92 research outputs found
Anisotropic Diffusion Partial Differential Equations in Multi-Channel Image Processing : Framework and Applications
We review recent methods based on diffusion PDE's (Partial Differential Equations) for the purpose of multi-channel image regularization. Such methods have the ability to smooth multi-channel images anisotropically and can preserve then image contours while removing noise or other undesired local artifacts. We point out the pros and cons of the existing equations, providing at each time a local geometric interpretation of the corresponding processes. We focus then on an alternate and generic tensor-driven formulation, able to regularize images while specifically taking the curvatures of local image structures into account. This particular diffusion PDE variant is actually well suited for the preservation of thin structures and gives regularization results where important image features can be particularly well preserved compared to its competitors. A direct link between this curvature-preserving equation and a continuous formulation of the Line Integral Convolution technique (Cabral and Leedom, 1993) is demonstrated. It allows the design of a very fast and stable numerical scheme which implements the multi-valued regularization method by successive integrations of the pixel values along curved integral lines. Besides, the proposed implementation, based on a fourth-order Runge Kutta numerical integration, can be applied with a subpixel accuracy and preserves then thin image structures much better than classical finite-differences discretizations, usually chosen to implement PDE-based diffusions. We finally illustrate the efficiency of this diffusion PDE's for multi-channel image regularization - in terms of speed and visual quality - with various applications and results on color images, including image denoising, inpainting and edge-preserving interpolation
Discretization schemes and numerical approximations of PDE impainting models and a comparative evaluation on novel real world MRI reconstruction applications
While various PDE models are in discussion since the last ten years and are widely applied nowadays in image processing and computer vision tasks, including restoration, filtering, segmentation and object tracking, the perspective adopted in the majority of the relevant reports is the view of applied mathematician, attempting to prove the existence theorems and devise exact numerical methods for solving them. Unfortunately, such solutions are exact for the continuous PDEs but due to the discrete approximations involved in image processing, the results yielded might be quite unsatisfactory. The major contribution of This work is, therefore, to present, from an engineering perspective, the application of PDE models in image processing analysis, from the algorithmic point of view, the discretization and numerical approximation schemes used for solving them. It is of course impossible to tackle all PDE models applied in image processing in this report from the computational point of view. It is, therefore, focused on image impainting PDE models, that is on PDEs, including anisotropic diffusion PDEs, higher order non-linear PDEs, variational PDEs and other constrained/regularized and unconstrained models, applied to image interpolation/ reconstruction. Apart from this novel computational critical overview and presentation of the PDE image impainting models numerical analysis, the second major contribution of This work is to evaluate, especially the anisotropic diffusion PDEs, in novel real world image impainting applications related to MRI
Image Denoising via L
The L0 gradient minimization (LGM) method has been proposed for image smoothing very recently. As an improvement of the total variation (TV) model which employs the L1 norm of the gradient, the LGM model yields much better results for the piecewise constant image. However, just as the TV model, the LGM model also suffers, even more seriously, from the staircasing effect and the inefficiency in preserving the texture in image. In order to overcome these drawbacks, in this paper, we propose to introduce an effective fidelity term into the LGM model. The fidelity term is an exemplar of the moving least square method using steering kernel. Under this framework, these two methods benefit from each other and can produce better results. Experimental results show that the proposed scheme is promising as compared with the state-of-the-art methods
Anisotropic Diffusion Stencils: From Simple Derivations over Stability Estimates to ResNet Implementations
Anisotropic diffusion processes with a diffusion tensor are important in
image analysis, physics, and engineering. However, their numerical
approximation has a strong impact on dissipative artefacts and deviations from
rotation invariance. In this work, we study a large family of finite difference
discretisations on a 3 x 3 stencil. We derive it by splitting 2-D anisotropic
diffusion into four 1-D diffusions. The resulting stencil class involves one
free parameter and covers a wide range of existing discretisations. It
comprises the full stencil family of Weickert et al. (2013) and shows that
their two parameters contain redundancy. Furthermore, we establish a bound on
the spectral norm of the matrix corresponding to the stencil. This gives time
step size limits that guarantee stability of an explicit scheme in the
Euclidean norm. Our directional splitting also allows a very natural
translation of the explicit scheme into ResNet blocks. Employing neural network
libraries enables simple and highly efficient parallel implementations on GPUs
Optimization Techniques for Image Restoration
Many fields of study use images to make discoveries about the past, decisions for the present and predictions for the future. Images often acquire degradations such as a blur due to a patient moving during an x-ray or noise picked up through remote sensing imaging equipment. Images may also lose information through compression or
transmission. In this thesis, diffusion based models were used to solve the image restoration problem as these models can simultaneously remove noise, preserve edges and restore lost information. Specifically, numerical schemes were developed and tested for denoising via nonstandard diffusion that are more computationally efficient than the current method. Furthermore, a new model for digital inpainting is proposed based on the nonstandard diffusion model. Numerical results illustrate the effectiveness of both the denoising and inpainting models in image restoration
Tensor field interpolation with PDEs
We present a unified framework for interpolation and regularisation of scalar- and tensor-valued images. This framework is based on elliptic partial differential equations (PDEs) and allows rotationally invariant models. Since it does not require a regular grid, it can also be used for tensor-valued scattered data interpolation and for tensor field inpainting. By choosing suitable differential operators, interpolation methods using radial basis functions are covered. Our experiments show that a novel interpolation technique based on anisotropic diffusion with a diffusion tensor should be favoured: It outperforms interpolants with radial basis functions, it allows discontinuity-preserving interpolation with no additional oscillations, and it respects positive semidefiniteness of the input tensor data
The jump set under geometric regularisation. Part 1: Basic technique and first-order denoising
Let u \in \mbox{BV}(\Omega) solve the total variation denoising problem
with -squared fidelity and data . Caselles et al. [Multiscale Model.
Simul. 6 (2008), 879--894] have shown the containment of the jump set of in that of . Their proof
unfortunately depends heavily on the co-area formula, as do many results in
this area, and as such is not directly extensible to higher-order,
curvature-based, and other advanced geometric regularisers, such as total
generalised variation (TGV) and Euler's elastica. These have received increased
attention in recent times due to their better practical regularisation
properties compared to conventional total variation or wavelets. We prove
analogous jump set containment properties for a general class of regularisers.
We do this with novel Lipschitz transformation techniques, and do not require
the co-area formula. In the present Part 1 we demonstrate the general technique
on first-order regularisers, while in Part 2 we will extend it to higher-order
regularisers. In particular, we concentrate in this part on TV and, as a
novelty, Huber-regularised TV. We also demonstrate that the technique would
apply to non-convex TV models as well as the Perona-Malik anisotropic
diffusion, if these approaches were well-posed to begin with
Image restoration: Wavelet frame shrinkage, nonlinear evolution PDEs, and beyond
In the past few decades, mathematics based approaches have been widely adopted in various image restoration problems; the partial differential equation (PDE) based approach (e.g., the total variation model [L. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259-268] and its generalizations, nonlinear diffusions [P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intel., 12 (1990), pp. 629-639; F. Catte et al., SIAM J. Numer. Anal., 29 (1992), pp. 182-193], etc.) and wavelet frame based approach are some successful examples. These approaches were developed through different paths and generally provided understanding from different angles of the same problem. As shown in numerical simulations, implementations of the wavelet frame based approach and the PDE based approach quite often end up solving a similar numerical problem with similar numerical behaviors, even though different approaches have advantages in different applications. Since wavelet frame based and PDE based approaches have all been modeling the same types of problems with success, it is natural to ask whether the wavelet frame based approach is fundamentally connected with the PDE based approach when we trace them all the way back to their roots. A fundamental connection of a wavelet frame based approach with a total variation model and its generalizations was established in [J. Cai, B. Dong, S. Osher, and Z. Shen, J. Amer. Math. Soc., 25 (2012), pp. 1033-1089]. This connection gives the wavelet frame based approach a geometric explanation and, at the same time, it equips a PDE based approach with a time frequency analysis. Cai et al. showed that a special type of wavelet frame model using generic wavelet frame systems can be regarded as an approximation of a generic variational model (with the total variation model as a special case) in the discrete setting. A systematic convergence analysis, as the resolution of the image goes to infinity, which is the key step in linking the two approaches, is also given in Cai et al. Motivated by Cai et al. and [Q. Jiang, Appl. Numer. Math., 62 (2012), pp. 51-66], this paper establishes a fundamental connection between the wavelet frame based approach and nonlinear evolution PDEs, provides interpretations and analytical studies of such connections, and proposes new algorithms for image restoration based on the new understandings. Together with the results in [J. Cai et al., J. Amer. Math. Soc., 25 (2012), pp. 1033-1089], we now have a better picture of how the wavelet frame based approach can be used to interpret the general PDE based approach (e.g., the variational models or nonlinear evolution PDEs) and can be used as a new and useful tool in numerical analysis to discretize and solve various variational and PDE models. To be more precise, we shall establish the following: (1) The connections between wavelet frame shrinkage and nonlinear evolution PDEs provide new and inspiring interpretations of both approaches that enable us to derive new PDE models and (better) wavelet frame shrinkage algorithms for image restoration. (2) A generic nonlinear evolution PDE (of parabolic or hyperbolic type) can be approximated by wavelet frame shrinkage with properly chosen wavelet frame systems and carefully designed shrinkage functions. (3) The main idea of this work is beyond the scope of image restoration. Our analysis and discussions indicate that wavelet frame shrinkage is a new way of solving PDEs in general, which will provide a new insight that will enrich the existing theory and applications of numerical PDEs, as well as those of wavelet frames
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