Though the above captures the characteristics of the term shape fairly well; this thesis will adapt the definition by D.G. Kendall [20] and define shape as:''A collection of corresponding border points.''[62]

''The characteristic surface configuration of a thing;[1]

an outline or a contour.''

''Something distinguished from its surroundings by its outline.''[1]

The term shape is - in other words - invariant to Euclidean transformations. This is reflected in figure 4.1. The next question that naturally arises is: How should one describe a shape? In everyday conversation, unknown shapes are often described as references to known shapes - e.g.

One way to describe a shape is by locating a finite number of points on the outline. Consequently, the concept of a

Dryden & Mardia further more discriminates landmarks into three subgroups [20]:

**Anatomical landmarks**Points assigned by an expert that corresponds between organisms in some biologically meaningful way.

**Mathematical landmarks**Points located on an object according to some mathematical or geometrical property, i.e. high curvature or an extremum point.

**Pseudo-landmarks**Constructed points on an object either around the outline or between landmarks.

Synonyms for landmarks include homologous points, nodes, vertices, anchor points, fiducial markers, model points, markers, key points etc. A mathematical representation of an

(4.1) |

Notice that the above representation does not contain any explicit information about the point connectivity.

Obtaining Landmarks

Another substantial problem in obtaining landmarks is that some object classes lack points, which can be classified as corresponding across examples. This is especially true for many biological shapes and is treated in depth by Bookstein [6]. Another source for this type of problems is occlusions in the 3D to 2D projections in perspective images. Annihilation of points can also be observed in malformation of organic shapes. All examples in the remains of this part of the thesis are based on annotations of a bone in the human hand. The image modality is radiographs and the precise name of the bone is

If

If a relationship between the distance in shape space and Euclidean distance in the original plane can be established, the set of shapes actually forms a Riemannian manifold containing the object class in question. This is also denoted as theProofInitially we havekndimensions. The translation removeskdimensions, the uniform scaling one dimension and the rotation dimensions.

- 1.
- Compute the centroid of each shape.
- 2.
- Re-scale each shape to have equal size.
- 3.
- Align w.r.t. position the two shapes at their centroids.
- 4.
- Align w.r.t. orientation by rotation.

Mathematically the squared Procrustes distance between two shapes,

(4.3) |

The

(4.4) |

To perform step 2 we obviously need to establish a

In the following the

(4.5) |

Another often used scale metric is the

(4.6) |

To filter out the rotational effects the following

- 1.
- Arrange the size and position aligned
**x**_{1}and**x**_{2}as matrices^{4.6}. - 2.
- Calculate the SVD,
**UDV**^{T}, of**x**_{1}^{T}**x**_{2} - 3.
- Then the rotation matrix needed to optimally superimpose
**x**_{1}upon**x**_{2}is**VU**^{T}. In the planar case:

(4.7)

As an alternative Cootes et al. suggest a variation on Procrustes distance-based alignment by minimizing the closed form of |T(

(4.8) |

The term |T(

- 1.
- Choose the first shape as an estimate of the mean shape.
- 2.
- Align all the remaining shapes to the mean shape.
- 3.
- Re-calculate the estimate of the mean from the aligned shapes
- 4.
- If the mean estimate has changed return to step 2.

(4.9) |

This is also referred to as the Frechét mean.

As an example figure 4.5 shows the landmarks of a set of 24 unaligned shapes. The result of the shape alignment can be seen as a scatter plot on figure 4.6 (a) where the mean shape is superimposed as a fully drawn shape. This is called the

Modelling Shape Variation

Conceptually the PCA performs a

(4.10) |

The maximum likelihood (ML) estimate of the covariance matrix can thus be given as:

(4.11) |

To prove the assumption of point correlation right, the correlation matrix of the training set of 24 metacarpal-2 bones is shown in figure 4.8. In the case of completely uncorrelated variables, the matrix would be uniformly gray except along its diagonal. Clearly, this is not the case.

The point correlation effect can be emphasized by normalizing the covariance matrix by the variance. Hence the

(4.12) |

(4.13) |

Recalling the shape vector structure;

The principal axes of the 2

(4.14) |

Where denotes a diagonal matrix of eigenvalues

(4.15) |

corresponding to the eigenvectors in the columns of .

(4.16) |

A shape instance can then be generated by deforming the mean shape by a linear combination of eigenvectors:

where

As a further example of such modal deformations, the first three - most significant - eigenvectors are used to deform the mean metacarpal shape in figure 4.11.

What remains is to determine how many modes to retain. This leads to a trade-off between the accuracy and the compactness of the model. However, it is safe to consider small-scale variation as noise. It can be shown that the variance along the axis corresponding to the

(4.18) |

Notice that this step basically is a regularization of the solution space. In the metacarpal case 95% of the shape variation can be modeled using 12 parameters. A rather substantial reduction since the shape space originally had a dimensionality of . To give an idea of the decay rate of the eigenvalues a percentage plot is shown in figure 4.12.

To further investigate the distribution of the

(4.19) |

No clear structure is observed in figure 4.13, thus concluding that the variation of the metacarpal shapes can be meaningfully described by the linear PCA transform. This however is not a general result for organic shapes due to the highly non-linear relationships observed in nature.

An inherently problem with PCA is that it is linear, and can thus only handle data with linear behavior. An often seen problem with data given to a PCA is the so-called

This section is concluded by remarking that the use of the PCA as a statistical reparametrisation of the shape space provides a compact and convenient way to deform a mean shape in a controlled manner similar to what is observed in a set of training shapes. Hence the shape variation has been modeled by obtaining a compact shape representation. Furthermore the PCA provides a simple way to compare a new shape to the training set by performing the orthogonal transformation into

The projection into tangent space align all rectangles with corners on straight lines (see fig. 4.16) thus enabling modeling of the training set using only linear displacements. Notice how the mean shape is contained in the training set since the PCA now only uses one parameter, , to model the change in aspect ratio. In this way, the distribution of PCA-parameters can be kept more compact and non-linearities can be reduced. This leads to better and simpler models.

**Articulated shapes**Shapes with pivotal rotations around one or more points are inherently non-linear.

**Bad landmarks**Manually placed landmarks can easily cause non-linearies.

**Bending**Can also be interpreted as a piece-wise rotation.

Another way to view the problem, is that the PCA-approach is based on the assumption, that all shapes of the class ends on the same manifold. More precisely as a hyper ellipsoid cluster in the new basis spanned by the PCA. However when dealing with non-linearity the ellipsoid changes into a more structured form. Dealing with objects with discreetized behavior