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
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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. In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion. In general the resulting diffusion tensor is anisotropic, and we give necessary and sufficient conditions under which it is isotropic. We also give a method to construct approximations of arbitrary high order for large times. In a second paper (Part II) we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion. Depending on the strength of the bias, it may lead to an anisotropic diffusion equation, to a drift term in the flux, or to both. Our analysis generalizes and simplifies previous derivations that lead to the classical Patlak--Keller--Segel--Alt... In this paper, we introduce a feature-preserving denoising algorithm. It is built on the premise that the underlying surface of a noisy mesh is piecewise smooth, and a sharp feature lies on the intersection of multiple smooth surface regions. A vertex close to a sharp feature is likely to have a neighborhood that includes distinct smooth segments. By defining the consistent subneighborhood as the segment whose geometry and normal orientation most consistent with those of the vertex, we can completely remove the influence from neighbors lying on other segments during denoising. Our method identifies piecewise smooth subneighborhoods using a robust density-based clustering algorithm based on shared nearest neighbors. In our method, we obtain an initial estimate of vertex normals and curvature tensors by robustly fitting a local quadric model. An anisotropic filter based on optimal estimation theory is further applied to smooth the normal field and the curvature tensor field. This is followed by second-order bilateral filtering, which better preserves curvature details and alleviates volume shrinkage during denoising. The support of these filters is defined by the consistent subneighborhood of a vertex. We have applied this algorithm to both generic and CAD models, and sharp features, such as edges and corners, are very well preserved. Fourth-order nonlinear diffusion filters used for image noise removal are mainly isotropic filters. It means that the spatially varying diffusivity determined by a diffusion function is applied on the image regardless of the orientation of its local features. However, the optimal choice of parameters in the numerical solver of these filters for having a minimal distortion of the image features results in forming speckle noise on the denoised image and a very slow convergence rate especially when the noise level is moderately high. In this paper, a new fourth-order nonlinear diffusion filter is introduced, which has an anisotropic behavior on the image features. In the proposed filter, it is shown that a suitable choice for a set of diffusivity functions to unevenly control the strength of the diffusion on the directions of the level set and gradient leads to a good edge preservation capability compared to the other diffusion or regularization filters. The comparison of the results obtained by the proposed filter with those of the other second and fourth-order filters shows that the proposed method produces a noticeable improvement in the quality of denoised images evaluated subjectively and quantitatively.