# Introduction

This package contains a set of tools to classify the pixels of digital images into colour categories arbitrarily defined by the user. It contains functions to

• visualize the distribution of the pixel colours in the images,
• define classification rules
• classify the pixels and to store this information in R objects,
• save these as image files.

It is a simple version of the multivariate technique known as Support Vector Machine (Cortes and Vapnik, 1995; Bennet and Campbell, 2000), adapted to this particular use.

### The procedure

The basic steps of the procedure are the following:

• One or more digital images in JPEG or TIFF format is imported into R. The categories to identify are represented in this set (the test set).
• The values of the three three colour variables (or bands) that compose each image (R, G, and B) are transformed into proportions (r, g and b).
• The pixels of the image are plotted in the plane defined by two of the transformed variables (the user can select them arbitrarily) and, hopefully, they would form separate clusters (pixel categories).
• The user then traces straight lines that separate the pixel clusters. Using the mathematical expression for these rules and the rgb values, each pixel can be tested for membership in each category (see below).
• Recording the results of the tests as 1 or 0 (pass/fail), an incidence matrix is build for that rule. This is the result of the procedure, which can be submitted to posterior analysis or used to create a new version of the original image showing the category of each pixel.

The second step simplifies the problem because it makes one of the variables dependent on the other two (as r + g + b = 1). Moreover, the transformation eliminates colour variations due to differences in illumination.

The expressions for classification rules are the same as the expression for a straight line but using one of the comparison operators $$<$$, $$\leq$$, $$>$$ or $$\geq$$. For example: $$r \geq a g +c$$, being $$a$$ and $$c$$ the slope and intercept of the line, and $$r$$ and $$g$$ the colour variables selected for the classification. A single line can produce two classification rules.

### Using several rules per category

When there are more than two categories, or when the cluster of points has a complex shape, a single rule is not enough. In these cases the procedure has additional steps:

• several rules are defined for each category,
• incidence matrices are created for each rule,
• the incidence matrices are combined with the & operator to obtain the category incidence matrix.

The last step is equivalent to estimate the union of the incidence matrices, i e $$\mathbf{M} = \mathbf{M}_{1} \cap \mathbf{M}_{2} \cap \ldots \cap \mathbf{M}_{p}$$, being p the number of rules.

### Concave category shapes

A caveat of the method is that the rules must delimit a convex polygon to combine the individual rule results successfully (in a convex polygon, a line joining any two internal points is contained in the polygon). Not all clusters have convex shape. In these cases, the cluster must be divided in convex sub-polygons (subcategories) for which rules are defined as before. The incidence matrices of the subcategories are combined using the | operator, i.e.Â $$\mathbf{M} = \mathbf{M}_{1} \cup \mathbf{M}_{2} \cup \ldots \cup \mathbf{M}_{s}$$, being s the number of subcategories. Note that any polygon, convex or not, can be subdivided in triangles and, as triangles are convex polygons, it is always possible to solve this problem. Note that the goal is to obtain a minimal set of convex polygons, not a complete triangulation. The example presented below is one of such cases.

# The session

What follows is a sample session illustrating both the method and the use of the package functions. It uses an example image and a test set created by cutting small areas out of the example image. It is not a good test set, see below, but it is enough to show how the method works, and its problems.

library(pixelclasser)