General Framework for Rotation Invariant Texture Classification Through Co-occurrence of Patterns

Abstract

The use of co-occurrences of patterns in image analysis has been recently suggested as one of the possible strategies to improve on the bag-of-features model. The intrinsically high number of features of the method, however, is a potential limit to its widespread application. Its extension into rotation invariant versions also requires careful consideration. In this paper we present a general, rotation invariant framework for co-occurrences of patterns and investigate possible solutions to the dimensionality problem. Using local binary patterns as bag-of-features model, we experimentally evaluate the potential advantages that co-occurrences can provide in comparison with bag-of-features. The results show that co-occurrences remarkably improve classification accuracy in some datasets, but in others the gain is negligible, or even negative. We found that this surprising outcome has an interesting explanation in terms of the degree of association between pairs of patterns in an image, and, in particular, that the higher the degree of association, the lower the gain provided by co-occurrences in comparison with bag-of-features.

Contingency Tables and Measures of Association

Given a population and two sets of unordered, categorical attributes (polytomies) $\mathcal {A} = \{A_0,A_1,\ldots ,A_{\alpha -1}\}$ and $\mathcal {B} = \{B_0,B_1,\ldots ,B_{\beta -1}\}$ a contingency table is an $\alpha \times \beta$ matrix of which each $(\rho _a,\rho _b)$ entry ($a \in \{0,\ldots ,\alpha -1\}$; $b \in \{0,\ldots ,\beta -1\}$), reports the fraction of the population that is classified as both $A_a$ and $B_b$ [12]. The degree of association between the attributes can be estimated through a number of parameters, some of which are breifly recalled here below.

1.1 Pearson’s Chi-squared Coefficient and Cramér’s $V$

A traditional measure of association is Pearson’s chi-squared coefficient, which estimates the bias of cross-classification from statistical independence [26]. For a population of $\nu$ members it is defined as follows:

$$\begin{aligned} \chi ^2 = \nu \sum _{a=0}^{\alpha -1} \sum _{b=0}^{\beta -1} \frac{\left( \rho _{ab} - \rho _{a \cdot } \rho _{\cdot b}\right) ^2}{\rho _{a \cdot } \rho _{\cdot b}} \end{aligned}$$

(15)

where $\rho _{a \cdot }$ and $\rho _{\cdot b}$ are the proportions of the population classified as $A_a$ and $B_b$, respectively.

A normalized version of Pearson’s $\chi ^2$ is Cramér’s $V$, which takes value between 0 and 1 (where 0 means ‘independence’ and 1 ‘complete association’) and is defined as follows:

$$\begin{aligned} V = \sqrt{\frac{\chi ^2}{\nu \left[ \text {max}\left( \alpha -1,\beta -1\right) \right] }} \end{aligned}$$

(16)

1.2 Goodman’s $\lambda$

Suppose a situation in which we had to guess the $B$-class (or the $A$-class) of an individual chosen at random from the population and were given: (1) no further information about the individual, or (2) its A class (or its B class). Goodman’s $\lambda$ estimates the relative decrease in the error probability that we experience when switching from case (1) to case (2):

$$\begin{aligned} \lambda = \frac{\frac{1}{2} \left( \sum _{a=0}^{\alpha -1} \rho _{am} + \sum _{b=0}^{\beta -1} \rho _{bm} - \rho _{\cdot m} -\rho _{m \cdot } \right) }{1 - \frac{1}{2} \left( \rho _{\cdot m} + \rho _{m \cdot } \right) } \end{aligned}$$

(17)

where

$$\begin{aligned} \rho _{\cdot m}&= \mathop {\mathrm{max}}\limits _{b} \rho _{\cdot b}, \quad \rho _{am} = \mathop {\mathrm{max}}\limits _{b} \rho _{ab} \end{aligned}$$

(18)

$$\begin{aligned} \rho _{m \cdot }&= \mathop {\mathrm{max}}\limits _{a} \rho _{a \cdot }, \quad \rho _{mb} = \mathop {\mathrm{max}}\limits _{a} \rho _{ab} \end{aligned}$$

(19)

1.3 Theil’s $U$

The last measure of association considered here is based on the concept of cross-information and was discussed by Theil [43]. It considers on the amount of information conveyed about $A$ by $B$ and vice-versa. This is properly scaled to give a value between 0 and 1. In formulas:

$$\begin{aligned} U^2 = \frac{2I}{H(A) + H(B)} \end{aligned}$$

(20)

where

$$\begin{aligned} I&= \sum _{a=0}^{\alpha -1} \sum _{b=0}^{\beta -1} \left( \rho _{ab} \right) \text {log} \left( \frac{\rho _{ab}}{ \rho _{a \cdot } \rho _{\cdot b}} \right) \end{aligned}$$

(21)

$$\begin{aligned} H(A)&= -\sum _{a=0}^{\alpha -1} \left( \rho _{a \cdot } \right) \text {log} \left( \rho _{a \cdot } \right) \end{aligned}$$

(22)

$$\begin{aligned} H(B)&= -\sum _{b=0}^{\beta -1} \left( \rho _{\cdot b} \right) \text {log} \left( \rho _{\cdot b} \right) \end{aligned}$$

(23)