Delineation of vessel structures in human vasculature forms the precursor to a number of clinical applications. Typically, the delineation is performed using both 2D (DSA) and 3D techniques (CT, MR, XRay Angiography). However the decisions are still made using a maximum intensity projection (MIP) of the data. This is problematic since MIP is also affected by other tissues of high intensity, and low intensity vasculature may never be fully realized in the MIP compared to other tissues. This calls for a need for a type of vessel enhancement which can be applied prior to MIP to ensure MIP of the imaging have significant representation of low intensity vessels for detection. It can also facilitate volumetric views of vasculature and enable quantitative measurements. 
 
 To this end, Frangi et al. propose a vessel enhancement method which defines a "vesselness measure" by using eigenvalues of the Hessian matrix as indicators. The eigenvalue analysis of Hessian provides the direction of the smallest curvature (along the tubular vessel structure). The eigenvalue decomposition of a Hessian on a spherical neighbourhood around a point $x_0$ maps an ellipsoid with the axis represented by the eignevectors and their magnitude represented by their corresponding eigenvalues. The method provides a framework with three eigenvalues $|\lambda_1| <= |\lambda_2| <= |\lambda_3|$ with heuristic rules about their absolute magnitude in the scenario where a vessel is present. Particularly, in order to derive a well-formed ``vessel measure" as a function of these eigenvalues, it is assumed that for a vessel structure, $\lambda_1$ will be very small (or zero). The authors also add prior information about the vessel in the sense that the vessels appear as bright tubes in a dark background in most images. Hence they indicate that a vessel structure of this sort must have the following configuration of $\lambda$ values $|\lambda_1| \approx 1$, $|\lambda_1| << |\lambda_2|$, $|\lambda_2| \approx |\lambda_3|$. Using a combination of these $\lambda$ values, as well as a Hessian-based function, the authors propose the following vessel measure:
$\mathcal{V}_0(s) = \begin{cases} 
       0 \quad \text{if} \quad \lambda_2 > 0 \quad \text{or} \quad \lambda_3 > 0\\
      (1 - exp\left(-\dfrac{\mathcal{R}_A^2}{2\alpha^2}\right))exp\left(-\dfrac{\mathcal{R}_B^2}{2\beta^2}\right)(1 - exp\left(-\dfrac{S^2}{2c^2}\right))
   \end{cases}$
The three terms that make up the measure are $\mathcal{R}_A$, $\mathcal{R}_B$, and $S$. The first term $\mathcal{R}_A$ refers to the largest area cross section of the ellipsoid represented by the eigenvalue decomposition. It distinguishes between plate-like and line-like structures. The second term $\mathcal{R}_B$ accounts for the deviation from a blob-like structure, but cannot distinguish between line- and a plit-like pattern. The third term $S$ is simply the Frebenius norm of the Hessian matrix which accounts for lack of structure in the background, and will be high when there is high contrast compared to background. The vesselness measure is then analyzed at different scales to ensure that vessels of all sizes get detected. 

The method was applied on 2D DSA images which are obtained from X-ray projection before and after contrast agent is injected. The method was also applied to 3D MRA images. The results showed promising background suppression when vessel enhancement filtering was applied before performing MIP.