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113
Topographic Maps and Local Contrast Changes in Natural Images
 Int. J. Comp. Vision
, 1999
"... . We call "natural" image any photograph of an outdoor or indoor scene taken by a standard camera. We discuss the physical generation process of natural images as a combination of occlusions, transparencies and contrast changes. This description fits to the phenomenological description of ..."
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Cited by 82 (9 self)
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. We call "natural" image any photograph of an outdoor or indoor scene taken by a standard camera. We discuss the physical generation process of natural images as a combination of occlusions, transparencies and contrast changes. This description fits to the phenomenological description of Gaetano Kanizsa according to which visual perception tends to remain stable with respect to these basic operations. We define a contrast invariant presentation of the digital image, the topographic map, where the subjacent occlusiontransparency structure is put into evidence by the interplay of level lines. We prove that each topographic map represents a class of images invariant with respect to local contrast changes. Several visualization strategies of the topographic map are proposed and implemented and mathematical arguments are developed to establish stability properties of the topographic map under digitization. Keywords: topographic map, mathematical morphology, level set, junctions, contrast changes, digitization 1.
Connected Components of Sets of Finite Perimeter and Applications to Image Processing
 JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
, 1999
"... This paper contains a systematic analysis of a natural measure theoretic notion of connectedness for sets of finite perimeter in R^N, introduced by H. Federer in the more general framework of the theory of currents. We provide a new and simpler proof of the existence and uniqueness of the decomposit ..."
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Cited by 37 (8 self)
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This paper contains a systematic analysis of a natural measure theoretic notion of connectedness for sets of finite perimeter in R^N, introduced by H. Federer in the more general framework of the theory of currents. We provide a new and simpler proof of the existence and uniqueness of the decomposition into the socalled Mconnected components. Moreover, we study carefully the structure of the essential boundary of these components and give in particular a reconstruction formula of a set of finite perimeter from the family of the boundaries of its components. In the two dimensional case we show that this notion of connectedness is comparable with the topological one, modulo the choice of a suitable representative in the equivalence class. Our strong motivation for this study is a mathematical justification of all those operations in image processing that involve connectedness and boundaries. As an application, we use this weak notion of connectedness to provide a rigorous mathemati...
Geometry and Color in Natural Images
, 2000
"... Most image analysis algorithms are defined for the grey level channel, particularly when geometric information is looked for in the digital image. We propose an experimental procedure in order to decide whether this attitude is sound or not. We adopt the hypothesis that the essential geometric conte ..."
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Cited by 34 (6 self)
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Most image analysis algorithms are defined for the grey level channel, particularly when geometric information is looked for in the digital image. We propose an experimental procedure in order to decide whether this attitude is sound or not. We adopt the hypothesis that the essential geometric contents of an image is contained in its level lines. The set of all level lines, or topographic map, is a complete contrast invariant image description : it yields a line structure by far more complete than any edge description, since we can fully reconstruct the image from it, up to a local contrast change. We then design an algorithm constraining the color channels of a given image to have the same geometry (i.e. the same level lines) as the grey level. If the assumption that the essential geometrical information is contained in the grey level is sound, then this algorithm should not alter the colors of the image or its visual aspect. We display several experiments confirming this hypothesis. Conversely, we also show the effect of imposing the color of an image to the topographic map of another one : it results, in a striking way, in the dominance of grey level and the fading of a color deprived of its geometry. We finally give a mathematical proof that the algorithmic procedure is intrinsic, i.e. does not depend asymptotically upon the quantization mesh used for the topographic map. We also prove its contrast invariance.
A quasilinear algorithm to compute the tree of shapes of nD images
"... Abstract. To compute the morphological selfdual representation of images, namely the tree of shapes, the stateoftheart algorithms do not have a satisfactory time complexity. Furthermore the proposed algorithms are only effective for 2D images and they are far from being simple to implement. That ..."
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Cited by 21 (12 self)
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Abstract. To compute the morphological selfdual representation of images, namely the tree of shapes, the stateoftheart algorithms do not have a satisfactory time complexity. Furthermore the proposed algorithms are only effective for 2D images and they are far from being simple to implement. That is really penalizing since a selfdual representation of images is a structure that gives rise to many powerful operators and applications, and that could be very useful for 3D images. In this paper we propose a simpletowrite algorithm to compute the tree of shapes; it works for nD images and has a quasilinear complexity when data quantization is low, typically 12 bits or less. To get that result, this paper introduces a novel representation of images that has some amazing properties of continuity, while remaining discrete. 1
Connected operators
"... Connected operators are filtering tools that act by merging elementary regions called flat zones. Connecting operators cannot create new contours nor modify their position. Therefore, they have very good contourpreservation properties and are capable of both lowlevel filtering and higherlevel obj ..."
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Cited by 20 (2 self)
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Connected operators are filtering tools that act by merging elementary regions called flat zones. Connecting operators cannot create new contours nor modify their position. Therefore, they have very good contourpreservation properties and are capable of both lowlevel filtering and higherlevel object recognition. This article gives an overview on connected operators and their application to image and video filtering. There are two popular techniques used to create connected operators. The first one relies on a reconstruction process. The operator involves first a simplification step based on a “classical ” filter and then a reconstruction process. In fact, the reconstruction can be seen as a way to create a connected version of an arbitrary operator. The simplification effect is defined and limited by the first step. The examples we show include simplification in terms of size or contrast. The second strategy to define connected operators relies on a hierarchical regionbased representation of the input image, i.e., a tree, computed in an initial step. Then, the simplification is obtained by pruning the tree, and, third, the output image is constructed from the pruned tree. This article presents the most important trees that have been used to create connected operators and also discusses important families of simplification or pruning criteria. We also give a brief overview on efficient implementations of the reconstruction process and of tree construction. Finally, the [ Philippe Salembier and Michael H.F. Wilkinson] [A review of regionbased morphological image processing techniques]
Concurrent Computation of Attribute Filters on Shared Memory Parallel Machines
, 2008
"... Morphological attribute filters have not previously been parallelized mainly because they are both global and nonseparable. We propose a parallel algorithm that achieves efficient parallelism for a large class of attribute filters, including attribute openings, closings, thinnings, and thickenings, ..."
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Cited by 17 (0 self)
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Morphological attribute filters have not previously been parallelized mainly because they are both global and nonseparable. We propose a parallel algorithm that achieves efficient parallelism for a large class of attribute filters, including attribute openings, closings, thinnings, and thickenings, based on Salembier’s MaxTrees and Mintrees. The image or volume is first partitioned in multiple slices. We then compute the Maxtrees of each slice using any sequential MaxTree algorithm. Subsequently, the Maxtrees of the slices can be merged to obtain the Maxtree of the image. A Cimplementation yielded good speedups on both a 16processor MIPS 14000 parallel machine and a dualcore Opteronbased machine. It is shown that the speedup of the parallel algorithm is a direct measure of the gain with respect to the sequential algorithm used. Furthermore, the concurrent algorithm shows a speed gain of up to 72 percent on a singlecore processor due to reduced cache thrashing.
P.: Binary partition trees for object detection
 IEEE Trans. on Image Processing
"... Abstract—This paper discusses the use of Binary Partition Trees (BPTs) for object detection. BPTs are hierarchical regionbased representations of images. They define a reduced set of regions that covers the image support and that spans various levels of resolution. They are attractive for object d ..."
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Cited by 17 (5 self)
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Abstract—This paper discusses the use of Binary Partition Trees (BPTs) for object detection. BPTs are hierarchical regionbased representations of images. They define a reduced set of regions that covers the image support and that spans various levels of resolution. They are attractive for object detection as they tremendously reduce the search space. In this paper, several issues related to the use of BPT for object detection are studied. Concerning the tree construction, we analyze the compromise between computational complexity reduction and accuracy. This will lead us to define two parts in the BPT: one providing accuracy and one representing the search space for the object detection task. Then we analyze and objectively compare various similarity measures for the tree construction. We conclude that different similarity criteria should be used for the part providing accuracy in the BPT and for the part defining the search space and specific criteria are proposed for each case. Then we discuss the object detection strategy based on BPT. The notion of node extension is proposed and discussed. Finally, several object detection examples illustrating the generality of the approach and its efficiency are reported. Index Terms—Binary partition tree, hierarchical representation, image region analysis, image representations, image segmentation, object detection. I.
Morphological filtering in shape spaces: Applications using treebased image representations
, 2012
"... Connected operators are filtering tools that act by merging elementary regions of an image. A popular strategy is based on treebased image representations: for example, one can compute an attribute on each node of the tree and keep only the nodes for which the attribute is sufficiently strong. This ..."
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Cited by 15 (10 self)
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Connected operators are filtering tools that act by merging elementary regions of an image. A popular strategy is based on treebased image representations: for example, one can compute an attribute on each node of the tree and keep only the nodes for which the attribute is sufficiently strong. This operation can be seen as a thresholding of the tree, seen as a graph whose nodes are weighted by the attribute. Rather than being satisfied with a mere thresholding, we propose to expand on this idea, and to apply connected filters on this latest graph. Consequently, the filtering is done not in the space of the image, but on the space of shapes build from the image. Such a processing is a generalization of the existing treebased connected operators. Indeed, the framework includes classical existing connected operators by attributes. It also allows us to propose a class of novel connected operators from the leveling family, based on shape attributes. Finally, we also propose a novel class of selfdual connected operators that we call morphological shapings †. 1.
CONTEXTBASED ENERGY ESTIMATOR: APPLICATION TO OBJECT SEGMENTATION ON THE TREE OF SHAPES
"... Image segmentation can be defined as the detection of closed contours surrounding objects of interest. Given a family of closed curves obtained by some means, a difficulty is to extract the relevant ones. A classical approach is to define an energy minimization framework, where interesting contours ..."
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Cited by 14 (9 self)
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Image segmentation can be defined as the detection of closed contours surrounding objects of interest. Given a family of closed curves obtained by some means, a difficulty is to extract the relevant ones. A classical approach is to define an energy minimization framework, where interesting contours correspond to local minima of this energy. Active contours, graph cuts or minimum ratio cuts are instances of such approaches. In this article, we propose a novel efficient ratiocut estimator which is both contextbased and can be interpreted as an active contour. As a first example of the effectiveness of our formulation, we consider the tree of shapes, which provides a family of level lines organized in a tree hierarchy through an inclusion relationship. Thanks to the tree structure, the estimator can be computed incrementally in an efficient fashion. Experimental results on synthetic and real images demonstrate the robustness and usefulness of our method. Index Terms — Ratiocut, Tree of shapes, Level lines, Active contours, Image segmentation.
Volumetric Attribute Filtering and Interactive Visualization using the MaxTree Representation
"... Abstract—The MaxTree, designed for morphological attribute filtering in image processing, is a data structure in which the nodes represent connected components for all threshold levels in a data set. Attribute filters compute some attribute describing the shape or size of each connected component, ..."
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Cited by 13 (5 self)
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Abstract—The MaxTree, designed for morphological attribute filtering in image processing, is a data structure in which the nodes represent connected components for all threshold levels in a data set. Attribute filters compute some attribute describing the shape or size of each connected component, and then decide which components to keep or to discard. In this paper, we augment the basic MaxTree data structure such that interactive volumetric filtering and visualization becomes possible. We introduce extensions that allow (i) direct, splattingbased, volume rendering, (ii) representation of the MaxTree on graphics hardware, and (iii) fast active cell selection for isosurface generation. In all three cases, we can use the MaxTree representation for visualization directly, without needing to reconstruct the volumetric data explicitly. We show that both filtering and visualization can be performed at interactive frame rates, ranging between 2.4 and 32 frames per seconds. In contrast, a standard texturebased volume visualization method manages only between 0.5 and 1.8 frames per second. For isovalue browsing, the experimental results show that the performance is comparable to the performance of an interval tree, where our method has the advantage that both filter threshold browsing and isolevel browsing are fast. It is shown that the methods using graphics hardware can be extended to other connected filters. Index Terms—MaxTree, nonlinear filtering, mathematical morphology, volume visualization, connected filters I.