Table 1.
Variables and notation used to compute the Haralick texture features.
The two left-most columns show the original definitions, and the two right-most columns show the modifications needed to make the feature invariant to the number of gray-levels.
Table 2.
The texture features computed from GLCMs.
The middle column shows the original definitions, and the right column shows the modifications needed to make the features invariant to the number of gray-levels. There was an error in the definition of Sum variance in Haralick et al. [4], which has been corrected. λ2(Q(i, j)) denotes the second largest eigenvalue of a matrix Q(i, j). Note, however, that this feature is computationally unstable, and was therefore not included in the examples in this work. For symmetric GLCMs, μx and μy is identical, and is represented by μ in the expression for Cluster prominence and Cluster shade.
Fig 1.
An illustration of how the Haralick texture features are computed.
In a 4 × 4 image, three gray-levels are represented by numerical values from 1 to 3. The GLCM is constructed by considering the relation of each voxel with its neighborhood. In this example, for simplicity, we only look at the neighbor to the right. The GLCM acts like a counter for every combination of gray-level pairs in the image. For each voxel, its value and the neighboring voxel value are counted in a specific GLCM element. The value of the reference voxel determines the column of the GLCM and the neighbor value determines the row. In this ROI, there are two instances when a reference voxel of 3 “co-occurs” with a neighbor voxel of 2 (indicated in solid, blue), and there is one instance of a reference voxel of 3 with a neighbor voxel of 1 (indicated in dashed, red). The normalized GLCM represents the estimated probability of each combination to occur in the image. The Haralick texture features are functions of the normalized GLCM, where different aspects of the gray-level distribution in the ROI are represented. For example, diagonal elements in the GLCM represent voxels pairs with equal gray-levels. The texture feature “contrast” gives elements with similar gray-level values a low weight but elements with dissimilar gray-levels a high weight. Reprinted from Brynolfsson et al. [37] under a CC BY license, with permission from Nature Publishing Group, original copyright 2017.
Fig 2.
A comparison between a original GLCM and an invariant GLCM for different numbers of bins.
Synthetic GLCMs generated as bivariate Gaussian distributions, for two different quantization levels, 16 and 24 gray-levels. The elements of the standard GLCM sums to 1, which means that the GLCM element values will decrease with increasing GLCM size. The invariant GLCMs are normalized to keep the total volume of the GLCM at 1, where each element is ascribed an area of 1/N2 units.
Fig 3.
The image data used to test the invariant features in classification problems.
Texture features were computed for a range of quantization gray-levels from a region in the cerebellum (a) and a region in the prefrontal cortex (b), and from benign (c) and malignant (d) colorectal glandular structures.
Fig 4.
Example of original and invariant GLCMs for different quantization levels from Dataset 1.
The left column shows the original GLCMs created from the cerebellum region in Fig 3, for different quantization levels. The right column shows the invariant GLCMs for the same quantization levels. The original GLCMs are normalized so that the sum is 1, whereas the invariant features are normalized so that the volume of the GLCM is 1.
Fig 5.
The original Haralick feature values, and the proposed invariant feature values.
Texture feature values computed from a benign glandular structure in the gland dataset. The upper left plot shows the original Haralick features that increase rapidly with the number of gray-levels. The middle left plot shows original Haralick features that increase modestly, or reach a plateau with increasing number of gray-levels. The bottom left plot shows original Haralick features that decrease with increasing number of gray-levels. The right column shows the corresponding invariant features. Note that the original features are plotted on a log-scale to accommodate the wide range of values. The Information measure of Correlation 2 and Correlation overlap in the original features. The Information Measure of Correlation 1 is negative in the original features, but the absolute values are displayed in the graph to accommodate the log scale.
Fig 6.
Accuracy of classifiers trained on multiple quantization gray-levels.
To evaluate the performance of classifiers that were trained on a wide range of quantizations, 100 logistic regression models were trained for each data set and method. Each image in the training data was randomly quantized to one of 32 quantization gray-levels between 8 and 256 for every model. The markers show the mean accuracy and the error bars show the standard deviations of the accuracy for each quantization of the test data. The dashed line shows the accuracy obtained by assigning all predictions to the most common class, which is 0.5 for either the cerebellum or prefrontal cortex in Dataset 1 and 0.615 for benign glandular structures in Dataset 2.
Fig 7.
Accuracy of classifiers trained on one quantization gray-level.
The heatmaps show the accuracy of logistic regression models trained on one quantization gray-level and tested on all quantization gray-levels in the range 8–256 in steps of 8. The colormap is scaled to show a neutral gray for the accuracy obtained by assigning all predictions to the most common class, which is 0.5 for either the cerebellum or prefrontal cortex in Dataset 1 and 0.615 for benign glandular structures in Dataset 2. The upper row shows the result from Dataset 1 and the lower row shows the results from Dataset 2. The left column shows the original features and the right column shows the invariant features. The diagonal elements show the accuracies where the same quantization gray-levels were used for the training and test data.