## 2 Methods
### Multiscale Gabor Filters  
The Gabor filter is a Gaussian curve modulated by a sinusoid function which presents an anisotropic property capable of detecting directional features [18]. [PERSON] et al. [12] proposed the use of Gabor filters for the detection of blood vessels of the retina. The Gabor wavelet is employed to detect the edges of blood vessels in the space domain by identifying singularities in the frequency domain. This method was originally proposed to work in a single scale, which increase the contrast of features that share a similar diameter and orientation. The real part of the single-scale Gabor filter can be defined as follows:  
\[G(x,y)=\frac{1}{2\pi\sigma_{x}\sigma_{y}}\exp\left[-\frac{1}{2}\left(\frac{x^ {2}}{\sigma_{x}^{2}}+\frac{y^{2}}{\sigma_{y}^{2}}\right)\right]\cos(2\pi f_{0 }x)\, \tag{3}\]  
where \(\sigma_{x}\) and \(\sigma_{y}\) control the standard deviation of a two dimensional Gaussian curve in the \(x\) and \(y\) axis, respectively. The value of \(f_{0}\) determines the frequency at which the Gaussian curve is modulated. The filter design was simplified by [PERSON] et al. [13; 14] in the sense of using only two parameters to define the Gabor template. The simplification states that the spreads \(\sigma_{x}\) and \(\sigma_{y}\) can be defined as \(\sigma_{x}=\tau/2\sqrt{2\ln 2}\) and \(\sigma_{y}=l\sigma_{y}\). In addition, the frequency \(f_{0}\) is set to modulate the curve under a period of \(\tau\). The reformulated Gabor filter depends only on two parameters: \(\tau\) and \(l\). The diameter of the features to be detected by the Gabor filter is controlled by \(\tau\), while \(l\) governs the elongation of a segment that shares a similar orientation. For the detection of vessel-like structures in multiple directions, the Gabor filter is rotated into \(\kappa\) evenly spaced angles in the range of \(\theta\in[-\pi/2,\pi/2]\).  
Under the same principle that the single-scale Gabor filter was proposed to detect blood vessels in the retina, this method has been successfully applied to the detection of coronary arteries in X-ray angiograms [16]. Although single-scale Gabor filters are capable of detecting the main vessels in coronary angiograms, the response obtained tends to ignore vessels of diameters lower than \(\tau\). To avoid this problem, a multiscale approach was proposed by [PERSON] et al. [14] for the detection of blood vessels in the retina, and later [PERSON] et al. [16] applied this method to segment coronary arteries.  
In Figure 2, two templates defined with different width of the Gabor filter are presented in order to illustrate the detection process carried out by this method. The main difference between single-scale and multiscale Gabor filters is the number of templates at different width employed for the directional filter bank construction, which is illustrated in Figure 2d,g. The variation of the filter width allows to enhance blood vessels of different diameters in the coronary angiogram, as shown in Figure 2f,i. The multiscale analysis with Gabor filters provides an improved vessel detection quality at expenses of increasing the computational time, which is mostly because of the number of image transformations between the spatial and frequency domains.  
Figure 1: (**a**) An original X-ray angiogram, and (**b**) the corresponding ground-truth delineated by a specialist. (**c**) The three-dimensional plot of the GMF filter proposed by [PERSON] et al. [6]. (**d**) The GMF template defined with \(\sigma=0.5\), (**e**) the oriented template at \(45^{\circ}\), and (**f**) the corresponding detection response. (**g**) The GMF template defined with \(\sigma=2\), (**h**) the template rotated \(45^{\circ}\), and (**i**) the corresponding detection result.