Curvilinear structure detections

As a structure-based detector, PCBR does not use edges, instead, it uses curvilinear structures, also called ridges. Curvilinear structures detection generates a single response for both lines and edges, producing a clearer structural sketch of an image than is usually provided by the gradient magnitude image. The Steger's algorithm ^[2] is modified to get the curvilinear images. As only the first step of this algorithm is used which is to calculate the principal curvature images, the principal curvature is adopted as the name of this detector. To get the principal curvature, the Hessian matrix is calculated:

$H(mathbf{x}) = egin{bmatrix} L_{xx}(mathbf{x}) & L_{xy}(mathbf{x})\ L_{xy}(mathbf{x}) & L_{yy}(mathbf{x})\ end{bmatrix}$

where $L_{aa}(mathbf{x})$ is second partial derivative of the image evaluated at point x in the $a$ direction and $L_{ab}(mathbf{x})$ is the mixed partial second derivative of the image evaluated at point x in the $a$ and $b$ directions. The maximum and minimum eigenvalues of this matrix form two images which correspond to white lines on black background and black lines on white background.