## 2 Methods
### Proposed Method  
To enhance and segment coronary arteries in X-ray angiograms, the proposed method consists of four different steps, which are illustrated in Figure 4.  
The first step enhance vessel-like structures in the original angiogram by using multiscale Gaussian matched filters. The parameters of the GMF are directly related to spatial features found in coronary angiograms, such as length, diameter, and orientation of vessel segments. In order to detect  
Figure 4: Diagram of the proposed method, involving the steps of multiscale vessel detection, and coronary arteries segmentation by binary classification. Figure 3: Diagram of a four layered perceptron network with two hidden layers and one neuron in the output layer. any coronary artery efficiently, the parameters of the GMF have to be selected from a search space properly defined. The combination of the parameters \(\sigma\) and \(T\) control the coronary arteries diameter that can be detected, for this reason the value of \(T\) is selected from the range \([8,9,\ldots,15]\) pixels, and \(\sigma\) in the range \([1.5,1.6,\ldots,2.5]\). The different vessel segments frequently present lengths in the range \([8,9,\ldots,15]\) pixels, which is governed in the GMF by the parameter \(L\). The orientation of the vessel segments is controlled by the parameter \(K\), which is defined to cover the angular range \([-\pi/2,\pi/2]\) and is commonly fixed to sample 12 evenly spaced orientations. To perform the multiscale analysis in the spatial domain, only \(\sigma\) is varied within its corresponding search space in order to enhance blood vessels of different diameters, while \(T\), \(L\), and \(K\) are kept fixed. The multiscale GMF response is conformed by 11 detection response images, corresponding to the different scales of \(\sigma\), and can be considered as a three-dimensional image of size \(300\times 300\times 11\) pixels. The second step consists of the analysis of the coronary angiograms using multiscale Gabor filters. For the simplified parameterization of Gabor filters, the parameter \(l\) governs the elongation of the vessel-like structures detector in a search space of \([1,1.1,1.2,\ldots,20]\). The parameter \(\tau\) controls the width of the detector in a range of \([1,3,\ldots,19]\). The value of \(\kappa\) determines the number of directional filter templates, which is commonly fixed to 45 evenly spaced orientations in the range \([-\pi/2,\pi/2]\). The multiscale analysis of the coronary angiograms in the frequency domain is performed by varying the value of \(\tau\) within its corresponding search space, in order to enhance vessel-like structures of different diameters, while the parameters \(l\) and \(\kappa\) are kept unchanged. The resulting multiscale Gabor filter response is conformed by 10 different images, one per each scale of \(\tau\), that can be considered as a three-dimensional image of size \(300\times 300\times 10\) pixels. In the third step, a four-layered perceptron network provides the coronary artery detection. The multiscale filter responses obtained from the first and second steps are concatenated to form a three-dimensional image of size \(300\times 300\times 21\). The MLP performs the vessel detection by using each \(1\times 1\times 21\) pixel-wise inputs from the concatenated multiscale detection response, forming the input layer of the artificial neural network with 21 neurons. The second and third layers of the MLP are hidden layers, where the number of neurons in each is selected by following a statistical hypothesis test for stochastic domination between a set of candidate architectures. The sigmoid function is selected as activation function for both hidden layers to provide a nonlinear transformation of the neuron responses. The set of all possible architectures is large; for that reason, in this work the search space has been limited to \(i,j=1,2,\ldots,10\), where \(i\) and \(j\) are the number of neurons in the first and second hidden layers, respectively.