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.
@cite_13: This contribution aims to give a basic introduction to diffusion-like methods. There are many different methods commonly used for regularization tasks. Some of them will be briefly introduced and their connection to diffusion shown. In addition to this we will go into some detail for diffusion-like methods in a narrower sense, i.e. methods based on PDEs similar to diffusion PDEs known from physics. Main issues highlighted here are which PDE to use, how diffusivities in such a PDE are constructed, and which discretization is suitable for a given task.
@cite_9: A number of imaging techniques are being used for diagnosis and treatment of vascular pathologies like stenoses, aneurysms, embolisms, malformations and remodelings, which may affect a wide range of anatomical sites. For computer-aided detection and highlighting of potential sites of pathology or to improve visualization and segmentation, angiographic images are often enhanced by Hessian based filters. These filters aim to indicate elongated and or rounded structures by an enhancement function based on Hessian eigenvalues. However, established enhancement functions generally produce a response, which exhibits deficiencies such as poor and non-uniform response for vessels of different sizes and varying contrast, at bifurcations and aneurysms. This may compromise subsequent analysis of the enhanced images. This paper has three important contributions: i) reviews several established enhancement functions and elaborates their deficiencies, ii) proposes a novel enhancement function, which overcomes the deficiencies of the established functions, and iii) quantitatively evaluates and compares the novel and the established enhancement functions on clinical image datasets of the lung, cerebral and fundus vasculatures.
@cite_29: 
@cite_6: The multiscale second order local structure of an image (Hessian) is examined with the purpose of developing a vessel enhancement filter. A vesselness measure is obtained on the basis of all eigenvalues of the Hessian. This measure is tested on two dimensional DSA and three dimensional aortoiliac and cerebral MRA data. Its clinical utility is shown by the simultaneous noise and background suppression and vessel enhancement in maximum intensity projections and volumetric displays.
@cite_24: Many image processing problems require the enhancement of crossing elongated structures. These problems cannot easily be solved by commonly used coherence-enhancing diffusion methods. Therefore, we propose a method for coherence-enhancing diffusion on the invertible orientation score of a 2D image. In an orientation score, the local orientation is represented by an additional third dimension, ensuring that crossing elongated structures are separated from each other. We consider orientation scores as functions on the Euclidean motion group, and use the group structure to apply left-invariant diffusion equations on orientation scores. We describe how we can calculate regularized left-invariant derivatives, and use the Hessian to estimate three descriptive local features: curvature, deviation from horizontality, and orientation confidence. These local features are used to adapt a nonlinear coherence-enhancing, crossing-preserving, diffusion equation on the orientation score. We propose two explicit finite-difference schemes to apply the nonlinear diffusion in the orientation score and provide a stability analysis. Experiments on both artificial and medical images show that preservation of crossings is the main advantage compared to standard coherence-enhancing diffusion. The use of curvature leads to improved enhancement of curves with high curvature. Furthermore, the use of deviation from horizontality makes it feasible to reduce the number of sampled orientations while still preserving crossings.
@cite_15: Estimation of local orientation in images is often posed as the task of finding the minimum variance axis in a local neighborhood. The solution is given as the eigenvector belonging to the smaller eigenvalue of a 2 spl times 2 tensor. Ideally, the tensor is rank-deficient, i.e., the smaller eigenvalue is zero. A large minimal eigenvalue signals the presence of more than one local orientation. We describe a framework for estimating such superimposed orientations. Our analysis of superimposed orientations is based on the eigensystem analysis of a suitably extended tensor. We show how to carry out the eigensystem analysis efficiently using tensor invariants. Unlike in the single orientation case, the eigensystem analysis does not directly yield the orientations, rather, it provides so-called mixed orientation parameters. We therefore show how to decompose the mixed orientation parameters into the individual orientations. These, in turn, allow the superimposed patterns to be separated.
@cite_10: Segmenting dendritic trees and corneal nerve fibres is challenging due to their uneven and irregular appearance in the respective image modalities. State-of-the-art approaches use hand-crafted features based on local assumptions that are often violated by tortuous and point-like structures, e.g., straight tubular shape. We propose a novel ridge detector, SCIRD, which is simultaneously rotation, scale and curvature invariant, and relaxes shape assumptions to achieve enhancement of target image structures. Experimental results on three datasets show that our approach outperforms state-of-the-art hand-crafted methods on tortuous and point-like structures, especially when captured at low resolution or limited signal-to-noise ratio and in the presence of other non-target structures.
Despite the long history of research in this area, improved filtering and detection of ridges continues to be an active topic in medical image analysis. Our work on improving localization through fourth-order diffusion complements recent advances. For example, the SCIRD ridge detector by @cite_10 , or the vesselness measure by @cite_9 that gives better responses for vessels of varying contrasts, could replace the vessel segmentation by @cite_6 that we use as a prefiltering step. Several recent works @cite_24 @cite_29 @cite_13 @cite_15 have addressed diffusion in crossings and bifurcations, and could be combined with our work to improve the performance of our filter in such cases.