## Vector-Valued Image Regularization with PDEs: A Common Framework for Different Applications (2003)

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Venue: | IEEE Transactions on Pattern Analysis and Machine Intelligence |

Citations: | 125 - 8 self |

### BibTeX

@INPROCEEDINGS{Tschumperlé03vector-valuedimage,

author = {David Tschumperlé and R. Deriche},

title = {Vector-Valued Image Regularization with PDEs: A Common Framework for Different Applications},

booktitle = {IEEE Transactions on Pattern Analysis and Machine Intelligence},

year = {2003},

pages = {506--517}

}

### Years of Citing Articles

### OpenURL

### Abstract

We address the problem of vector-valued 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 vector-valued images. They are finally applied on a wide variety of image processing problems, including color image restoration, inpainting, magnification and flow visualization.

### Citations

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Stochastic relaxation, Gibbs distribution and Bayesian restoration of images
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(Show Context)
Citation Context ...ms are crucial. For these reasons, a lot of regularization frameworks have already been proposed in the literature. Pioneering works in this area have been initiated, for instance, in [1], [3], [18], =-=[19]-=-, [21], [34]. In the late 1980s, the framework of nonlinear PDEs (partial differential equations) led to strong improvements in the formalization of regularization methods. First created to describe p... |

1378 | Scale-space and edge detection using anisotropic diffusion
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Citation Context ...ssical image-related issues (restoration, segmentation). Thus, many regularization schemes have been presented so far in the literature, particularly for the case of 2D scalar images I : Ω ⊂ R2 ��=-=� R ( [1, 17, 18, 28] and r-=-eferences therein). Extensions of these algorithms to vector-valued images I : Ω → Rn have been recently proposed, leading to more elaborated diffusion PDE’s : a coupling between image channels ... |

716 | Bilateral filtering for gray and color images
- Tomasi, Manduchi
- 1998
(Show Context)
Citation Context ...formed pointwise by the tracebased PDE (6). This local filtering concept makes the link between a generic form of vector-valued diffusion PDE’s (6) and Bilateral filtering techniques, as described i=-=n [2, 23]. -=-A similar approach with non-Gaussian kernels has been also recently proposed for the Beltrami Flow framework [21]. • Trace-based and Divergence-based tensors : Differences between divergence tensors... |

693 |
The structure of images
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Citation Context ...tands for the convolution operator and G (T,t) is an oriented gaussian kernel, defined by : G (T,t) (x) = 1 4πt exp � − xT T −1 � x 4t with x = (x y) T It is a generalization of the Koenderin=-=k’s idea [13], who -=-proved this property for the isotropic diffusion tensor T = Id, resulting in the well-known heat flow equation : ∂Ii ∂t = ∆Ii. The top row of Fig.1 illustrates a gaussian kernel G (T,t) (x, y) o... |

560 |
Scale-space filtering
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Citation Context ... many PDE-based schemes have been presented so far in the literature, particularly for the regularization of 2D scalar images I : IR 2 ! IR (see, for instance, [2], [4], [27], [30], [34], [37], [51], =-=[53]-=-, [54] and references therein). Another interesting property of nonlinear regularization PDEs such as @I @t Ris the notion of scale-space behind: The data are gently regularized step-by-step and a co... |

502 | Scale-space theory in computer vision
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Citation Context ...on, registration, etc.). Thus, many PDE-based schemes have been presented so far in the literature, particularly for the regularization of 2D scalar images I : IR 2 ! IR (see, for instance, [2], [4], =-=[27]-=-, [30], [34], [37], [51], [53], [54] and references therein). Another interesting property of nonlinear regularization PDEs such as @I @t Ris the notion of scale-space behind: The data are gently reg... |

398 | Image inpainting
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Citation Context ..., bloc effects are classical JPEG drawbacks. Using our flow (12) significantly improves the quality of such degraded images (Fig.3b). • Color image inpainting : Image inpainting, recently proposed i=-=n [4, 7, 8, 9] c-=-onsists in filling undesired holes (defined by the user) in images by interpolating the data located at the hole’s neighborhood. It is possible to do that by applying our regularization PDE (12) onl... |

260 |
Geometric Partial Differential Equations and Image Analysis
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(Show Context)
Citation Context ...ssical image-related issues (restoration, segmentation). Thus, many regularization schemes have been presented so far in the literature, particularly for the case of 2D scalar images I : Ω ⊂ R2 ��=-=� R ( [1, 17, 18, 28] and r-=-eferences therein). Extensions of these algorithms to vector-valued images I : Ω → Rn have been recently proposed, leading to more elaborated diffusion PDE’s : a coupling between image channels ... |

241 | Deterministic edge-preserving regularization in computed imaging
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Citation Context ...Then, the minimization of (1) is performed through a gradient descent (PDE), coming from the Euler-Lagrange equations of EðIÞ. This technique has been widely used in the context of scalar images [4], =-=[15]-=-, [16], [24], [25], [54], for instance, by minimizing the area of a surface representing the image (Fig. 1). Corresponding references for vector-valued images are: [10], [22], [33], [37], [39], [42], ... |

213 |
Oscillating patterns in image processing and nonlinear evolution equations, The fifteenth Dean Jacqueline B. Lewis memorial lectures
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Citation Context ...of iterations. In this article, we are mainly interested in the regularization term behavior rather than the fidelity term. For an interesting mathematical study about fidelity terms, please refer to =-=[29]-=-, [31]. Extensions of these nonlinear regularization PDEs to vector-valued images I : ! IR n have been recently proposed, leading to more elaborated expressions: A coupling between image channels gene... |

211 | Anisotropic diffusion in image processing
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Citation Context ...ssical image-related issues (restoration, segmentation). Thus, many regularization schemes have been presented so far in the literature, particularly for the case of 2D scalar images I : Ω ⊂ R2 ��=-=� R ( [1, 17, 18, 28] and r-=-eferences therein). Extensions of these algorithms to vector-valued images I : Ω → Rn have been recently proposed, leading to more elaborated diffusion PDE’s : a coupling between image channels ... |

164 | Nonlinear anisotropic filtering of mri data
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Citation Context ...rization are [5, 12, 16, 18, 20, 22, 26]. (2) Divergence expressions : A regularization process may be also designed more locally, as the diffusion of pixel values - viewed as chemical concentrations =-=[11, 28] - driv-=-en by a 2×2 diffusion tensor D (symmetric and positive matrix) : ∂Ii ∂t = div (D∇Ii) (i = 1..n) (2) It is generally assumed that the spectral elements of D give the two weights and directions o... |

149 |
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Citation Context ...egularization may be finally seen as the juxtaposition of two oriented 1D heat flows, i.e two monodimensional gaussian smoothing along orthonormal directions u⊥v, with corresponding weights c1 and c=-=2 [14, 19, 25, 26] : ∂I ∂t = c-=-1 ∂2I ∂u2 + c2 ∂2I ∂v2 = c1 Iuu + c2 Ivv (3) Like divergence expressions, c1, c2 and u, v are usually designed from the spectral elements λ± and θ± of G, in order to perform edge-preservin... |

116 | Color TV: total variation methods for restoration of vector-valued images
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Citation Context ...) = � λ+ + λ− = trace (G) 1 2 . The minimization of (1) is performed with a gradient descent (PDE) given by the Euler-Lagrange equations of E(I). Useful references for vector image regularizatio=-=n are [5, 12, 16, 18, 20, 22, 26].-=- (2) Divergence expressions : A regularization process may be also designed more locally, as the diffusion of pixel values - viewed as chemical concentrations [11, 28] - driven by a 2×2 diffusion ten... |

101 | Images as embedding maps and minimal surfaces: Movies, color, texture, and volumetric medical images
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Citation Context ...) = � λ+ + λ− = trace (G) 1 2 . The minimization of (1) is performed with a gradient descent (PDE) given by the Euler-Lagrange equations of E(I). Useful references for vector image regularizatio=-=n are [5, 12, 16, 18, 20, 22, 26].-=- (2) Divergence expressions : A regularization process may be also designed more locally, as the diffusion of pixel values - viewed as chemical concentrations [11, 28] - driven by a 2×2 diffusion ten... |

95 | Euler’s elastica and curvature based inpainting
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Citation Context ..., bloc effects are classical JPEG drawbacks. Using our flow (12) significantly improves the quality of such degraded images (Fig.3b). • Color image inpainting : Image inpainting, recently proposed i=-=n [4, 7, 8, 9] c-=-onsists in filling undesired holes (defined by the user) in images by interpolating the data located at the hole’s neighborhood. It is possible to do that by applying our regularization PDE (12) onl... |

87 | Computational Fluid Dynamics - Wesseling - 2000 |

84 | Non-texture inpainting by curvature-driven diffusions (CDD
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Citation Context ..., bloc effects are classical JPEG drawbacks. Using our flow (12) significantly improves the quality of such degraded images (Fig.3b). • Color image inpainting : Image inpainting, recently proposed i=-=n [4, 7, 8, 9] c-=-onsists in filling undesired holes (defined by the user) in images by interpolating the data located at the hole’s neighborhood. It is possible to do that by applying our regularization PDE (12) onl... |

82 | A theoretical framework for convex regularizers in PDE–based computation of image motion - Weickert, Schnörr - 2001 |

68 |
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Citation Context ...fiffiffi p þ þ for vector-valued images [7], [10], [33], [41], [46], [47], but other norms are possible such as NðIÞ p ffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ [9], =-=[35]-=-, [36], or NðIÞ þ [38], [49], [50]. For scalar images I : ! IR, these norms naturally reduce to the same expression NðIÞ krIk. Then, the minimization of (1) is performed through a gradient descent ... |

57 |
The application of constrained least squares estimation to image restoration by digital computer
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Citation Context ... crucial. For these reasons, a lot of regularization frameworks have already been proposed in the literature. Pioneering works in this area have been initiated, for instance, in [1], [3], [18], [19], =-=[21]-=-, [34]. In the late 1980s, the framework of nonlinear PDEs (partial differential equations) led to strong improvements in the formalization of regularization methods. First created to describe physica... |

56 | Diffusion of general data on non-flat manifolds via harmonic maps theory: The direction diffusion case
- Tang, Sapiro, et al.
(Show Context)
Citation Context ...) = � λ+ + λ− = trace (G) 1 2 . The minimization of (1) is performed with a gradient descent (PDE) given by the Euler-Lagrange equations of E(I). Useful references for vector image regularizatio=-=n are [5, 12, 16, 18, 20, 22, 26].-=- (2) Divergence expressions : A regularization process may be also designed more locally, as the diffusion of pixel values - viewed as chemical concentrations [11, 28] - driven by a 2×2 diffusion ten... |

56 |
Restoring with maximum likelihood and maximum entropy
- Frieden
- 1972
(Show Context)
Citation Context ...problems are crucial. For these reasons, a lot of regularization frameworks have already been proposed in the literature. Pioneering works in this area have been initiated, for instance, in [1], [3], =-=[18]-=-, [19], [21], [34]. In the late 1980s, the framework of nonlinear PDEs (partial differential equations) led to strong improvements in the formalization of regularization methods. First created to desc... |

50 |
Les EDP en Traitement des images et Vision par Ordinateur
- Deriche, Faugeras
- 1995
(Show Context)
Citation Context ...the minimization of (1) is performed through a gradient descent (PDE), coming from the Euler-Lagrange equations of EðIÞ. This technique has been widely used in the context of scalar images [4], [15], =-=[16]-=-, [24], [25], [54], for instance, by minimizing the area of a surface representing the image (Fig. 1). Corresponding references for vector-valued images are: [10], [22], [33], [37], [39], [42], [44]. ... |

44 |
Anisotropic diffusion in vector field visualization on euclidean domains and surfaces
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(Show Context)
Citation Context ...n scalespace of F is constructed. Here, our used regularization equation (13) ensures that the smoothing of the pixels is done exactly in the direction of the flow F (Fig.3f). This is not the case in =-=[3, 6, 10]-=-, where the authors based their equations on divergence expressions. Using similar divergencebased techniques would raise a risk of smoothing the image in false directions, as this has been pointed ou... |

38 | Regularization, scale-space and edge detection filters
- Nielsen, Florack, et al.
- 1997
(Show Context)
Citation Context ...gistration, etc.). Thus, many PDE-based schemes have been presented so far in the literature, particularly for the regularization of 2D scalar images I : IR 2 ! IR (see, for instance, [2], [4], [27], =-=[30]-=-, [34], [37], [51], [53], [54] and references therein). Another interesting property of nonlinear regularization PDEs such as @I @t Ris the notion of scale-space behind: The data are gently regulariz... |

35 | Diffusion tensor regularization with constraints preservation
- Tschumperlé, Deriche
(Show Context)
Citation Context ...ted to local image variations and :IR! IR is an increasing function. One often chooses NðIÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p þ þ for vector-valued images [7], [10], [33], [41], =-=[46]-=-, [47], but other norms are possible such as NðIÞ p ffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ [9], [35], [36], or NðIÞ þ [38], [49], [50]. For scalar images I : ! I... |

34 |
Mathematical models for local deterministic inpaintings
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(Show Context)
Citation Context |

30 | Nonlinear Operators in Image Restoration
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Citation Context ...egularization may be finally seen as the juxtaposition of two oriented 1D heat flows, i.e two monodimensional gaussian smoothing along orthonormal directions u⊥v, with corresponding weights c1 and c=-=2 [14, 19, 25, 26] : ∂I ∂t = c-=-1 ∂2I ∂u2 + c2 ∂2I ∂v2 = c1 Iuu + c2 Ivv (3) Like divergence expressions, c1, c2 and u, v are usually designed from the spectral elements λ± and θ± of G, in order to perform edge-preservin... |

30 | Diffusions and confusions in signal and image processing
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Citation Context ...or-valued diffusion PDE’s (6) and Bilateral filtering techniques, as described in [2, 23]. A similar approach with non-Gaussian kernels has been also recently proposed for the Beltrami Flow framewor=-=k [21]. • -=-Trace-based and Divergence-based tensors : Differences between divergence tensors D in (2) and trace tensors T in (6) can be understood as follows. We develop (2) as : div (D∇Ii) = trace (DHi) + ∇... |

28 |
PDEs Based Regularization of Multivalued Images and Applications
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(Show Context)
Citation Context ... as : T = c1uu T + c2vv T , characterized by its two eigenvalues c1, c2 and its corresponding eigenvectors u⊥v. Suppose that T is a constant tensor over the definition domain Ω. Then, it can be sh=-=own [24, 27] tha-=-t the formal solution of the PDE (6) is : Ii (t) = Ii (t=0) ∗ G (T,t) (i = 1..n) (7) where ∗ stands for the convolution operator and G (T,t) is an oriented gaussian kernel, defined by : G (T,t) (x... |

28 | Modified curvature motion for image smoothing and enhancement
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Citation Context ...PDE-based schemes have been presented so far in the literature, particularly for the regularization of 2D scalar images I : IR 2 ! IR (see, for instance, [2], [4], [27], [30], [34], [37], [51], [53], =-=[54]-=- and references therein). Another interesting property of nonlinear regularization PDEs such as @I @t Ris the notion of scale-space behind: The data are gently regularized step-by-step and a continuo... |

22 | Bilateral Filtering and Anisotropic Diffusion: Towards a Unified Viewpoint,” technical report, HP Laboratories
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Citation Context ...formed pointwise by the tracebased PDE (6). This local filtering concept makes the link between a generic form of vector-valued diffusion PDE’s (6) and Bilateral filtering techniques, as described i=-=n [2, 23]. -=-A similar approach with non-Gaussian kernels has been also recently proposed for the Beltrami Flow framework [21]. • Trace-based and Divergence-based tensors : Differences between divergence tensors... |

21 | Estimating the jacobian of the singular value decomposition: Theory and applications - Papadopoulo, Lourakis |

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Total variation methods for restoration of vector valued images
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Citation Context ...iffiffiffiffi p þ þ for vector-valued images [7], [10], [33], [41], [46], [47], but other norms are possible such as NðIÞ p ffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ =-=[9]-=-, [35], [36], or NðIÞ þ [38], [49], [50]. For scalar images I : ! IR, these norms naturally reduce to the same expression NðIÞ krIk. Then, the minimization of (1) is performed through a gradient de... |

17 |
Diffusion pde’s on vector-valued images
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Citation Context ... by the spectral elements λ+, λ− (positive eigenvalues) and θ+, θ− (orthogonal eigenvectors) of the 2 × 2 symmetric and semi positive-definite matrix G = �n j=1 ∇Ij∇IT j (also called st=-=ructure tensor [25, 26, 28, 29]). The λ±-=- respectively define the local min/max vector-valued variations of I in corresponding spatial directions θ±, i.e. the local configuration of the image discontinuities. (note that λ+=�∇I� and ... |

16 | Geometric partial dierential equations and image analysis - Sapiro - 2001 |

16 | Regularization of orthonormal vector sets using coupled PDE’s
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Citation Context ... local image variations and :IR! IR is an increasing function. One often chooses NðIÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p þ þ for vector-valued images [7], [10], [33], [41], [46], =-=[47]-=-, but other norms are possible such as NðIÞ p ffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ [9], [35], [36], or NðIÞ þ [38], [49], [50]. For scalar images I : ! IR, the... |

14 |
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Citation Context |

14 | Curve evolution and segmentation functionals: Applications to color images
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14 | Constrained and unconstrained pde’s for vector image restoration
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Citation Context ...ctor geometry, given pointwise by the spectral elements þ; (positive eigenvalues) and þ; (orthogonal eigenvectors) of the 2 2 symmetric and semi-positive-definite matrix, also called structure tensor =-=[45]-=-, [48], [51], [55]: G Xn j1 rIjrI T j : Each rIj corresponds to the spatial gradient of the jth channel (i.e., vector component) of the vector-valued image I. As demonstrated in [55], the structure... |

10 | and D L Ringach, Anisotropic Diusion of multivalued images with applications to color - Sapiro - 1996 |

9 | Pde methods in flow simulation post processing
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Citation Context ...n scalespace of F is constructed. Here, our used regularization equation (13) ensures that the smoothing of the pixels is done exactly in the direction of the flow F (Fig.3f). This is not the case in =-=[3, 6, 10]-=-, where the authors based their equations on divergence expressions. Using similar divergencebased techniques would raise a risk of smoothing the image in false directions, as this has been pointed ou... |

8 |
S.: A note on the gradient of a multi-image, Comput. Vision Graphics Image Process
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Citation Context ...en pointwise by the spectral elements þ; (positive eigenvalues) and þ; (orthogonal eigenvectors) of the 2 2 symmetric and semi-positive-definite matrix, also called structure tensor [45], [48], [51], =-=[55]-=-: G Xn j1 rIjrI T j : Each rIj corresponds to the spatial gradient of the jth channel (i.e., vector component) of the vector-valued image I. As demonstrated in [55], the structure tensor G is parti... |

8 | Nonlinear anisotropic ltering of mridata - Gerig, Kubler, et al. - 1992 |

8 |
Contributions à la Restauration d’Images et à l’Analyse de Séquences: Approches Variationnelles et Solutions de Viscosité,” PhD thesis, Université de Nice-Sophia Antipolis
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Citation Context ...nimization of (1) is performed through a gradient descent (PDE), coming from the Euler-Lagrange equations of EðIÞ. This technique has been widely used in the context of scalar images [4], [15], [16], =-=[24]-=-, [25], [54], for instance, by minimizing the area of a surface representing the image (Fig. 1). Corresponding references for vector-valued images are: [10], [22], [33], [37], [39], [42], [44]. 1.2 Di... |

8 |
Color snakes,” Computer Vision and
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(Show Context)
Citation Context ...fi p þ þ for vector-valued images [7], [10], [33], [41], [46], [47], but other norms are possible such as NðIÞ p ffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ [9], [35], =-=[36]-=-, or NðIÞ þ [38], [49], [50]. For scalar images I : ! IR, these norms naturally reduce to the same expression NðIÞ krIk. Then, the minimization of (1) is performed through a gradient descent (PDE),... |

7 | Bilateral _ltering for gray and color images - Tomasi, Manduchi - 1998 |

6 |
Constrained and unconstrained PDE’s for vector image restoration
- Tschumperl, Deriche
- 2001
(Show Context)
Citation Context ... by the spectral elements λ+, λ− (positive eigenvalues) and θ+, θ− (orthogonal eigenvectors) of the 2 × 2 symmetric and semi positive-definite matrix G = �n j=1 ∇Ij∇IT j (also called st=-=ructure tensor [25, 26, 28, 29]). The λ±-=- respectively define the local min/max vector-valued variations of I in corresponding spatial directions θ±, i.e. the local configuration of the image discontinuities. (note that λ+=�∇I� and ... |