Survey on liver CT image segmentation methods <s> Probability atlas based automatic methods <s> The nearest neighbor decision rule assigns to an unclassified sample point the classification of the nearest of a set of previously classified points. This rule is independent of the underlying joint distribution on the sample points and their classifications, and hence the probability of error R of such a rule must be at least as great as the Bayes probability of error R^{\ast} --the minimum probability of error over all decision rules taking underlying probability structure into account. However, in a large sample analysis, we will show in the M -category case that R^{\ast} \leq R \leq R^{\ast}(2 --MR^{\ast}/(M-1)) , where these bounds are the tightest possible, for all suitably smooth underlying distributions. Thus for any number of categories, the probability of error of the nearest neighbor rule is bounded above by twice the Bayes probability of error. In this sense, it may be said that half the classification information in an infinite sample set is contained in the nearest neighbor. <s> BIB001 </s> Survey on liver CT image segmentation methods <s> Probability atlas based automatic methods <s> In this paper the authors present a new approach for the nonrigid registration of contrast-enhanced breast MRI. A hierarchical transformation model of the motion of the breast has been developed. The global motion of the breast is modeled by an affine transformation while the local breast motion is described by a free-form deformation (FFD) based on B-splines. Normalized mutual information is used as a voxel-based similarity measure which is insensitive to intensity changes as a result of the contrast enhancement. Registration is achieved by minimizing a cost function, which represents a combination of the cost associated with the smoothness of the transformation and the cost associated with the image similarity. The algorithm has been applied to the fully automated registration of three-dimensional (3-D) breast MRI in volunteers and patients. In particular, the authors have compared the results of the proposed nonrigid registration algorithm to those obtained using rigid and affine registration techniques. The results clearly indicate that the nonrigid registration algorithm is much better able to recover the motion and deformation of the breast than rigid or affine registration algorithms. <s> BIB002 </s> Survey on liver CT image segmentation methods <s> Probability atlas based automatic methods <s> We have implemented and validated an algorithm for three-dimensional positron emission tomography transmission-to-computed tomography registration in the chest, using mutual information as a similarity criterion. Inherent differences in the two imaging protocols produce significant nonrigid motion between the two acquisitions. A rigid body deformation combined with localized cubic B-splines is used to capture this motion. The deformation is defined on a regular grid and is parameterized by potentially several thousand coefficients. Together with a spline-based continuous representation of images and Parzen histogram estimates, our deformation model allows closed-form expressions for the criterion and its gradient. A limited-memory quasi-Newton optimization algorithm is used in a hierarchical multiresolution framework to automatically align the images. To characterize the performance of the method, 27 scans from patients involved in routine lung cancer staging were used in a validation study. The registrations were assessed visually by two expert observers in specific anatomic locations using a split window validation technique. The visually reported errors are in the 0- to 6-mm range and the average computation time is 100 min on a moderate-performance workstation. <s> BIB003 </s> Survey on liver CT image segmentation methods <s> Probability atlas based automatic methods <s> A fully automatic system for segmentation of the liver from CT scans is presented. The core of the method consists of a voxel labeling procedure where the probability that each voxel is part of the liver is estimated using a statistical classifier (k-nearest-neighbor) and a set of features. Several features encode positional information, obtained from a multi-atlas registration procedure. In addition, pre-processing steps are carried out to determine the vertical scan range of the liver and to rotate the scan so that the subject is in supine position, and post-processing is applied to the voxel classification result to smooth and improve the final segmentation. The method is evaluated on 10 test scans and performs robustly, as the volumetric overlap error is 12.5% on average and 15.3% for the worst case. A careful inspection of the results reveals, however, that locally many errors are made and the localization of the border is often not precise. The causes and possible solutions for these failures are briefly discussed. <s> BIB004
The approach proposed by BIB004 is based on voxel classification in combination with a multi atlas registration. K nearest neighbor (kNN) classifier BIB001 ) is employed to label each voxel within an automatically detected candidate region as liver tissue or background. In preprocessing, each image is re-sampled to isotropic voxels. To detect and correct rotations around the Z-axis, bones are detected by thresholding and the spread of the resulting binary mask in X-direction is maximized by applying different rotations. After that, lungs are detected by thresholding and the potential liver region is limited to a fixed height around the lower lung rim. Twelve selected training scans are then registered to the new image using an affine transform followed by a B-splines approach BIB002 in multiple resolutions. For this purpose, a negative mutual information cost function proposed by BIB003 is optimized by a stochastic gradient optimizer. The resulting transformation fields are used to map individual training segmentations to the new image. This yields probabilistic atlas segmentation  ) on which the three spatial features are based: they represent the percentage of the probabilistic segmentation above, left, and behind the voxel in question. After classifying each voxel in the vicinity of the mask with a 15-nearest-neighbor classifier, results are post-processed by smoothing and morphological operations proposed by  .