Basic Morphology Concepts

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• Mathematical Morphology is based on the algebra of non-linear operators operating on object shape and in many respects supersedes the linear algebraic system of convolution.

• It performs in many tasks – pre-processing, segmentation using object shape, and object quantification – better and more quickly than the standard approach.

• Mathematical morphology tool is different from the usual standard algebra and calculus.

• Morphology tools are implemented in most advanced image analysis.

• Mathematical morphology is very often used in applications where shape of objects and speed is an issue—example: analysis of microscopic images, industrial inspection, optical character recognition, and document analysis.

• The non-morphological approach to image processing is close to calculus, being based on the point spread function concept and linear transformations such as convolution.

• Mathematical morphology uses tools of non-linear algebra and operates with point sets, their connectivity and shape.

• Morphology operations simplify images, and quantify and preserve the main shape characteristics of objects.

• Morphological operations are used for the following purpose:

° Image pre-processing (noise filtering, shape simplification).

° Enhancing object structure (skeleton zing, thinning, thickening, convex hull, object marking).

° Segmenting objects from the background.

° Quantitative description of objects (area, perimeter, projections, Euler- Poincare characteristics).

• Mathematical morphology exploits point set properties, results of integral geometry, and topology.

• The real image can be modeled using point sets of any dimension; the Euclidean 2D space E2 and its system of subsets is a natural domain for

planar shape description.

• Set difference is defined by

X\Y = X ∩ Yc

• Computer vision uses the digital counterpart of Euclidean space – sets of integer pairs (∈ Z2) for binary image morphology or sets of integer triples(∈ Z3) for gray-scale morphology or binary 3D morphology.

• Discrete grid can be defined if the neighborhood relation between points is well defined. This representation is suitable for both rectangular and hexagonal grids.

• A morphological transformation l/J is given by the relation of the image with another small point set B called structuring element. B is expressed with respect to a local origin 0.

• Structuring element is a small image-used as a moving window-- whose support delineates pixel neighborhoods in the image plane.

• It can be of any shape, size, or connectivity (more than 1 piece, have holes).

• To apply the morphological transformation w(X) to the image X means that the structuring element B is moved systematically across the entire image.

• Assume that B is positioned at some point in the image; the pixel in the image corresponding to the representative point O of the structuring element is called the current pixel.

• The result of the relation between the image X and the structuring element B in the current position is stored in the output image in the current image pixel position.

clip_image001

Fig 1: Typical structuring elements.

• The duality of morphological operations is deduced from the existence of the set complement; for each morphological transformation 1/(X) there

image

• The translation of the point set X by the vector ℎ is denoted by Xh ; it is defined by

image

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