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Active contours with shape priors

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Once this minimum is achieved, the mean computed over the transformed shapes is

called the empirical mean. An empirical covariance is then computed from it. A

Principal Component Analysis (PCA) is applied to generate the characteristic Eigen

modes of deformations.

The shape prior term is modeled as the search for the optimum rigid transformations

that minimize a distance between evolving contour and the characteristic deformation

mode. The authors have suggested that the empirical mean can also be used as prior.

In that case the shape prior term will try to reduce the distance between the evolving

contour and the empirical mean by finding the rigid transforms that generates the best

fit. The difference between the two formulations is that, using empirical mean as shape

statistics will result in more rigid priors, while by using the characteristic Eigen modes

statistics the shape priors will allow for variations and will result in more flexible

priors.

x Fourier-based shape descriptors provide quite an efficient and powerful way of

contour representation. Such a representation can be particularly useful in the context

of explicit active contours with shape priors. In this regard, one method (Staib &

Duncan 1992) proposes the contour representation using elliptical Fourier descriptors.

The prior shapes are considered as a set of outline boundaries corresponding to each

shape prior class. For each class, the shapes are defined in terms of the elliptical

Fourier descriptor parameters. Using the mean and variance of these parameters, for

each class, a Gaussian probability density function is defined. To embed the shape

prior information in the curve evolution, first the shape of the curve has to be matched

to one of the existing classes of shapes. This is done by expressing the evolving curve

also as elliptical Fourier descriptor parameters and estimating the maximum a

posteriori (minimum error). This is accomplished by comparing the parameters of

evolving curve with the Gaussian approximation of shape distribution for existing

shape classes. The class which increases the maximum a posteriori will be chosen and

the shape constraints of the particular class will be enforced by using the elliptical

descriptors of that class and adjusting them to the pose and scale of evolving curve. To

attract the evolving curve to the image information, the image gradient is used. This

method is very sensitive to initial shape of the deformable curve.

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Active contours with shape priors

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x Recently, Fourier based geometric shape priors have been used with the variational

setup for snakes (Charmi et al. 2008). Both the template and the deformable model are

represented by a set of Fourier descriptors. A force based approach is then applied to

guide the deformable contour towards the template by minimizing the difference

between the reconstructions of the template and the deformable curve. However, this

method is sensitive to the starting point and needs the reconstruction of the template

and deformable contours in the spatial domain in order to compute the force that

guides the deformable model towards the template.

x /HJHQGUHV PRPHQWV are used as shape priors in region-based active contours

(Foulonneau et al. 2006). This is achieved by minimizing a distance between shape

descriptors GHILQHG E\ /HJHQGUHV PRPHQWV DQG WKDW RI HYROYLQJ FRQWRXU . One main

limitation of this method is that the shape priors are not invariant to rotation changes.

The method can be theoretically extended for rotational invariance, but this leads to

the increase in complexity by many folds.

x In one recent technical report (Park 2010), invariant shape priors have been added in

the polygonal implementation of Mumford Shah formulation. To embed the shape

prior information, the reference shape is represented in the form of the inter-vertex

distance. The inter-vertex distance of a polygon is a matrix in which each row

corresponds to the distance of a vertex from all the other vertices. In its crude form,

the inter-vertex distance description is invariant to the translation.

The prior information is added in the curve evolution process by computing the inter-

vertex distance for the deforming curve and comparing it to that of the reference

shape. The shape prior term is then modeled as the minimization of the distance

between inter-vertex representation of the reference shape and the evolving curve.

Both the reference shape and deforming curve must have the same number of points in

order for this formulation to work. The best scale and rotation for prior shape is

estimated during the evolution. The scale is derived from the scale of the evolving

contour. To achieve the rotation invariance and estimate the correct orientation of the

object to be segmented, the reference shape is dynamically rotated during each step to

find the best fit between the orientation of the evolving contour and that of the prior

shape. The best fit is the orientation that minimizes the prior energy term.

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Active contours with shape priors

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To adjust the influence of the shape prior term, a weight parameter is associated with

it. For the evolution under shape prior, the authors, propose two schemes to balance

the influence of the prior shape term with respect to the other terms of the Mumford

Shah model. As a first strategy, they suggest that an initial segmentation should be

carried out without using the prior energy terms. This segmentation can be then used

as an initialization step for the model with shape prior energy. As a second strategy,

the authors propose gradual increase in the parameter of shape energy term during the

curve evolution.

One limitation of this method is that during the evolution, the evolving contour cannot

be resampled (i.e. vertices cannot be added or deleted). This limitation comes from the

fact that inter-vertex distance matrices of reference shape and evolving contour needs

to be of the same size for prior term calculation. Therefore, the number of vertices of

reference and evolving contour needs stay the same during the entire process. The

other limitation is that the inter-vertex distance based description can only be applied

to convex polygonal shapes.

In this context we propose an improved version of the greedy algorithm of explicit active

contour models with shape priors. These priors enable the deformable model to converge

towards the desired shape even in the presence of occlusion and context noise. We introduce

these shape priors through the use of stable and complete Fourier descriptors (Bartolini et al.

2005). These priors are invariant to the translation, scaling, rotation factors and starting point.

We will present this contribution in its full details in chapter 2 of this thesis.

Here, we will present a brief discussion regarding our choice of active contour model and

shape descriptors. The greedy algorithm was chosen because it is fast and it achieves better

segmentation performance when compared to the variational and dynamic programming

approaches. Moreover, it is more stable (Kim & Lee 2003). Another motivation for the use of

the greedy algorithm came from the fact that its energy based setup is quite intuitive for

adding more energy terms. Thus, we can introduce shape based information in greedy

algorithm through one of such energy terms. The shape prior energy term tries to minimize

the distance between the shape of the evolving contour and a reference contour. For the

description of these contours, a stable set of Fourier descriptors (Bartolini et al. 2005) is used.

The choice of Fourier descriptors was made because they are very effective for representing

contours and their efficiency in contour-based shape matching is high when compared to the

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The segmentation results by active contour methods suffer seriously when there is occlusion,

context noise and object overlapping or missing parts of the object. The main objective of this

thesis is to improve the segmentation results of active contours in case of such problems. Our

hypothesis is that if some prior information about the shape of the object to be segmented is

available then embedding such information in the active contour model can improve the

segmentation results. We base our hypothesis on the notion that any missing or occluded parts

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work, we address some difficulties that arise when introducing the shape priors in the active

contour models. These include: which shape descriptors should be used? How and when to

integrate the shape priors in the energy equation? How to achieve the segmentation with

shape priors in a computationally efficient manner? How to balance the influence of the prior

based energy in the presence of other energy terms?

In this chapter, we will be presenting our main contribution by adding such shape priors to the

segmentation process of active contours to improve the segmentation results.

We propose to add shape prior based energy term in the greedy algorithm of active contours.

Such an energy term should constraint the deformation of active contour with respect to a

prior shape but should at the same time balance the influence of the shape prior term in the

presence of other energy terms. This energy term should be able to guide the segmentation

with respect to a prior shape by minimizing a distance between the prior shape and the

evolving contour. The greedy algorithm was chosen because of its speed, stability and its

adaptability for permitting the integration of additional energy terms quite intuitively.

To introduce any such energy term, we need a robust and compact shape description method

that is invariant to at least the basic transformations such as translation, rotation and scaling

factors. In this regard, many shape descriptors have been discussed in chapter 1, based on

which Fourier based shape descriptors have been chosen as the shape description method.

This choice was made on multiple criteria. Firstly, the Fourier based descriptors are naturally

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