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106
The image foresting transform: Theory, algorithms, and applications
 IEEE TPAMI
, 2004
"... The image foresting transform (IFT) is a graphbased approach to the design of image processing operators based on connectivity. It naturally leads to correct and efficient implementations and to a better understanding of how different operators relate to each other. We give here a precise definiti ..."
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Cited by 59 (21 self)
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The image foresting transform (IFT) is a graphbased approach to the design of image processing operators based on connectivity. It naturally leads to correct and efficient implementations and to a better understanding of how different operators relate to each other. We give here a precise definition of the IFT, and a procedure to compute it—a generalization of Dijkstra’s algorithm—with a proof of correctness. We also discuss implementation issues and illustrate the use of the IFT in a few applications.
Multilabel Random Walker Image Segmentation Using Prior Models
 In: IEEE Comp. Soc. Conf. Comp. Vision Pattern Recog
, 2005
"... 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 userspecified labels and produces a segmentation where each segment is connected ..."
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Cited by 40 (4 self)
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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 userspecified 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 userspecified labels. Finally, we show that this formulation leads to a deep connection with the popular graph cuts method of [8, 24]. 1
Persistent Homology  a Survey
 CONTEMPORARY MATHEMATICS
"... Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multiscale 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. ..."
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Cited by 36 (1 self)
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Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multiscale 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.
On topological watersheds
 JOURNAL OF MATHEMATICAL IMAGING AND VISION
, 2005
"... 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 algorit ..."
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Cited by 35 (10 self)
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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.
Quasilinear algorithms for the topological watershed
 JOURNAL OF MATHEMATICAL IMAGING AND VISION
, 2005
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Perceptionbased 3d triangle mesh segmentation using fast marching watersheds
 in Proceedings of the International Conference on Computer Vision and Pattern Recognition, II
, 2003
"... 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 vert ..."
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Cited by 32 (2 self)
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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.
Efficient Computation of MorseSmale Complexes for ThreeDimensional Scalar Functions
, 2007
"... The MorseSmale 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 MorseSmale complex in a series of sweeps through the data, ..."
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Cited by 23 (11 self)
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The MorseSmale 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 MorseSmale 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.
Watershed algorithms and contrast preservation
 DISCRETE GEOMETRY FOR COMPUTER IMAGERY, FRANCE
, 2003
"... 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 co ..."
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Cited by 17 (4 self)
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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.
A Practical Approach to MorseSmale Complex Computation: Scalability and Generality
"... Abstract—The MorseSmale (MS) complex has proven to be a useful tool in extracting and visualizing features from scalarvalued data. However, efficient computation of the MS complex for large scale data remains a challenging problem. We describe a new algorithm and easily extensible framework for co ..."
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Cited by 16 (3 self)
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Abstract—The MorseSmale (MS) complex has proven to be a useful tool in extracting and visualizing features from scalarvalued data. However, efficient computation of the MS complex for large scale data remains a challenging problem. We describe a new algorithm and easily extensible framework for computing MS complexes for large scale data of any dimension where scalar values are given at the vertices of a closurefinite and weak topology (CW) complex, therefore enabling computation on a wide variety of meshes such as regular grids, simplicial meshes, and adaptive multiresolution (AMR) meshes. A new divideandconquer strategy allows for memoryefficient computation of the MS complex and simplification onthefly to control the size of the output. In addition to being able to handle various data formats, the framework supports implementationspecific optimizations, for example, for regular data. We present the complete characterization of critical point cancellations in all dimensions. This technique enables the topology based analysis of large data on offtheshelf computers. In particular we demonstrate the first full computation of the MS complex for a 1 billion/1024 3 node grid on a laptop computer with 2Gb memory. Index Terms—Topologybased analysis, MorseSmale complex, large scale data. 1