1. Introduction
Fractal geometry, proposed by B. Mandelbrot in [
1], triggered the computer-based analysis of self-similar and scale-independent objects called fractals and enabled their application in many domains. The fundamental fractal measure is the fractal dimension, defined to assess the roughness or complexity of such objects. To be more specific, the fractal dimension objectively quantifies the variations of a fractal object or a signal exhibiting fractal properties along the analysis scales [
2]. The resulting fractal dimension is a scalar comprising the interval
, where
E is the topological dimension of a scalar-value object. For a grayscale image, the fractal dimension is between 2 and 3, taking into account that the topological dimension of the image support is
. For an RGB color image, the color fractal dimension should belong to the interval
, which is between 2 and 5 according to [
3]. By generalization, for multidimensional signals and in particular for multi-spectral images, the fractal dimension should be within
, where
M is the number of image spectral bands [
4]. The fractal dimension has been used in a plethora of applications for the classification of signals or patterns exhibiting fractal properties, such as texture images [
5,
6], or for image segmentation [
7,
8]. In the fields of remote sensing and Earth observation, fractal analysis was used for noise characterization in SAR sea-ice images [
9], while the fractal dimension was used to correct the scale [
10].
The theoretical fractal dimension is the Hausdorff dimension [
11], which cannot be used in practice due to its definition for continuous objects. Consequently, various estimators were proposed in order to allow the fractal analysis for digital images with fractal properties: the similarity dimension [
1], the probability measure [
12,
13], the Minkowski–Bouligand dimension, also known as the Minkowski dimension or box-counting dimension [
14], the
-parallel body method, also known as the covering blanket approach, morphological covers or Minkowski sausage [
15], the gliding box-counting algorithm based on the box-counting approach [
16], the fuzzy logic-based approaches [
17,
18], and the pyramidal decomposition-based approach [
19]. There also exist various surveys on fractal estimators, such as [
20,
21], as well as an attempt to unify several existing approaches into a single one [
22]. However, all these approaches were designed for binary and grayscale images, and they are usually used without calibration or referencing to fractal images with known fractal dimensions.
Various attempts were made to extend the fractal dimension estimation approaches to the multivariate image domain, starting with color and extending to the multi-spectral images. The initial approaches for defining the fractal measures for color images were marginal, considering each color channel independently [
23]. The probabilistic box-counting approach was extended for the complexity assessment of color fractal images with independent color components, and its validity was proven first mathematically and then experimentally in [
3]. Some limitations of this latter approach were underlined in [
24]. In [
25], the authors proposed an approach based on the box-counting paradigm by dividing the image in non-overlapping blocks and considering the pixel count in the RGB color domain for both synthetic and natural images. In [
26], extensions of the differential box-counting approach were proposed for RGB color images without a mathematical proof or calibration. The approach proposed in [
27] allows for an extension to the multi-spectral image domain. Recently, the fractal generation and fractal dimension estimation were extended to the multi-spectral image case [
4] without a mathematical proof of validity of the multi-spectral fractal image model.
The domain of multi-spectral and hyper-spectral imaging, which experienced great development recently, requires the adaptation of existing tools or even the definition of new tools for image analysis. Multi-spectral and hyper-spectral imaging allows for capturing higher-resolution spectral information for a scene, sometimes covering both the visible and infrared wavelength spectra. A better spectral resolution can provide a deeper understanding of the materials and surfaces in the scene, particularly for the land cover type in an Earth observation scenario [
28]. Spectral imaging, in a more general sense, is used in a wide variety of applications, such as agriculture [
29,
30], forest management [
31,
32], and geology [
33,
34].
In this article, we embrace the approach in [
4]. We describe it extensively, mathematically prove the conjecture in [
4], and add more experimental results for both synthetic and real multi-spectral images. More specifically, in
Section 2, we first propose the extension of the midpoint displacement generation technique to the case of multi-spectral images with seven spectral bands and then visualize the generated images using three different techniques. We then prove mathematically the validity of the fractal model for the generated synthetic multi-spectral fractal bands with statistically independent bands. In the end, we extend to the domain of multi-spectral images the probabilistic box-counting approach for the estimation of the fractal dimension. In
Section 3, we tune the proposed estimation approach on the generated synthetic multi-spectral images with seven statistically-independent spectral bands in an attempt to reach the theoretical fractal dimensions of the respective images. In
Section 4, we estimate the fractal dimensions of real satellite images, and in
Section 5, we draw our conclusions.
3. Fine-Tuning the Estimator
In order to experimentally test and validate the proposed approach, we considered the three generated multi-spectral fractal data cubes or images with seven spectral bands in
Figure 4, having a spatial resolution of
pixels, of varying fractal complexity (i.e., low, medium, and high, which translate into Hurst coefficients of
,
, and
, respectively). As we mentioned in the theoretical considerations, the fractal dimension of such a multi-spectral fractal image should comprise between 2 (the complexity of a plane for a
uni-image or an image having the same color in every pixel) and 9 (the highest value achievable for a nine-dimensional image (i.e.,
, with two spatial coordinates plus seven spectral coordinates)). For the three synthetic multi-spectral images, we ran the proposed probabilistic box-counting fractal dimension estimation adapted to the multi-spectral case. The maximum analysis window size
was varied for all three images from 41 to 101 in steps of 10. However, the maximum analysis window was set to smaller values for the low- and mid-complexity images, as a maximum window of 31 proved to be very large, especially for the low-complexity image. The threshold for the standard deviation
was varied from
to
in steps of
. This standard deviation refers to the extent to which the regression line slope estimation approaches should agree on the
measure (represented in a log-log space), which has a direct impact on the fractal dimension estimation. For the three multi-spectral images we obtained the numerical results presented in
Table 2,
Table 3 and
Table 4 for the low, middle and high generated complexity, respectively.
For the lowest-complexity image, the highest achievable fractal dimension was
for
and for the standard deviation comprised between
and
. For the mid-complexity image, the highest achievable fractal dimension was
for
and for the standard deviation comprised between
and
. It is important to mention the fact that when setting a threshold to such small values, the estimated fractal dimension was estimated based only on three points in the
measure. A more reliable estimation would be
for
and for the standard deviation comprised between
and
. For the high-complexity image, the highest achievable fractal dimension was
for
and for the standard deviation comprised between
and
. Making the same observation as before, a more confident estimation would be
for
and for the standard deviation comprised between
and
. As a general observation, the estimated fractal dimensions indicated the correct ranking of the generated image complexity. In addition, as expected, the parameter
has to be adapted to the complexity of the image, which in practical application of the fractal estimation approach leads to a paradoxical situation: the fractal dimension which is desired to be estimated and thus unknown should be known in order to set the correct parameter values for the estimator. Another important observation is that, when comparing the current obtained results to the one obtained for color fractal images in [
27], the extra information due to the additional four spectral bands, compared with the color RGB case, led to higher complexity values.
In order to graphically observe the evolution of the estimated multi-spectral fractal dimension, we present the corresponding plots in
Figure 10,
Figure 11 and
Figure 12 (the evolution as a function of
) and
Figure 13,
Figure 14 and
Figure 15 (the evolution as a function of
) for the low, medium and high complexities, respectively, for the common interval of parameter values (
from 41 to 101 and
from
to
). As a general observation, for the low- and mid-complexity images, the tendency of the estimated multi-spectral fractal dimension was to decrease with the increase in the maximum analysis window and the increase in precision for the agreement of regression line estimators (decrease in the standard deviation). However, this behavior was observed outside the most pertinent interval of values for
. A possible explanation for the low performance of the estimator for large values of the maximum analysis box size is the less statistically significant data deployed in the regression line estimation as a consequence of the smaller effective image area for which the fractal analysis was performed (for
, approximately 37% of the pixels of the generated images were disregarded). If the image’s spatial resolution (i.e., the image size) allows it, increasing the size of the maximum analysis box makes sense, given that the current estimator disregards the small boxes and allocates more weight to the larger boxes, especially for the high generated complexity fractal images, where the variations of the signals can be very important and thus need to adapt the maximum analysis window. For the high generated complexity image in our experiments, the variation of the analysis box size
showed that the middle range of values was the most pertinent one for the estimation. For the appropriate values of
, increasing the precision of the slope agreement in the regression line estimators (thus diminishing the standard deviation) clearly improved the estimation, as the estimated multi-spectral fractal dimension increased.
4. Experimental Results
The multi-spectral images used in our experiments were two crops (left upper corner and right lower corner) of a Pavia University hyper-spectral image downsampled in the spectral domain to only seven spectral bands. The Pavia University data set is a 610 × 340 image with a spectral resolution of 4 nm and a spatial resolution of 1.3 m. The image has 103 bands in the 430–860 nm range. The scene in the image contains a total of nine materials according to the provided ground truth, both natural and man-made. We selected 7 spectral bands from the hyper-spectral data: 1, 14, 26, 39, 51, 64, and 76, corresponding to the 430, 482, 530, 582, 630, 682, and 730 nm wavelengths, respectively. We cropped the left upper corner and the right lower corner of the image so that the spatial resolution was
pixels, similar to the one of the synthetic fractal images used for validation (see
Figure 16 and
Figure 17). The estimated multi-spectral fractal dimensions of the seven spectral bands in the Pavia University multi-spectral image crops are presented in
Table 5 and
Table 6 for
varying between 31 and 71 in steps of 10 and
varying from
to
in steps of
(i.e., the settings for the most confident estimation results, considering a parameter setting of the estimation tool for low-to-mid-complexity images, as for the considered Pavia University multi-spectral images).
For the left upper corner crop of the Pavia University image, the maximum estimated multi-spectral fractal dimension was , while for the right lower corner crop, it was . The relative difference in complexity was obvious due to the image content; the more complex image contained more colors and variations with more objects present in the scene, while the less complex image contained less colors, less objects, and a larger area of small signal variations. Consequently, the estimator clearly indicates the relative ranking of images as a function of their complexity. However, both images, through their assessed complexities, were in the mid-to-low complexity range.
5. Conclusions
We proposed both a fractal generator and a fractal dimension estimator for multi-spectral images. The proposed estimator allows for fully vector-based fractal analysis of multi-spectral images with fractal properties, compared with all the other existing methods which work only as marginal approaches for each spectral band, considered independently or on color images, thus limiting the application domain and disregarding the rich information in a multi-spectral image. The proposed generator allows for the generation of multi-spectral fractal images with known generated complexity, thus enabling the calibration of the fractal dimension estimator before using it on real-life images in practical use cases.
The generator is based on the midpoint displacement algorithm used for generating fractional Brownian motion, and the estimator is based on the classical probabilistic box-counting approach. The model for the generated multi-spectral fractal images was proven mathematically and illustrated for the case of seven statistically independent spectral bands. The model can be extended theoretically to an arbitrary number of spectral bands, as long as the hypothesis of statistical independence between bands holds (which may not be the case for high spectral resolution images, such as hyper-spectral images). For a qualitative evaluation, the resulting synthetic multi-spectral data sets were visualized as color RGB composites using three different approaches: the widely used band selection, using a linear model for the color formation, and deploying an artificial neural network which was previously trained to learn the correspondences between the multi-spectral pixel signatures and colors specified in the RGB color space. The fractal dimension estimator was adapted to work on nine-dimensional fractal objects, and we estimated the multi-spectral fractal dimension of the generated synthetic multi-spectral fractal images. The estimation requires setting the values for the parameters and , as they should be adapted to the envisaged complexity range of the analyzed images. We presented and interpreted the numerical results obtained in the process of fine-tuning the estimator. However, for the highest generated complexity image, the desirable multi-spectral dimension of has not yet been achieved.
Furthermore, we used the proposed multi-spectral fractal dimension estimator for the fractal complexity assessment of real images. We chose for the experiments the widely known Pavia University hyper-spectral data set, which was first downsampled in the spectral domain from 103 spectral bands to only 7 spectral bands in order to fit to the spectral capabilities of the designed estimator. Secondly, the image was cropped so that the spatial resolution of the resulting images would be identical to one of the generated synthetic multi-spectral fractal images (). The dynamic range was also scaled to the – interval in order to have the same variation of values on all seven bands and in the same range as the spatial domain. The obtained results are in accordance with the perceived complexity of the two scenes. The usefulness of the proposed multi-spectral fractal dimension estimator can be proven in two types of applications: image classification and image segmentation, where the multi-spectral fractal dimension can be used as a global or local feature, respectively, for multi-spectral texture characterization. The proposed model and estimator can be applied on remotely sensed data, such as the multi-spectral images from the Sentinel 2 satellites of the Copernicus Earth Observation program.