Implementing continuous-scale morphology via curve evolution

Sapiro, Guillermo; Kimmel, Ron; Shaked, Doron; Kimia, Benjamin B.; Bruckstein, Alfred M.

doi:10.1016/0031-3203(93)90142-j

Cited by 123 publications

(82 citation statements)

References 18 publications

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“…In ( [34,33]) nonlinear partial differential equations were proposed that mimic the process of dilation and erosion. For flat morphology with a ball as structuring element this diffusion equation reads…”

Section: Continuous Morphologymentioning

confidence: 99%

Morphology for matrix data: Ordering versus PDE-based approach

Burgeth

Bruhn

Didas

et al. 2007

Image and Vision Computing

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Matrix fields are becoming increasingly important in digital imaging. In order to perform shape analysis, enhancement or segmentation of such matrix fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the setting of matrices, in the literature sometimes referred to as tensors despite the fact that matrices are only rank two tensors. The goal of this paper is to introduce and explore two approaches to mathematical morphology for matrix-valued data: One is based on a partial ordering, the other utilises nonlinear partial differential equations (PDEs). We start by presenting definitions for the maximum and minimum of a set of symmetric matrices since these notions are the cornerstones of the morphological operations. Our first approach is based on the Loewner ordering for symmetric matrices, and is in contrast to the unsatisfactory component-wise techniques. The notions of maximum and minimum deduced from the Loewner ordering satisfy desirable properties such as rotation invariance, preservation of positive semidefiniteness, and continuous dependence on the input data. These properties are also shared by the dilation and erosion processes governed by a novel nonlinear system of PDEs we are proposing for our second approach to morphology on matrix data. These PDEs are a suitable counterpart of the nonlinear equations known from scalar continuous-scale morphology. Both approaches incorporate information simultaneously from all matrix channels rather than treating them independently. In experiments on artificial and real medical positive semidefinite matrix-valued images we contrast the resulting notions of erosion, dilation, opening, closing, top hats, morphological derivatives, and shock filters stemming from these two alternatives. Using a ball shaped structuring element we illustrate the properties and performance of our ordering-or PDE-driven morphological operators for matrix-valued data.

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Section: Continuous Morphologymentioning

confidence: 99%

Morphology for matrix data: Ordering versus PDE-based approach

Burgeth

Bruhn

Didas

et al. 2007

Image and Vision Computing

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“…Inspired by the use of the classic heat PDE to model the linear (Gaussian) scale-space [8], in 1992 three teams of researchers (Alvarez et al [1], Brockett and Maragos [5], and Van den Boomgaard and Smeulders [28]) independently published nonlinear PDEs that model the nonlinear scale-space of elementary morphological operators; each team focused on different aspects of the problem. The PDEs for flat dilations and erosions by disks were numerically implemented by Arehart et al [3] and Sapiro et al [24] using the Osher and Sethian [20] algorithm for solving Hamilton-Jacobi PDEs of the curve evolution type. These implementations demonstrated the superiority of the performance of the PDE approach over that of discrete morphology in terms of isotropy and subpixel accuracy.…”

Section: A Pde Generating Levelingsmentioning

confidence: 99%

Nonlinear Scale-Space Representation with Morphological Levelings

Meyer¹,

Maragos²

2000

Journal of Visual Communication and Image Representation

107

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In this paper we present a nonlinear scale-space 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 scale-space 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 scale-space by using the latter as an initial seed to generate multiscale levelings that have a superior preservation of image edges. C 2000 Academic Press

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“…Many other morphological processes such as openings, closings, top hats and morphological derivative operators can be derived from them. While dilation/erosion are frequently realised using a set-theoretical framework, an alternative formulation is available via partial differential equations (PDEs) [10,11,13,14,15]. Compared to the set-theoretical approach, the latter offers the conceptual advantages of digital scalability and subpixel accuracy.…”

Section: Introductionmentioning

confidence: 99%

Highly Accurate PDE-Based Morphology for General Structuring Elements

Breuß

Weickert

2009

Lecture Notes in Computer Science

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Abstract. Modelling the morphological processes of dilation and erosion with convex structuring elements with partial differential equations (PDEs) allows for digital scalability and subpixel accuracy. However, numerical schemes suffer from blur by dissipative artifacts. In our paper we present a family of so-called flux-corrected transport (FCT) schemes that addresses this problem for arbitrary convex structuring elements. The main characteristics of the FCT-schemes are: (i) They keep edges very sharp during the morphological evolution process, and (ii) they feature a high rotational invariance. Numerical experiments with diamonds and ellipses as structuring elements show that FCT-schemes are superior to standard schemes in the field of PDE-based morphology.

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Implementing continuous-scale morphology via curve evolution

Cited by 123 publications

References 18 publications

Morphology for matrix data: Ordering versus PDE-based approach

Morphology for matrix data: Ordering versus PDE-based approach

Nonlinear Scale-Space Representation with Morphological Levelings

Highly Accurate PDE-Based Morphology for General Structuring Elements

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