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25
An Algorithm for Total Variation Minimization and Applications
, 2004
"... We propose an algorithm for minimizing the total variation of an image, and provide a proof of convergence. We show applications to image denoising, zooming, and the computation of the mean curvature motion of interfaces. ..."
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Cited by 634 (8 self)
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We propose an algorithm for minimizing the total variation of an image, and provide a proof of convergence. We show applications to image denoising, zooming, and the computation of the mean curvature motion of interfaces.
Edge Direction Preserving Image Zooming: A Mathematical and Numerical Analysis
- SIAM Journal on Numerical Analysis
, 2000
"... We focus in this paper on some reconstruction/restoration methods which aim is to improve the resolution of digital images. The main point is here to study the ability of such methods to preserve 1D structures. Indeed such structures are important since they are often carried by the image "edge ..."
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Cited by 54 (5 self)
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We focus in this paper on some reconstruction/restoration methods which aim is to improve the resolution of digital images. The main point is here to study the ability of such methods to preserve 1D structures. Indeed such structures are important since they are often carried by the image "edges". We first focus on linear methods, give a general framework to design them and show that the preservation of 1D structures pleads in favor of the cancellation of the periodization of the image spectrum. More precisely, we show that preserving 1D structures implies the linear methods to be written as a convolution of the "sinc interpolation". As a consequence, we can not cope linearly with Gibbs effects, sharpness of the results and the preservation of the 1D structure. Secondly, we study variational nonlinear methods and in particular the one based on total variation. We show that this latter permits to avoid these shortcomings. We also prove the existence and consistency of an approximate sol...
A partial differential equation approach to image zoom
- Image Processing, 2004. ICIP ’04. 2004 International Conference on, 1:649–652
, 2004
"... We propose a new model for zooming digital image. This model, driven by a partial differential equation, will balance between linear zooming on homogenous zones to anisotropic diffusion near edges. This allows to combines advantages on linear zoom models and of some non linear ones while leaving out ..."
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Cited by 9 (0 self)
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We propose a new model for zooming digital image. This model, driven by a partial differential equation, will balance between linear zooming on homogenous zones to anisotropic diffusion near edges. This allows to combines advantages on linear zoom models and of some non linear ones while leaving out their drawbacks. 1
How to discretize the total variation of an image
- in CIAM07 Minisymposia - Part. Diff. Eq
, 2007
"... Because they are based on finite differences, usual discretizations of the Total Variation lead to aliased images. We propose a new discretization called spectral total variation that agrees with Shannon sampling principles and produces images that can be exactly interpolated. The quality improvemen ..."
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Cited by 6 (0 self)
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Because they are based on finite differences, usual discretizations of the Total Variation lead to aliased images. We propose a new discretization called spectral total variation that agrees with Shannon sampling principles and produces images that can be exactly interpolated. The quality improvement is illustrated experimentally in the case of image deblurring. 1
Vector-valued image interpolation by an anisotropic diffusion-projection PDE
- Scale Space and Variational Methods in Computer Vision, volume 4485 of Lecture Notes in Computer Science
, 2007
"... Abstract. We propose a nonlinear image interpolation method, based on an anisotropic diffusion PDE and designed for the general case of vector-valued images. The interpolation solution is restricted to the subspace of functions that can recover the discrete input image, after an appropriate smoothin ..."
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Cited by 5 (1 self)
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Abstract. We propose a nonlinear image interpolation method, based on an anisotropic diffusion PDE and designed for the general case of vector-valued images. The interpolation solution is restricted to the subspace of functions that can recover the discrete input image, after an appropriate smoothing and sampling. The proposed nonlinear diffusion flow lies on this subspace and its strength and anisotropy effectively adapt to the local variations and geometry of image structures. The derived model efficiently reconstructs the real image structures, leading to a natural interpolation, with reduced blurring, staircase and ringing artifacts of classic methods. This method also outperforms other existing PDE-based interpolation methods. We present experimental results that prove the potential and efficacy of the method as applied to graylevel and color images. 1
Total variation denoising using posterior expectation
- In Proceedings of the European Signal Processing Conference (Eusipco). Eurasip
, 2008
"... Total Variation image denoising, generally formulated in a variational setting, can be seen as a Maximum A Posteriori (MAP) Bayesian estimate relying on a simple explicit image prior. In this formulation, the denoised image is the most likely image of the posterior distribution, which favors regular ..."
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Cited by 5 (2 self)
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Total Variation image denoising, generally formulated in a variational setting, can be seen as a Maximum A Posteriori (MAP) Bayesian estimate relying on a simple explicit image prior. In this formulation, the denoised image is the most likely image of the posterior distribution, which favors regularity and produces staircasing artifacts: in regions where smooth-varying intensities would be expected, constant zones appear separated by artificial boundaries. In this paper, we propose to use the Least Square Error (LSE) criterion instead of the MAP. This leads to a new denoising method called TV-LSE, that produces more realistic images by computing the expectation of the posterior distribution. We describe a Monte-Carlo Markov Chain algorithm based on Metropolis scheme, and provide an efficient convergence criterion. We also discuss the properties of TV-LSE, and show in particular that it does not suffer from the staircasing effect. 1.
Reversible interpolation of vectorial images by an anisotropic diffusion-projection PDE
- Int. J. Comput. Vis
, 2009
"... Abstract In this paper, a nonlinear model for the interpolation of vector-valued images is proposed. This model is based on an anisotropic diffusion PDE and performs an interpolation that is reversible. The interpolation solution is restricted to the subspace of functions that can recover the discre ..."
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Cited by 2 (0 self)
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Abstract In this paper, a nonlinear model for the interpolation of vector-valued images is proposed. This model is based on an anisotropic diffusion PDE and performs an interpolation that is reversible. The interpolation solution is restricted to the subspace of functions that can recover the discrete input image, after an appropriate smoothing and sampling. The proposed nonlinear diffusion flow lies on this subspace while its strength and anisotropy adapt to the local variations and geometry of image structures. The derived method effectively reconstructs the real image structures and yields a satisfactory interpolation result. Compared to classic and other existing PDE-based interpolation methods, our proposed method seems to increase the accuracy of the result and to reduce the undesirable artifacts, such as blurring, ringing, block effects and edge distortion. We present extensive experimental results that demonstrate the potential of the method as applied to graylevel and color images.
Super-Resolution Using Sub-band Constrained Total Variation
"... Abstract. Super-resolution of a single image is a severely ill-posed prob-lem in computer vision. It is possible to consider solving this problem by considering a total variation based regularization framework. The choice of total variation based regularization helps in formulating an edge pre-servi ..."
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Abstract. Super-resolution of a single image is a severely ill-posed prob-lem in computer vision. It is possible to consider solving this problem by considering a total variation based regularization framework. The choice of total variation based regularization helps in formulating an edge pre-serving scheme for super-resolution. However, this scheme tends to re-sult in a piece-wise constant resultant image. To address this issue, we extend the formulation by incorporating an appropriate sub-band con-straint which ensures the preservation of textural details in trade off with noise present in the observation. The proposed framework is extensively evaluated and the experimental results for the same are presented. 1
Regularized image up-sampling
, 2004
"... This thesis addresses the problem of performing image magnification to achieve higher perceived resolution for grey-scale and color images. A new perspective on the prob-lem is introduced through the new concept of a theoretical camera that can acquire an ideal high resolution image. A new formulati ..."
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Cited by 1 (0 self)
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This thesis addresses the problem of performing image magnification to achieve higher perceived resolution for grey-scale and color images. A new perspective on the prob-lem is introduced through the new concept of a theoretical camera that can acquire an ideal high resolution image. A new formulation of the problem is then introduced using two ingredients: a newly designed observation model and the total-variation regularizer. An observation model, that establishes a generalized relation between the desired magnified image and the measured lower resolution image, has been newly designed based on careful study of the physical acquisition processes that have generated the images. The result is a major contribution of this thesis: a closed-form solution for obtaining the observation model. This closed form has been implemented and observation models were obtained for different typical scenarios, and their performance was shown to outperform observation models used in the literature. Two new theorems for designing the theoretical camera, adapted to the display device used, on arbitrary lattices have been developed. The thesis presents new analysis with a signal
