user or... |

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Citation Context ...eature space [25, 60, 63]. Our approach to mode detection and clustering is based on the mean shift procedure, proposed in 1975 by Fukunaga and Hostetler [21] and largely forgotten till Cheng’s paper =-=[7]-=- rekindled the interest in it. In spite of its excellent qualities, the mean shift procedure does not seem to be known in the statistical literature. While the book [54, Sec.6.2.2] discusses [21], the... |

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Citation Context ...ineated based on the local structure of the feature space [25],[60],[63]. Our approach to mode detection and clustering is based on the mean shift procedure,proposed in 1975 by Fukunaga and Hostetler =-=[21]-=- and largely forgotten until Cheng's paper [7] rekindled interest in it. In spite of its excellent qualities,the mean shift procedure does not seem to be known in statistical literature. While the boo... |

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Citation Context ...stopping criterion and after a sufficiently large number of iterations, the processed image collapses into a flat surface. The connection between anisotropic diffusion and M-estimators is analyzed in =-=[5]-=-. A recently proposed noniterative discontinuity preserving smoothing technique is the bilateral filtering [59]. The relation between bilateral filtering and diffusion based techniques was analyzed in... |

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Citation Context ...s on the Laplacian of the unknown density being estimated,and its performance is not well understood [62,p. 108]. For the univariate case,a reliable method for bandwidth selection is the plug-in rule =-=[53]-=-,which was proven to be superior to leastsquares cross-validation and biased cross-validation [42],[55,p. 46]. Its only assumption is the smoothness of the underlying density. . The second bandwidth s... |

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Citation Context ...y a linear mapping property [65,p.166]. Our first image segmentation algorithm was a straightforward application of the feature space analysis technique to an L*u*v* representation of the color image =-=[11]-=-. The modularity of the segmentation algorithm enabled its integration by other groups to a large variety of applications like image retrieval [1],face tracking [6],object-based video coding for MPEG-... |

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Citation Context ...Psi j=1;2::: and n ^fh;K(j)o j=1;2::: converge, and n ^fh;K(j)oj=1;2::: is also monotonically increasing. The proof is given in the Appendix. The theorem generalizes the result derived differently in =-=[13]-=-, where K was the Epanechnikov kernel, and G the uniform kernel. The theorem remains valid when each data point xi is associated with a nonnegative weight wi. An example of nonconvergence when the ker... |

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Citation Context ...nally, since in most of the cases the decomposition is task dependent, top-down information provided by the user or by an upper-level module can be used to control the kernel bandwidth. We present in =-=[15]-=- a detailed analysis of the bandwidth selection problem. To solve the difficulties generated by the narrow peaks and the tails of the underlying density, two locally adaptive solutions are proposed. O... |

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Citation Context ...ature,we will mention only some whose basic processing relies on the joint domain. In each case,a vector field is defined over the sampling lattice of the image. The attraction force field defined in =-=[57]-=- is computed at each pixel as a vector sum of pairwise affinities between the current pixel and all other pixels,with similarity measured in both spatial and range domains. The region boundaries are t... |

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Citation Context ...ks retained for the final classification are marked with red dots. 14can be reliably supported by input domain information. The multimodal feature space analysis technique was discussed in detail in =-=[12]-=-. It was shown experimentally that for a synthetic, bimodal normal distribution the technique achieves a classification error similar to the optimal Bayesian classifier. The behavior of this feature s... |

64 | Edge Flow: A Framework of Boundary Detection and Image Segmentation
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Citation Context ...tor computed at that pixel and projected into the spatial domain. However,in contrast to [57],the mean shift procedure moves in the direction of this vector,away from the boundaries. The edge flow in =-=[34]-=- is obtained at each location for a given set of directions as the magnitude of the gradient of a smoothed image. The boundaries are detected at image locations which encounter two opposite directions... |

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Citation Context ...ch are based on in situ optimization. Under this paradigm the solution is obtained by using the input domain to define the optimization problem. The in situ optimization is a very powerful method. In =-=[23]-=- and [58] each input data point was associated with a local field (voting kernel) to produce a more dense structure from where the sought information (salient features, the hyperplane representing the... |

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Citation Context ...rs arising from the dominant colors, and a decomposition of the space into elliptical tiles will introduce severe artifacts. Enforcing a Gaussian mixture model over such data is doomed to fail, e.g., =-=[49]-=-, and even the use of a robust approach with contaminated Gaussian densities [67] cannot be satisfactory for such complex cases. Note also that the mixture models require the number of clusters as a p... |

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Citation Context ... representation of the color image [11]. The modularity of the segmentation algorithm enabled its integration by other groups to a large variety of applications like image retrieval [1],face tracking =-=[6]-=-,object-based video coding for MPEG-4 [22],shapedetectionandrecognition[33],andtextureanalysis [47],to mention only a few. However,since the feature space analysis can be applied unchanged to moderate... |

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Citation Context ...67] cannot be satisfactory for such complex cases. Note also that the mixture models require the number of clusters as a parameter,which raises its own challenges. For example,the method described in =-=[45]-=- proposes several different ways to determine this number. Arbitrarily structured feature spaces can be analyzed only by nonparametric methods since these methods do not have embedded assumptions. Num... |

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Citation Context ... [62, p.108]. For the univariate case a reliable method for bandwidth selection is the plug-in rule [53], which was proven to be superior to least squares cross validation and biased cross-validation =-=[42]-=-, [55, p.46]. Its only assumption is the smoothness of the underlying density. ffl The second bandwidth selection technique is related to the stability of the decomposition. The bandwidth is taken as ... |

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Citation Context ...never the feature space has more than (say) six dimensions,the analysis should be approached carefully. Employing projection pursuit,in which the density is analyzed along lower dimensional cuts,e.g.,=-=[27]-=-,is a possibility. To conclude,the mean shift procedure is a valuable computational module whose versatility can make it an important component of any computer vision toolbox. APPENDIX Proof of Theore... |

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Citation Context ...n algorithm enabled its integration by other groups to a large variety of applications like image retrieval [1],face tracking [6],object-based video coding for MPEG-4 [22],shapedetectionandrecognition=-=[33]-=-,andtextureanalysis [47],to mention only a few. However,since the feature space analysis can be applied unchanged to moderately higher dimensional spaces (see Section 5),we subsequently also incorpora... |

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Citation Context ...than practical importance. 2.6 Relation to Location M-estimators The M-estimators are a family of robust techniques which can handle data in the presence of severe contaminations, i.e., outliers. See =-=[26]-=-, [32] for introductory surveys. In our context only the problem of location estimation has to be considered. 11Given the data and the scale , will define , the location estimator as (30) where, is a... |

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Citation Context ...with the requirements to be satisfied by the objective function . The relation between location M-estimators and kernel density estimation is not well investigated in the statistical literature, only =-=[9]-=- discusses it in the context of an edge preserving smoothing technique. 3 Robust Analysis of Feature Spaces Multimodality and arbitrarily shaped clusters are the defining properties of a real feature ... |

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Citation Context ...ractical importance. 2.6 Relation to Location M-estimators The M-estimators are a family of robust techniques which can handle data in the presence of severe contaminations, i.e., outliers. See [26], =-=[32]-=- for introductory surveys. In our context only the problem of location estimation has to be considered. 11Given the data and the scale , will define , the location estimator as (30) where, is a symme... |

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Citation Context ...lat surface. The connection between anisotropic diffusion and M-estimators is analyzed in [5]. A recently proposed noniterative discontinuity preserving smoothing technique is the bilateral filtering =-=[59]-=-. The relation between bilateral filtering and diffusion-based techniques was analyzed in [3]. The bilateral filters also work in the joint spatial-range domain. The data is independently weighted in ... |

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Citation Context ...4. Optional: Eliminate spatial regions containing less than M pixels. The cluster delineation step can be refined according to a priori information and,thus,physics-based segmentation algorithms,e.g.,=-=[2]-=-,[35],can be incorporated. Since this process is performed on region adjacency graphs,hierarchical techniques like [36] can provide significant speed-up. The effect of the cluster delineation step is ... |

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Citation Context ... the space. The problem of color representation will be discussed in Section 4,but the employed parameterization has to be carefully examined even in a simple case like the Hough space of lines,e.g., =-=[48]-=-,[61]. The presence of a Mahalanobis metric can be accommodated by an adequate choice of the bandwidth matrix (2). In practice,however,it is preferable to have assured that the metric of the feature s... |

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Citation Context ... A recently proposed noniterative discontinuity preserving smoothing technique is the bilateral filtering [59]. The relation between bilateral filtering and diffusion-based techniques was analyzed in =-=[3]-=-. The bilateral filters also work in the joint spatial-range domain. The data is independently weighted in the two domains and the center pixel is computed as the weighted average of the window. The f... |

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Citation Context ...to be known in the statistical literature. While the book [54, Sec.6.2.2] discusses [21], the advantages of employing a mean shift type procedure in density estimation were only recently rediscovered =-=[8]-=-. As will be proven in the sequel a computational module based on the mean shift procedure is an extremely versatile tool for feature space analysis and can provide reliable solutions for many vision ... |

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Citation Context ... of approaches to achieve this goal,from adaptive Wiener filtering [31],to implementing isotropic [50] and anisotropic [44] local diffusion processes, a topic which recently received renewed interest =-=[19]-=-,[37], [56]. The diffusion-based techniques,however,do not have a straightforward stopping criterion and,after a sufficiently large number of iterations,the processed image collapses into a flat surfa... |

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Citation Context ...ma of the p.d.f., that is,to the modes of the unknown density. Once the location of a mode is determined,the cluster associated with it is delineated based on the local structure of the feature space =-=[25]-=-,[60],[63]. Our approach to mode detection and clustering is based on the mean shift procedure,proposed in 1975 by Fukunaga and Hostetler [21] and largely forgotten until Cheng's paper [7] rekindled i... |

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Citation Context ...ptional: Eliminate spatial regions containing less than M pixels. The cluster delineation step can be refined according to a priori information and,thus,physics-based segmentation algorithms,e.g.,[2],=-=[35]-=-,can be incorporated. Since this process is performed on region adjacency graphs,hierarchical techniques like [36] can provide significant speed-up. The effect of the cluster delineation step is shown... |

12 |
Finding Groups in Data: An Introduction to Cluster Analysis
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Citation Context ...izes an objective function that expresses the quality of the decomposition (i.e., the index of cluster validity). The objective function typically compares the inter- versus intra-cluster variability =-=[30, 28]-=- or evaluates the isolation and connectivity of the delineated clusters [43]. 15Finally, since in most of the cases the decomposition is task dependent, top-down information provided by the user or b... |

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Citation Context ...pproaches to achieve this goal,from adaptive Wiener filtering [31],to implementing isotropic [50] and anisotropic [44] local diffusion processes, a topic which recently received renewed interest [19],=-=[37]-=-, [56]. The diffusion-based techniques,however,do not have a straightforward stopping criterion and,after a sufficiently large number of iterations,the processed image collapses into a flat surface. T... |

9 | Nonparametric robust methods for computer vision - Comaniciu - 2001 |

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6 |
ªPfinder: Real-time Tracking of the Human Body,º
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(Show Context)
Citation Context ...4c. The segmentation step does not add a significant overhead to the filtering process. The region representation used by the mean shift segmentation is similar to the blob representation employed in =-=[64]-=-. However,while the blob has a parametric description (multivariate Gaussians in both spatial and color domain),the partition generated by the mean shift is characterized by a nonparametric model. An ... |

5 |
Finding Salient Regions
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(Show Context)
Citation Context ...,the index of cluster validity). The objective function typically compares the inter- versus intra-cluster variability [30],[28] or evaluates the isolation and connectivity of the delineated clusters =-=[43]-=-. . Finally,since in most of the cases the decomposition is task dependent,top-down information provided by the user or by an upper-level module can be used to control the kernel bandwidth. We present... |