Abstract
The quantitative categorization of textures according to their visual appearances is an important area of research in computer vision and image understanding, because texture analysis and its applications are found useful in many areas of health, medicine, sciences, and engineering. For the first time, the theory of chaos and fuzzy sets are applied in this paper to measure the spatial dynamics of the texture spectrum. Experiments carried out on the well-known Brodatz texture database suggest the promising application of the method proposed for texture quantification.
Introduction
Texture analysis is a major area of image processing and computer vision for image retrieval and object detection. Texture classification has been widely adopted in many fields of research and development, such as medicine [1, 2], biology [3], life science [4], engineering and technology [5–9]. Methods developed for texture analysis, texture classification, texture synthesis, and texture naming are numerous [10–14]. Developments in this area of research are to continue to provide more effective solutions to the spectrum of texture characterization whose definition is subject to human perception as well as the large variation of texture at different scales.
A texture spectrum has been suggested [15] to cover a range from regular (periodic patterns), near-regular (statistical distortion of regular texture), irregular, near-stochastic, and stochastic, where most images of near-regular texture are of nature and artificial synthesis. Most patterns of texture are often found to be a continuous variation from deterministic (regular) to random (stochastic) appearances. The naming of texture for data visualization is also suggested to be dependent on the image properties of repetition, contrast, direction, granularity, coarseness, fineness, and complexity [16–18]. The mixture of global regularity and local randomness in texture set challenging tasks for the-state-of-the art advancement in methods for texture analysis and synthesis. Furthermore, the naming of visual texture involves in high-level semantics, which is important for content-based image retrieval systems [19] as well as ontology-based cognitive vision [20]. Given many efforts spent on texture classification, little research has been carried out to obtain better ways for the objective grouping of texture into categories. Thus, finding a good approach for the categorization of texture is a challenging task and must be very helpful for texture analysis and synthesis.
This paper addresses a development toward objective texture categorization using the method of fuzzy metric entropy [21]. The fuzzy metric entropy is an extension of the metric entropy of dynamical systems [22], which is also known as measure-theoretic entropy, or Kolmogorov-Sinai (K-S) entropy, with the notion of fuzzy sets [23]. Due to the inherently complex and imprecise properties of texture, the implementation of fuzzy sets via the fuzzy c-means algorithm [24] into the formulation of the metric entropy for texture categorization is natural. Using representative texture patterns of the Brodatz database [25], which is a widely used texture benchmark database, experimental results suggest the usefulness of the fuzzy metric entropy that can provide insights into the properties of the texture spectrum.
The rest of this paper is organized as follows. Section 2 presents the method of the fuzzy metric entropy. Experimental results are reported, and discussion on the results addressed in Section 3. Finally, Section 4 is the conclusion of the research findings, and suggestion for further investigation.
Method
The formulation of the fuzzy metric or fuzzy K-S entropy [21] is described as follows. Metric entropy, denoted as H
KS
, is a measure of the entropy difference of a dynamical system as time approaches infinity and bin size ɛ that subdivides the phase space of the system reduces to zero [26, 27]:
Let X = {x} be a collection of points. A fuzzy set A ∈ X is characterized by a membership function μ
A
(x), which maps x ∈ X to a real number in the interval [0, 1] [23]. The entropy of the fuzzy set A, denoted by D (A), is a measure of the degree of its fuzziness, which has the following three properties [28]: D (A) = 0 if μ
A
(x) = 0 or 1 (non-fuzzy). D (A) is maximum if and only if μ
A
(x) =0.5 (most fuzzy). D (A) ≥ D (A*) where is any “sharpened” (less fuzzy) version of μ
A
(x), such that if μ
A
(x) ≥0.5 and if μ
A
(x) ≤0.5.
Based on the definition of the entropy of a fuzzy set, the uncertainty of a fuzzy system in the context of the K-S entropy measured with a sequence of observations after m units of time can be defined as
Based on Equation (1), the metric entropy of fuzzy sets is expressed as
On the sequence probabilities for calculating H
KS
, which are the likelihoods that the system will follow each of the various possible routes, for successive time intervals; these probabilities are computed using the product rule. For the sequence fuzzy membership values associating with Equation (3), which are the degrees expressing possible routes the system will pass through, the product rule for combining conditional probabilistic events is replaced with the intersection of fuzzy sets. Let A and B be two fuzzy sets, each of which associates with each x in a Euclidean n-space R
n
. The intersection of A and B is defined as [23]
For the formulation of the metric entropy of fuzzy sets, the first alternative is to divide each entropy by the associated time to get an entropy rate, and the plot of the entropy rates versus time allows the estimate of the asymptotic entropy rate as time increases. However, this procedure can be impractical because the data must be large enough for the time events to reach the asymptotic limit or convergence. The other alternative resorts to an entropy difference [29], denoted by Δ
m
, as follows:
The metric entropy of fuzzy sets estimated by the method of the entropy difference becomes
The fuzzy membership grades required for the calculation of the fuzzy K-S entropy to quantify uncertainty in a texture image, which is inherently imprecise, can be obtained by the partition of the image intensities using the fuzzy c-means (FCM) algorithm [24]. Thus, given fuzzy c-partitions, the FCM assigns the intensity value of each pixel to the c clusters with its respective membership grades. Such a fuzzy partition can be readily used for calculating the sequential fuzzy entropy differences.
On ten samples of Brodatz texture
The fuzzy metric entropy was applied to study the dynamical behavior of texture using ten samples of the Brodatz database [25], which is one of the most widely adopted benchmark databases for texture analysis. The ten images are 215 × 215 sub-images of 640 × 640 orginal images D1, D3, D5, D9, D15, D20, D40, D42, D49, and D56, obtained from the public repository described in [30]. Figure 1 shows the ten Brodatz images. These images were selected to reflect typical nature of the texture spectrum, with an assumption that the images have two main classes: background and foreground. Thus, the specified number of clusters for the FCM analysis was two. The fuzzy membership grades obtained from the two fuzzy partitions were used to compute the entropy differences in the formulation of the metric entropy.
There are two fuzzy metric entropy values for each cluster: one for row-wise orientation, and the other for column-wise orientation. These values are taken as the average ones of the last 10 points of the converging entropy-difference curves. As a visual illustration of the convergence of the calculation of the fuzzy metric or fuzzy K-S (FKS) entropy, Fig. 2 shows the plots of the entropy-difference vs. iteration in row-wise and column-wise directions for one cluster for Brodatz texture images D1 and D3.
Tables 1 and 2 present the fuzzy metric entropy values and the corresponding variances of the ten texture images. The fuzzy metric entropy values or entropy values of the images are lower in one cluster and higher in the other cluster.
D1 and D2 have the lowest entropy values that are consistent in both orientations (rwo-wise and column-wise) and both clusters. D42 has the highest entropy values (33.7 and 25.7) in one cluster, and low entropy values (2.4 and 3.5) in the other cluster. D1 and D2 are of near-regular texture pattern with low contrast, while D42 of a combination of regular (inner part) and irregular (mainly along the left, bottom, and right edges) texture patterns. The fuzzy metric entropy with the modeling of two clusters can capture such a dynamical behavior of the texture of D42. D49 has the lowest entropy values (0.4 and 0.5) in row-wise orientation, while one value is high in the column-wise (1.1 and 10.2). D49 has a regular texture pattern but is of high anisotropy, which may explain its inconsistent results. D20 is of near-regular texture with high contrast, which has low entropy values (1.5 and 4.3) in one cluster, and higher entropy values in the other cluster (8.0 and 12.6). D56, which has a regular or near-regular pattern with a mixture of contrasts and shapes, is characterized with an entropy range from relatively medium c (column-wise) to high values (row-wise). The entropy values are high for D5, which is of between highly irregular and stochastic texture. D15 are of stochastic and has consistently high entropy values in both clusters. The uncertainly measure of D9, which is of random-like texture with a uniform distribution, results in high entropy values in one cluster (16.7, 15.7) but low in the other one (0.9, 0.005). Similarly, D40, which is both regular (background) and irregular (object), has both relatively low (1.2, 1.3) and high (12.1, 7.3) entropy values.
For the metric entropy, H KS = 0 for a deterministic system that is not chaotic, H KS is a positive constant for a chaotic system, and H KS is infinite for a random process [26]. Here, the fuzzy metric entropy applied to texture images suggests a small constant for a regular pattern, and a high value for an irregular or random-like one. The mixture of results given by the fuzzy clusters indicate a corresponding mixture of texture patterns.
On forty samples of Brodatz texture
This experiment on forty selected images of the Brodatz texture database aimed to quantify the categorization of four common types of texture: fine-periodic, fine-aperiodic, coarse-periodic, and coarse-aperiodic. The ground-truth naming of these four types of texture of the Brodatz images was obtained by manual inspection and listed in 18]. Ten images for each of the four texture names were selected. The image indices of the fine-periodic texture are: D3, D6, D14, D17, D21, D34, D36, D38, D49, and D52. The image indices of the fine-aperiodic texture are: D4, D9, D16, D19, D24, D26, D28, D29, D32, and D39. For the coarse-periodic texture, the ten selected images are: D1, D8, D10, D11, D18, D20, D22, D25, D35, and D47. For the coarse-aperiodic texture, the ten selected images are: D2, D5, D7, D12, D13, D15, D23, D27, and D31. Figures 3–6 show the images of the four texture categories.
Given the number of fuzzy clusters c=2, two values of the fuzzy metric entropy or fuzzy K-S entropy (FKS) (row-wise and column-wise orientations of the image) for each of the two clusters were calculated. Table 3 shows the means of the row-wise and column-wise FKS values of the four texture categories. A matter of interest is to know the validity of the FKS measures of the texture categories. One possible way is to study the relationships of these four texture categories by constructing the “phylogenetic” tree, which is described in [31], based on the mean values given in Table 3. Before constructing the trees, the classification of the texture images were carried out to study the reliability of the FKS features. Using the k-NN algorithm, where k=1, to classify the texture-class pair, without training, the correct classification rates of 60% and 55% were obtained for the row-wise and column-wise orientations, respectively. Such classification rates, which are above 50% without using training data, suggest the usefulness of the FKS values for building the relationship trees of the four texture categories.
Figures 7, 8, and 9 are the trees of the four texture categories using the FKS values obtained from the row-wise, column-wise, and the combination of the row-wise and column-wise directions, respectively. The two trees shown in Figs. 8 and 9 indicate a clear separation between the fine-periodic texture and coarse-aperiodic texture, which are of opposite texture properties. The fine-aperiodic texture is well separated as the outgroup in the topology shown in Fig. 7. In general, all the three trees show a consensus that periodicity and aperiodicity appear to play a stronger influence than fineness and coarseness for distinguishing texture characteristics.
Figures 10 and 11 show the plots of the FKS values of the fine-periodic texture and coarse-aperiodic texture for cluster 1 vs. those for cluster 2, obtained from the row-wise and column-wise orientations of the images, respectively. For the row-wise orientation, the outliers of the fine-periodic texture are D34 and D36; while the outlier of the coarse-aperiodic texture is D23. As for the column-wise orientation, thelocation of D36 can be observed being far away from the rest of the fine-periodic texture group; whereas D7 and D23 are distant from the other images of the coarse-aperiodic texture samples.
Conclusion
The fuzzy metric entropy or fuzzy K-S entropy has been presented and discussed as a measure of texture dynamics in images for texture categorization. This study appears to be the first of its kind in the utilization of the theories of chaos and fuzzy sets for characterizing texture properties. Such numerical quantification can be useful toward an objective taxonomy of texture patterns. Through the illustrations of selected ten images of the Brodatz texture database, and with an assumption that an image approximately has a background and a foreground, the entropy is low for a low-contrast regular or near-regular texture, higher for irregular, and both low and high for a mixture of texture patterns. On the experiment using forty Brodatz texture images belonging to four ground-truth texture classes, the fuzzy K-S entropy can be useful by providing an insight into the influence of periodicity and aperiodicity that is stronger than the effect of coarseness and fineness of texture. Relaxing the assumption of two clusters in the FCM analysis is possible for the fuzzy metric entropy to study a wider range of the texture spectral dynamics. The proposed approach can also be extended to study texture in color images.
Acknowledgement
This journal article is the extended version of the paper presented at the 12th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD’15) [32].