## A Mixture Model for Representing Shape Variation (1997)

Venue: | Image and Vision Computing |

Citations: | 98 - 7 self |

### BibTeX

@INPROCEEDINGS{Taylor97amixture,

author = {Cootes And Taylor and T. F. Cootes and C. J. Taylor},

title = {A Mixture Model for Representing Shape Variation},

booktitle = {Image and Vision Computing},

year = {1997},

pages = {110--119},

publisher = {BMVA Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

The shape variation displayed by a class of objects can be represented as a probability density function, allowing us to determine plausible and implausible examples of the class. Given a training set of example shapes we can align them into a common co-ordinate frame and use kernel based density estimation techniques to represent this distribution. Such an estimate is complex and expensive, so we generate a simpler approximation using a mixture of gaussians. We show how to calculate the distribution, and how it can be used in image search to locate examples of the modelled object in new images.

### Citations

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(Show Context)
Citation Context ...les of the object in new images. This paper appears in: Image and Vision Computing 17, No.8, 1999, pp 567-574 1 A general approach is to use a density estimation technique such as the kernel method [=-=9]-=-. This represents the distribution as a sum of gaussians, one placed at every original data point. However, when there are many points this becomes far too expensive (in both time and memory). It is n... |

1046 |
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(Show Context)
Citation Context ... it can be used in image search to locate examples of the modelled object in new images. 1 Introduction Deformable models have proved effective for interpreting images of objects whose shape can vary =-=[2]-=-. Where a training set of example images is available, a successful approach is to build a statistical model of the shape variation seen in the training set. Such a model can be used for image search ... |

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Citation Context ...bution up to arbitrary accuracy, assuming sufficient components are used. The hope is that a small number of components will give a `good enough' estimate. The Expectation Maximisation (EM) algorithm =-=[8]-=- is the standard method of fitting such a mixture to a set of data. However, if we were to use as many components as samples (m = N ), the optimal fit of the standard EM algorithm is to have a delta f... |

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Citation Context ...mmon co-ordinate frame. 3.1 Aligning a Set of Shapes There is considerable literature on methods of aligning shapes into a common coordinate frame, the most popular approach being Procrustes Analysis =-=[4-=-]. This aligns each shape so that the sum of distances of each shape to the mean (D = P jx i xj 2 ) is minimised. It is poorly defined unless constraints are placed on the alignment of the mean (for ... |

59 | Active Shape Models: Evaluation of a Multi-resolution Method for Improving Image Search
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(Show Context)
Citation Context ...an be obtained by using a training set to build a statistical model to represent the image structure expected at each point. During search we use the model at each point to find the best nearby match =-=[3]-=-. In the second step we first update M to minimise the errors in the image plane, jX 0 M(x)j 2 . We then invert M to project X 0 into the model frame using x 0 = M 1 (X 0 ). If our model space is in t... |

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Citation Context ...at unconstrained search would lead to illegal parameter combinations. A common source of non-linearity is the rotation of sub-parts or of 3D objects viewed in 2D. Heap and Hogg [6] and Bowdenset. al. =-=[1]-=- describe applications where the training examples form a highly nonlinear space, which can be successfully represented by piece-wise linear sub-models. These are essentially the same as the approach ... |

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Citation Context ...y noisy or ambiguous that unconstrained search would lead to illegal parameter combinations. A common source of non-linearity is the rotation of sub-parts or of 3D objects viewed in 2D. Heap and Hogg =-=[6]-=- and Bowdenset. al. [1] describe applications where the training examples form a highly nonlinear space, which can be successfully represented by piece-wise linear sub-models. These are essentially th... |

27 | Non-linear point distribution modelling using a multi-layer perceptron
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Citation Context ... are changes in viewing position of a 3D object. There have been several non-linear extensions to the PDM, either using polynomial modes [11], using a multi-layer perceptron to perform non-linear PCA =-=[10]-=- or using polar co-ordinates for rotating sub-parts of the model [5]. However, all these approaches assume that varying the parameters b within given limits will always generate plausible shapes, and ... |

17 | Automated pivot location for the cartesian-polar hybrid point distribution model
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Citation Context ...ral non-linear extensions to the PDM, either using polynomial modes [11], using a multi-layer perceptron to perform non-linear PCA [10] or using polar co-ordinates for rotating sub-parts of the model =-=[5]-=-. However, all these approaches assume that varying the parameters b within given limits will always generate plausible shapes, and that all plausible shapes can be so generated. This is not always th... |

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Citation Context ...ameter space (assuming the rest to be independent normal), we may not need too many more training examples. We anticipate that this requirement can in part be addressed by automatic labelling schemes =-=[7]-=-, which, given a set of contours, choose the optimal positions of the landmarks required for the shape model. Using a mixture model representation of the p.d.f.introduces more constraints on valid par... |

3 |
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Citation Context ...ch as those generated when parts of the object rotate, or there are changes in viewing position of a 3D object. There have been several non-linear extensions to the PDM, either using polynomial modes =-=[11]-=-, using a multi-layer perceptron to perform non-linear PCA [10] or using polar co-ordinates for rotating sub-parts of the model [5]. However, all these approaches assume that varying the parameters b ... |