## Control Points Estimation

In the segmentation result obtained by the QHCF algorithm, the following information is available for further processing: region distribution, region intensity related properties (such as mean and standard deviation), and region boundaries. Although an MRF model can take into account the intensity continuity among neighboring pixels during the segmentation process, it imposes no constraint along the contour direction. Therefore, this problem of the QHCF method that there is no curve continuity constraint of object's contour during the optimization process makes the segmented object contour to be easily distorted due to noise. The experiment results in Fig. 8.5 have shown this drawback (rugged object boundary) that is unacceptable in some practical applications, such as quantitative medical image analysis and measurement.

In the proposed framework, a further fine-tune of region's boundary is accomplished by employing the MPA contour model [23]. To have an accurate initialization, it first needs to find the control points automatically.

As mentioned previously, the labeling process of each pixel in an MRF model is decided by the MAP, max{p(xi | y), i = 1, 2,..., N, where N is the number of labels}. Based on this segmentation, the contour of an object can be easily found by searching region's boundary points. However, the experimental analysis of the a posteriori probabilities of these contour pixels exhibit very large variations, indicating that the region boundary points found by the QHCF are not equally believed as the contour points. To reach the optimal contour tracking, it is necessary to select those most reliable ones as control points and search the other object boundary points by adhering to MPA constraints. The proposed solution for control point searching has the following steps:

Step 1. Locate the boundary points of the desired region based on the QHCF segmentation.

Step 2. Divide the region's boundary into sections.

Step 3. For each section, select the most reliable boundary point as a control point for MPA.

The selection process of step three is crucial to the success of the algorithm. Assume the boundary of the object of interest is divided into M sections and section m, 0 < m = M, contains im total points. To simplify the problem formulation, we considered only the boundary points that have one adjacent region (they belong to another region). Suppose a particular boundary point is labeled p and its adjacent region's label is q, the a posterior probability of this point with label p and q can be expressed, respectively, as p(xs = p | y) a exp j -(yim - fp)2 - ^ Un(x = p) + Ue(x = p)U ,

p(xs = q | y) a exp | -(yim - fq)2 - ^ [Un(x = q) + Ue(x = q)] | •

Assume this point belongs to region with label p, it is obvious that its a posteriori probability with label p should always have higher value than that with its adjacent region's label q . In a real image, like MR images or ultrasound image, noise affects the capturing process in boundary regions making the above assumption invalid. Distortion due to noise can blur edges and create a lack of separation in the a posterior probabilities of the true "edge points." To assure good measurement of the probability difference, we introduce the reliability of boundary points as r(s) = 1 - [p(xs = p | y) - p(xs = q | y)]^ (8.35)

The value of the reliability is within the range [0, 1]. If s from the segmented contour is more likely to be a boundary point, its a posteriori probability with label p and q will be quite similar. It then leads to the value of r(s) being closer to 1 and makes point s more reliable. Therefore, in the control point searching process, we use maximum reliability as a criterion, which can be expressed as s = max {r(si)}. (8.36)

The above criterion can be applied directly to the boundary points obtained with the QHCF algorithm because of the location and shape accuracy of the found object region. A further advantage of this accuracy is the solid foundation from which to do further work. This foundation is similar to the manual outline provided by the human operator for the traditional Snake algorithm. Consequently, use of the MRF-based segmentation result and the MAP criterion allows for an automatic initialization process that is relatively free of traps due to noise and spurious edges and has consistent reproducibility.

Step 2 addresses the selection problem of section number M and size of each section. Image quality and confidence of the contour points are determining factors in finding the solutions. For example, in our carotid lumen segmentation of MR images shown in Fig. 8.18, typical images generally needed 3-6 sections for contour tracking, while low-quality images required 8-10 splitting sections to track the whole blood vessel boundary. Object boundary corruption by noise results in more splits in the attempt to attain higher accuracy. The size of the object also is an important factor. Most of our studies contain objects sequestered within a square the size of 128 by 128 pixels. Division of the contour is accomplished by equal-length splitting. A more dynamic approach can be used in the case of a contour with noisier pixels resulting in more control points for ACM. The resulting curve will be noise-resistant and reliable. In addition, processing speed will be increased.

Step 1 is the most flexible and application-dependent of the three steps. It may also be totally eliminated in cases that target regions are already known. However, in most situations lack of advance knowledge of the exact location and spurious knowledge of the object's properties can be referenced as an additional constraint during the segmentation process. When this occurs an identification process can be designed based on the QHCF segmented regions to extract the boundary of the region-of-interest, which can then be used in further contour fine-tuning. An example is the lumen segmentation in a sequence of MR images.

The lumen may often be almost circular in shape and have a dark intensity. Applications in step one provide for the design of a decision tree to better identify the dark lumen from other regions in the QHFC segmentation.

In summary, the search for control points is the crucial step of this proposed framework. This step provides the bridge between the MRF segmentation algorithm and the active contour model. It decides the initialization accuracy, the key to the success of the minimal path approach. Finally, Step 1, being flexible, allows space for prior knowledge and integration of target constraints.

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