Image dependent colour morphology

This page contains colour figures and additional material for the paper "Mathematical morphology for color images: an image dependent approach"
by X. Benavent, E. Dura, F. Vegara and J . Domingo, submitted to Mathematical Problems in Engineering.

Introduction

In this work we propose one possibility to generalise the morphological operations (particularly, dilation, erosion,
opening and closing) to colour images.

We have tried three different attempts to do it:

• Morphology based on plain histogram is based on a total ordering of the colours in an image induced by its colour histogram.
• Morphology based on smoothed histogram in which we consider the 3-D colour histogram as the probability density of the appearance of colours in the image. It is smoothed so that each colour exerts influence over neighbour colours. We have done the experiments using both the
  • Euclidean distance and the
  • Mahalanobis distance
• Morphology for a class of similar images is to build the histogram not for a single image but for a set of similar ones, accumulating their pixels in a global 3-D histogram, appropriately weighted. Again, experiments have been done using both the
  • Euclidean distance and the
  • Mahalanobis distance

  For a class of similar images more cross tests have been made with various Miro´s paintings. (see at MoreMiroCrossedTests.htm)

• We have also calculated the morphological gradient.
• Usual morphological gray level operations on the intensity image and conversion to gray of the results for the color image. This experiment is done to compare the usual gray morphological operations with those implemented with this approach. In order to do that the same colour image has been converted to gray, getting only the intensity band, and the four basic morphological operations have been applied to it. Also, the operations have been applied according to our proposal, using the Eucledian distance and the RGB color space (this has already been done in the last experiment) but the results have then been converted to gray, again keeping only the intensity component.

Morphology based on plain histogram

	This picture is a map of the Valencian Community (east of Spain) with very few colours and a very simple structure.
Map (a) with its erosion (b), dilation (c), opening (d) and closing (e) with a disk of radius 3 pixels (from left to right). Method of the plain histogram.

Morphology based on smoothed histogram

In this approach we consider the $3-D$ colour histogram as the probability density of the appearance of colours
in the image. It is smoothed so that each colour exerts influence over neighbour colours.

We have done the experiments calculating the distance between two colours with two different distances:

1. The Euclidean distance
2. The Mahalanobis distance

Euclidean distance

Miro-la-chanteuse painting

	This is a painting from Joan Miro that is composed mostly by planar patches, but also contains thin lines and graded tonalities.
Miro painting (a) with its erosion (b/f), dilation (c/g), opening (d/h) and closing (e/i) with a disk of radius 3 pixels. Cube of potentials constructed with the image itself in the RGB (b to e) and L*a*b* (f to i) spaces using Euclidean distance.

Dragon sculpture from The Güell Park (Barcelona)

Real image dragon (a) with its erosion (b/f), dilation (c/g), opening (d/h) and closing (e/i) with a disk of radius 3 pixels. Cube of potentials constructed with the image itself in the RGB (b to e) and L*a*b* (f to i) spaces using Euclidean distance.

Mahalanobis distance

Miro-la-chanteuse painting

Miro painting (a) with its erosion (b/f), dilation (c/g), opening (d/h) and closing (e/i) with a disk of radius 3 pixels. Cube of potentials constructed with the image itself in the RGB (b to e) and L*a*b* (f to i) spaces using Mahalanobis distance.

Dragon sculpture

Dragon image (a) with its erosion (b/f), dilation (c/g), opening (d/h) and closing (e/i) with a disk of radius 3 pixels. Cube of potentials constructed with the image itself in the RGB (b to e) and L*a*b* (f to i) spaces using Mahalanobis distance.

Morphology for a class of similar images

Databases used for the two experiments

1. Miro's paintings

Database used to create the cube of potentials for all similar Miro's paintings.

2. Dragon sculpture from the Güell Park in Barcelona.

Euclidean distance

Miro-la-chanteuse painting

Miro painting erosion (a/e), dilation (b/f), opening (c/g) and closing (d/h) with a disk of radius 3 pixels. Method of cube with other images in the RGB (a to d) and L*a*b* (e to h) colour spaces, using Euclidean distance.

Dragon sculpture

Mahalanobis distance

Miro-la-chanteuse painting

Miro painting erosion (a/e), dilation (b/f), opening (c/g) and closing (d/h) with a disk of radius 3 pixels. Method of cube with other images in the RGB (a to d) and L*a*b* (e to h) colour spaces, using Mahalanobis distance.

Dragon sculpture

Morphological gradient

A possible application of these colour morphology definitions is the calculation of morphological gradient.
In gray level images, morphological gradient is normally defined as the difference between dilation and erosion
of the same image with the same structuring element. As the colour difference is not defined, we cannot
extend this definition in a trivial manner. We suggest that, since the potential calculated for each colour is a
measure of its importance, difference between the potentials associated to each colour in the dilated and eroded
image can be a reasonable generalisation of the morphological gradient.

Miro painting morphological gradient in RGB (a/b) and L*a*b* (c/d), using a disk shaped SE with radius 3 pixels(a and c) and 5 pixels (b and d).

Morphological gray operations

Dragon sculpture

Dragon image erosion (a/e), dilation (b/f), opening (c/g) and closing (d/h) with a disk of radius 3 pixels. Usual morphological gray level operations on the intensity image (a to d) and conversion to gray of the results for the color image (e to h).