Tipos de Indicadores

The correspondence establishment can be divided into many categories in terms of different standards, such as manual landmarking, feature-based approach, param- eterization based approach, physical properties based method, and image based correspondence [79], or using another categorizing criterion such as the mesh to mesh approach, mesh to volume, volume to volume, parameterization to param- eterization approaches etc.[88]. However, all the approaches are not individually applied, they are always combined for a better result. Besides, the correspondence establishment is related largely with image registration methods, both rigid and non-rigid image registration. As a result we will not make an explicit distinction between these methods. Here we just pick some typical methods for a brief review. Early methods focused more on rigid registration. Some were based on equally spacing labeling on a mesh surface to get the landmarks and build the correspon- dence, such as the method of Bastuscheck et al. [72]. They used fast Fourier transform to determine the least square difference between the sequences of points

sampled in equal spacing on the 3D curves. Then they used the returned rotation and translation to register the curves. Besl et al.[71] create the iterative closest points (ICP) algorithm to align the unlabelled points together by means of minimiz- ing the distances between pair of points iteratively with similarity transformation. This algorithm also works for surfaces with potential different number of vertices before the initialization of building the correspondence. Many derived simpler algorithms and modified algorithms have been implemented. However, because of the restrictions to these algorithms using similarity transformation and based on distance between the equivalent points, these approaches are not suitable for a non-rigid registration.

For non-rigid registration, Bookstein [76] proposed thin-plate spline (TPS). Soon Bookstein [77] used it to adjust the landmarks along the tangential directions of the shape contour to minimize the correspondence error defined by thin-plate bending energy to control the correspondence. Richardsonet al. [78] improved this method with sliding, inserting and deleting landmarks to minimize the landmark correspondence error and representation error, and to improve the representation compactness.

Declercket al. [74] and Subsol et al. [73] adapted ICP algorithm together with a B-spline transformation to minimize the influence of an erroneous matched points. Similarly, Rueckertet al. [75] have proposed the B-splines based free-form defor- mation, applied on the breast MR images. They applied the global transformation with rigid image registration algorithm such as ICP, but local deformation using B-spline. In contrast to TPS, B-splines are locally controlled, resulting in an efficient computation for large amount of control points. The free-form deformation registration is robust to noise and popular in medical imaging, but not accurately invertible. So for generating a diffeomorphic deformation, a novel interpolating spline, the clamped-plate spline (CPS) has been introduced by Twininget al. [85]. This approach can guarantee diffeomorphic warps between pairs of images with a dense pixel to pixel correspondence, which overcomes the drawbacks of B-splines and TPS. Marslandet al. [87] has implemented a polyharmonic CPS to perform the non-rigid registration group-wise.

An alternative is based on shape context registration proposed by Belongie et al. [81]. This method also allows the input points with different number of vertices. It firstly constructs the feature vector in each data set, then matches these vectors by minimizing the defined cost function. Kroonet al. [55] has applied this approach to the correspondence establishment of MR cartilage images and further to the mandibular image registration [56] with the histogram enhancement and B-splines approximation, and so did Maanet al. [22].

The approaches above manipulate the correspondence directly on the object surface, no matter if they are based on distance or features. Another group of distinctive methods are based on the parameterization of shapes, which manipulate the

correspondence in the parameterization instead of the shape itself. Kelemen et al.

[82] proposed a spherical harmonics (SPHARM) representation of shape surface providing the parametric descriptions of object shapes, where the coefficients of a parameterized basis function are fit to surface. Then the correspondence of object shapes are consistent with the parameterization. The parameterizations are developed mainly based on the work of Brechb¨uhler et al. [83] or the developed version with a hierarchical optimization by Quicken et al. [84]. These methods were later used for the initialization of correspondence. Davies et al. [86] used the spherical parameterization to initialize the correspondence, and optimized them using a cauchy kernel representation with respect to minimize the description length. Heimann et al. [88] used a conformal mapping based on the spherical parameterization and minimum description length (MDL) as the objective function1 with a gradient descent optimization to evaluate the correspondence. Davies et al.

[89] have also proposed an automatic model building by direct optimization of de- scription length where the CPS was applied for manipulating the parameterization2. Kirschneret al. [90] suggested to use the ICP alignment before parameterization, and to remove the area distortion [91] to correct and refine the parameterization for correspondence. They also investigated the work from Davieset al. and made a comparison. Vincent et al. have adopted the parameterization based group-wise MDL registration to build the correspondence of landmarks to utilize the active appearance models for cartilage [54] and prostate [23] segmentation respectively. Some approaches above are performed pairwise, such as the ICP, B-splines based free-form deformation [75], shape context approach [81] etc. But some are group- wise performed, such as the polyharmonic CPS non-rigid registration [87] and MDL based approaches [86]. Actually, the pairwise algorithm is problematic for some cases, even though most of them work fine. From a theoretical viewpoint, the match between pair of shapes needs to be determined within the wider context of the group [79], since some ambiguity can be removed by the information introduced from the group instead of the pair. The group-wise registration mainly differs in the objective function where the global information would be considered. The group-wise objective functions are featured to be based on the models, such as the determinant of the model covariance, or based on the information theoretical measurement such as MDL. Because we apply the AAM which requires a consis- tent correspondence across the data set, a group-wise registration algorithm for correspondence establishment and optimization would be preferable.

The correspondence optimization can be regarded as an optimization problem since it can be seen that the aim of the approaches above are to minimize some objective functions. Therefore, we can generalize the framework of correspondence optimization into three phases [79]: manipulating the correspondence, calculating

1_{MDL is similar with the cost function, energy function, erroneous function or distance function}

in the former methods. This is also used to evaluate the shape correspondence

2

The parameterization as a meshed sphere can also be manipulated using CPS, so all the methods for manipulating the shape mesh can also be applied to control the parameterization mesh

the objective functions, and optimizing the correspondence. Figure 4.2 illustrates this framework. This flow can be conducted iteratively until we get a satisfying result. There are multiple choices of approaches in each phase. For instance, CPS, symmetric theta transformation can both achieve the correspondence manipulation; either MDL or the model covariance determinant can be the objective function.

Figure 4.2: Framework for correspondence optimization

We choose the group-wise correspondence optimization based on parameterization of shapes with the MDL objective function. The following sections will interpret the whole procedure in detail.