The recently introduced random walker segmentation algorithm of [14] has been shown to have desirable theoretical properties and to perform well on a wide variety of images in practice. However, this algorithm requires user-specified labels and produces a segmentation where each segment is connected to a labeled pixel. We show that incorporation of a nonparametric probability density model allows for an extended random walkers algorithm that can locate disconnected objects and does not require user-specified labels. Finally, we show that this formulation leads to a deep connection with the popular graph cuts method of [8, 24]. 1

In this paper, we describe an algorithm called Fast Marching Watersheds that segments a triangle mesh into visual parts. This computer vision algorithm leverages a human vision theory known as the minima rule. Our implementation computes the principal curvatures and principal directions at each vertex of a mesh, and then our hillclimbing watershed algorithm identifies regions bounded by contours of negative curvature minima. These regions fit the definition of visual parts according to the minima rule. We present evaluation analysis and experimental results for the proposed algorithm. 1.

Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization we frequently observe in nature into a mathematical formalism. Here we give a record of the short history of persistent homology and present its basic concepts. Besides the mathematics we focus on algorithms and mention the various connections to applications, including to biomolecules, biological networks, data analysis, and geometric modeling.

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In this paper, we investigate topological watersheds [1]. One of our main results is a necessary and sufficient condition for a map G to be a watershed of a map F, this condition is based on a notion of extension. A consequence of the theorem is that there exists a (greedy) polynomial time algorithm to decide whether a map G is a watershed of a map F or not. We introduce a notion of “separation between two points ” of an image which leads to a second necessary and sufficient condition. We also show that, given an arbitrary total order on the minima of a map, it is possible to define a notion of “degree of separation of a minimum ” relative to this order. This leads to a third necessary and sufficient condition for a map G to be a watershed of a map F. At last we derive, from our framework, a new definition for the dynamics of a minimum.

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This paper is devoted to the study of watershed algorithms behavior. Through the introduction of a concept of pass value, we show that most classical watershed algorithms do not allow the retrieval of some important topological features of the image (in particular, saddle points are not correctly computed). An important consequence of this result is that it is not possible to compute sound measures such as depth, area or volume of basins using most classical watershed algorithms. Only one watershed principle, called topological watershed, produces correct watershed contours.

Our research deals with a semi-automatic region-growing segmentation technique. This method only needs one seed inside the region of interest (ROI). We applied it for spinal cord segmentation but it also shows results for parotid glands or even tumors. Moreover, it seems to be a general segmentation method as it could be applied in other computer vision domains then medical imaging. We use both the thresholding simplicity and the spatial information. The gray-scale and spatial distances from the seed to all the other pixels are computed. By normalizing and subtracting to 1 we obtain the probability for a pixel to belong to the same region as the seed. We will explain the algorithm and show some preliminary results which are encouraging.

The Morse-Smale complex is an efficient representation of the gradient behavior of a scalar function, and critical points paired by the complex identify topological features and their importance. We present an algorithm that constructs the Morse-Smale complex in a series of sweeps through the data, identifying various components of the complex in a consistent manner. All components of the complex, both geometric and topological, are computed, providing a complete decomposition of the domain. Efficiency is maintained by representing the geometry of the complex in terms of point sets.

We describe a multiresolution extension to maximum intensity projection (MIP) volume rendering, allowing progressive refinement and perfect reconstruction. The method makes use of morphological adjunction pyramids. The pyramidal analysis and synthesis operators are composed of morphological 3-D erosion and dilation, combined with dyadic downsampling for analysis and dyadic upsampling for synthesis. In this case the MIP operator can be interchanged with the synthesis operator. This fact is the key to an efficient multiresolution MIP algorithm, because it allows the computation of the maxima along the line of sight on a coarse level, before applying a two-dimensional synthesis operator to perform reconstruction of the projection image to a finer level. For interpolation and resampling of volume data, which is required to deal with arbitrary view directions, morphological sampling is used, an interpolation method well adapted to the nonlinear character of MIP. The structure of the resulting multiresolution algorithm is very similar to wavelet splatting, the main differences being that (i) linear summation of voxel values is replaced by maximum computation, and (ii) linear wavelet filters are replaced by (nonlinear) morphological filters.