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16
Multiresolution Segmentation of Natural Images: From Linear to NonLinear ScaleSpace Representations
 IEEE TRANS. IMAGE PROCESS
, 2003
"... In this paper, we introduce a framework that merges classical ideas borrowed from scalespace and multiresolution segmentation with nonlinear partial differential equations. A nonlinear scalespace stack is constructed by means of an appropriate diffusion equation. This stack is analyzed and a ..."
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Cited by 21 (0 self)
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In this paper, we introduce a framework that merges classical ideas borrowed from scalespace and multiresolution segmentation with nonlinear partial differential equations. A nonlinear scalespace stack is constructed by means of an appropriate diffusion equation. This stack is analyzed and a tree of coherent segments is constructed based on relationships between different scale layers. Pruning this tree proves to be a very efficient tool for unsupervised segmentation of different classes of images (e.g.
Generalized scale: Theory, algorithms, and application to image inhomogeneity correction
 Computer Vision and Image Understanding 101
, 2006
"... and to the memory of my dear late friend Chirag Doshi. asatho maa sadgamaya thamaso maa jyothirgamaya mrityour maa amritham gamaya From the unreal, lead me to the real; From darkness, lead me to the light; From death, lead me to immortality. From the Brihadaranyaka Upanishad ii ..."
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Cited by 11 (4 self)
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and to the memory of my dear late friend Chirag Doshi. asatho maa sadgamaya thamaso maa jyothirgamaya mrityour maa amritham gamaya From the unreal, lead me to the real; From darkness, lead me to the light; From death, lead me to immortality. From the Brihadaranyaka Upanishad ii
α scale spaces on a bounded domain
 In Preparation
, 2003
"... Abstract. We consider α scale spaces, a parameterized class (α ∈ (0, 1]) of scale space representations beyond the wellestablished Gaussian scale space, which are generated by the αth power of the minus Laplace operator on a bounded domain using the Neumann boundary condition. The Neumann boundar ..."
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Cited by 10 (2 self)
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Abstract. We consider α scale spaces, a parameterized class (α ∈ (0, 1]) of scale space representations beyond the wellestablished Gaussian scale space, which are generated by the αth power of the minus Laplace operator on a bounded domain using the Neumann boundary condition. The Neumann boundary condition ensures that there is no greyvalue flux through the boundary. Thereby no artificial greyvalues from outside the image affect the evolution proces, which is the case for the α scale spaces on an unbounded domain. Moreover, the connection between the α scale spaces which is not trivial in the unbounded domain case, becomes straightforward: The generator of the Gaussian semigroup extends to a compact, selfadjoint operator on the Hilbert space L2(Ω) and therefore it has a complete countable set of eigen functions. Taking the αth power of the Gaussian generator simply boils down to taking the αth power of the corresponding eigenvalues. Consequently, all α scale spaces have exactly the same eigenmodes and can be implemented simultaneously as scale dependent Fourier series. The only difference between them is the (relative) contribution of each eigenmode to the evolution proces. By introducing the notion of (nondimensional) relative scale in each α scale space, we are able to compare the various α scale spaces. The case α = 0.5, where the generator equals the square root of the minus Laplace operator leads to Poisson scale space, which is at least as interesting as Gaussian scale space and can be extended to a (Clifford) analytic scale space. 1
The Application of Catastrophe Theory to Image Analysis
, 2001
"... In order to investigate the deep structure of Gaussian scale space images, one needs to understand the behaviour of critical points under the influence of blurring. We show how the mathematical framework of catastrophe theory can be used to describe the various different types of annihilations and t ..."
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Cited by 9 (5 self)
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In order to investigate the deep structure of Gaussian scale space images, one needs to understand the behaviour of critical points under the influence of blurring. We show how the mathematical framework of catastrophe theory can be used to describe the various different types of annihilations and the creation of pairs of critical points and how this knowledge can be exploited in a scale space hierarchy tree for the purpose of presegmentation. We clarify the theory with an artificial image and a simulated MR image.
The Relevance of NonGeneric Events in Scale Space Models
, 2001
"... In order to investigate the deep structure of Gaussian scale space images, one needs to understand the behaviour of spatial critical points under the influence of blurring. We show how the mathematical framework of catastrophe theory can be used to describe and model the behaviour of critical poi ..."
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Cited by 9 (2 self)
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In order to investigate the deep structure of Gaussian scale space images, one needs to understand the behaviour of spatial critical points under the influence of blurring. We show how the mathematical framework of catastrophe theory can be used to describe and model the behaviour of critical point trajectories when various different types of generic events, viz. annihilations and creations of pairs of spatial critical points, (almost) coincide. Although such events are nongeneric in mathematical sense, they are not unlikely to be encountered in practice. Furthermore the behaviour leads to the observation that finetocoarse tracking of critical points doesn't suffice, since trajectories can form closed loops in scale space. The modelling of the trajectories include these loops. We apply the theory to an artificial image and a simulated MR image and show the occurrence of the described behaviour.
Logical Filtering in Scale Space
 Utrecht University
, 2002
"... Using a Gaussian scale space, one can use the extra dimension, viz. scale, for investigation of "builtin" properties of the image in scale space. We show that one of such induced properties is the nesting of special isointensity manifolds, that yield an implicit present hierarchy of t ..."
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Cited by 6 (1 self)
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Using a Gaussian scale space, one can use the extra dimension, viz. scale, for investigation of "builtin" properties of the image in scale space. We show that one of such induced properties is the nesting of special isointensity manifolds, that yield an implicit present hierarchy of the critical points and regions of their influence, in the original image. Its very nature allows one not only to segment the original image automatically, but also to apply "logical filters" to it, obtaining simplified images. We give an algorithm deriving this hierarchy and show its effectiveness on two different kinds of images, both with respect to segmentation and simplification.
Multiscale Hierarchical Segmentation
, 2002
"... In this article three different methods for building a hierarchical graph from scale space images are discussed. The first method has been suggested by Lifshitz and Pizer [8]. It creates graph structures for image description by defining a linking relationship between pixels in successively blurred ..."
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Cited by 3 (0 self)
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In this article three different methods for building a hierarchical graph from scale space images are discussed. The first method has been suggested by Lifshitz and Pizer [8]. It creates graph structures for image description by defining a linking relationship between pixels in successively blurred versions of the initial image. The second method has been proposed by Kuijper [6]. It uses isointensity surfaces through scale space critical points to find a scale space hierarchy and a ”presegmentation ” of the initial image. The third method is a method with similarities to the ”scale space primal sketch ” described by Lindeberg [9]. This algorithm is based on so called attraction areas and it combines the strong points from the two other previously mentioned methods. 1
On Image Reconstruction from Multiscale Top Points
"... Abstract. Image reconstruction from a fiducial collection of scale space interest points and attributes (e.g. in terms of image derivatives) can be used to make the amount of information contained in them explicit. Previous work by various authors includes both linear and nonlinear image reconstruc ..."
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Cited by 1 (0 self)
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Abstract. Image reconstruction from a fiducial collection of scale space interest points and attributes (e.g. in terms of image derivatives) can be used to make the amount of information contained in them explicit. Previous work by various authors includes both linear and nonlinear image reconstruction schemes. In this paper, the authors present new results on image reconstruction using a top point representation of an image.A hierarchical ordering of top points based on a stability measure is presented, comparable to feature strength presented in various other works. By taking this into account our results show improved reconstructions from top points compared to previous work. The proposed top point representation is compared with previously proposed representations based on alternative feature sets, such as blobs using two reconstruction schemes (one linear, one nonlinear). The stability of the reconstruction from the proposed top point representation under noise is also considered. 1
Exploiting Deep Structure
"... Blurring an image with a Gaussian of width σ and considering σ as an extra dimension, extends the image to an Gaussian scale space (GSS) image. In this GSSimage the isointensity manifolds behave in an nicely predetermined manner. As a result of that, the GSSimage directly generates a hierarchy i ..."
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Blurring an image with a Gaussian of width σ and considering σ as an extra dimension, extends the image to an Gaussian scale space (GSS) image. In this GSSimage the isointensity manifolds behave in an nicely predetermined manner. As a result of that, the GSSimage directly generates a hierarchy in the form of a binary ordered rooted tree, that can be used for segmentation, indexing, recognition and retrieval. Understanding the geometry of the manifolds allows fast methods to derive the hierarchy. In this paper we discuss the relevant geometric properties of GSS images, as well as their implications for algorithms used for the tree extraction. Examples show the applicability and increased speed of the proposed method compared to traditional ones.