#### DMCA

## A vectorial self-dual morphological filter based on total variation minimization (2005)

Venue: | In Proceedings of the First International Conference on Visual Computing |

Citations: | 4 - 2 self |

### Citations

2094 | Fast Approximate Energy Minimization via Graph Cuts
- Boykov, Veksler, et al.
- 2001
(Show Context)
Citation Context ...ile success �= false Fig. 1. Pseudo-code for our minimization algorithm. 3.1 Minimimum Cost Cut Based Minimization Our algorithm relies on the α-expansion moves algorithm proposed by Boykov et al. in =-=[3]-=-. An α-expansion move from a current labeling is defined as follows: given a value α, every pixel can either keep its current value or take α. We are interested in finding the optimal α-expansion move... |

1590 |
Image Analysis and Mathematical Morphology
- Serra
- 1982
(Show Context)
Citation Context ... main issue for mathematical morphology is its extension to the vectorial case. The difficulty arises because a total order between elements is required using the classical approach of lattice theory =-=[19]-=-. Extensions of this approach to the vectorial case have been tackled using vector ranking concepts [10, 20]. In this paper, we propose a new, self dual morphological filter for vectorial images, base... |

1263 | Measure Theory and Fine Properties of Functions. CRC–Press (Boca Raton - Evans, Gariepy - 1992 |

220 |
Vector median filters
- Astola, Haavisto, et al.
- 1990
(Show Context)
Citation Context ...s considered by Caselles et al. in [4] who present a morphological operator on color images. Vector median filters are another approach for vector filtering, originally introduced by Astola et al. in =-=[1]-=-. The process consists of replacing the pixel value by the median of the pixels contained in a window around it. The median is defined as the value that minimizes the L 1 -norm between all pixels in a... |

155 | Color TV: total variation methods for restoration of vector-valued images
- Blomgren, Chan
- 1998
(Show Context)
Citation Context ... coefficient β. Note that the gradient is taken in the distributional sense. An efficient algorithm is proposed in [7] to perform an exact minimization of (1), i.e, it provides a global minimizer. In =-=[2]-=-, Blomgren et al. propose some extensions of the total variation definition to the vectorial case. They study them for image restoration purposes. However, no relation with mathematical morphology is ... |

105 |
Color image processing and applications
- Plataniotis, Venetsanopoulos
- 2000
(Show Context)
Citation Context ...esults on color images. Since, in this paper, we focus on demonstrating the effectiveness of our filter, we applied our model to the RGB space. We are aware that many other color spaces are available =-=[18]-=- and we are currently studying more suitable spaces. Figure 2 presents some results on the image hand. We note that the higher the coefficient β, the more the image is simplified. Details of the textu... |

44 |
Morphological operators on the unit circle
- Hanbury, Serra
(Show Context)
Citation Context ...ign of morphological operators for color images as a specific case of vectorial images. Main approaches consist in choosing a suitable color space representation and defining an ordering relationship =-=[8, 13]-=-. In [5], Chambolle proposes a definition for contrast invariance of operators on colors. This definition is considered by Caselles et al. in [4] who present a morphological operator on color images. ... |

37 |
Morphological operators for image sequences
- Goutsias, Heijmans, et al.
- 1995
(Show Context)
Citation Context ...cause a total order between elements is required using the classical approach of lattice theory [19]. Extensions of this approach to the vectorial case have been tackled using vector ranking concepts =-=[10, 20]-=-. In this paper, we propose a new, self dual morphological filter for vectorial images, based on total variation minimization. Our approach does not require any order relation between vectors. A lot o... |

33 | Geometry and color in natural images
- Caselles, Coll, et al.
- 2002
(Show Context)
Citation Context ...presentation and defining an ordering relationship [8, 13]. In [5], Chambolle proposes a definition for contrast invariance of operators on colors. This definition is considered by Caselles et al. in =-=[4]-=- who present a morphological operator on color images. Vector median filters are another approach for vector filtering, originally introduced by Astola et al. in [1]. The process consists of replacing... |

33 | Adaptive vector median filtering - Lukac - 2003 |

28 | A fast and exact algorithm for total variation minimization
- Darbon, Sigelle
- 2005
(Show Context)
Citation Context ...(x)|dx + β |∇u| , (1) Ω where the last term is the TV of u weighted by a non-negative coefficient β. Note that the gradient is taken in the distributional sense. An efficient algorithm is proposed in =-=[7]-=- to perform an exact minimization of (1), i.e, it provides a global minimizer. In [2], Blomgren et al. propose some extensions of the total variation definition to the vectorial case. They study them ... |

21 |
Partial differential equations and image processing
- Chambolle
- 1994
(Show Context)
Citation Context ...ological operators for color images as a specific case of vectorial images. Main approaches consist in choosing a suitable color space representation and defining an ordering relationship [8, 13]. In =-=[5]-=-, Chambolle proposes a definition for contrast invariance of operators on colors. This definition is considered by Caselles et al. in [4] who present a morphological operator on color images. Vector m... |

11 | N.: Selection weighted vector directional filters - Lukac, Smolka, et al. - 2004 |

9 | S (2005) Total variation based image cartoon-texture decomposition
- Yin, Goldfarb, et al.
(Show Context)
Citation Context ...is kept. Moreover, the colors of the background of the hand do not merge. Finally, the color of the ring is well preserved. This result gives a very good initialization for a segmentation process. In =-=[21]-=-, the authors perform the minimization of the L 1 + T V model in order to decompose the image into two parts: the first one contains the geometry while the second one contains the textures. Figure 2 d... |

8 |
Morphological algorithms for color images based on a generic-programming approach
- d'Ornellas, Boomgaard, et al.
- 1998
(Show Context)
Citation Context ...ign of morphological operators for color images as a specific case of vectorial images. Main approaches consist in choosing a suitable color space representation and defining an ordering relationship =-=[8, 13]-=-. In [5], Chambolle proposes a definition for contrast invariance of operators on colors. This definition is considered by Caselles et al. in [4] who present a morphological operator on color images. ... |

7 | Mathematical morphology “almost everywhere
- Guichard, Morel
(Show Context)
Citation Context ...l mathematical morphology In this section we briefly review the L 1 + T V model. Then we show how to generalize the approach to the vectorial case. We define a continous change of contrast as follows =-=[12]-=-: any continuous non-decreasing function on IR is called a continuous change of contrast. The following theorem is proved in [6] and in appendix A. Theorem 1 Let v be an observed image and g be a cont... |

5 | Total variation minimization with L 1 data fidelity as a constrast invariant filter
- Darbon
- 2005
(Show Context)
Citation Context ...ogical operators and partial differential equations. They consider a lexicographic order to obtain these connections. Complementary results on these links for the scalar case can be found in [11]. In =-=[6]-=-, the author deals with the scalar case and shows that minimization of the total variation (TV) under the L1-norm as data fidelity yields a morphological filter. This model will be referred to as L1 +... |

4 |
Discrete Convex Optimization
- Murota
(Show Context)
Citation Context ...hin one α−expansion move of u, such that E(u) > E(u α ) . Proof: Before giving to the proof, we recall that for a one dimensional discrete convex function f : ZZ ↦→ IR, the following inequality holds =-=[17]-=- : ∀x ∀y∀d | (y ≥ x) ∧ (0 ≤ d ≤ (y − x)), f(x) + f(y) ≥ f(x + d) + f(y − d) . (3) Let û be a global minimizer of E, i.e., E(û) = infu ′ E(u′ ) Given a value α ∈ C, we define an image δ as follows: ⎧ ⎨... |

3 |
Image Iterative Smoothing and PDE s. downloadable manuscript: please write email to fguichard@poseidon-tech.com
- Guichard, Morel
- 2000
(Show Context)
Citation Context ...ntroduction It is well known that morphological filters on gray-scale images feature the property of being invariant with respect to any change of contrast and that they do not create new gray levels =-=[11]-=-. One of the main issue for mathematical morphology is its extension to the vectorial case. The difficulty arises because a total order between elements is required using the classical approach of lat... |

1 | Classification based adaptive vector filter for color image restoration - Ma, Wu - 2005 |