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39
Efficient and reliable schemes for nonlinear diffusion filtering
 IEEE Transactions on Image Processing
, 1998
"... AbstractNonlinear diffusion filtering is usually performed with explicit schemes. They are only stable for very small time steps, which leads to poor efficiency and limits their practical use. Based on a recent discrete nonlinear diffusion scalespace framework we present semiimplicit schemes whi ..."
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Cited by 231 (21 self)
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AbstractNonlinear diffusion filtering is usually performed with explicit schemes. They are only stable for very small time steps, which leads to poor efficiency and limits their practical use. Based on a recent discrete nonlinear diffusion scalespace framework we present semiimplicit schemes which are stable for all time steps. These novel schemes use an additive operator splitting (AOS), which guarantees equal treatment of all coordinate axes. They can be implemented easily in arbitrary dimensions, have good rotational invariance and reveal a computational complexity and memory requirement which is linear in the number of pixels. Examples demonstrate that, under typical accuracy requirements, AOS schemes are at least ten times more efficient than the widely used explicit schemes.
VectorValued Image Regularization with PDEs: A Common Framework for Different Applications
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2003
"... We address the problem of vectorvalued image regularization with variational methods and PDE's. From the study of existing formalisms, we propose a unifying framework based on a very local interpretation of the regularization processes. The resulting equations are then specialized into new reg ..."
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Cited by 181 (8 self)
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We address the problem of vectorvalued image regularization with variational methods and PDE's. From the study of existing formalisms, we propose a unifying framework based on a very local interpretation of the regularization processes. The resulting equations are then specialized into new regularization PDE's and corresponding numerical schemes that respect the local geometry of vectorvalued images. They are finally applied on a wide variety of image processing problems, including color image restoration, inpainting, magnification and flow visualization.
A Common Framework for Nonlinear Diffusion, Adaptive Smoothing, Bilateral Filtering and Mean Shift
, 2004
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Fast Anisotropic Smoothing of MultiValued Images using CurvaturePreserving PDE’s
 Research Report “Les Cahiers du GREYC”, No 05/01. Equipe IMAGE/GREYC (CNRS UMR 6072), Février
, 2005
"... We are interested in PDE’s (Partial Differential Equations) in order to smooth multivalued images in an anisotropic manner. Starting from a review of existing anisotropic regularization schemes based on diffusion PDE’s, we point out the pros and cons of the different equations proposed in the liter ..."
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Cited by 65 (3 self)
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We are interested in PDE’s (Partial Differential Equations) in order to smooth multivalued images in an anisotropic manner. Starting from a review of existing anisotropic regularization schemes based on diffusion PDE’s, we point out the pros and cons of the different equations proposed in the literature. Then, we introduce a new tensordriven PDE, regularizing images while taking the curvatures of specific integral curves into account. We show that this constraint is particularly well suited for the preservation of thin structures in an image restoration process. A direct link is made between our proposed equation and a continuous formulation of the LIC’s (Line Integral Convolutions by Cabral and Leedom [11]). It leads to the design of a very fast and stable algorithm that implements our regularization method, by successive integrations of pixel values along curved integral lines. Besides, the scheme numerically performs with a subpixel accuracy and preserves then thin image structures better than classical finitedifferences discretizations. Finally, we illustrate the efficiency of our generic curvaturepreserving approach in terms of speed and visual quality with different comparisons and various applications requiring image smoothing: color images denoising, inpainting and image resizing by nonlinear interpolation.
A Generative Model for Image Segmentation Based on Label Fusion
 IEEE TRANSACTIONS IN MEDICAL IMAGING
, 2010
"... We propose a nonparametric, probabilistic model for the automatic segmentation of medical images, given a training set of images and corresponding label maps. The resulting inference algorithms rely on pairwise registrations between the test image and individual training images. The training labels ..."
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Cited by 62 (5 self)
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We propose a nonparametric, probabilistic model for the automatic segmentation of medical images, given a training set of images and corresponding label maps. The resulting inference algorithms rely on pairwise registrations between the test image and individual training images. The training labels are then transferred to the test image and fused to compute the final segmentation of the test subject. Such label fusion methods have been shown to yield accurate segmentation, since the use of multiple registrations captures greater intersubject anatomical variability and improves robustness against occasional registration failures. To the best of our knowledge, this manuscript presents the first comprehensive probabilistic framework that rigorously motivates label fusion as a segmentation approach. The proposed framework allows us to compare different label fusion algorithms theoretically and practically. In particular, recent label fusion or multiatlas segmentation
Spherical Demons: Fast Diffeomorphic LandmarkFree Surface Registration
 IEEE TRANSACTIONS ON MEDICAL IMAGING. 29(3):650–668, 2010
, 2010
"... We present the Spherical Demons algorithm for registering two spherical images. By exploiting spherical vector spline interpolation theory, we show that a large class of regularizors for the modified Demons objective function can be efficiently approximated on the sphere using iterative smoothing. B ..."
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Cited by 25 (5 self)
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We present the Spherical Demons algorithm for registering two spherical images. By exploiting spherical vector spline interpolation theory, we show that a large class of regularizors for the modified Demons objective function can be efficiently approximated on the sphere using iterative smoothing. Based on one parameter subgroups of diffeomorphisms, the resulting registration is diffeomorphic and fast. The Spherical Demons algorithm can also be modified to register a given spherical image to a probabilistic atlas. We demonstrate two variants of the algorithm corresponding to warping the atlas or warping the subject. Registration of a cortical surface mesh to an atlas mesh, both with more than 160k nodes requires less than 5 minutes when warping the atlas and less than 3 minutes when warping the subject on a Xeon 3.2GHz single processor machine. This is comparable to the fastest nondiffeomorphic landmarkfree surface registration algorithms. Furthermore, the accuracy of our method compares favorably to the popular FreeSurfer registration algorithm. We validate the technique in two different applications that use registration to transfer segmentation labels onto a new image: (1) parcellation of invivo cortical surfaces and (2) Brodmann area localization in exvivo cortical surfaces.
On local region models and a statistical interpretation of the piecewise smooth MumfordShah functional
 Int. J. Comput. Vision
"... The MumfordShah functional is a general and quite popular variational model for image segmentation. In particular, it provides the possibility to represent regions by smooth approximations. In this paper, we derive a statistical interpretation of the full (piecewise smooth) MumfordShah functional ..."
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Cited by 24 (3 self)
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The MumfordShah functional is a general and quite popular variational model for image segmentation. In particular, it provides the possibility to represent regions by smooth approximations. In this paper, we derive a statistical interpretation of the full (piecewise smooth) MumfordShah functional by relating it to recent works on local region statistics. Moreover, we show that this statistical interpretation comes along with several implications. Firstly, one can derive extended versions of the MumfordShah functional including more general distribution models. Secondly, it leads to faster implementations. Finally, thanks to the analytical expression of the smooth approximation via Gaussian convolution, the coordinate descent can be replaced by a true gradient descent. 1.
On the statistical interpretation of the piecewise smooth Mumford–Shah functional
 in Scale Space and Variational Methods in Computer Vision, Lecture Notes in Comput. Sci. 4485
, 2007
"... Abstract. In regionbased image segmentation, two models dominate the field: the MumfordShah functional and statistical approaches based on Bayesian inference. Whereas the latter allow for numerous ways to describe the statistics of intensities in regions, the first includes spatially smooth approx ..."
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Cited by 20 (2 self)
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Abstract. In regionbased image segmentation, two models dominate the field: the MumfordShah functional and statistical approaches based on Bayesian inference. Whereas the latter allow for numerous ways to describe the statistics of intensities in regions, the first includes spatially smooth approximations. In this paper, we show that the piecewise smooth MumfordShah functional is a first order approximation of Bayesian aposteriori maximization where region statistics are computed in local windows. This equivalence not only allows for a statistical interpretation of the full MumfordShah functional. Inspired by the Bayesian model, it also offers to formulate an extended MumfordShah functional that takes the variance of the data into account. 1
Splines in Higher Order TV Regularization
, 2006
"... Splines play an important role as solutions of various interpolation and approximation problems that minimize special functionals in some smoothness spaces. In this paper, we show in a strictly discrete setting that splines of degree m−1 solve also a minimization problem with quadratic data term a ..."
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Cited by 16 (3 self)
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Splines play an important role as solutions of various interpolation and approximation problems that minimize special functionals in some smoothness spaces. In this paper, we show in a strictly discrete setting that splines of degree m−1 solve also a minimization problem with quadratic data term and mth order total variation (TV) regularization term. In contrast to problems with quadratic regularization terms involving mth order derivatives, the spline knots are not known in advance but depend on the input data and the regularization parameter λ. More precisely, the spline knots are determined by the contact points of the m–th discrete antiderivative of the solution with the tube of width 2λ around the mth discrete antiderivative of the input data. We point out that the dual formulation of our minimization problem can be considered as support vector regression problem in the discrete counterpart of the Sobolev space W m 2,0. From this point of view, the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines.
Recursive Separable Schemes for Nonlinear Diffusion Filters
, 1997
"... Poor efficiency is a typical problem of nonlinear diffusion filtering, when the simple and popular explicit (Eulerforward) scheme is used: for stability reasons very small time step sizes are necessary. In order to overcome this shortcoming, a novel type of semiimplicit schemes is studied, socall ..."
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Cited by 14 (4 self)
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Poor efficiency is a typical problem of nonlinear diffusion filtering, when the simple and popular explicit (Eulerforward) scheme is used: for stability reasons very small time step sizes are necessary. In order to overcome this shortcoming, a novel type of semiimplicit schemes is studied, socalled additive operator splitting (AOS) methods. They share the advantages of explicit and (semi)implicit schemes by combining simplicity with absolute stability. They are reliable, since they satisfy recently established criteria for discrete nonlinear diffusion scalespaces. Their efficiency is due to the fact that they can be separated into onedimensional processes, for which a fast recursive algorithm with linear complexity is available. AOS schemes reveal good rotational invariance and they are symmetric with respect to all axes. Examples demonstrate that, under typical accuracy requirements, they are at least ten times more efficient than explicit schemes.