Decay and Global Existence for Nonlinear Wave Equations with Localized Dissipations in General Exterior Domains

IEEE Trans. on Image Process.

2011

Abstract-The conflicting demands for simultaneous low-pass and high-pass processing, required in image denoising and enhancement, still present an outstanding challenge, although a great deal of progress has been made by means of adaptive diffusion-type algorithms. To further advance such processing methods and algorithms, we introduce a family of second-order (in time) partial differential equations. These equations describe the motion of a thin elastic sheet in a damping environment. They are also derived by a variational approach in the context of image processing. The new operator enables better edge preservation in denoising applications by offering an adaptive lowpass filter, which preserves high-frequency components in the pass-band better than the adaptive diffusion filter, while offering slower error propagation across edges. We explore the action of this powerful operator in the context of image processing and exploit for this purpose the wealth of knowledge accumulated in physics and mathematics about the action and behavior of this operator. The resulting methods are further generalized for color and/or texture image processing, by embedding images in multidimensional manifolds. A specific application of the proposed new approach to superresolution is outlined.

“…Convergence of the continuous TeD was proven by Nakao [17]. Initial analysis, supported by computational results, indicates that the proposed discrete image-processing scheme is stable.…”

Section: Discussionmentioning

confidence: 64%

Section: B Elastic Sheet Approachmentioning

confidence: 99%

Denoising-Enhancing Images on Elastic Manifolds

IEEE Trans. on Image Process.

2011

“…(2.1) not only for small a, but also after very long time [10], [11]. Given positive and bounded coefficients, the TeD converges to a unique bounded solution ( [16]). It is important to note that although (2.3) is a wave equation, in most cases discussed here the wave-like nature is suppressed by over-damping (i.e.…”

Section: Telegraph Diffusionmentioning

confidence: 96%

“…This parabolic-hyperbolic equation is often encountered in various fields, such as description of random motion of particles ([22], [23]), transmission of signals over telegraph wires (hence the terminology) and others ([24], [25]. It has also been thoroughly investigated from a mathematical viewpoint ( [11], [16], [26], [27]). …”

Section: Telegraph Diffusionmentioning

confidence: 99%

The dynamics of image processing viewed as deformation of elastic sheet

2009 16th International Conference on Digital Signal Processing

2009

Diffusion-type algorithms have been integrated successfully into the toolbox used in image processing and computer vision. We introduce in the context of digital signal and image processing a new more flexible and powerful family of parabolic-hyperbolic partial differential equations (PDEs) that somewhat resembles the structure of the parabolic diffusion equation, but incorporates the second order derivative in time. It is instructive intuitively to consider in this context the dynamics of image processing as the deformation of an 'elastic sheet'. Indeed, our parabolichyperbolic PDE models elastic deformation. This analogy between a well-known physical system and process on one hand, and the dynamics of an image processing scheme on the other hand, contributes interesting and important insight about images and their processing. We explore and demonstrate the capabilities and advantages afforded by the application of the proposed family of equations in image enhancement. Efficient numeric schemes are also presented.Index Terms-Parabolic-hyperbolic equations, PDE image processing, image enhancement, denoising, image edge analysis

“…A method proposed by Weickert [9] achieves it by manipulating eigenvalues of a smoothed structure tensor (7), enhancing diffusion along edges and reducing, or even reversing it across them. This is achieved by replacing the member in the diffusion and TeD equations by the following: k u (5) .…”

Section: Tensor Telegraph-diffusionmentioning

confidence: 99%

Image representation and enhancement on elastic manifolds

2008 Ph.D. Research in Microelectronics and Electronics

2008

-Enhancing detail and contours of objects in low resolution noisy natural images is still one of the most outstanding challenges in image processing. This study is devoted to a new approach to adaptive image enhancement, based on the application of a damped elastic deformation process to single images. The proposed approach results in a Telegraph-Diffusion (TeD) denoising-and-sharpening filtering scheme which enhances fine details in images. Three efficient numerical schemes are presented. Advantages of the algorithm are discussed with reference to computational results.(2) , where is the spatial frequency and t is time (Fig. 1). Varying diffusivity, k, over an image in an adaptive manner allows feature dependant smoothing. This, in turn, facilitates selective denoising, while preserving important image features, e.g. strong smoothing of flat areas (where most fluctuations in image contrast are due to noise), and almost no smoothing around edges [6].In [2], Gilboa et. al. extended the diffusion based processing to the ill-posed negative time regime. They have shown that when properly localized, forward-and-backward (FAB) diffusion results in edge enhancement.Weickert ([9]) proposed a fully anisotropic diffusion filtering scheme, where the application of structure tensor instead of scalar coefficient k, allows control also over orientation of smoothing and not only its strength.