Rotation invariant co-occurrence features based on digital circles and discrete Fourier transform☆
Introduction
Grey-level co-occurrence matrices are among the most long-standing texture descriptors in use, their origin dating back to the pioneering work of Haralick et al. [23]. Though many other methods have been proposed since their appearance – see Ref. [44] for a comprehensive overview – GLCM continue to be very common and widely adopted still today. Bibliometric data reveal that the number of relevant scientific papers has even increased during the last years (see Table 1). GLCM features are particularly appealing for their conceptual simplicity, ease of implementation and the low number of features they produce. A recent comparative experiment on image classification under non-ideal conditions [27] showed that GLCM features tend to perform better when few classes (10 or less) are involved, a situation in which they can compete with newer and more powerful methods. Besides, co-occurrence features can be combined with other descriptors that convey complementary information through suitable fusion schemes [13]. Recent applications of GLCM span very diverse areas of image processing, including surface inspection [17], [7], environmental monitoring [3], [29], content-based image retrieval [39] and image reconstruction [4]. Among the numerous application areas, co-occurrence matrices seem to be particularly common in medical image analysis [24], [28], [6], [37], [20] and remote sensing [8], [42], [30], [26].
Co-occurrence matrices have been extended in various directions, leading to several variations such as generalised co-occurrence matrices [16], which consider the distribution of local maxima; integrative co-occurrence matrices [36], which operate on colour images and, more recently, pattern co-occurrence matrices [40], [21], which analyse the co-occurrence of local patterns. By contrast, the original formulation has not changed significantly since its appearance. This is not uncommon: when a method matures and new ones appear, scientific interest tends to switch from the former to the latter. Newer methods receive more attention, and the older becomes frozen, somewhat immutable, with few chances of improvement. Something of this type we believe has happened with co-occurrence matrices, at least for what it concerns rotation invariant features.
Motivated by the wide diffusion of the method – even in very critical areas like medical image analysis and computer-assisted diagnosis, we wished to investigate whether it was possible to improve robustness and accuracy of the method in rotation invariant classification tasks. This is a major concern, for in many applications images can occur in different and uncontrolled rotation angles. The common approach to obtaining rotationally-invariant features from co-occurrence matrices consists of averaging [23], [1], [11] or – equivalently – summing up [38, p. 215] the matrices corresponding to the same distance and different directions. We believe that this procedure reduces significantly and somewhat unnecessarily the discrimination capability of the resulting features. We therefore propose some improvements to compute more efficient rotationally-invariant features from GLCM. Our study considers the effects of two design factors that determine how GLCM features are computed. These are: (1) the spatial arrangement of pairs of pixels; and (2) the way to convert GLCM features into rotation invariant ones. In the remainder of the paper we first discuss such design factors (Section 2), then evaluate their effects through an image classification experiment (Section 3). We present and analyse the results in Section 4 and conclude with final considerations in Section 5.
Section snippets
Design factors
Grey-level co-occurrence matrices estimate the joint occurrence probability of grey levels at a given distance and direction. The method is intrinsically directional, hence sensitive to rotation. In order to achieve rotation invariant descriptors, we need to remove the dependence on direction and obtain features that depend on distance only. Such a goal can be obtained through the following steps: (1) for each pixel in the image, consider all pixels that are located approximately at a given
Experiments
We performed a set of image classification experiments to evaluate accuracy and robustness against rotation of the descriptors presented in the preceding section. Datasets and procedure used in the experiments are detailed in the following subsections.
Results and discussion
The overall results of the classification experiment are summarised in Table 3. For each descriptor and dataset the table reports the results in the form , where and are, respectively, the mean and standard deviation of the classification accuracy (in %) over the different rotation angles. The figures in boldface highlight the highest accuracy values. Numeric subscripts indicate the distance (d) at which the treatments have been computed. The complete results for each rotation angle are
Conclusions
Co-occurrence matrices are among the most used image descriptors, with applications covering almost any area of image analysis. They are also one of the oldest – they first appeared on the scene forty years ago. In this study we investigated whether it was possible to improve robustness and accuracy of the method for rotation invariant image classification. The results of our study are in the affirmative: there is room for improvement.
As for the type of neighbourhood, digital circles proved
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This paper has been recommended for acceptance by Edwin Hancock.