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.
Perona-Malik diffusion turns smoothly shaded surfaces into piecewise constant profiles, an effect that is often referred to as a staircasing artifact, and that can be seen in the central Pepper'' image in Figure . To avoid this effect, higher-order diffusion replaces the two first-order spatial derivatives in the heat equation with second-order derivatives. More recently, higher-order PDEs were also generalized to implicit surfaces @cite_27 and image colorization @cite_28 .
@cite_28 @cite_27
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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. 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. In this paper, we propose to conduct inpainting and upsampling for defective depth maps when aligned color images are given. These tasks are referred to as the guided depth enhancement problem. We formulate the problem based on the heat diffusion framework. The pixels with known depth values are treated as the heat sources and the depth enhancement is performed via diffusing the depth from these sources to unknown regions. The diffusion conductivity is designed in terms of the guidance color image so that a linear anisotropic diffusion problem is formed. We further cast the steady state problem of this diffusion into the famous random walk model, by which the enhancement is achieved efficiently by solving a sparse linear system. The proposed algorithm is quantitatively evaluated on the Middlebury stereo dataset and is applied to inpaint Kinect data and upsample Lidar's range data. Comparisons to the commonly used bilateral filter and Markov Random Field based methods are also presented, showing that our algorithm is competent. We extend a recently introduced method for numerically solving partial differential equations on implicit surfaces M. Bertalmio, L.T. Cheng, S. Osher, G. Sapiro. Variational problems and partial differential equations on implicit surfaces, J. Comput. Phys. 174 (2) (2001) 759-780] to fourth order PDEs including the Cahn-Hilliard equation and a lubrication model for curved surfaces. By representing a surface in R N as the level set of a smooth function, ?, we compute the PDE using only finite differences on a standard Cartesian mesh in R N . The higher order equations introduce a number of challenges that are of less concern when applying this method to first and second order PDEs. Many of these problems, such as time-stepping restrictions and large stencil sizes, are shared by standard fourth order equations in Euclidean domains, but others are caused by the extreme degeneracy of the PDEs that result from this method and the general geometry. We approach these difficulties by applying convexity splitting methods, ADI schemes, and iterative solvers. We discuss in detail the differences between computing these fourth order equations and computing the first and second order PDEs considered in earlier work. We explicitly derive schemes for the linear fourth order diffusion, the Cahn-Hilliard equation for phase transition in a binary alloy, and surface tension driven flows on complex geometries. Numerical examples validating our methods are presented for these flows for data on general surfaces.