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_28: The work of (2004) popularized stroke-based methods that add color to gray value images according to a small amount of user-specified color samples. Even though such reconstructions from sparse data suggest a possible use in compression, only few attempts were made so far in this direction. Diffusion-based compression methods pursue a similar idea: they store only few image pixels and inpaint the missing regions. Despite this close relation and a lack of diffusion-based color codecs, colorization ideas were so far only integrated into transform-based approaches such as JPEG. We address this missing link with two contributions. First, we show the relation between the discrete colorization of and continuous diffusion-based inpainting in the YCbCr color space. It decomposes the image into a luma (brightness) channel and two chroma (color) channels. Our luma-guided diffusion framework steers the diffusion inpainting in the chroma channels according to the structure in the luma channel. We show that making the luma-guided colorization anisotropic outperforms the method of significantly. Second, we propose a new luma preference codec that invests a large fraction of the bit budget into an accurate representation of the luma channel. This allows a high-quality reconstruction of color data with our colorization technique. Simultaneously, we exploit the fact that the human visual system is more sensitive to structural than to color information. Our experiments demonstrate that our new codec outperforms the state of the art in diffusion-based image compression and is competitive to transform-based codecs.
@cite_27: 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.
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 .