Feature extraction

Detection of tube-like structures using multiscale analysis has been carried out by other researchers [16,17,18]. The main purpose of these works is to develop a line-enhancement filter based on the eigenvalue analysis of the Hessian matrix. These filters were applied to 2-D and 3-D medical images such as digital subtraction angiography or magnetic resonance angiography of blood vessels and computer tomography of airways. We use similar information in combination with gradient information to segment blood vessels rather than to enhance them.

Gradient magnitude. The magnitude of the gradient , represents the slope of the image intensity for a particular value of the parameter s. Figure 3 shows the gradient magnitude at different scales for the subimages shown in Figure 2.

Figure 3: Magnitude of the gradient along the scales for: $s=2$, $s=8$, and $s=14$ pixels.

Principal curvature. The second directional derivatives describe the variation in the gradient of intensity in the neighbourhood of a point. Since vessels appear as ridge-like structures in the images, we look for pixels where the intensity image has a local maximum in the direction for which the gradient of the image undergoes the largest change (largest concavity) [19]. The second derivative information is derived from the Hessian of the intensity image $I(x,y)$:

$$\displaystyle H = \left( \begin{array}{cc} \partial_{xx}I &... ...xy}I \\ \partial_{yx}I & \partial_{yy}I \end{array} \right )$$ (4)

Since $\partial_{xy}I=\partial_{yx}I$ the Hessian matrix is symmetrical with real eigenvalues and orthogonal eigenvectors which are rotation invariant. The eigenvalues, $\lambda _{+}$ and $\lambda _{-}$, where we take $\lambda_{+} \ge \lambda_{-}$, measure convexity and concavity in the corresponding eigendirections. Figure 4 shows the profile across a red-free vessel and the corresponding eigenvalues of the Hessian matrix, where $\lambda _{+} \approx 0$ and $\lambda _{-} << 0$ for pixels in the vessel. For a fluorescein vessel profile, where vessels are darker than the background, the eigenvalues are $\lambda_{-}\approx 0$ and $\lambda_{+} >> 0$ for vessel pixels.

Figure 4: a) Intensity profile across a blood vessel from a red-free image. b) Eigenvalues, $\lambda _{+}$ (dashed line) and $\lambda _{-}$ (solid line). Ridges are regions where $\lambda _{+} \approx 0$ and $\lambda _{-} << 0$.

In order to analyse both red-free and fluorescein images with the same algorithm, we define $\lambda_{1}=\min(\left \vert \lambda_{+} \right \vert,\left \vert \lambda_{-} \right \vert)$ and $\lambda_{2}=\max(\left \vert \lambda_{+} \right \vert,\left \vert \lambda_{-} \right \vert)$. The maximum eigenvalue, $\lambda _{2}$, corresponds to the maximum principal curvature of the Hessian tensor, which we will refer to as maximum principal curvature. Thus, a pixel belonging to a vessel region will be weighted as a vessel pixel if $\lambda_{2} \gg 1$, for both red-free and fluorescein images. Figure 5 shows the maximum principal curvature $\lambda _{2}$ at different scales for the subimages shown in Figure 2.

Figure 5: Maximum principal curvature $\lambda _{2}$ at different scales for: $s=2$, $s=8$, and $s=14$ pixels.