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Anatomical Landmarks

Anatomical landmarks are the anatomical structures that can be reliably detected on given images of any patients and that can be used for uniquely identifying the location in each given image [70]. It is very important for the medical image segmentation to detect anatomical landmarks accurately because they are useful for anatomically identifying body regions in the images and for registering organ models to the images (see Sect. 2.3.5). One of the difficulties of automatic landmark detection is finding a set of local structures of whose features are similar from patient to patient. Many local structures have similar appearances, and it is difficult to find a set of landmarks each of which can signify a single location. One approach for handling this difficulty is to detect a set of landmarks simultaneously by using not only the local image features of each landmark but also a model of their spatial distributions. In the following, the methodology of this approach is described [70].

The algorithm of a method [70] that detects a set of anatomical landmarks consists of two steps: A set of candidates for each of the anatomical landmarks is first detected by using only its local image features and then false candidates are rejected using the model of the relative locations between the landmarks at the second step. At the first step, a detector for each of the landmarks is constructed by using a set of training data. Let the location of j-th landmark (j = in the i-th training image, I (i = 1,2, ••• , M/ be denoted by pj, which is entered manually.

Let I] denote a local image pattern observed in a small region centered at p in I,-. In cases where the region is a cube and its size is L x L x L, then is represented by a L3-vector. The detector for the j-th landmark is constructed from a set of the local images, Sj = {IJii = 1,2, ••• ,M}, by applying a PCA to Sj. Let a set of eigenvectors obtained by the PCA be denoted by {vjjk = 1,2 , ..., Tj}, which consists of Tj largest eigenvalues, {Aj > Aj2 > ... > Ajtj}. Let a L3-vector, I(p/, denote a local appearance observed around a point p in a given new image. Then the log likelihood of the location p for the j-th landmark, p(I p/ = Lj(p/, can be computed:

where V is the average of {Vjji = 1,2, ••• ,M}: V = P tjM. The likelihood of p, l(p/, is proportional to exp{Lj(p/}:

This computation of the likelihood can be interpreted as the extraction of a feature that is observed around the landmark, p. Unfortunately, the likelihood distribution cannot identify the location of the j-th landmark in many cases because one can find other anatomical structures that have the appearances similar to that of the target. A set of multiple candidates for the j-th landmark, hence, would be obtained by detecting the local maximums of Lj(p/ (or of V(p/). Let the detected candidates be denoted by ck (k = 1,2 , ..., Kj/.

At the second step, a true point is selected from the candidates {cJkk =

1.2 , ..., Kj} in the given new image by using a model that represents the relative relationships between the anatomical landmarks. Let a true position of the j-th landmark be denoted by a three-vector, p j, and the distance between the j-th landmark and the j'-th landmark be denoted by djj'. In this method [70], a model of djj is constructed for each pair of the landmarks by using the training set of the landmarks manually extracted from all of the training images, { pij i =

1.2 , ..., M, j = 1,2 , ..., N}. Let djj = pk — pj || (j ф j'). Then, it is straightforward to construct a Gaussian model of djj' in (2.151) from the training set:

where dj and aj are the average and the standard deviation of {df ji = 1,2, ••• , Mg, respectively. A true point, cj, is selected for each j (j = 1,2, ••• , N) from the candidate set, {cJk jk = 1,2,..., Kjg, by finding the number к for each j that maximizes the following simultaneous probability:

where djj' = cy — cj ||. The final estimated location of the j-th landmark can be

obtained as c1 = cj. The model shown in (2.152) is a Markov random field (MRF) model that can be used to estimate the arguments that maximize the probability.

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