Multivariate Image Analysis in Mineral Processing - Advanced Control and Supervision of Mineral Processing Plants - page 109

icant distance above or below the hyperplane defined by the latent variable space
( i.e. observations showing a different correlation structure, inconsistent with that
of the latent variable space).

The latent variable method used to compute T F (and other related vectors) de-
pends upon the existence of a response matrix Y , as well as the nature of the re-
sponse variables ( i.e. , categorical or not). The following discusses which method to
use for three generic types of machine vision problems.

• Unsupervised classification. This situation arises when no response matrix Y is

available, or when one simply wants to assess the natural clustering pattern of the
image based on their features. This is accomplished by applying PCA on the fea-
ture matrix X F . An example of such a situation will be presented in Section 3.5.2.

• Supervised classification. When the machine vision objective is to classify new,

incoming images in pre-defined categories, and when the N images of the train-
ing set can be assigned to a known class, then a supervised classification problem
may be formulated. In this case, the response matrix Y is filled with binary num-
bers assigning each image to a particular class. It has as many columns as the
number of categories. If the n th image belongs to the m th class, then a value of
1 will appear in position ( n

m )inmatrix Y and zeros elsewhere. When such a
categorical response matrix is available, PLS may be used to extract the infor-
mation in the feature matrix X F , which is the most highly correlated with the
classes in Y ,thatis,tofind the combinations of features that maximize class
discrimination. Applying PLS regression to a categorical response matrix Y is
called PLS-DA, for PLS-discriminant analysis. A supervised classification prob-
lem using PLS-DA is illustrated in Section 3.5.3.

,

• Regression . Image regression can be used for rapid and non-intrusive prediction

of key process variables ( i.e. , Y ), when these are difficult to measure on-line or
when they are obtained after long dead-times, such as assays requiring the use
of analytical instruments. In this case, PLS regression is used to build prediction
models for Y based on image features X F . An example image regression problem
is presented in Section 3.5.1. Note that one may need to account for the dynam-
ics between X F and Y . This can be accomplished by augmenting X F with past
lags ( i.e. , time-shifts) of image features and process data before using PLS ( i.e. ,
finite impulse response model). There are other ways of introducing dynamics
in the model, but this subject is outside the scope of this chapter (see system
identification literature as well as [57-59]).

3.5 Case Studies

Three case studies are presented in this section to illustrate the various multivari-
ate imaging concepts presented in this chapter. The first two cases investigate froth