Multiscale Vessel Enhancement Filtering

Introduction

This paper presents a method for vessel enhancement filtering which relies on local structure. Using information about the second order ellipsoid, this is an improvement over previous works that exhibits robustness to noise and background for vessel enhancement in the experiments with two clinical image modalities - 2D DSA images and 3D MRA images.

Method

Given an image $L$, the Taylor series expansion of the image in the neighborhood of a point $x_0$ can be used to represent the approximate structure of the image up to the second order and can be written as $$ L(x_0 + \delta x_0, s) \approx L(x_0, s) + \delta x^T_0 \nabla_{0,s} + \delta x^T_0 H_{0,s} \delta x_0 $$ where $\nabla_{0,s}$ and $H_{0,s}$ represent the gradient vector and the Hessian metric of the image computed in $x_0$ at scale $s$.

Using linear scale space theory, the differentiation becomes a convolution operation with derivatives of Gaussians, $$ \frac{\partial}{\partial x} L(x,s) = s^\gamma L(x) * \frac{\partial}{\partial x} G(x,s) $$ where $G(x, s)$ is a D-dimensional Gaussian defined as $$ G(x,s) = \frac{1}{\sqrt{(2 \pi s^2)}^D} e^{- \frac{|x|^2}{2s^2}} $$ where $\gamma$ represents a family of normalized derivatives.

The second order derivative at scale $s$ (represented by the Hessian) is used to measure the contrast between the regions inside and outside the range $(-s, s)$. Eigenvalue analysis of the Hessian is performed in order to find the principal directions in which the local second order structure can be decomposed, and this directly finds the direction of the smallest curvature along the vessel.

Using eigenvalue decomposition, we have

$$ \hat{u}^T_{s,k} H_{0,s} \hat{u}_{s,k} = \lambda_{s,k} $$

where $\lambda_{s,k}$ represents the eigenvalue corresponding to the $k$-th normalized eigenvector $\hat{u}_{s,k}$ of the Hessian $H_{0,s}$ computed at scale $s$.

Eigenvalue decomposition results in three orthonormal directions which are invariant to a scaling factor when mapped by the Hessian matrix.

For an ideal tubular structure in a 3D image, $$ |\lambda_1| \approx 0 $$$$ |\lambda_1| \ll |\lambda_2| $$$$ \lambda_2 \approx \lambda_3 $$ The respective eigenvectors point out directions - $\hat{u}_1$ points the direction of the minimum intensity variation (since $|\lambda_1| \leq |\lambda_2| \leq |\lambda_3|$) and therefore the direction of the vessel, and $\hat{u}_2$ and $\hat{u}_3$ form a base for the orthogonal plane.

The paper also proposes 2 ratios based on the second order ellipsoid which happen to be grey-level invariant and are therefore constant when subjected to intensity rescalings. $$ R_B = \frac{|\lambda_1|}{\sqrt{|\lambda_2 \lambda_3}}, \ and $$$$ R_A = \frac{|\lambda_2|}{|\lambda_3|} $$

The ratio $R_B$ is a “blob-ness” measure and is maximum for a blob-like structure. The ratio $R_A$ represents the aspect ratio of the two largest second order derivatives, effectively distinguishing between plate-like structures and line-like structures.

Let $S$ represent the Frobenius norm of the Hessian of a D-dimensional image, and we have

$$ S = \|H\|_F = \sqrt{\sum_{j \leq D} \lambda_j^2} $$

Representing the two ratios in one expression for a 3D image, we have $$ \nu_0(s) = \bigg(1 - exp\Big(- \frac{R_A^2}{2\alpha^2} \Big)\bigg) exp\Big(- \frac{R_B^2}{2\beta^2} \Big) \bigg(1 - exp\Big(- \frac{S^2}{2c^2} \Big)\bigg) $$ if $\lambda_2 \leq 0 \ or \ \lambda_3 \leq 0$.

Similarly for a 2D image, we have $$ \nu_0(s) = exp\Big(- \frac{R_B^2}{2\beta^2} \Big) \bigg(1 - exp\Big(- \frac{S^2}{2c^2} \Big)\bigg) $$ if $\lambda_2 \leq 0$. This expression is similar to the previous one except for the fact that $R_A$ is not present in 2D images, and $R_B$ represents the eccentricity of the second order ellipse.

In these expressions, $\alpha$, $\beta$, and $c$ represent threshold values to control the sensitivity of the line filter to the measures $R_A$, $R_B$, and $S$. For the experiments in this paper, $\alpha$ and $\beta$ were chosen to be 0.5, and $c$ was chosen as half the value of the maximum Hessian norm.