MULTIVARIATE MATHEMATICAL MORPHOLOGY BASED ON PRINCIPAL 
COMPONENT ANALYSIS: INITIAL RESULTS IN BUILDING EXTRACTION 
Jonathan Li*, Yu Li 
CFI Virtual Environment Lab, Department of Civil Engineering, Ryerson University, 350 Victoria Street, Toronto, 
Ontario, M5B 2K3, Canada - {junli, yuli}(@ryerson.ca 
Inter-Commission IC, WG III/IV 
KEY WORDS: Multichannel image processing, colour morpholo 
gy, vector ordering, principal component analysis, urban analysis, 
building extraction, Ikonos, QuickBird, aerial imagery. 
ABSTRACT: 
Today, colour or multichannel satellite and aerial images are increasingly becoming available due to the commercial availability of 
multispectral digital sensors and pansharpening function of the commercial remote sensing software tools. Comparing to their 
monochromic counterparts, colour image data can offer not only more useful information about landscape but also the correlations 
among channels. Recently, multivariate mathematical morpholog 
y has received increased attention due to its rigorous mathematical 
theory and its powerful utility in multichannel image analysis. In this paper, a new morphological method for multichannel remotely 
sensed image processing is presented and analyzed. The proposed method utilizes a multivariate ordering principle based on 
principal component analysis. To define the colour morphology the colour vectors are ordered by using the first principal component 
analysis. On the basis of this ordering, new infimum and supremum are defined. Using the new infimum and supremum, the 
fundamental erosion and dilation operations are defined. Two series of experiments have been prepared to test the performance of 
the proposed method by using Ikonos and QuickBird pansharpened images and colour aerial images acquired over a built-up area. 
1. INTRODUCTION 
As a methodology analyzing spatial structures in remotely 
sensed image data, mathematical morphology has become more 
and more popular in the image processing community not only 
due to its rigorous mathematical theory but also its powerful 
utility in image analysis. Generally speaking, mathematical 
morphology uses the morphological operations to analyze and 
recognize geometrical properties and structure of objects in 
images. So far, mathematical morphology has been developed 
as a complete and efficient tool for analyzing the spatial 
organization in binary and grayscale images (Serra, 1982). It is 
categorized into binary morphology and grayscale morphology. 
Initially, mathematical morphology was proposed by Matheron 
(1975) for investigating the geometry of the objects of a binary 
image in his classical book on random set. In the grayscale 
case, complete lattices are used as the mathematical basis for 
the grayscale morphology. The basic idea behind the grayscale 
morphology is on the assumption that the set of all possible 
images forms a complete lattice. Based on this assumption, the 
set of all operators mapping one grayscale image into another 
also constitutes a complete lattice. 
Both from a practical and theoretical point of view, colour 
mathematical morphology can be of great interest. First, colour 
is known to play a significant role in human visual perception 
and is becoming more and more relevant to computer vision as 
colour sensors become more widely available. It is well known 
that in an image a great deal of extra information may be 
contained in the colour, and this extra information can then be 
used to simplify image analysis, e.g., object identification and 
feature extraction based on colour. So it is necessary to develop 
a new effective technique to analyze colour images. Secondly, 
since binary mathematical morphology and  grayscale 
mathematical morphology are intended to analyze binary 
images and gray-scale images, respectively, it would be 
  
interesting, from a pure theoretical point of view, to extend 
morphological theory to process colour images. 
Although some techniques developed for grayscale 
mathematical morphology can be extended to colour images by 
applying the operators to each channel of a colour image 
separately, for example, the most straightforward scheme for 
the extension is to treat a colour image as an independent 
monochrome image and the grayscale morphological operator Is 
directly applied to each colour component separately. 
Unfortunately, this procedure has some drawbacks, e.g., 
producing new colours that are not contained in the original 
image and may lead missing of the correlations between 
components (Astola et al., 1990; Goutsias et al., 1995). 
The extension of concepts from grayscale morphology to colour 
morphology raises some important problem (Louverdis, 2002; 
Vardavoulia, 2002). First, an appropriate colours ordering must 
be found to define colours morphological operations that will 
retain the basic properties of their grayscale counterparts. 
Secondly, a colour space that determines the way in which 
colours are represented must be chosen. Third, an infimum and 
a supremum operator in the selected colour space should be 
defined well. It would be perfect for the two operators to be 
vector preserving, so that they do not introduce new colours 
that do not exist in the original image. 
In this paper, a new reduced ordering based on fuzzy first 
principal component in RGB colour space is proposed. On the 
basis of the vectors ordering, new infimum and supremum 
operators that are both vector preserving are defined. Then, 
colour morphology, which takes into consideration the vector 
nature of colours, is introduced. Using new infimum and 
supremum operators, the basic morphological operations in 
RGB colour space: erosion, dilation, opening, and closing, are 
defined. Last, as an example, the proposed colour morphology 
* Corresponding author. Dr. Jonathan Li, P.Eng., O.L.S., Assistant Professor of Geomatics Engineering, Ryerson University 
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