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33
Geodesic Active Contours
, 1997
"... A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both in ..."
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Cited by 1073 (43 self)
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A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved, allowing stable boundary detection when their gradients suffer from large variations, including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. The scheme was implemented using an efficient algorithm for curve evolution. Experimental results of applying the scheme to real images including objects with holes and medical data imagery demonstrate its power. The results may be extended to 3D object segmentation as well.
HamiltonJacobi Skeletons
, 1999
"... The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shapefromshading and for recent dynamic theories of shape. Its numerical simulation can be delicate, ..."
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Cited by 119 (12 self)
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The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shapefromshading and for recent dynamic theories of shape. Its numerical simulation can be delicate, owing to the formation of singularities in the evolving front and is typically based on level set methods. However, there are more classical approaches rooted in Hamiltonian physics which have yet to be widely used by the computer vision community. In this paper we review the Hamiltonian formulation, which offers specific advantages when it comes to the detection of singularities or shocks. We specialize to the case of Blum's grass fire flow and measure the average outward ux of the vector field that underlies the Hamiltonian system. This measure has very different limiting behaviors depending upon whether the region over which it is computed shrinks to a singular point or a nonsingular one. Hence, it is an effective way to distinguish between these two cases. We combine the ux measurement with a homotopy preserving thinning process applied in a discrete lattice. This leads to a robust and accurate algorithm for computing skeletons in 2D as well as 3D, which has low computational complexity. We illustrate the approach with several computational examples.
Shapes, Shocks, and Deformations I: The Components of TwoDimensional Shape and the ReactionDiffusion Space
 International Journal of Computer Vision
, 1994
"... We undertake to develop a general theory of twodimensional shape by elucidating several principles which any such theory should meet. The principles are organized around two basic intuitions: first, if a boundary were changed only slightly, then, in general, its shape would change only slightly. Th ..."
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Cited by 64 (5 self)
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We undertake to develop a general theory of twodimensional shape by elucidating several principles which any such theory should meet. The principles are organized around two basic intuitions: first, if a boundary were changed only slightly, then, in general, its shape would change only slightly. This leads us to propose an operational theory of shape based on incremental contour deformations. The second intuition is that not all contours are shapes, but rather only those that can enclose "physical" material. A theory of contour deformation is derived from these principles, based on abstract conservation principles and HamiltonJacobi theory. These principles are based on the work of Sethian [82, 86], the OsherSethian level set formulation [65], the classical shock theory of Lax [53, 54], as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of GageHamiltonGrayson [32, 37]. The result is a characterization of th...
Shape Recovery Algorithms Using Level Sets in 2D/3D Medical Imagery: A StateoftheArt Review
, 2001
"... The class of geometric deformable models, socalled level sets, has brought tremendous impact to medical imagery due to its capability to preserve topology and fast shape recovery. In an effort to facilitate a clear and full understanding of these powerful stateoftheart applied mathematical tools ..."
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Cited by 36 (2 self)
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The class of geometric deformable models, socalled level sets, has brought tremendous impact to medical imagery due to its capability to preserve topology and fast shape recovery. In an effort to facilitate a clear and full understanding of these powerful stateoftheart applied mathematical tools, this paper is an attempt to explore these geometric methods, their implementations and integration of regularization terms to improve the robustness of these topologically independent propagating curves/surfaces. This paper first presents the origination of the level sets, followed by the taxonomy tree of level sets. We then derive the fundamental equation of curve/surface evolution and zerolevel curves/surfaces. The paper then focuses on the first core class of level sets, the socalled level sets "without regularizers". The next section is devoted on a second kind, socalled level sets "with regularizers". In this class, we present four kinds of systems on the design of the regularizers. Next, the paper presents a third kind of level sets, socalled the "bubblebased" techniques. An entire section is dedicated to optimization and quantification techniques for shape recovery when used with the level sets. Finally, the paper concludes with 22 general merits and four demerits on level sets and the future of level sets in medical image segmentation. We present the applications of level sets to complex shapes likethehuman cortex acquired via MRI for neurological image analysis.
Subpixel Distance Maps and Weighted Distance Transforms
 JOURNAL OF MATHEMATICAL IMAGING AND VISION
, 1994
"... A new framework for computing the Euclidean distance and weighted distance from the boundary of a given digitized shape is presented. The distance is calculated with subpixel accuracy. The algorithm is based on an equal distance contour evolution process. The moving contour is embedded as a level ..."
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Cited by 30 (8 self)
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A new framework for computing the Euclidean distance and weighted distance from the boundary of a given digitized shape is presented. The distance is calculated with subpixel accuracy. The algorithm is based on an equal distance contour evolution process. The moving contour is embedded as a level set in a time varying function of higher dimension. This representation of the evolving contour makes possible the use of an accurate and stable numerical scheme, due to Osher and Sethian [22]. The relation between the classical shape from shading problem and the weighted distance transform is presented, as well as an algorithm that calculates the geodesic distance transform on surfaces.
Minimal surfaces: a geometric three dimensional segmentation approach
, 1997
"... A novel geometric approach for three dimensional object segmentation is presented. The scheme is based on geometric deformable surfaces moving towards the objects to be detected. We show that this model is related to the computation of surfaces of minimal area (local minimal surfaces). The space w ..."
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Cited by 26 (6 self)
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A novel geometric approach for three dimensional object segmentation is presented. The scheme is based on geometric deformable surfaces moving towards the objects to be detected. We show that this model is related to the computation of surfaces of minimal area (local minimal surfaces). The space where these surfaces are computed is induced from the three dimensional image in which the objects are to be detected. The general approach also shows the relation between classical deformable surfaces obtained via energy minimization and geometric ones derived from curvature flows in the surface evolution framework. The scheme is stable, robust, and automatically handles changes in the surface topology during the deformation. Results related to existence, uniqueness, stability, and correctness of the solution to this geometric deformable model are presented as well. Based on an efficient numerical algorithm for surface evolution, we present a number of examples of object detection in real and synthetic images.
Nonlinear scalespace representation with morphological levelings
 J. of Visual Comm. and Image Representation
, 2000
"... In this paper we present a nonlinear scalespace representation based on a general class of morphological strong filters, the levelings, which include the openings and closings by reconstruction. These filters are very useful for image simplification and segmentation. From one scale to the next, det ..."
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Cited by 19 (7 self)
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In this paper we present a nonlinear scalespace representation based on a general class of morphological strong filters, the levelings, which include the openings and closings by reconstruction. These filters are very useful for image simplification and segmentation. From one scale to the next, details vanish, but the contours of the remaining objects are preserved sharp and perfectly localized. Both the lattice algebraic and the scalespace properties of levelings are analyzed and illustrated. We also develop a nonlinear partial differential equation that models the generation of levelings as the limit of a controlled growth starting from an initial seed signal. Finally, we outline the use of levelings in improving the Gaussian scalespace by using the latter as an initial seed to generate multiscale levelings that have a superior preservation of image edges. C ○ 2000 Academic Press Key Words: scalespace; mathematical morphology; levelings; multiscale representation; differential equations.
A unified geometric model for 3D confocal image analysis in cytology
 Lawrence Berkeley National Laboratory, Univ. California, Berkeley
, 1998
"... Abstract—In this paper, we use partialdifferentialequationbased filtering as a preprocessing and post processing strategy for computeraided cytology. We wish to accurately extract and classify the shapes of nuclei from confocal microscopy images, which is a prerequisite to an accurate quantitativ ..."
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Cited by 13 (0 self)
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Abstract—In this paper, we use partialdifferentialequationbased filtering as a preprocessing and post processing strategy for computeraided cytology. We wish to accurately extract and classify the shapes of nuclei from confocal microscopy images, which is a prerequisite to an accurate quantitative intranuclear (genotypic and phenotypic) and internuclear (tissue structure) analysis of tissue and cultured specimens. First, we study the use of a geometrydriven edgepreserving image smoothing mechanism before nuclear segmentation. We show how this filter outperforms other widelyused filters in that it provides higher edge fidelity. Then we apply the same filter, with a different initial condition, to smooth nuclear surfaces and obtain subpixel accuracy. Finally we use another instance of the geometrical filter to correct for misinterpretations of the nuclear surface by the segmentation algorithm. Our prefiltering and post filtering nicely complements our initial segmentation strategy, in that it provides substantial and measurable improvement in the definition of the nuclear surfaces. Index Terms—Cytology, differential geometry, dynamic surfaces, image processing, level sets, Riemannian geometry, segmentation, surface evolution. I.
Efficient Dilation, Erosion, Opening and Closing Algorithms
 Mathematical Morphology and Its Applications to Image and Signal Processing
, 2000
"... Abstract—We propose an efficient and deterministic algorithm for computing the onedimensional dilation and erosion (max and min) sliding window filters. For a pelement sliding window, our algorithm computes the 1D filter using 1:5 þ oð1Þ comparisons per sample point. Our algorithm constitutes a de ..."
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Cited by 13 (1 self)
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Abstract—We propose an efficient and deterministic algorithm for computing the onedimensional dilation and erosion (max and min) sliding window filters. For a pelement sliding window, our algorithm computes the 1D filter using 1:5 þ oð1Þ comparisons per sample point. Our algorithm constitutes a deterministic improvement over the best previously known such algorithm, independently developed by van Herk [25] and by Gil and Werman [12] (the HGW algorithm). Also, the results presented in this paper constitute an improvement over the Gevorkian et al. [9] (GAA) variant of the HGW algorithm. The improvement over the GAA variant is also in the computation model. The GAA algorithm makes the assumption that the input is independently and identically distributed (the i.i.d. assumption), whereas our main result is deterministic. We also deal with the problem of computing the dilation and erosion filters simultaneously, as required, e.g., for computing the unbiased morphological edge. In the case of i.i.d. inputs, we show that this simultaneous computation can be done more efficiently then separately computing each. We then turn to the opening filter, defined as the application of the min filter to the max filter and give an efficient algorithm for its computation. Specifically, this algorithm is only slightly slower than the computation of just the max filter. The improved algorithms are readily generalized to two dimensions (for a rectangular window), as well as to any higher finite dimension (for a hyperbox window), with the number of comparisons per window remaining constant. For the sake of concreteness, we also make a few comments on implementation considerations in a contemporary programming language. Index Terms—Mathematical morphology, running maximum filter, minmax filter, computational efficiency. æ 1
Curvature Vector Flow to Assure Convergent Deformable Models for Shape Modelling
 In EMMCVPR
, 2003
"... Poor convergence to concave shapes is a main limitation of snakes as a standard segmentation and shape modelling technique. The gradient of the external energy of the snake represents a force that pushes the snake into concave regions, as its internal energy increases when new inexion points are ..."
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Cited by 12 (3 self)
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Poor convergence to concave shapes is a main limitation of snakes as a standard segmentation and shape modelling technique. The gradient of the external energy of the snake represents a force that pushes the snake into concave regions, as its internal energy increases when new inexion points are created. In spite of the improvement of the external energy by the gradient vector ow technique, highly non convex shapes can not be obtained, yet. In the present paper, we develop a new external energy based on the geometry of the curve to be modelled. By tracking back the deformation of a curve that evolves by minimum curvature ow, we construct a distance map that encapsulates the natural way of adapting to non convex shapes. The gradient of this map, which we call curvature vector ow (CVF), is capable of attracting a snake towards any contour, whatever its geometry. Our experiments show that, any initial snake condition converges to the curve to be modelled in optimal time.