Morphological Image Processing

Overview

Morphological image processing is a tool for extracting or modifying information

on the shape and structure of objects within an image. Morphological operators,

such as dilation, erosion and skeletonization, are particularly useful for the

analysis of binary images, although they can be extended for use with grayscale

images. Morphological operators are non-linear, and common usages include

filtering, edge detection, feature detection, counting objects in an image, image

segmentation, noise reduction, and finding the mid-line of an object.

9.1 Mathematical Morphology

The field of mathematical morphology contributes a wide range of operators to

image processing, all based around a few simple mathematical concepts from

set theory and, in the case of binary images, (Boolean) logic operations such as

“AND”, “OR”, “XOR” (exclusive OR) and “NOT”. The “union” operation, AB, for

example, is equivalent to the “OR” operation for binary images; and the

“intersection” operator, AB, is equivalent to the “AND” operation for binary

images (Appendix B).

9.1.1 Connectivity

In binary images an object is defined as a connected set of pixels. With two-

dimensional images connectivity can be either 4-connectivity or 8-connectivity

(Fig. 9.1). In 4-connectivity, each pixel (P) has four connected neighbors (N) –

top, bottom, right and left. The diagonally touching pixels are not considered to

be connected. In 8-connectivity, each pixel (P) has eight connected neighbors

(N) – including the diagonally touching pixels. For three-dimensional images

neighborhoods can be 6-connected, 18-connected or 26-connected.

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Figure 9.1 Connectivity in two-dimensional images. (i) 4-connectivity - each pixel

(P) has four connected neighbors (●) (ii) 8-connectivity - each pixel (P) has eight

connected neighbors (●).

This leads to different ideas of distance. In a 4-connected neighborhood, N4, the

distance is known as the city-block, taxicab or Manhattan distance by analogy

with a city based on an orthogonal grid of roads. It is the distance a taxicab would

drive in Manhattan (if there were no one way streets!). The distance in a 4-

connected neighborhood is given by

(9.1)

A diagonal step has a distance of two since it requires a horizontal and a vertical

step. Equal distances from a certain position would form diamonds centered on

it. In an 8-connected neighborhood, N8, the distance is known as the Chebyshev

or chessboard distance, by analogy with the moves available to a king in chess.

The distance in an 8-connected neighborhood is given by

(9.2)

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A diagonal step has a distance of one, the same as a horizontal or vertical step.

Equal distances from a certain position would form squares centered on it.

Neither is the same as Euclidean distance, which is given by

(9.3)

A diagonal step is given by a distance of 1/√2, and equal distances from a certain

position form circles centered on it. In physical space the Euclidean distance is

the most natural distance, because the length of a rigid body does not change

with rotation. Alternating the two metrics (N4-N8 or N8-N4) is an approximation to

Euclidean distance.

9.2 Morphological Operators

There are a number of morphological operators, but two most fundamental

operations are dilation and erosion; all other morphological operations are built

from a combination of these two.

9.2.1 Dilation and Erosion

In binary images dilation is an operation that increases the size of foreground

objects, generally taken as white pixels although in some implementations this

convention is reversed. It can be defined in terms of set theory, although we will

use a more intuitive algorithm. The connectivity needs to be established prior to

operation, or a structuring element defined (Fig. 9.2).

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Figure 9.2 Structuring elements corresponding to (i) 4-connectivity (ii) 8-

connectivity.

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