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...rely on image representation based on morphological trees to extract meaningful objects. In [6], objects are selected when their level lines match some criterion based on compactness and contrast. In =-=[2]-=-, objects are spotted as minima of the number of false alarms estimated on every level lines thanks to an a contrario model. We have introduced in [13] an energy functional, computed on image componen...

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...f those methods. In [9], the author proposes a segmentation algorithm that selects the perceptible level lines matching some criteria: number of Tjunctions, compactness, and contrast. Papers [10] and =-=[11]-=- are the closest ones to what we propose here. The authors define the meaningfulness of a given level line using the a contrario model. Only the smallest gradient along a level line and its length are...

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...ions. In [3] Morel et al. define the notion of significative level line (a well contrasted level-line) and propose a parameter-less algorithm to compute those level-lines. In a later work, Cao et al. =-=[4]-=- improved the model of Morel et al. [3] by using a local concept of significative level-line. Finally, Cao [5] proposes an algorithm that selects level lines using the good continuation principle, whi...

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...s invariant to contrast changes. Image simplification or segmentation can then be defined by selecting meaningful level lines in that tree. That subject has been investigated in the past ten years by =-=[3, 4, 5, 6]-=-. In [7] Lu et al. propose also a tree simplification method for image simplification purpose using the binary partition tree [8] and a knee function. Following the seminal work of Mumford and Shah [9...

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...gh. In addition, we want to obtain the regularizing effect of the boundary term in the M-S functional, which favors short curves. For this reason, we use a definition similar to the one by Cao et al. =-=[8]-=-, in the sense that we say a curve is meaningful if it has an extraordinarily high contrast. However, we also penalize long curves by using the accumulated contrast along the curve instead of the mini...

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...he mean gradient value along the segment to be less sensitive to outliers. The gradient magnitude is computed using 2x2 finite differences to avoid creating artificial correlation between pixels (see =-=[2]-=- for detailed explanations). The a contrario model we choose is a white uniform noise with the same gradient magnitude distribution as in the original image, but where pixels are spatially independent...

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...out applications can be found in [13]). The reason why the tree of shapes is interesting is reported by several authors who claim that, in gray level images, object contours coincide with level lines =-=[14, 15, 16]-=-. Indeed, in [17] we have shown that object detection based on the tree of shapes can outperform the Chan and Vese method. Yet obtaining a tree of shapes for color images is challenging. In a gray lev...

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...very simple and efficient method to select the most meaningful level lines in the topographic map has been introduced by Desolneux, Moisan and Morel in [13]. Recent improvements have been achieved in =-=[9]-=- but, for the sake of simplicity, we only describe the original arguments which rely on an a contrario detection principle which will be detailed in Sect. 3. Let u : R2 → R be a differentiable grey le...

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... and local features matching [10]. The first step in a contrario methods is to define a set of possibly meaningful events. In the context of edge detection, two approaches have been proposed. In [6], =-=[11]-=-, [12], candidate edges are extracted from the level lines of the image. Given a grey level image u : Ω → R, level lines are defined as closed Jordan curves contained in the boundary of a level set wi...