## 2 Materials and Methods
### Proposed Method Application for Evaluation with Clinical Data
#### 2.2.1 Detection of Cerebral Aneurysms in Head MRA Images with Proposed Feature Extraction  
This cerebral aneurysm detection method is applicable to MRA images scaled to 0.6 mm isotropic voxels.  
First, brain artery regions are extracted by applying adaptive thresholding, morphological opening and closing, and labeling processes. Next, aneurysm candidates are detected by the sliding window method with a cubic window of 32 voxels per side and 16-voxel stride. When more than 10% of the cubic window area overlaps with the extracted brain artery regions, the cubic area is detected as a candidate 3D image patch. Finally, FP candidate patches are classified and eliminated by the lesion candidate classification process with the proposed unsupervised feature extraction and an Ada-Boosted classifier ensemble [16] using a feature set extracted by the proposed unsupervised feature extraction method.  
The proposed unsupervised feature extraction with multiple CAEs is applied to exclude FP candidate patches from the true aneurysm patches. The feature extraction is pre-trained using normal vascular image patches obtained from 252 cases of normal head MRA image datasets, as shown in Table 2. These images were scanned for screening purposes at the University of Tokyo Hospital.  
In contrast, the Ada-Boosted classifier ensemble for FP candidate elimination is pre-trained in a supervised fashion, using a large number of normal patches and a small number of true aneurysm patches. The output of the Ada-Boosted classifier ensemble, \(H(\mathbf{x})\), is shown in the following Equation (3), where \(\mathbf{x}\) is the feature vector, \(h_{i}(\cdot)\) is the \(i\)-th weak classifier, \(\alpha_{i}\) is the weight coefficient of \(h_{i}(\cdot)\), and \(\theta\) is the control bias.  
\[H(\mathbf{x})=\sum_{i}\alpha_{i}h_{i}(\mathbf{x})+\theta \tag{3}\]  
The weight, \(\alpha_{i}\), is positively correlated with the corresponding \(h_{i}(\cdot)\) classification performance. When the function \(H(\mathbf{x})\) appears positive or negative, the lesion candidate patch from which the feature vector \(\mathbf{x}\) is extracted is classified as a true lesion or FP.  
In the supervised learning of the classifier ensemble, weak classifiers are sequentially added to improve the classification accuracy of the ensemble, following the Ada-Boost algorithm [16]. This learning also introduces cost-sensitive learning [17], which assigns a significant penalty if a minor number of lesion candidate data are misidentified in the learning process. The cost \(\chi(\cdot)\) for true lesion candidates and FP candidates are calculated using Equations (4) and (5). Cost-sensitive learning helps avoid the generation of a biased classifier that can only identify FP candidates.  
\[\chi(true\; lesion\; candidate)=\frac{n(true\; lesion\; class)+n(FP\; class)}{n(FP\;class)} \tag{4}\]  
\[\chi(FP\; candidate)=\frac{n(true\; lesion\; class)+n(FP\; class)}{n( true\; lesionclass)} \tag{5}\]  
In this study, decision stumps are employed as weak classifiers. The decision stump is a single-feature thresholding. In other words, this learning is regarded as a sequential feature selection process suitable for the classification problem.  
The performance is evaluated in three-fold cross-validation with a dataset of 378 head MRA images containing at least 1 cerebral artery, as shown in Table 2. These images were also scanned for screening purposes at the University of Tokyo Hospital. The dataset was assigned lesion teacher labels defined by a consensus of at least two experienced radiologists who perform MRI image readings as part of their daily work. Both radiologists had over 10 years of MRA-reading experience. The dataset sorted for training in this cross-validation is used to train the Ada-Boosted classifier ensemble used in the lesion candidate identification process.