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## Local scale measure from the topographic map and application to remote sensing images. Multiscale Model

Venue: | Simul |

Citations: | 8 - 6 self |

### Citations

8954 | Distinctive image features from scale-invariant keypoints
- Lowe
- 2004
(Show Context)
Citation Context ...e computed by studying extrema of differential operators in the linear scale-space. This method has been widely applied in computer vision to select the optimal analysis scale of structures, see e.g. =-=[27, 33]-=-. Similar to this approach, the methods presented in [19, 44, 45, 50] propose to estimate the salient scale of an image by considering extrema of various information theoretic operators in the linear ... |

2267 | Nonlinear total variation based noise removal algorithms
- Rudin, Osher, et al.
- 1992
(Show Context)
Citation Context ...riational definitions of scale. 4.1.1. Definition based on total variation regularization. Strong and Chan have proposed in [49] to define the scales in an image by using the Rudin-Osher-Fatemi model =-=[39]-=- (ROF). Recall that the ROF model (or total variation regularization) consists, given an image f , in finding the solution u of: inf u (∫ |Du|+ 1 2T ‖f − u‖2L2 ) (4.1) It is shown in [49] that if the ... |

1467 | Scale and affine invariant interest point detectors.
- Mikoljczyk, Schmid
- 2004
(Show Context)
Citation Context ...e computed by studying extrema of differential operators in the linear scale-space. This method has been widely applied in computer vision to select the optimal analysis scale of structures, see e.g. =-=[27, 33]-=-. Similar to this approach, the methods presented in [19, 44, 45, 50] propose to estimate the salient scale of an image by considering extrema of various information theoretic operators in the linear ... |

1247 | On the statistical analysis of dirty pictures,”
- Besag
- 1986
(Show Context)
Citation Context ...er k-means, P (x; lx) = |E(x) − ĉ(lx)| , β is a non-positive weighting parameter, Nx is the 4-neighborhood of x, and δ(lx, ly) = { −1, lx = ly 1, otherwise (7.4) The ICM (Iterative Conditional Mode) =-=[6]-=- algorithm is finally used to minimize the energy. On Figure 7.2 is shown a segmentation result obtained with this approach. The original image of Toulouse is displayed on Figure 7.2 (a), and the scal... |

1069 |
Functions of bounded variation and free discontinuity problems
- Ambrosio, Fusco, et al.
- 2000
(Show Context)
Citation Context ...yer in [32]. Indeed, a possible definition of the G norm of an image f with zero mean is [48]: ‖f‖G = sup E⊂Ω ∫ E f P (E,Ω) (4.4) where P (E) stands for the perimeter of E, as defined for instance in =-=[2]-=-. The G norm of an image can therefore be seen as an area divided by a perimeter. These relations have been used in [48] to propose a variant of the ROF model where the user gives as input of the algo... |

621 |
Stochastic Geometry and its Applications.
- Stoyan, WS, et al.
- 1995
(Show Context)
Citation Context ...C−1), where ⊕ stands for the Minkowski1 addition and D(r) is a disk of radius r centered at the origin. On the other hand, assuming that fi is a convex set, the area of fi⊕D(qC−1) is (Steiner formula =-=[47]-=-) S(fi ⊕D( q C )) = S(fi) + pi ( q C )2 + q C P (fi). This suggests that Formula (3.2) enables one to group level lines corresponding to the same edge as soon as λ > qC−1. The values of C ensuring tha... |

289 |
Cluster analysis of multivariate data: efficiency versus interpretability of classifications.”
- Forgy
- 1965
(Show Context)
Citation Context ...tion algorithms. The segmentation scheme we use here is based on the following steps. We first compute a scale map of the image using Equation (3.7). A pre-segmentation is then performed by a k-means =-=[16]-=- method using the local scale of each pixel, resulting in labels lx ∈ {1, . . . , k} at each pixel x. The initial cluster centers (c0(l))l=1,...,k are chosen as c0(l) = l × supx∈Ω(E(x)k . 22 (a) (b) (... |

202 |
Image Analysis and
- Serra
- 1988
(Show Context)
Citation Context ...r scale. Independently, the mathematical morphology school has long ago proposed to characterize materials by looking at the size distribution of grains, through the use of the so-called granulometry =-=[17, 41]-=-. Following this idea, it is proposed in [30] to use the pattern spectrum of images (roughly, the derivative of granulometry) to index gray-scale images. In the framework of remote sensing imaging, [5... |

191 | An Affine Invariant Salient Region Detector”,
- Kadir, Zisserman, et al.
- 2004
(Show Context)
Citation Context ... disregard structures without significant gradients, for which no scale is computed. In order to obtain spatially accurate scale measures, it is quite natural to look toward non-linear approaches. In =-=[20, 21]-=- it is proposed to measure the significant scales of structures by computing the entropy of the joint distribution of pixels in a fixed neighborhood. It has been shown, see [21], that such an approach... |

189 | Detecting salient blob-like image structures and their scales with a scale-space primal sketch - A method for focus-of-attention,
- Lindeberg
- 1993
(Show Context)
Citation Context ...t this idea is feasible and yields spatially accurate features for describing remote sensing images. The most classical approach to estimate the scale of structures in an image has been introduced by =-=[24, 25]-=-. Local scales are computed by studying extrema of differential operators in the linear scale-space. This method has been widely applied in computer vision to select the optimal analysis scale of stru... |

175 | Flat zones filtering, connected operators, and filters by reconstruction
- Salembier, Serra
- 1995
(Show Context)
Citation Context ...lobally invariant with respect to contrast changes. Each of these family, upper sets on the one hand and lower sets on the other hand, has a tree structure with respect to inclusion. Several authors (=-=[40, 10, 11, 18]-=-) have proposed the connected components of level sets as an efficient way to represent images. This representation enjoys the same contrast invariant and hierarchical structure as level sets represen... |

162 |
Pattern spectrum and multiscale shape representation
- Maragos
- 1989
(Show Context)
Citation Context ...gy school has long ago proposed to characterize materials by looking at the size distribution of grains, through the use of the so-called granulometry [17, 41]. Following this idea, it is proposed in =-=[30]-=- to use the pattern spectrum of images (roughly, the derivative of granulometry) to index gray-scale images. In the framework of remote sensing imaging, [5, 12] have proposed, in view of the classific... |

99 | Edge-preserving and scale-dependent properties of total variation regularization,”
- Strong, Chan
- 2003
(Show Context)
Citation Context ...mainly based on non-linear scale spaces, either in a morphological or variational framework. Recently, several scale measures based on non-linear partial differential equations have been proposed. In =-=[49, 48]-=-, a local definition of scale based on total variation regularization is introduced. In [8], it is proposed to estimate the local scales of structures from their evolution speed under the total variat... |

89 | On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and SIDEs
- Steidl, Weickert, et al.
- 2004
(Show Context)
Citation Context ...ave also been used in [7, 8] to introduce a definition of scale in images. Let us recall that the solution u of the total variation diffusion satisfies{ u(., 0) = f ∂u ∂t = div ( Du |Du| ) (4.5) 8 In =-=[46]-=-, the authors have proved the equivalence for 1-dimensional signal of total variation regularization (ROF model) and total variation diffusion. They have derived the same type of results as in [49] (w... |

82 | Topographic maps and local contrast changes in natural images.
- Caselles, Coll, et al.
- 1999
(Show Context)
Citation Context ... extensions to the proposed definition of scale. 2. Topographic map. In this section, we introduce the main tool to be used in this paper, the topographic map of an image as introduced and studied in =-=[10, 11, 35, 36]-=-. It is made of the connected components of the topological boundaries of level sets, thereafter called level lines. It gives a complete representation of the image, invariant to local contrast change... |

76 | Classification and feature extraction for remote sensing images from urban areas based on morphological transformation.
- Benediktsson, Pesaresi, et al.
- 2003
(Show Context)
Citation Context ...1]. Following this idea, it is proposed in [30] to use the pattern spectrum of images (roughly, the derivative of granulometry) to index gray-scale images. In the framework of remote sensing imaging, =-=[5, 12]-=- have proposed, in view of the classification of satellite images, to compute size distributions (called derivative morphological profile) at each pixel. A closely related approach to the extraction o... |

57 |
Advances in mathematical morphology applied to geoscience and remote sensing,
- Soille, Pesaresi
- 2002
(Show Context)
Citation Context ...ed to connected filters [40]. In [26], it is proposed to use such connected filters to compute the size distribution of structures in the image, in view of the classification of urban area. Similarly =-=[42, 12]-=- have proposed to use the differential morphological profile (DMP) to classify satellite images. Starting from an image I, a series of images Ir (respectively Ir) are first obtained by applying openin... |

51 |
Oscillating Patterns in Image Processing and in some Nonlinear Evolution Equations.
- Meyer
- 2001
(Show Context)
Citation Context ...nds on the parameter T . It was later noticed in [48] that in fact this notion of scale is related to the polar semi-norm associated to the total variation, the so-calledG norm introduced by Meyer in =-=[32]-=-. Indeed, a possible definition of the G norm of an image f with zero mean is [48]: ‖f‖G = sup E⊂Ω ∫ E f P (E,Ω) (4.4) where P (E) stands for the perimeter of E, as defined for instance in [2]. The G ... |

32 | The discontinuity set of solutions of the TV denoising problem and some extensions. Multiscale Model. - Caselles, Chambolle, et al. - 2007 |

29 |
The total variation flow in RN
- Bellettini, Caselles, et al.
(Show Context)
Citation Context ...(4.7), as well as to clarify the link with the definition of scale given in the present paper, Formula (3.7). These results have been proved by V. Caselles and his collaborators in a series of papers =-=[3, 4, 9, 1]-=-. In particular, it is shown that, if an image f is the characteristic function of a convex set C, i.e. f = 1C , then total variation regularization is equivalent to total variation flow. In both case... |

29 | A TV flow based local scale measure for texture discrimination
- Brox, Weickert
- 2004
(Show Context)
Citation Context ...reserved (smaller scales are considered as noise and therefore wiped out). 4.1.2. Definition based on the total variation diffusion. The properties of total variation diffusion have also been used in =-=[7, 8]-=- to introduce a definition of scale in images. Let us recall that the solution u of the total variation diffusion satisfies{ u(., 0) = f ∂u ∂t = div ( Du |Du| ) (4.5) 8 In [46], the authors have prove... |

28 | Classification of remote sensing images from urban areas using a fuzzy probabilistic model.
- Chanussot, Benediktsson, et al.
- 2006
(Show Context)
Citation Context ...1]. Following this idea, it is proposed in [30] to use the pattern spectrum of images (roughly, the derivative of granulometry) to index gray-scale images. In the framework of remote sensing imaging, =-=[5, 12]-=- have proposed, in view of the classification of satellite images, to compute size distributions (called derivative morphological profile) at each pixel. A closely related approach to the extraction o... |

24 | Uniqueness of the Cheeger set of a convex body,
- Caselles, Chambolle, et al.
- 2007
(Show Context)
Citation Context ...(4.7), as well as to clarify the link with the definition of scale given in the present paper, Formula (3.7). These results have been proved by V. Caselles and his collaborators in a series of papers =-=[3, 4, 9, 1]-=-. In particular, it is shown that, if an image f is the characteristic function of a convex set C, i.e. f = 1C , then total variation regularization is equivalent to total variation flow. In both case... |

20 |
Scale-space from a level lines tree.
- Monasse, Guichard
- 1999
(Show Context)
Citation Context ... extensions to the proposed definition of scale. 2. Topographic map. In this section, we introduce the main tool to be used in this paper, the topographic map of an image as introduced and studied in =-=[10, 11, 35, 36]-=-. It is made of the connected components of the topological boundaries of level sets, thereafter called level lines. It gives a complete representation of the image, invariant to local contrast change... |

20 | The entropy of scale-space
- Sporring
- 1996
(Show Context)
Citation Context ...he linear scale-space. This method has been widely applied in computer vision to select the optimal analysis scale of structures, see e.g. [27, 33]. Similar to this approach, the methods presented in =-=[19, 44, 45, 50]-=- propose to estimate the salient scale of an image by considering extrema of various information theoretic operators in the linear scale space. For remote-sensing images, [29] has also proposed to rel... |

15 |
Heijmans, “Connected morphological operators for binary images
- M
- 1999
(Show Context)
Citation Context ...lobally invariant with respect to contrast changes. Each of these family, upper sets on the one hand and lower sets on the other hand, has a tree structure with respect to inclusion. Several authors (=-=[40, 10, 11, 18]-=-) have proposed the connected components of level sets as an efficient way to represent images. This representation enjoys the same contrast invariant and hierarchical structure as level sets represen... |

14 |
Morphological representation of digital images and application to registration
- Monasse
- 2000
(Show Context)
Citation Context ...en the smaller one is chosen. We conclude this section by noticing that a method to group level lines relying on criteria similar to Formula (3.2) (but using no perimeter information) was proposed in =-=[34]-=- as an efficient alternative to shock filters, in the framework of image restoration. 3.2. Level lines, edges and blur. In this section, we investigate the validity of the use of Formula (3.2) for gro... |

14 |
Scale recognition, regularization parameter selection and Meyer’s G– norm in total variation regularization,” Multiscale Model
- Strong, Aujol, et al.
- 2006
(Show Context)
Citation Context ...mainly based on non-linear scale spaces, either in a morphological or variational framework. Recently, several scale measures based on non-linear partial differential equations have been proposed. In =-=[49, 48]-=-, a local definition of scale based on total variation regularization is introduced. In [8], it is proposed to estimate the local scales of structures from their evolution speed under the total variat... |

11 |
Morphologie mathématique et granulométries en place. Annales des Mines
- Haas, Matheron, et al.
- 1967
(Show Context)
Citation Context ...r scale. Independently, the mathematical morphology school has long ago proposed to characterize materials by looking at the size distribution of grains, through the use of the so-called granulometry =-=[17, 41]-=-. Following this idea, it is proposed in [30] to use the pattern spectrum of images (roughly, the derivative of granulometry) to index gray-scale images. In the framework of remote sensing imaging, [5... |

9 | Image restoration involving connectedness
- Masnou, Morel
- 1997
(Show Context)
Citation Context ...on of QuickBird images, shapes smaller than 16 pixels are not taken into consideration. This is equivalent to the application of a grain filter of size 16 before the computation of the scale map, see =-=[31]-=-. The scale map is shown in Figure 5.5. Again it can be observed that for most structures, such as the big buildings on the top left, computed scales are spatially accurate. However, for the city bloc... |

8 |
Scale in geography, in
- Montello
- 2001
(Show Context)
Citation Context ... hand, it is a crucial information to tune the spatial extent of analysis tools. In remote sensing imaging, the best observation scale is closely related to the concept of Analysis Scale in geography =-=[37]-=-. The NATO standard STANAG 3769 [38] gives some examples of the best scales for interpreting certain objects in remote-sensing images. In computer vision, most object recognition or detection methods ... |

8 | An Original Multi-Sensor Approach to Scale-Based Image Analysis for Aerial and Satellite Images, in IEEE-ICIP-97, vol - Winter, Mâıtre, et al. - 1997 |

7 |
Analyse de texture par méthodes markoviennes et par morphologie mathématique : application à l’analyse des zones urbaines sur des images satellitales
- Lorette
- 1999
(Show Context)
Citation Context ...ation of satellite images, to compute size distributions (called derivative morphological profile) at each pixel. A closely related approach to the extraction of urban area was previously proposed in =-=[26]-=-. In these works, the proposed local features contain some information about scale. This information is well localized thanks to the use of connected morphological filtering of features. In this paper... |

7 |
Resolution independent characteristic scale dedicated to satellite images
- Luo, Aujol, et al.
(Show Context)
Citation Context ...presented in [19, 44, 45, 50] propose to estimate the salient scale of an image by considering extrema of various information theoretic operators in the linear scale space. For remote-sensing images, =-=[29]-=- has also proposed to rely on a linear scale-space to estimate a resolution invariant characteristic scale. The invariance is achieved by studying the effect of the image acquisition process on the li... |

7 |
A topdown algorithm for computation of level line trees
- Song
(Show Context)
Citation Context ...ciently computed using the algorithm presented in [36]. Figure 2.1 shows the result obtained with the FLST algorithm on a synthetic image. Notice that another implementation has also been proposed in =-=[43]-=-. We end this section by giving some notations for the attribute of shapes that are used in the sequel. For a pixel x of an image u, we denote by {fi(x)}i∈A(x) the set of shapes that contain x, A(x) b... |

6 |
Total Variation Minimization by the Fast Level Sets Transform
- Dibos, Koepfler
- 2001
(Show Context)
Citation Context ...8] for experimental comparisons. We use two different numerical approaches. The first one is a classical finite difference scheme, while the second implementation is based on the FLST, as proposed in =-=[14]-=-. Using this second implementation, the evolution speed of a given pixel is the area of the shape f0 containing this pixel, divided by its perimeter. That is, the gray value u(x) of the pixel at locat... |

6 |
SPOT5 THR mode
- Latry, Rouge
- 1998
(Show Context)
Citation Context ... definition of scale relying on a linear scale-space approach, even though some approximations in this direction are possible, see [29]. Figure 5.4 shows the scale map obtained from a SPOT5 THR image =-=[23]-=- of Marseille with resolution 2.5m. In this clever imaging system, two images captured by two different CCD line arrays are interpolated to generate the high resolution image. The PSF of SPOT5 THR ima... |

4 |
Scale space versus topographic map for natural images
- Caselles, Coll, et al.
- 1997
(Show Context)
Citation Context ...al approaches to scale computation. The first contribution is a method to compute a local scale measure (defining a characteristic scale at each pixel of a digital image) by using the topographic map =-=[10]-=- of the image. The main idea is that, for each pixel, we associate the scale of the most significant structure containing it. The definition of this structure relies on the topographic map, which is m... |

4 |
Saliency maps and attention selection in scale and spatial coordinates: An information theoretic approach
- Jägerstand
- 1995
(Show Context)
Citation Context ...he linear scale-space. This method has been widely applied in computer vision to select the optimal analysis scale of structures, see e.g. [27, 33]. Similar to this approach, the methods presented in =-=[19, 44, 45, 50]-=- propose to estimate the salient scale of an image by considering extrema of various information theoretic operators in the linear scale space. For remote-sensing images, [29] has also proposed to rel... |

4 |
Flou et quantification dans les images numériques
- Ladjal
- 2005
(Show Context)
Citation Context ...g kernel for the whole image is realistic in the case of satellite images, but is less consistent in the case of natural images for which the blur associated to an object depends on its position (see =-=[22]-=-). We thus recursively define the cumulated contrast as: C̄(fi) = { C̄(fi−1) + C(fi) if S(fi)− S(fi−1) < λP (fi−1) C(fi) otherwise 5 In other words the cumulated contrast of a shape fi is defined as C... |

3 |
Echelle et résolution en imagerie de télédétection
- Luo
- 2007
(Show Context)
Citation Context ...s observation more precise in the case of a one-dimensional Gaussian kernel, which constitutes an approximation of the PSF (Point Spread Function) of satellite captors such as those of SPOT5 HMA, see =-=[28]-=-. In Figure 5.2 is displayed a SPOT5 HMA image of Los Angeles (mainly urban area) together with its computed scale map. It can be observed that the computed scales are spatially very accurate (e.g. at... |

2 | Edge detection by helmholtz principle, Int - Desolneux, Moisan, et al. |

2 |
Total variation minimization for scalar/vector regularization
- Dibos, Koepfler, et al.
(Show Context)
Citation Context ...set smoothed by the acquisition kernel. Writing q for the quantization step and neglecting sampling, we have, for some gray level l, fi = ∂Ψl = {x ∈ Ω / u(x) = l} and fi+1 = ∂Ψl+q. Now, as noticed in =-=[15]-=-, if x(s) is a parameterization of ∂Ψl, then ∂Ψl+q can be approximated, for small q by: x̃(s) = x(s) + q ∇u |∇u|2 (3.5) If we now assume that |∇u| ≥ C for some C > 0, then fi+1 ⊂ fi⊕D(qC−1), where ⊕ s... |

2 |
saliency and image description, Int
- Kadir, Brady, et al.
(Show Context)
Citation Context ... disregard structures without significant gradients, for which no scale is computed. In order to obtain spatially accurate scale measures, it is quite natural to look toward non-linear approaches. In =-=[20, 21]-=- it is proposed to measure the significant scales of structures by computing the entropy of the joint distribution of pixels in a fixed neighborhood. It has been shown, see [21], that such an approach... |

2 |
computation of a contrast-invariant image representation
- Fast
(Show Context)
Citation Context ...structure relies on the topographic map, which is made of the connected component of the boundaries of level sets of the image. More precisely, we make use of the digital image transform presented in =-=[36]-=-, an efficient tool to compute the topographic map, representing an image by a hierarchical structure (an inclusion tree) of shapes. From this tree we search, for a given pixel, the most contrasted sh... |

2 |
3769: Minimum resolved object sizes and scales for imagery interpretation
- NATO
- 1993
(Show Context)
Citation Context ...o tune the spatial extent of analysis tools. In remote sensing imaging, the best observation scale is closely related to the concept of Analysis Scale in geography [37]. The NATO standard STANAG 3769 =-=[38]-=- gives some examples of the best scales for interpreting certain objects in remote-sensing images. In computer vision, most object recognition or detection methods includes a scale computation while o... |