An Approach to Computational Microtexture Perceptual Detection with Management of Uncertainty Domènec Puig†
Eduard Montseny‡
†Dept. Enginyeria Informàtica i Matemàtiques Universitat Rovira i Virgili Autovia Salou s/n, 43006 Tarragona, SPAIN e-mail:
[email protected]
Abstract The aim of this work is determining both a minimum set of perceptual features allowing microtexture description, in the scope of textured aerial images, and adequate computational methods for the evaluation of those features. Moreover, fuzzy techniques are introduced in the features evaluation process to be faithful to principle of least commitment, and to make easier the management of the uncertainty inherent to perceptual features of microtextures. Choosing aerial images for the fulfillment of the work is based on the wide range of microtextures contained in this kind of images, their uncertainty, variability and inherent complexity. Our goal has been attained considering the most usual microtexture features, based on human texture perception, and taking into account the way in which microtextures of aerial images are usually described. Combining the results of: an Analysis of Principal Components, the correlation among considered features and known texture segmentation computational measures, and fuzzy techniques, a robust computational method is obtained to evaluate the three independent components representing the set of microtexture perceptual features. Keywords: attentive microtexture perception, perceptual features, feature extraction, principal component analysis, fuzzy rules.
1. Introduction Texture is related to qualitative properties of surfaces, but due to its complexity and great variety, there exists neither a unique definition of texture nor an accepted computational representation of it. One of the widely accepted definitions of texture was given by Pickett [21]. He states that a texture is an optical pattern that contains a large number of elements (spatial variations in intensity or wavelength), each visible to some degree, and, on the whole, densely and evenly arrayed over the field of view. Several researchers, as Haralick [5], Tomita [28], and Vanrell [31], have classified textures into two large groups
Pilar Sobrevilla
‡Dept. Eng. Sistemes Automàtica Informàtica Industrial e-mail:
[email protected] Dept. Matemàtica Aplicada II e-mail:
[email protected] Universitat Politècnica de Catalunya Pau Gargallo 5, 08028 Barcelona, SPAIN
according to their internal composition. The first group is constituted by macrotextures, which are made up of primitives with specific shape, meted out following some placement rules. The second group is constituted by microtextures, without distinguishable primitives and characterized by qualitative properties that can be observed in small regions. Moreover, from a human point of view, texture perception can be preattentive or attentive. Preattentive perception takes place when two textures in an image can be easily sorted out by human vision. This is possible when two textures are constituted by components clearly identifiable by their shape or size. On the contrary, attentive perception happens when a detailed observation is required for discriminating between two different textures in an image, as it is usually the case of microtextures, where no structural components are visible. Preattentive texture analysis is based on the texton theory, developed by Julesz [8]. This theory considers textures as formed by local components called textons, characterized by their contrast, orientation, length and width. In this way, a recent work developed by Vanrell [31] shows that these textures can be represented in a fourdimensional feature space. Unfortunately, there are textures that can’t be preattentively discriminated but require higher level features, such as coarseness or directionality. So, attentive perception is necessary, for example, in biomedical image analysis, manufactured products inspection or aerial image segmentation. The study of perceptual features used by humans for attentive analysis of textures started in the seventies (Haralick [4], Nevatia [16], Tamura [27] and Weszka [33]), when the first texture features and some ad-hoc computational measures for their evaluation were introduced. Even though more related work has been carried out in the eighties and nineties (Grau [2], Nevatia [17], and Rao [23], [24]), the study of the relevance of perceptual features, and the way in which these features can be evaluated are currently open problems. Research in this field can be grouped in three lines: (1) Analysis of high level perceptual features, based on qualitative properties of texture, such as coarseness, granularity, linearity, etc.
(2) Analysis of shapes and spatial organization. It consists of extracting basic primitives and their relationships. Often, those primitives extraction is difficult to achieve and it is not applicable to microtextures. (3) Analysis of receptive channels of human visual system, which is based on photometry and spatial characteristics used in preattentive perception. According to this classification, present work is included in first line, because of the fitting of its analysis to the human mind analysis and objects description. In addition, this line has been widely used and is suitable for all kinds of textures, particularly microtextures. To carry out any microtexture feature analysis the uncertainty treatment must be taken into consideration. This uncertainty is due to the ambiguity in the texture features definitions, the variability of natural textures and their features, and other factors due to the presence of noise in the images, the dispersion of the light and the digitalization process. In this sense, since some works in eighties (Pal [18], Pao [20]), introduced the presence of a non-random vagueness component in the variables taking part in computer vision systems, many researchers, as Gupta [3], have used fuzzy techniques to improve vagueness treatment. So, our approach after fixing a set of relevant high level perceptual features, and computational methods to evaluate them, make use of fuzzy techniques to take into account images uncertainty. Thus, in the following section, a microtexture features set is proposed taking into account the aforementioned works on analysis of microtexture perceptual features. Section 3 presents a study showing the relationship between perceptual features defined in previous section, and well-known computational measures, in order to obtain a computational description and evaluation of textures. Section 4 presents some results about microtextures feature evaluation in aerial images. Finally, conclusions are given in section 5.
2. Determination of Microtexture Feature Set
the
Minimum
Although coarseness and linearity are the main features used by perceptual approach to describe textures ([4], [27], and [30]), they are not enough to identify a texture out of the wide range of them. So, some researchers ( [1], [4], [14], [16], [17], [26], [27], and [33] ) have used many other features related to the two previous ones. In this sense, and focusing on microtextures, the majority of studies define groups of features which number goes from 5 to 12. Since many of these groups of features, used with texture segmentation aims, have common components, most voted perceptual features in those previous psychological studies ([2], [24], and [27]) will constitute
our starting perceptual feature set. These features are granularity, linearity homogeneity, coarseness, continuity, abruptness and contrast. Once the initial features set has been defined, we can proceed to reduce it. Following three restrictions has been imposed to the minimum perceptual features set. 1st.- It must contain a number of features as small as possible to reduce the complexity of automatic segmentation algorithms based on them. 2nd.- As many features, as necessary to segment any pair of microtextures must be included. 3rd.- Chosen set must allow the identification of microtextures in a particular context, such as aerial images. Taking into account the aim of the work, every initial feature have to be defined in such a way that intuitive and imprecise concepts can be translated, as clear as possible, to a computational implementation. According to this principle, our proposed definitions are the following: 1.- Granularity: Denotes the presence of either a large number of isolated pixels or many small groups of pixels, forming spots in a region, with similar intensity, different from the background intensity. 2.- Linearity: Points out the presence of neighboring pixels with similar, but different from background intensity, forming lineal structures in a region. 3.- Homogeneity: Denotes presence of smooth intensity changes in any direction in a region. 4.- Coarseness: Is given when groups of pixels with variable size presenting sudden changes of intensity in any direction appear at a region. 5.- Continuity: Denotes presence of neighboring pixels with similar intensity, different from background intensity, forming continuous lineal structures in a region. 6.- Abruptness: Points out presence either sudden transitions or strong changes in the direction of lineal structures in a region. 7.- Contrast: Denotes the presence of a wide range of gray levels in a region, being a meaningful representation of every gray level of the range.
2.1. Human Perception of Microtextures As our minimum set of perceptual textures must allow microtexture description within the framework of aerial images, next we study the implication of initial features set within some representative aerial images of natural environments. Haralick [4] considers as prototypes different elements appearing in natural texture such as: forest, scrub, railroad, urban, suburb, lake, swamp, marsh, orchard, . . , and other researchers ( [1], [14], [7], and [33]) have used some of them at their works. Our prototypes have been selected from among the aforementioned surfaces, following next conditions
− The number of selected elements must be as small as possible (redundant elements could distort subsequent studies). − Each element has to represent as many types of elements as possible, attending to their features. − A new element is selected if its type of texture is different from the one of already considered elements. These restrictions have led us to consider forests, scrub, road, orchard, sown field and urban area as prototype surfaces. With regard to perceptual features and their presence in a given microtexture, being faithful to human mind behavior, next considerations have been taken into account: • Any feature can be observed in a larger or lower proportion. • Low proportion of a feature means high proportion of the opposite concept. • All features have been defined in a generic way, with the aim of having a general purpose in the microtexture scope. • These features have been observed to be sufficient to identify microtextures in a certain context. • Features have been defined to be the most independent possible. Following this general outline, the level of human perception for the seven selected features, has been studied for each one of the six surfaces or texture prototypes. To do it, presence of each feature at a texture has been rated from 0 to 3, with 0 meaning absence of a feature and 3 meaning total presence of the feature. Results of rating process are shown at next table. Sown Urban Forest Scrub Road Orchard Field Area Granularity Linearity Homogeneity Coarseness Continuity Abruptness Contrast
3 0 1 3 0 0 1
2 0 1 2 0 0 1
0 3 2 1 3 1 1
2 2 1 2 2 1 2
0 0 2 1 0 0 1
1 3 1 2 2 3 3
Table 1: Feature values obtained by human perception.
Looking at Table 1 can be noted that, there aren't textures with the same feature estimation values, which is favorable to the discrimination purpose of this microtexture feature set. However, a more attentive observation suggests certain correlation between some features. Taking into account the aforementioned data analysis considerations, a Principal Component Analysis will be performed at next subsection, in order to achieve the next goals: • Obtaining the correlation among the 7 considered features. • Grouping the 7 initial features into a few no correlated components according to the correlation previously obtained. It can make easier the
interpretation of the final set of microtexture perceptual features. • Obtaining the proportion of the variance, among textures, which is explained by every initial feature.
2.2. Principal Components Microtexture’s Feature Space
Analysis
of
the
The PCA is a descriptive variable-directed technique, which main benefit lies in allowing us to reduce the dimensionality of the problem so as to simplify subsequent analyses. Applying this method we'll transform the original set of features to a new set of non correlated ones called principal components. These new features will be lineal combinations of the originals and derived in decreasing order of importance so that, for example, the first principal component accounts as much as possible of the information in the original data. Therefore, this method is a useful tool to improve the results of the segmentation and identification processes based on initial features. To get round the scaling problem a normalized PCA has been performed. This means that the principal components are gotten from the correlation matrix of the previous 7 perceptual features (table 2), and the components turn out to be the eigenvectors of this matrix. As this matrix shows a strong correlation between some couples of features, the fitting of the PCA to get a lower number of features allowing improving the results of the segmentation and identification processes based on initial features can be guaranteed.
Granularity Linearity Homogeneity Coarseness Continuity Abruptness Contrast
Gran. Line. Hom.
Coar.
Cont. Abru. Contr.
1.0 -0.40 -0.85 0.95 -0.41 -0.24 0.0
1.0 -0.37 -0.04 0.16
1.0 0.67 0.45
1.0 0.08 1.0 -0.29 -0.86 0.96 0.19 0.83 -0.22 0.64 -0.46
1.0 0.92
1.0
Table 2: Correlation between the 7 features.
After calculating the eigenvalues and principal components of the correlation matrix, we look at the first components, which account for a large proportion of the total information. In order to do it, we have opted for retain as many components as to account for about the 95% of total information but making possible to explain the differences among microtextures. These objectives have been attained retaining the first three components. Looking at Tables 3 and 4 can be observed that, the correlation among the components and the original features are very great, and which out of the 7 initial features have more influence in the interpretation of each selected independent component. Also an analysis of Table 4
reveals that almost all the information of initial features is accounted for the three components.
Granularity Linearity Homogeneity Coarseness Continuity Abruptness Contrast
Component C1 -0.376 0.490 0.195 -0.318 0.468 0.415 0.298
Component C2 0.395 0.172 -0.542 0.454 0.087 0.341 0.437
Component C3 0.348 0.348 0.028 0.195 0.613 -0.301 -0.502
Table 3: Eigenvectors for the first 3 components. Component Component C1 C2 -0.707 0.66 Granularity 0.92 0.287 Linearity 0.366 -0.905 Homogeneity -0.597 0.758 Coarseness 0.88 0.145 Continuity 0.779 0.57 Abruptness 0.56 0.73 Contrast 50.449 39.844 Inertia Percentage of supported information
Component C3 0.25 0.25 0.02 0.14 0.44 -0.216 -0.36 7.355 97.648
Next, a summary of these methods and how are used in this work is given. •
•
•
Table 4: Component correlation for the first 3 components.
This result is in the line of the works developed by Rao [24] and Vanrell [31], who tried to identify a representation space for textures as general as, for example, the wellknown three dimensional representations used for codifying color
3. Computational Evaluation of the AerialImage-Based Microtexture Set Obtaining a method to evaluate the independent components which considers uncertainty factors, has been carried out following three steps: 1. Main methods used during the last years for microtexture analysis and segmentation are summarized. 2. A study of the fitting of every method to evaluate the features is performed in order to get the most suitable method for evaluating each feature. 3. Methods selected at previous step are used to get a fuzzy computational method to evaluate the three independent components.
3.1. Computational Methods for Microtexture Analysis Some of the most significant computational methods can be classified, in a similar way to the one given by Reed [25], Van Gool [30], and Wechsler [32], in four groups: Linear Filtering Operators, Statistical Methods, Transform Domain Methods and Methods based on Texture Models.
•
Linear Filtering Operators. They are based on local texture energy measures formulated by Laws [10]. Energy using R5R5, E5L5, L5E5, E5E5 convolution masks has been computed in our work. The choice of these masks is because have been shown to be the most effective ones [10], [30]. Moreover, two measures have been calculated from the local linear transforms proposed by Unser [29], using 2x2 Hadamard masks. Statistical Methods. Are based on spatial gray level dependence analysis. This yields a co-occurrence matrix of dimension equal to the number of gray levels in the image, considering every distance and orientation. Many measures have been extracted from these matrix [5], [11], and [30]. From among these measures we have selected the more representative of each group. These are contrast, entropy, homogeneity and variance. Transform Domain Methods. They are based on local energy measures obtained from frequency domain, which is represented by means of the discrete Fourier transform, Gabor functions, and wavelet transforms. Some important measures have been obtained from the Fourier Power Spectrum distribution in annularring geometry [6], [28], and [30]. Other methods rely more on Gabor energy approaches [21], which consist of spatially localized modifications of Fourier transform. Also, energy measures obtained from different wavelet transforms [13], [21] have recently been defined. After an analysis of the methods attending to their response in front of the kind of texture and window (neighbor) size, seven measures have been selected. Four of them are based on Fourier Power Spectrum (r1, r2, r1-r2, r2-r3), one is based on Gabor function, and two are measures obtained from the Daubechies and Haar wavelets transforms, respectively. Methods based on Texture Models. Were originally developed in the texture synthesis field, in order to define models being capable of discriminate or characterize among a wide range of textures. These methods assume some kind of dependence between a pixel and its neighborhood. This dependence can be linear, as is the case of autoregressive models, or based on joint probability, as in the Markov fields [9]. Moreover, fractal functions have received a high attention since Pentland saw a great degree of correlation between fractal dimension and human perception of coarseness. In this way, fractal functions have been proposed as textural models, and fractal dimension is used as a measure for texture segmentation [25]. In this work a measure from Gaussian Markov Random Fields and a fractal function [9], [25], and [30] are considered.
3.2. Selection of Computacional Methods As can be deduced from previous section, many existing methods could be used to evaluate the features' set. Our approach for getting a computational method to evaluate independent components is based on selecting the more suitable method to evaluate each feature. A first proposal to perform aforementioned selection is presented which consists in computing, for each perceptual feature, the distances between the values obtained by human perception (Table 1) and those obtained applying the computational methods. Table 5 depicts the computed distances. So, the lower distance values between a computational method and a perceptual feature the more precisely this method captures the concept denoted by the feature.
To understand better the values appearing at table 5, the steps followed to get them are explained for the particular case of the feature "granularity" and the Linear Filtering Operator "Laws R5R5". 1.
This method is separately evaluated for each different kind of texture, obtaining five probability density functions (PDF), one for each prototype texture. In order to leave out problems due to noise, the 10% of extreme values are eliminated in all the PDF's (Figure 1).
0.20 Probability
All these methods have been evaluated by using the microtexture set defined in section 2.1.
0.15 0.10 0.05
Energy
0.00 0
PERCEPTUAL FEATURES Gran.
Line.
Hom.
Coar. Conti. Abru.
Laws (E5E5)
12,86
3,69
5,31
10,11
3,01
1,46
2,10
Laws (E5L5)
14,39
3,15
4,40
10,81
2,32
2,00
2,48
Laws (L5E5)
12,62
4,39
5,84
10,18
3,87
1,48
1,84
Laws (R5R5)
6,48
6,93
4,43
3,59
6,18
4,58
1,55
Unser
12,63
11,47
8,49
14,93
8,47
4,55
7,20
15,27
2,61
4,84
10,29
2,66
1,92
1,68
10,10
21,49
2,30
8,46
17,01
16,31
9,58
11,74
4,20
5,49
9,18
3,66
1,36
1,68
Statistic (Entropy)
12,50
7,09
5,21
5,00
7,65
8,81
3,05
Fourier (r1)
13,61
4,06
4,92
11,38
2,75
1,85
3,07
Fourier (r2)
12,82
3,26
4,48
9,53
2,45
2,06
2,22
Fourier (r1-r2)
12,64
5,20
4,48
11,07
3,40
2,44
3,43
Fourier
11,66
4,03
3,97
8,97
2,75
2,56
2,50
9,91
4,12
4,65
5,68
3,81
3,92
1,83
9,73
4,24
5,20
5,68
4,16
3,45
1,50
Gabor
13,25
9,12
2,39
11,08
6,56
5,35
4,25
Markov
10,22
4,81
4,84
8,05
3,88
1,87
1,78
Fractal
6,50
20,13
4,02
1,79
17,84
15,64
6,35
(Variance vert.)
Statistic (Variance)
Statistic (Homogeneity)
Statistic (Contrast)
(r2-r3)
Wavelet (Daubechies 4)
Wavelet (Haar 2)
5
10
2.
20
25
30
Then, the resultant PDF's (from now on PDF´) are brought together obtaining a new PDF, and its interval of energy values (axis of abscissas) is transformed, by means a linear transformation, in the continuous interval [0,3] (Figure 2).
0.15 0.10 0.05 0.00 0
5 0
Table 5: Distances between perceptual features and computational measures. The range of the distances is [0, 45]. Lower distances are in bold-faced type.
15
Figure 1: LawsR5R5 Probability Density Function for the texture "forest"
Contr.
Probability
COMPUT. METHODS
10%
10%
10
15
20
1
2
3
Figure 2: Approximated LawsR5R5 Probability Density Function and its projection over the continuous interval [0,3]
3.
Considering the new axis of abscissas, the mean of the PDF´ is calculated for each texture. So, the mean for the texture "forest" is 1,49. This process is repeated for each texture, obtaining the values shown at Table 6.
Laws (R5R5)
Forest
Scrub
Road
Sown Field
Urban Area
1,49
1,12
1,41
0,40
2,13
Table 6: Means for “LawsR5R5” PDF’s
4.
Next, the square of the differences between the level of response given by the computational method (Table 6), and the level of granularity human perception (Table 1), is calculated for every texture. All these values and their addition are displayed at next table for our particular case. Forest Scrub Granularity Laws (R5R5) Square of differences
Road
Sown Field
Urb. Area
3
2
0
0
1
1,49
1,12
1,41
0,40
2,13
2,28
0,77
1,99
0,16
1,28
Total Difference
6,48
Table 7: Results for obtaining the distance between the perceptual feature Granularity and the computational measure Laws R5R5.
5.
The value of the total difference represents the fitting of the computational method "Laws R5R5" to evaluate the perceptual feature "granularity".
This process is repeated for each computational method and each feature, obtaining the values of Table 5. Looking at columns of this table, methods giving the lower distance value are chosen to evaluate the features (Table8).
Table 8: Computational Methods given the lower distance value evaluating the perceptual features.
Later on, the vector of values obtained evaluating the features with the corresponding methods is projected over the axes defined by the components to evaluate them.
3.3. Evaluating the Independent Components Using Fuzzy Techniques
Low
Medium
High
C1
FS1,1
FS1,2
FS1,3
FS1,4
C2
FS2,1
FS2,2
FS2,3
FS2,4
C3
FS3,1
FS3,2
FS3,3
FS3,4
Table 9: Fuzzy sets associated to each activation level per component.
These fuzzy sets will be defined making use of the probability density functions, because as they are obtained from the data, ought to reflect its variability and uncertainty. Obtaining the Fuzzy Sets So as to obtain the fuzzy sets, starting from the probability density functions, our algorithm proceeds as follows: 1.- The activation perceptual levels of the components in the prototype textures are evaluated. It is performed projecting the vectors of feature values obtained by human perception (Table 1) over each component. Here Orchard feature has been considered in order to get a complete model reflecting all the activity levels. Resultant values are presented at Table 10.
Perceptual Features Gran. Line. Hom. Coars. Conti Abru. Contr Laws Statistic Statistic Laws Statistic Wavelet Compt. Fractal E5L5 Contrast Haar2 Method R5R5 Variance Homog.
Very Low
Forest C1 C2 C3
Scrub
-1.593 -0.897 2.441 1.591 1.151 0.610
Next, in order to fuzzify the information supplied by the components a fuzzy set is associated to each activation level per component, so having the twelve fuzzy sets given at Table 9.
Sown Field
Urban Area
3.654 0.933 2.316
0.372 -0.197 -0.250
3.717 3.793 0.610
1.726 2.896 1.720
Table 10: Activation perceptual levels for the components in the perceptual prototype textures.
2.- Next, the interval of variability of each component, say ICi = [aCi, bCi], is divided into four subintervals ICi,1, ICi,2, ICi,3, ICi,4, having the same length and related to the aforementioned activation levels like this: ICi,1↔Very Low, ICi,2↔Low,
To carry out the independent component evaluation taking into account features and textures uncertainty, and trying to be faithful to principle of least commitment, a fuzzy treatment is introduced. As our aim is to get an image segmentation algorithm based on pixel classification, first of all the activation level -but not the value- of the components at each pixel must be inspected. To do it our approach considers that a pixel can present four activation levels for each component, socalled: Very Low, Low, Medium and High.
Road Orchard
ICi,3↔Medium, ICi,4↔High. Table 11 summarizes the activation levels for each feature after applying previous transformation.
Road
Orchard
Sown Field
Urban Area
Very Low
High
Medium
Low
High
C2 Medium
Low
Very Low
High
C3 Medium
Low
High
High
Forest Scrub C1
Very Low
Very Low Very Low
High Low
Table 11: Activation levels after the transformation.
3.- Then, the probability density functions, connected with the 12 fuzzy sets (pdfkl; 1≤ k ≤3, 1≤ l ≤4) are gotten computing the Ci over the pattern features, as expound in section 3.2. When a Ci has the same activation level for two textures (as C1 for Forest and Scrub), the probability density functions are put together. 4.- After adjusting the probability density functions by normal distributions (Npdfkl, 1≤ k ≤3, 1≤ l ≤4), the fuzzification process is performed. So, if Xk (i, j) is the activation level of Ck for the pixel (i, j) and [infkl, supkl] is the domain of Npdfkl, our fuzzy approach considers next rules: •
R1: The more close to the maximum Npdfkl (Xk(i,j)) is, the bigger the membership degree µFSk,l (i,j) is.
•
R2: The more close to infkl and supkl the value Xk(i,j) is, the less µFSk,l(i,j) is.
Implementing these rules as explained at [15], the twelve fuzzy sets are obtained.
First set of results To show how well the selected computational methods work evaluating the perceptual features, some results for Granularity, Linearity and Abruptness are presented. So, for each feature, three images owning the feature at different level are considered. Then three operators (computational methods) are used to evaluate the feature; the first will be those shown at Table 7 and the second and third are chosen from table 5 having intermediated and the worst adjusting levels respectively. Figure 4 shows the results for Granularity. Looking at this figure can be observed that, applying Laws R5R5 (Figure 4-(a)) to images having decreasing granularity level the result retains this decreasing degree after evaluate the feature. Nevertheless, this property isn't fulfilled when Markov (Figure 4-(b)) or Statistic (Variance) (Figure 4-(c)) are applied.
So, the steps followed to get the activation level degree of the components in a given pixel (i, j) are: 1st.- Computational method given in 3.2 is applied to evaluate the components, providing three real numbers: X1(i,j), X2(i,j), X3(i,j). 2nd.- This values are used to get the activation level degree as a membership degree to the defined fuzzy sets.
4. Experimental Results
forest-I (3)
forest-I
scrub-I
sown field-I
road-I
orchard-I
forest-II
road-II
urban-I
Figure 3: Textured aerial images used for the tests.
sown field-I (0)
4-(a) .- Computational results using Laws R5R5
forest-I (3)
In accordance with the process followed by the design of the algorithm, two sets of results are presented. The first set shows the adjusting of the selected computational methods to evaluate the considered perceptual features. Results in second set show that microtextures can be quite well described, in the sense of the activation levels given at Table 11, by the fuzzy sets associated to the components. Even though many images have been used to test the efficiency of the algorithm, in this section only the results of the tests performed over eight textured aerial images owning some perceptual features, at different levels, are presented. Figure 3 depicts the images used for the tests.
scrub-I (2)
scrub-I (2)
sown field-I (0)
4-(b).- Computational results using Markov.
forest-I (3)
scrub-I (2)
sown field-I (0)
4-(c).- Computational results using Statistic (Variance). Figure 4: Results of Granularity tests.
Results for Abruptness are shown at Figure 5. As can be observed at Figure 5-(a), when Statistic (Contrast) operator is applied over images with decreasing Abruptness level, the result retains this decreasing degree after evaluate the feature. Nevertheless, applying the Statistic (Homogeneity) operator (Figure 5-(b)) to these images an increasing degree is produced. Moreover, as shown at Figure 5-(c), when the Laws R5R5 operator is applied the maximum degree is obtained for the image with the smallest abruptness degree.
Second set of results
urban-I (3)
road-I (1)
forest-I (0)
5-(a).- Computational results using Statistic (Contrast).
urban-I (3)
road-I (1)
forest-I (0)
5-(b).- Computational results using Laws R5R5.
urban-I (3)
road-I (1)
forest-I (0)
5-(c).- Computational results using Statistic (Homogeneity). Figure 5: Abruptness tests.
Next the results for Linearity are presented. As can be observed looking at Figure 6-(a), if Statistic (Variance) is applied over images with decreasing linearity level, resultant images retain this decreasing degree. Must be remarked that this feature is present only within an area of road images, whereas Orchard image has the feature distributed on the entire image. When Gabor operator is applied, certain decreasing level is observed in the results (Figure 6-(b)), but isn't in harmony with the evaluated perceptual value. Finally, applying the Statistic (Homogeneity) operator (Figure 6(c)) an increasing degree is obtained evaluating the feature.
In order to represent the set of microtexture perceptual features, the three components must be able to discern them. The images here presented - figure 7- will show that it is possible using the fuzzy sets defined at previous section. Although a great number of tests have been performed, because of space problems, one case for the textured aerial image "forest" is the only presented here. To show that microtextures in this image can be quite well described by the fuzzy sets associated to the components, the greatest membership degrees of the pixels in the image must be obtained for FS1,1, FS2,1, and FS3,1. That is to agree with the components activation levels given at table 11. C1 ↔ Very Low, C2 ↔ Medium, C3 ↔ Medium In these images, the greatest the membership degree is, the clearest the representation is. Having a look at figure.7-(a) can be observed that the clearest image is a-1, which agrees with the activation level of forest for the component C1. This agreement is extendable to figures 7-(b) and 7-(c), because the clearest images are b-3 and c-3, associated to Medium activation levels.
µFS1,1(i,j) a-1:µ
µFS1,2(i,j) a-2:µ
µFS1,3(i,j) a-3:µ
µFS1,4(i,j) a-4:µ
7-(a).- Membership degrees to fuzzy sets corresponding to C1
µFS2,1(i,j) b-1:µ
µFS2,2(i,j) b-2:µ
µFS2,3(i,j) b-3:µ
µFS2,4(i,j) b-4:µ
7-(b).- Membership degrees to fuzzy sets corresponding to C2 road-I (3)
orchard-I (2)
forest-I (0)
6-(a).- Computational results using Statistic (Variance).
µFS3,1(i,j) c-1:µ
c-2:µFS3,2(i,j)
µFS3,3(i,j) c-3:µ
µFS3,4(i,j) c-4:µ
7-(c).- Membership degrees to fuzzy sets corresponding to C3 road-I (3)
orchard-I (2)
forest-I (0)
6-(b).- Computational results using Gabor.
Figure 7: Components evaluation for the textured aerial image forest.
5. Conclusions
road-I (3)
orchard-I (2)
forest-I (0)
6-(c).- Computational results using Statistic (Homogeneity). Figure 6: Linearity tests.
The study performed in this paper introduces a computational method to evaluate perceptual features. Moreover, an approach to reduce the number of necessary features to identify the several textures is presented, and a computational method to evaluate them is proposed.
On the other hand, the use of fuzzy techniques allows to introduce a robust method to describe the textures appearing within the aerial images, which takes into account the vagueness connected with this kind of images. The feasibility of the proposed approach has been proved by the numerous tests performed.
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