Point distribution model: Difference between revisions

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The Point'''point 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. It has been developped by Cootes, Taylor et al [1][3] and became a standard in computer vision for statistical study of shape [2] and for segmentation of medical images [1] 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).

==Background==

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

Point Distribution Models rely on [[Landmark 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 in a training set of 2D hands outlines. [[Principal Component Analysis]] (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.

|author = T. F. Cootes

|title = Statistical models of appearance for computer vision

|date=May 2004

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

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

|title = "Active shape models - theirmodels—their training and application",

|journal = "Computer Vision and Image Understanding",

|pages = "38--59",38–59

|year = 1995

author |author1= "D.H. Cooper and |author2=T.F. Cootes and |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>{{cite conference

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

|year = 2003,

|conference = "IMPI"

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

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

|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 [[pixel]]s/[[voxel]]s. The latter point leads to [[active shape model]]s (ASM) and [[active appearance model]]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==

First, a set of training images are manually landmarked 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.

The implementation of the procedure is rouglhy the following:

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

* '''1:''' annotate the training set outlines with enough corresponding landmarks to sufficiently approximate the geometry of the original shapes

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

* '''2:''' 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.

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.

* '''3:''' 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

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.

* '''4:''' 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.

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>

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 nematod worm opens the road to Kernel PCA-based methods.

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.

Due to the PCA properties: eigenvectors are mutually orhtogonal, 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.

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>).

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 (ASD, Allowable Shape Domain [1])

==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 nematod[[nematode]] worm opensis used as an example in the roadteaching toof Kernel[[kernel PCA]]-based methods.

Due to the PCA properties: eigenvectors are mutually orhtogonal[[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 gaussianGaussian cloud of points (whatever their dimension): this suggests the concept of bounded variation.

==Some Interesting Articles==

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—[[allowable shape ___domain]] (ASD, Allowable Shape Domain [1]).<ref name=taylor/>

[1]:

@article{ cooper95asp_training,

author = "D.H. Cooper and T.F. Cootes and 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

}

[2]:

@inproceedings{davies03impi,

title = "Shape discrimination in the Hippocampus using an MDL Model",

author = "Rhodri H. Davies and Carole J. Twining and P. Daniel Allen and Tim F. Cootes and Chris J. Taylor",

year = 2003,

conference = "IMPI"

}

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

[3]: @techreport{ cootes04report, author = "T. F. Cootes ", title = "Statistical models of appearance for computer vision", year = "2004", month = "May"}

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

[4]: @techreport{ stegmann02pdmtut, author = {Stegmann, M. B. and Gomez, D. D.}, title = {A Brief Introduction to Statistical Shape Analysis}, year = {2002}, publisher = {Informatics and Mathematical Modelling, Technical University of Denmark, DTU}, address = {Richard Petersens Plads, Building 321, DK-2800 Kgs. Lyngby}, month = {mar}, note = {Images, annotations and data reports are placed in the enclosed zip-file.}}

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

==See also==

* [[Principal Component Analysis]]

* [[Procrustes analysis]]

* [[Generalized procrustes analysis]]

==References==

* [[Active shape model]]

<references/>

* [[Active Appearance Model]]

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

|title = A Brief Introduction to Statistical Shape Analysis

|year = 2002

|publisher = Informatics and Mathematical Modelling, Technical University of Denmark, DTU

|address = Richard Petersens Plads, Building 321, DK-2800 Kgs. Lyngby, |month = March

|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.

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

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

==External links==

* [https://web.archive.org/web/20080509041813/http://www.isbe.man.ac.uk/~bim//Models/index.html Description]Flexible ofModels PDM,for ASMComputer andVision], AAMsTim fromCootes, Manchester University.

* [http://www.isbeicaen.manuiowa.ac.ukedu/~bimdip/LECTURE/Understanding3.html TimA Cootes']practical homeintroduction page (one of the original co-inventors ofto PDM &and ASMs)].

Among his ressources, this report summarizes the state of the art in this discipline: http://www.isbe.man.ac.uk/~bim/Models/app_models.pdf

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

[[Category:Computer vision]]