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58
Variational Restoration Of Nonflat Image Features: Models And Algorithms
, 2000
"... We develop both mathematical models and computational algorithms for variational denoising and restoration of nonflat image features. Nonflat image features are those that live on Riemannian manifolds, instead of on the Euclidean spaces. Familiar examples include the orientation feature (from optica ..."
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Cited by 89 (13 self)
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We develop both mathematical models and computational algorithms for variational denoising and restoration of nonflat image features. Nonflat image features are those that live on Riemannian manifolds, instead of on the Euclidean spaces. Familiar examples include the orientation feature (from optical flows or gradient flows) that lives on the unit circle S&sup1;, the alignment feature (from fingerprint waves or certain texture images) that lives on the real projective line RP&sup1; and the chromaticity feature (from color images) that lives on the unit sphere S&sup2;. In this paper, we apply the variational method to denoise and restore general nonflat image features. Mathematical models for both continuous image domains and discrete domains (or graphs) are constructed. Riemannian objects such as metric, distance and LeviCivita connection play important roles in the models. Computational algorithms are also developed for the resulting nonlinear equations. The mathematical framework can be applied to restoring general nonflat data outside the scope of image processing and computer vision.
Anisotropic Diffusion of Surfaces and Functions on Surfaces
, 2002
"... We present a unified anisotropic geometric diffusion PDE model for smoothing (fairing) out noise both in triangulated 2manifold surface meshes in IR 3 and functions defined on these surface meshes, while enhancing curve features on both by careful choice of an anisotropic diffusion tensor. We combi ..."
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Cited by 75 (8 self)
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We present a unified anisotropic geometric diffusion PDE model for smoothing (fairing) out noise both in triangulated 2manifold surface meshes in IR 3 and functions defined on these surface meshes, while enhancing curve features on both by careful choice of an anisotropic diffusion tensor. We combine the C 1 limit representation of Loop’s subdivision for triangular surface meshes and vector functions on the surface mesh with the established diffusion model to arrive at a discretized version of the diffusion problem in the spatial direction. The time direction discretization then leads to a sparse linear system of equations. Iteratively, solving the sparse linear system, yields a sequence of faired (smoothed) meshes as well as faired functions.
Diffusion of General Data on NonFlat Manifolds via Harmonic Maps Theory: The Direction Diffusion Case
 Journal Computer Vision
, 2000
"... Abstract. In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to denoise directional data and to obtain multiscale representati ..."
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Cited by 65 (6 self)
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Abstract. In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to denoise directional data and to obtain multiscale representations of it. The basic idea is to apply and extend results from the theory of harmonic maps, and in particular, harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the intrinsic unit norm constraint of directional data. We show the corresponding variational and partial differential equations formulations for isotropic diffusion, obtained from an L2 norm, and edge preserving diffusion, obtained from an L p norm in general and an L1 norm in particular. In contrast with previous approaches, the framework is valid for directions in any dimensions, supports nonsmooth data, and gives both isotropic and anisotropic formulations. In addition, the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general nonflat manifolds, that is, functions between two general manifolds. We present a number of theoretical results, open questions, and examples for gradient vectors, optical flow, and color images.
An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise
 SIAM J. SCI. COMPUT
, 2009
"... We extend the alternating minimization algorithm recently proposed in [38, 39] to the case of recovering blurry multichannel (color) images corrupted by impulsive rather than Gaussian noise. The algorithm minimizes the sum of a multichannel extension of total variation (TV), either isotropic or anis ..."
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Cited by 50 (8 self)
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We extend the alternating minimization algorithm recently proposed in [38, 39] to the case of recovering blurry multichannel (color) images corrupted by impulsive rather than Gaussian noise. The algorithm minimizes the sum of a multichannel extension of total variation (TV), either isotropic or anisotropic, and a data fidelity term measured in the L1norm. We derive the algorithm by applying the wellknown quadratic penalty function technique and prove attractive convergence properties including finite convergence for some variables and global qlinear convergence. Under periodic boundary conditions, the main computational requirements of the algorithm are fast Fourier transforms and a lowcomplexity Gaussian elimination procedure. Numerical results on images with different blurs and impulsive noise are presented to demonstrate the efficiency of the algorithm. In addition, it is numerically compared to an algorithm recently proposed in [20] that uses a linear program and an interior point method for recovering grayscale images.
A fast algorithm for edgepreserving variational multichannel image restoration
"... Abstract. We generalize the alternating minimization algorithm recently proposed in [32] to efficiently solve a general, edgepreserving, variational model for recovering multichannel images degraded by within and crosschannel blurs, as well as additive Gaussian noise. This general model allows th ..."
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Cited by 45 (9 self)
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Abstract. We generalize the alternating minimization algorithm recently proposed in [32] to efficiently solve a general, edgepreserving, variational model for recovering multichannel images degraded by within and crosschannel blurs, as well as additive Gaussian noise. This general model allows the use of localized weights and higherorder derivatives in regularization, and includes a multichannel extension of total variation (MTV) regularization as a special case. In the MTV case, we show that the model can be derived from an extended halfquadratic transform of Geman and Yang [14]. For color images with three channels and when applied to the MTV model (either locally weighted or not), the periteration computational complexity of this algorithm is dominated by nine fast Fourier transforms. We establish strong convergence results for the algorithm including finite convergence for some variables and fast qlinear convergence for the others. Numerical results on various types of blurs are presented to demonstrate the performance of our algorithm compared to that of the MATLAB deblurring functions. We also present experimental results on regularization models using weighted MTV and higherorder derivatives to demonstrate improvements in image quality provided by these models over the plain MTV model.
Numerical Methods for pHarmonic Flows and Applications to Image Processing
 SIAM J. NUMER. ANAL
, 2002
"... We propose in this paper an alternative approach for computing pharmonic maps and flows: instead of solving a constrained minimization problem on S N i, we solve an unconstrained minimization problem on the entire space of functions. This is possible, using the projection on the sphere of any arbi ..."
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Cited by 43 (6 self)
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We propose in this paper an alternative approach for computing pharmonic maps and flows: instead of solving a constrained minimization problem on S N i, we solve an unconstrained minimization problem on the entire space of functions. This is possible, using the projection on the sphere of any arbitrary function. Then we show how this formulation can be used in practice, for problems with both isotropic and anisotropic diffusion, with applications to image processing, using a new finite difference scheme.
Orientation Diffusion or How to comb a Porcupine
 Journal of Visual Communication and Image Representation
, 2001
"... This paper addresses the problem of feature enhancement in noisy images, when the feature is known to be constrained to a manifold. As an example, we approach the orientation denoising problem via the geometric Beltrami framework for image processing. The feature (orientation) field is represented a ..."
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Cited by 37 (6 self)
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This paper addresses the problem of feature enhancement in noisy images, when the feature is known to be constrained to a manifold. As an example, we approach the orientation denoising problem via the geometric Beltrami framework for image processing. The feature (orientation) field is represented accordingly as the embedding of a two dimensional surface in the spatialfeature manifold. The resulted Beltrami flow is a selective smoothing process that respects the feature constraint. Orientation diffusion is treated as a canonical example where the feature (orientation in this case) space is the unit circle S1. Applications to color analysis are discussed and numerical experiments demonstrate again the power of the Beltrami framework for nontrivial geometries in image processing. C ○ 2002 Elsevier Science (USA) 1.
Solving variational problems and partial differential equations mapping into general target manifolds
 J. Comput. Phys
, 2004
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Variational models for image colorization via Chromaticity and Brightness decomposition
 IEEE TRANS. IMAGE PROC
, 2006
"... Colorization refers to an image processing task which recovers color of gray scale images when only small regions with color are given. We propose a couple of variational models using chromaticity color component to colorize black and white images. We first consider Total Variation minimizing (TV) ..."
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Cited by 19 (1 self)
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Colorization refers to an image processing task which recovers color of gray scale images when only small regions with color are given. We propose a couple of variational models using chromaticity color component to colorize black and white images. We first consider Total Variation minimizing (TV) colorization which is an extension from TV inpainting to color using chromaticity model. Secondly, we further modify our model to weighted harmonic maps for colorization. This model adds edge information from the brightness data, while it reconstructs smooth color values for each homogeneous region. We introduce penalized versions of the variational models, we analyze their convergence properties, and we present numerical results including extension to texture colorization.
SURELET Multichannel Image Denoising: Interscale Orthonormal Wavelet Thresholding
, 2008
"... We propose a vector/matrix extension of our denoising algorithm initially developed for grayscale images, in order to efficiently process multichannel (e.g., color) images. This work follows our recently published SURELET approach where the denoising algorithm is parameterized as a linear expansion ..."
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Cited by 18 (3 self)
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We propose a vector/matrix extension of our denoising algorithm initially developed for grayscale images, in order to efficiently process multichannel (e.g., color) images. This work follows our recently published SURELET approach where the denoising algorithm is parameterized as a linear expansion of thresholds (LET) and optimized using Stein’s unbiased risk estimate (SURE). The proposed wavelet thresholding function is pointwise and depends on the coefficients of same location in the other channels, as well as on their parents in the coarser wavelet subband. A nonredundant, orthonormal, wavelet transform is first applied to the noisy data, followed by the (subbanddependent) vectorvalued thresholding of individual multichannel wavelet coefficients which are finally brought back to the image domain by inverse wavelet transform. Extensive comparisons with the stateoftheart multiresolution image denoising algorithms indicate that despite being nonredundant, our algorithm matches the quality of the best redundant approaches, while maintaining a high computational efficiency and a low CPU/memory consumption. An online Java demo illustrates these assertions.