You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
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.

Cited Abstract(s):
[1]: This article introduces generalizations of the Perona-Malik (1990) equation. An edge enhancing functional is proposed for direct edge enhancement. A number of super diffusion operators is introduced for fast and effective smoothing. Statistical information is utilized for robust edge-stopping. Numerical integration is conducted by using a previously developed quasi-interpolating wavelet method. Computer experiments indicate that the present algorithm is very efficient for edge-detecting and noise-removing.

[2]: High contrast scenes are difficult to depict on low contrast displays without loss of important fine details and textures. Skilled artists preserve these details by drawing scene contents in coarseto-fine order using a hierarchy of scene boundaries and shadings. We build a similar hierarchy using multiple instances of a new low curvature image simplifier (LCIS), a partial differential equation inspired by anisotropic diffusion. Each LCIS reduces the scene to many smooth regions that are bounded by sharp gradient discontinuities, and a single parameter K chosen for each LCIS controls region size and boundary complexity. With a few chosen K values (K1 > K2 > K3:::) LCIS makes a set of progressively simpler images, and image differences form a hierarchy of increasingly important details, boundaries and large features. We construct a high detail, low contrast display image from this hierarchy by compressing only the large features, then adding back all small details. Unlike linear filter hierarchies such as wavelets, filter banks, or image pyramids, LCIS hierarchies do not smooth across scene boundaries, avoiding “halo” artifacts common to previous contrast reducing methods and some tone reproduction operators. We demonstrate LCIS effectiveness on several example images. CR Descriptors: I.3.3 [Computer Graphics]: Picture image generation Display algorithms; I.4.1 [Image Processing and Computer Vision]: Enhancement -Digitization and Image Capture
Two very similar higher order methods based on evolution equations without underlying variational formulation have been introduced by Tumblin and Turk [2] and Wei [1] . They use fourth-order evolution equations of the form where @math is the gradient norm [2] or the Frobenius norm of the Hessian [1] .