V. Caselles, B. Coll, and J. M Morel. Topographic maps and local contrast changes in natural images. International Journal of Computer Vision, 33(1):5-27, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....f are defined as: L fl (f) fx 2 S : f(x) flg; fl 2 R and their boundaries constitute the level lines of f . Gamma We define the bilevel sets of f with levels v and w, 0 v w, as the set of pixels x 2 S such as v f(x) w. Gamma BV (S) is the space of functions with bounded variations [33, 6]. The BV space is a very straightforward space for image segmentation. A situation for an image I not to be in the BV space is to have level lines with infinite length or the sum of finite perimeters of level lines tends to infinity. We assume hereafter that small objects are not too numerous in I ....

....The algorithm we propose is automatic and does require neither the number of regions nor any initial average values for regions and background. 4.1. Level sets and object boundaries The key ingredient of the procedure is the construction of objects whose boundaries are level lines in the image [6]. In the today s technology, we can traditionally associate with an image 256 level sets fL fl (f)g, fl 2 f0; 1; 2; Delta Delta Delta ; 255g. Let fl be a fixed level of the image, 0 fl 255, and let u fl be the binary image at level fl of the discretized original image f , defined by u fl (x) ....

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V. Caselles, B. Coll, and J.M. Morel. Topographic maps and local contrast changes in natural images. Int J. Computer Vision, 33(1):5--27, 1999.

....length of Omega . Equation 9 states a necessary condition which is essential to prove that a subset of level lines globally minimizes the energy. If f is of bounded variation, the connected components of level sets can be characterized by their boundaries, that is the so called level lines of f [3]. In consequence of Lemma 1, those curves constitute the borders f Omega i g of objects f Omega i g. 4 A stepwise greedy algorithm for image segmentation This section describes our algorithmic procedure for object boundaries estimation. Our recommendations for the concrete choice of the ....

....in this section. The algorithm we propose does require neither the number of regions nor any initial mean gray values for regions and background. 4. 1 Level sets and object boundaries The key ingredient of the procedure is the construction of objects whose boundaries are image level lines [3]. In practical imaging, we can associate with an image 255 level sets fL fl (f)g, 0 fl 255. We consider the scenario where a point x belongs to one single connected component at once within the image level sets. We take into account this fact and define the bilevel sets of f as the set of pixels ....

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V. Caselles, B. Coll, and J.M. Morel. Topographic maps and local contrast changes in natural images. Int J. Computer Vision, 33(1):5--27, 1999.

.... shown that perceptual edge curves can be represented by pieces of level lines: level lines may be considered as the atoms of the perception that is, the basic elements on Image Compression Through Level Lines and Wavelet Packets 5 which further representations may be built [Caselles et al. 1996, Caselles et al. 1999]. 1.1. MORPHOLOGICAL EDGES ARE PERCEPTIBLE LEVEL LINES In order to easily de ne the level lines, let us introduce the space of the functions of bounded variation. Let be an open bounded subset of IR 2 . The total variation of an continuous image u : IR can be simply obtained, if u 2 C 1( ....

....and their essential boundaries constitute the level lines of u. If we map the level lines of an image for a given set of levels f 1 2 : n = 1g; 8) we get a segmentation of the image with sets of type fx 2 = i 1 u(x) i g: 9) 6 Such segmentation is called a topographic map [Caselles et al. 1999]. Example of a topographic map is given in Fig. 1.3 B. More generally, one can consider a segmentation achieved using some connected components of lower levet sets ( u ] and upper level sets ( u ] only. From Fig. 1.3 B one can observe that pieces of some level lines are located at the ....

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Caselles, V., Coll, B., and Morel, J.-M. (1999). Topographic maps and local contrast changes in natural images. Int. J. Comp. Vision, 33(1):5-27.

....Processing. Number of Computer Vision algorithms are based on the comparison of edges between two images. If the necessary condition of a representation invariant under translation has been acknowledged in the past, the significance of the contrast invariant property has been pointed out recently [3] : the light captors of cameras are nonlinear increasing functions, and these functions differ from one camera to the other (the response of a given camera is even changing with time) In addition, lighting condition in an outdoor scene is continually changing, generating changes of contrast. An ....

....is continually changing, generating changes of contrast. An indoor scene may also be concerned by such modification : a moving object or a moving camera with exposure set to automatic will experience contrast changes. Topographic map of gray level images has been introduced in this framework [3], as a geometrical and contrast invariant representation of the information contained in natural images. First applications of topographic maps include extraction of shapes [9, 6] comparison of images [1, 9] structured compression [5] and disocclusion [8] recovery of hidden parts of an object ....

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V. Caselles, B. Coll, and J.-M. Morel. Topographic maps and local contrast changes in natural images. Int. J. Comp. Vision, 33(1):5--27, 1999.

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V. Caselles, B. Coll, and J. M Morel. Topographic maps and local contrast changes in natural images. International Journal of Computer Vision, 33(1):5-27, 1999.

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V. Caselles, B. Coll, and J.-M. Morel. Topographic maps and local contrast changes in natural images. Int. J. Comp. Vision, 33(1):5--27, 1999.

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V. Caselles, B. Coll, and J. M. Morel, Topographic maps and local contrast changes in natural images, Internat. J. Comput. Vision, 33 (1999), pp. 5--27.

....same morphology by the way of the evolutive pressure , two image analysis structures with very di erent origins, namely the variational snakes and the maximal meaningful level lines, arrive at almost exactly the same numerical results. Level lines for image representation have been proposed in [CCM99] as an ecient contrast invariant representation of any image. This representation stems from Mathematical Morphology [Ser82] where connected components of level sets are extensively used as image features and indeed are contrast invariant features (level lines are nothing but the boundaries of ....

V. Caselles, B. Coll, and J.-M Morel. Topographic maps and local contrast changes in natural images. Int. J. of Computer Vision, 33(1):5-27, 1999.

....formula R : x X # u . 1. 2) The basic postulate of Mathematical Morphology prescribes that the geometric information of the image u is contained in the family of its level sets [62] 38] or in a more local formulation, in the family of connected components of the level sets of u [62, 63] [24]. We shall refer to the family of connected components of the upper level sets of u as the topographic map of u. In case that u is a function of bounded variation in D , i.e. u BV (D) 2, 35, 77] its topographic map has a description in terms of Jordan curves [3] With an adequate ....

V. Caselles, B. Coll and J.M. Morel, Topographic Maps and Local Contrast Changes in Natural Images, Int. J. of Computer Vision, 33 (1999), pp. 5-27.

....this principle. For a classi cation of contrast invariant image multiscale smoothing operators we refer to [1] 17] 36] Level sets are therefore basic objects for image processing and analysis. In order to have a more local description of the basic objects of an image, several authors [41] [7] proposed to consider the connected components of (upper or lower) level sets as the basic objects of the image. They argue that contrast changes are local and depend upon the re ectance properties of objects. Thus, not only global contrast, but also local contrast is irrelevant. In [7] a notion ....

....[41] 7] proposed to consider the connected components of (upper or lower) level sets as the basic objects of the image. They argue that contrast changes are local and depend upon the re ectance properties of objects. Thus, not only global contrast, but also local contrast is irrelevant. In [7], a notion of local contrast change is de ned and it is proved that only connected components of level sets are invariant under such contrast changes. This led to the introduction of topographic maps, the family of connected components of upper (or lower) level sets [u ] resp. u ] More ....

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V. Caselles, B. Coll and J.M. Morel, Topographic Maps and Local Contrast Changes in Natural Images, To appear at Int. J. Comp. Vision.

....by the luminance component only. We present experiments which illustrate the advantages and the drawbacks of each method. 1. Introduction and Background This paper discusses the extension of topographic maps to color images. Topographic maps of gray level images have been recently introduced [5, 6] as a geometrical representation of the information contained in natural images. A lower (or upper) topographic map of a gray level image u : Omega ae IR 2 IR is the family of the connected components of the lower (or upper) level sets of u, a lower level set [u ] being the set of pixels x ....

....based on topographic maps can t afford to deal with all of these data, and the representation must be widely simplified. The right way to simplify a topographic map is to apply a morphological filtering on the gray level image u. A filter F is morphological if it is a contrast invariant operator [6], that is if, for any increasing continuous real function g, we have F (g(u) g(F (u) 2) The use of a morphological filter is justified by the property [F (g(u) F (u) g Gamma1 ( which ensures that the family of level sets of the filtered image is invariant under any change of ....

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V. Caselles, B. Coll, and J.-M. Morel. Topographic maps and local contrast changes in natural images. Int. J. Comp. Vision, 33(1):5--27, 1999.

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V. Caselles, B. Coll, and J-M. Morel, "Topographic maps and local contrast changes in natural images," Int. J. Comp. Vision, vol. 33, no. 1, pp. 5--27, 1999.

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V. Caselles, T. Coll, and J.M. Morel. Topographic maps and local contrast changes in natural images. International Journal of Computer Vision, 33(1):5--27, 1999.

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V. Caselles, T. Coll, and J.M. Morel. Topographic maps and local contrast changes in natural images. International Journal of Computer Vision, 33(1):5--27, 1999.

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