Last update: Aug. 8, 2001

BACK

Publications on Morphological Scale-Spaces (in reverse chronological order) :

• Diatom contour analysis using morphological curvature scale spaces, 2000.
• Morphological Scale Space for 2D Shape Smoothing, B.K. Jang & R.T. Chin, 1998.
• On the Scale-Space Theorem of Chen and Yan, Paul T. Jackway, PAMI 1998.
• Scale-Space Properties of the Multiscale Morphological Dilation-Erosion, P.T. Jackway & M.Deriche, PAMI 1996.
• Morphological Scale-Space with Application to 3D Object Recognition, Ph.D. thesis, Paul T. Jackway, 1995.
• Scale-Space Using Mathematical Morphology, K-R. Park & C-N. Lee, PAMI 1996.
• Shape Features Using Curvature Morphology, F. Leymarie & M.D. Levine, 1989.
• Curvature Morphology, F. Leymarie & M.D. Levine, 1988.

BibTeX references.

Web-link

A method for shape analysis of diatoms (single-cell algae with silica shells) based on extraction of features on the contour of the cells by multi-scale mathematical morphology is presented. After building a morphological contour curvature scale space, we present a method for extracting the most prominent features by unsupervised cluster analysis. The number of extracted features matches well with those found visually in 92% of the 350 diatom images examined.

In this paper, we describe a multiple-scale boundary representation based on morphological operations An object boundary is first progressively smoothed by a number of opening and closing operations using a structuring element of increasing size, generating a multiple scale representation of the object. Then, smooth boundary segments across a continuum of scales are extracted and linked together creating a pattern called the morphological scale space. Properties of this scale space pattern are investigated and contrasted with those of Gaussian scale space. A shape smoothing algorithm based on this scale space is proposed to show how the scale space representation could be applied to image analysis. Specifically, in line with Witkin's scale space filtering, boundary features that are explicitly related across scales by the morphological scale space are organized into global regions and local boundary features. From the organization, perceptually dominant features for a smooth boundary are determined without the requirement of prior knowledge of the object nor input parameters. Extensive experiments were conducted to show the performance of morphological scale space for 2D shape smoothing

In an earlier paper, Chen and Yan presented a theorem concerning zero crossings of boundary curvature under morphological openings. In this correspondence, we show by means of a counterexample a problem with this theorem and suggest how the theorem may be modified to make it correct.

A multiscale morphological dilation-erosion smoothing operation and its associated scale-space expansion for multidimensional signals are proposed. Properties of this smoothing operation are developed and, in particular a scale-space monotonic property for signal extrema is demonstrated. Scale-space fingerprints from this approach have advantages over Gaussian scale-space fingerprints in that they are defined for negative values of the scale parameter; have monotonic properties in two and higher dimensions, do not cause features to be shifted by the smoothing, and allow efficient computation. The application of reduced multiscale dilation-erosion fingerprints to the surface matching of terrain is demonstrated.

This thesis develops and demonstrates an original approach to scale-space theory. A new scale-space theory based on a unified multiscale morphological dilation-erosion smoothing operator is presented. The essential scale-space causality property for local extrema of a signal under this operation is proved. This result holds for signals on IR² and higher dimensions and for negative as well as positive scales. When applied to grayscale images we show that structuring functions from the ``elliptic poweroids'' lead to favourable dimensionality and semi-group properties. Paraboloids, in particular, allow efficient computation of the scale-space, and such an algorithm is presented.

The generalised frequency response of this signal smoother, which is similar to that of a Butterworth filter (with an amplitude dependent corner frequency), is obtained. The filter is statistically characterised by obtaining second-order statistical properties of the output signal with independent and identically distributed uniform noise input. Similar scale-space results are obtained for the multiscale morphological closing-opening operator, and we show that the resulting scale-space fingerprints are identical to those of the dilation-erosion.

To demonstrate the utility of the new theory, we present an approach for the recognition of multiple 3-D objects in range data via the local matching of surfaces. In this approach the reduced morphological scale-space fingerprint is used as the primitive for matching. The resulting recognition process is invariant to translation, rotation, limited scaling, and partial occlusion. The results of the proposed object recognition method showing the recognition of a scene containing nine faces at various positions, angles and scales is presented. In a second demonstration we show the recognition of 8 mountains in a digital elevation map.

In this paper, we prove that the scale-space of a one-dimensional gray-scale signal based on morphological filterings satisfies causality (no new feature points are created as scale gets larger). For this we refine the standard definition of zero-crossing so as to allow signals with certain singularity, and use them to define feature points. This new definition of zero-crossing agrees with the standard one in the case of functions with second order derivative. In particular, the scale-space based on the Gaussian kernel G does not need this concept because a filtered signal G * f is always infinitely differentiable. Using this generalized concept of zero-crossing, we show that a morphological filtering based on opening (and, hence, also closing by duality) satisfies causality. We note that some previous works have mistakes which are corrected in this paper. Our causality results do not apply to more general two-dimensional gray scale images. Causality results on alternating sequential filter, obtained as byproduct, are also included.

Download report :    

The notion of curvature of planar curves has emerged as one of the most powerful for the representation and interpretation of objects in an image. Although curvature extraction from a digitized object contour would seem to be a rather simple task, few methods exist that are at the same time easy to implement, fast, and reliable in the presence of noise. In this paper we first briefly present a scheme for obtaining the discrete curvature function of planar contours based on the chain-code representation of a boundary. Secondly, we propose a method for extracting important features from the curvature function such as extrema or peaks, and segments of constant curvature. We use mathematical morphological operations on functions to achieve this. Finally, on the basis of these morphological operations, we suggest a new scale-space representation for curvature named the Morphological Curvature Scale-Space. Advantages over the usual scale-space approaches are shown.

Index terms:

Shape Representation, Curvature Representation and Analysis, Morphological Operators, Multiscale Description, Multi-Dimensional Scale-Space.

Page created & maintained by Frederic Leymarie, 1998-2001.
Comments, suggestions, etc., mail to: leymarie@lems.brown.edu