Multivariate Image Analysis in Mineral Processing - Advanced Control and Supervision of Mineral Processing Plants - page 101

and error unless prior knowledge is available on the features of interest. Note that
only those feature extraction methods that will be used in Section 3.5 are presented
here. For more details or for a more extensive account of the various possibilities,
the reader is referred to Yu and MacGregor [32].
A final but important remark needs to be made about MIA. Applying PCA to
an image requires unfolding ( i.e. reorganizing) the original three-way array image
X into a matrix X (see Figure 3.5) by collecting the light intensities of each pixel
row-wise in X . Since interchanging rows in X does not affect the results of the
PCA decomposition, this unfolding operation destroys the information related to
the spatial organization of the pixel intensities ( i.e. , the image texture). Therefore,
MIA is a very useful technique for extracting spectral characteristics from images,
but cannot be used to extract textural features. The latter is performed using textural
methods, such as wavelet texture analysis (WTA).

3.4.1.2 Multiresolution Analysis for Texture Analysis

Images collected in various fields often show objects or structures of different sizes
[41]. For example, consider the brick wall shown in Figure 3.8. When looking at the
wall from a certain distance (Figure 3.8 (a)), one can clearly recognize the bricks
and mortar joints as well as the visual pattern defined by those. One could also
detect cracks in mortar joints at this level ( i.e. , distance from the wall). However,
when zooming on a single brick (see Figure 3.8 (b)), visual information at a much
finer level of details appear: the brick surface texture with pits and subtle gray color
variations. That is the brick wall image contains very different textural information
depending on the distance from the wall. If one was to use this image to detect joint
cracks ( i.e. , coarser texture) or to grade the quality of the brick surfaces, textural
information at a very different level will need to be extracted. This could be accom-
plished by taking several pictures of the wall at various distances to obtain the de-
sired resolution. However, when the optimal resolution is initially unknown, or when
several resolutions are required to obtain the desired information, then collecting
several images can be cumbersome. A multiresolution framework was jointly de-
veloped by Stephane Mallat [41-44] and Yves Meyer [45] in the late 1980s [46, 47]
for extraction of coarser to finer textural information based on a single version of the
image ( i.e. , image captured only once at a fixed distance), allowing scale-invariant
interpretation of the images [41].
In multiresolution analysis (MRA) of a 1-Dl signal, the latter is sequentially de-
composed at different levels, also called scales or frequencies. At each level of de-
composition j

= , ,...,

J , the signal is decomposed into two orthogonal signals:
an approximation signal containing coarser information, and a detail signal captur-
ing finer characteristics. This decomposition is performed such that the approxima-
tion signal at level j includes all detail signals at higher levels of decomposition
j

1 2

J ( i.e. , the detail signals at higher levels than j are embedded into
the approximation signals at level j ). Such a decomposition allows for reconstruc-
tion of the original signal by combining the approximation at the highest level J and

+ 1 , j + 2 ,...,