Multivariate Image Analysis in Mineral Processing - Advanced Control and Supervision of Mineral Processing Plants - page 96

been used in various areas such as in the forest products and pulp and paper fields
[30, 31], in the snack food industry [32, 33], in steam plants [34], and in the mineral

processing industry, in both ore concentration [26, 35, 36] and pyrometallurgical

areas [38, 39].
Multivariate image analysis consists of two steps: (1) a decomposition of the im-
age information using multi-way PCA (MPCA) which yields a lower-dimensional
feature space (or latent variable space); and (2) feature exploration by iterative seg-
mentation of regions-of-interest (ROI) within the feature space followed by mapping
of these selected ROIs back in the original image space for visual interpretation of
the identified spectral features. The MPCA decomposition of a multivariate image is
illustrated in Figure 3.5 using the froth image shown in Figure 3.3. First, it involves
reorganizing the image data cube ( i.e. , three-way array) X ( x × y × λ) into a matrix

X of dimensions ( λ) by collecting the light intensities at the various wave-

lengths row-wise for all pixels of the image. The image data X is then decomposed
using PCA into a set of A orthogonal score vectors t a and loading vectors p a to-
gether defining a lower dimensional subspace where most of the image information
lies:

( x × y )×

A
a = 1 t a p a + E .

X = (3.4)

When the number of principal components (PCs) A λ, E contains the projection

residuals of each row of X ( i.e. , distance of each observation in the original ℜ λ
space and their projection into the lower dimensional space). In image analysis, X
is generally not centered and scaled prior to applying PCA since the average color
intensities explained by the first PC is often meaningful and the light intensities
captured in each channel are measured in the same units and span the same range
(0-255 for an 8-bit camera). As mentioned in Section 3.2.1, the loading and score
vectors can be computed either by the iterative NIPALS algorithm or using SVD.
Since the number of rows of X is typically very large ( i.e. , equal to the number of
pixels of the image), it is more computationally efficient to compute the loading
vectors p a by applying SVD on the kernel matrix ( X X ) [29], which is only a 3

<

× 3

= =

matrix for a RGB image. The score vectors are then computed as t a Xp a , a

, ,..., =

1 2 is typically used since the explained

variance of X with two components is often in excess of 99%.
A shown in Figure 3.5, the results of the MPCA decomposition of X can also be
expressed in multi-way notation by reorganizing the score vectors t a (

2 A . In the analysis of RGB images, A

( x × y ) × 1)

into a matrix T a ( x × y × 1) according to the original location of each pixel of the

digital image:

A
a = 1 T a ⊗ p a + E ,

X = (3.5)

where symbol is the Kronecker product. The advantage of this representation is

that the score matrix T a can be displayed as univariate (gray-scale) images, which
enables direct visual interpretation of the information captured by each principal
component.

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