Point distribution model: Difference between revisions

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The '''Pointpoint Distributiondistribution Modelmodel''' is a model for representing the mean geometry of a shape and some statistical modes of geometric variation inferred from a training set of shapes.

==Background==

The point distribution model Itconcept has been developed by Cootes,<ref>{{citation

|author = T. F. Cootes

|title = Statistical models of appearance for computer vision

|year date=May 2004

|url=http://www.isbeface-rec.man.ac.ukorg/~bimalgorithms/ModelsAAM/app_models.pdf

|month = May

}}</ref> Taylor ''et al.''<ref name=taylor>{{citation

|url=http://www.isbe.man.ac.uk/~bim/Models/app_models.pdf

}}</ref> Taylor ''et al''<ref name=taylor>{{citation

|authors = D.H. Cooper, T.F. Cootes, C.J. Taylor and J. Graham

|title = Active shape models—their training and application

|journal = Computer Vision and Image Understanding

|number = 61

|pages = 38–59

|year = 1995

|authors author1= D.H. Cooper, |author2=T.F. Cootes, |author3=C.J. Taylor and |author4=J. Graham |issue = 61

}}</ref> and became a standard in [[computer vision]] for the [[statistical shape analysis|statistical study of shape]]<ref>{{citationcite conference

|title = Shape discrimination in the Hippocampus using an MDL Model

|authors = Rhodri H. Davies and Carole J. Twining and P. Daniel Allen and Tim F. Cootes and Chris J. Taylor

|year = 2003

|conference = IMPI

|url = http://www2.wiau.man.ac.uk/caws/Conferences/10/proceedings/8/papers/133/rhhd_ipmi03%2Epdf

|authorsauthor = Rhodri H. Davies and Carole J. Twining and P. Daniel Allen and Tim F. Cootes and Chris J. Taylor

}}</ref> and for segmentation of medical images<ref name=taylor/> where shape priors really help interpretation of noisy and low-contrasted pixels/voxels. The latter point leads to [[Active shape model]]s (ASM) and [[Active Appearance Model]]s (AAM).

|access-date = 2007-07-27

|archive-url = https://web.archive.org/web/20081008194350/http://www2.wiau.man.ac.uk/caws/Conferences/10/proceedings/8/papers/133/rhhd_ipmi03%2Epdf

|archive-date = 2008-10-08

|url-status = dead

}}</ref> and for [[image segmentation|segmentation]] of [[medical imaging|medical images]]<ref name=taylor/> where shape priors really help interpretation of noisy and low-contrasted pixels[[pixel]]s/voxels[[voxel]]s. The latter point leads to [[Activeactive shape model]]s (ASM) and [[Activeactive Appearanceappearance Modelmodel]]s (AAM).

Point Distributiondistribution Modelsmodels rely on [[Landmarklandmark point]]s. A landmark is an annotating point posed by an anatomist onto a given locus for every shape instance across the training set population. For instance, the same landmark will designate the tip of the [[index finger]] in a training set of 2D hands outlines. [[Principal Componentcomponent Analysisanalysis]] (PCA), for instance, is a relevant tool for studying correlations of movement between groups of landmarks among the training set population. Typically, it might detect that all the landmarks located along the same finger move exactly together across the training set examples showing different finger spacing for a flat-posed hands collection.

==Details==

The implementation of the procedure is roughly the following:

#First, Annotatea theset of training setimages are manually outlineslandmarked with enough corresponding landmarks to sufficiently approximate the geometry of the original shapes. These landmarks are aligned using the [[generalized procrustes analysis]], which minimizes the least squared error between the points.

# Align the clouds of landmark using the [[Generalized procrustes analysis]] (minimization of overall distance between landmarks of same label). The big idea is that shape information is not related to affine pose parameters, which need to be removed before any shape study. A mean shape can now be computed in averaging the aligned landmark positions.

# Now the shape outlines are reduced to sequences of n landmarks, we can see the training set as a 2n or 3n (2D/3D) space where any shape instance is a single dot. Assuming the scattering is gaussian in this space, PCA is supposedly the most straightforward tool to analyse the training set in this space

# PCA computes normalized eigenvectors and eigenvalues of the training set covariance matrix. Each eigenvector describe a principal mode of variation along the set, the corresponding eigenvalue indicating the importance of this mode in the shape space scattering. Since correlation was found between landmarks, the total variation of the space is concentrated on the very first eigenvectors, showing a very fast descent. Otherwise correlation was not found, suggesting the training set shows no variation or the landmarks are not properly posed.

<math>k</math> aligned landmarks in two dimensions are given as

An eigenvector, interpreted in euclidean space, can be seen as a sequence of n euclidean vectors associated to corresponding landmark and designating a compound move for the whole shape. Global nonlinear variation is usually well handled provided nonlinear variation is kept to a reasonable level. Typically, a twisting nematode worm is used as an example in the teaching of [[kernel PCA]]-based methods.

:<math>\mathbf{X} = (x_1, y_1, \ldots, x_k, y_k)</math>.

Due to the PCA properties: eigenvectors are mutually orthogonal, form a basis of the training set cloud in the shape space, and cross at the 0 in this space, which represents the mean shape. Also, PCA is a traditional way of fitting a closed ellipsoid to a Gaussian cloud of points (whatever their dimension): this suggests the concept of bounded variation.

It's important to note that each landmark <math>i \in \lbrace 1, \ldots k \rbrace </math> should represent the same anatomical ___location. For example, landmark #3, <math>(x_3, y_3)</math> might represent the tip of the ring finger across all training images.

The very big idea of PDM is that eigenvectors can be linearly combined to create an infinity of new shape instances that will 'look like' the one in the training set. The coefficients are bounded alike the values of the corresponding eigenvalues, so as to ensure the generated 2n/3n-dimensional dot will remain into the hyper-ellipsoïdal allowed ___domain—[[Allowable Shape Domain]] (ASD).<ref name=taylor/>

Now the shape outlines are reduced to sequences of <math>k</math> landmarks, so that a given training shape is defined as the vector <math>\mathbf{X} \in \mathbb{R}^{2k}</math>. Assuming the scattering is [[gaussian distribution|gaussian]] in this space, PCA is used to compute normalized [[eigenvectors]] and [[eigenvalues]] of the [[covariance matrix]] across all training shapes. The matrix of the top <math>d</math> eigenvectors is given as <math>\mathbf{P} \in \mathbb{R}^{2k \times d}</math>, and each eigenvector describes a principal mode of variation along the set.

Finally, a [[linear combination]] of the eigenvectors is used to define a new shape <math>\mathbf{X}'</math>, mathematically defined as:

:<math>\mathbf{X}' = \overline{\mathbf{X}} + \mathbf{P} \mathbf{b}</math>

where <math>\overline{\mathbf{X}}</math> is defined as the mean shape across all training images, and <math>\mathbf{b}</math> is a vector of scaling values for each principal component. Therefore, by modifying the variable <math>\mathbf{b}</math> an infinite number of shapes can be defined. To ensure that the new shapes are all within the variation seen in the training set, it is common to only allow each element of <math>\mathbf{b}</math> to be within <math>\pm</math>3 standard deviations, where the standard deviation of a given principal component is defined as the square root of its corresponding eigenvalue.

PDM's can be extended to any arbitrary number of dimensions, but are typically used in 2D image and 3D volume applications (where each landmark point is <math>\mathbb{R}^2</math> or <math>\mathbb{R}^3</math>).

==Discussion==

An eigenvector, interpreted in [[euclidean space]], can be seen as a sequence of n<math>k</math> euclidean vectors associated to corresponding landmark and designating a compound move for the whole shape. Global nonlinear variation is usually well handled provided nonlinear variation is kept to a reasonable level. Typically, a twisting [[nematode]] worm is used as an example in the teaching of [[kernel PCA]]-based methods.

Due to the PCA properties: eigenvectors are mutually [[orthogonal]], form a basis of the training set cloud in the shape space, and cross at the 0 in this space, which represents the mean shape. Also, PCA is a traditional way of fitting a closed ellipsoid to a Gaussian cloud of points (whatever their dimension): this suggests the concept of bounded variation.

The very big idea ofbehind PDMPDMs is that eigenvectors can be linearly combined to create an infinity of new shape instances that will 'look like' the one in the training set. The coefficients are bounded alike the values of the corresponding eigenvalues, so as to ensure the generated 2n/3n-dimensional dot will remain into the hyper-ellipsoïdalellipsoidal allowed ___domain—[[Allowableallowable Shapeshape Domain___domain]] (ASD).<ref name=taylor/>

==See also==

* [[Procrustes analysis]]

==References==

<references/>

<!-- Not referenced [4]{{citation

|authors = Stegmann, M. B. and Gomez, D. D.

|title = A Brief Introduction to Statistical Shape Analysis

|year = 2002

Line 45 ⟶ 68:

|url=http://www2.imm.dtu.dk/pubdb/views/publication_details.php?id=403

|quote=Images, annotations and data reports are placed in the enclosed zip-file.

|authors author1= Stegmann, M. B. and |author2=Gomez, D. D.

}} -->

|name-list-style=amp }} -->

==See also==

* [[Procrustes analysis]]

==External links==

* [https://web.archive.org/web/20080509041813/http://www.isbe.man.ac.uk/~bim/Models/index.html Flexible Models for Computer Vision], Tim Cootes, Manchester University.

* [http://www.icaen.uiowa.edu/~dip/LECTURE/Understanding3.html A practical introduction to PDM and ASMs].