Ridge and valley enhancing filters are widely used in applications such as vessel detection in medical image computing. When images are degraded by noise or include vessels at different scales, such filters are an essential step for meaningful and stable vessel localization. In this work, we propose a novel multi-scale anisotropic fourth-order diffusion equation that allows us to smooth along vessels, while sharpening them in the orthogonal direction. The proposed filter uses a fourth-order diffusion tensor whose eigentensors and eigenvalues are determined from the local Hessian matrix, at a scale that is automatically selected for each pixel. We discuss efficient implementation using a fast explicit diffusion scheme and demonstrate results on synthetic images and vessels in fundus images. Compared to previous isotropic and anisotropic fourth-order filters, as well as established second-order vessel enhancing filters, our newly proposed one better restores the centerlines in all cases.
We generalize this approach to a novel anisotropic fourth-order diffusion equation that smoothes along the crease, while creating a sharp peak in the orthogonal direction, to clearly indicate its center. We are aware of only one previous formulation of anisotropic fourth order diffusion, proposed by Hajiaboli @cite_18 . However, it has been designed to preserve edges, rather than enhance creases. Consequently, it is not well-suited for our purposes, as we will demonstrate in the results. Moreover, it differs from our approach in that it does not make use of a fourth-order diffusion tensor, and includes no mechanism for scale selection.
@cite_18
2122287830
A class of fourth-order partial differential equations (PDEs) are proposed to optimize the trade-off between noise removal and edge preservation. The time evolution of these PDEs seeks to minimize a cost functional which is an increasing function of the absolute value of the Laplacian of the image intensity function. Since the Laplacian of an image at a pixel is zero if the image is planar in its neighborhood, these PDEs attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image. Piecewise planar images look more natural than step images which anisotropic diffusion (second order PDEs) uses to approximate an observed image. So the proposed PDEs are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion, while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion. Although both approaches seem to be comparable in removing speckles in the observed images, speckles are more visible in images processed by the proposed PDEs, because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges. Speckles can be easily removed by simple algorithms such as the one presented in this paper.