The laplacian pyramid is a well-known technique for image processing in which local operators of many scales, but identical shape, serve as the basis functions. The required properties to the pyramidal filter produce a family of filters, which is unipara metrical in the case of the classical problem, when the length of the filter is 5. We pay attention to gaussian and fractal behaviour of these basis functions (or filters), and we determine the gaussian and fractal ranges in the case of single parameter
Image analysis involves many different tasks, such as identifying objects into images (segmentation), assigning labels to individual pixels by taking into account relevant information (classification), or extracting some meaning from the image as a whole (interpretation). The segmentation of soil images appears into Soil Science as a tool for the measurement of properties as well as for detecting and recognizing objects in soil [
Different methods have been used to segment soil images such as a simple binary threshold method [
Soil is not a continuous medium because soil is susceptible to changes from many influences: wetting, drying, compaction, plant growth, and so forth. So, the continuous soil models lead to approximate results only, and anomalous phenomena cannot be easily handled. It is known that pores in porous material are highly complex [
Soil is formed from many constituents, and to represent it as a two-phase material, solid and pore, is often an oversimplification. The behaviour of water, gas, and organisms can affect it. A classification of pore models could be [
Models of soil physical structure have been developed since the 1950s. Childs and Collisgeorge [
More sophisticated approaches are [
Our goal is to calculate the porosity of soil images. The proposed procedure for segmentation of soil micromorphological images is based on Laplacian pyramid algorithm [
Another objective of this study is also to compare the results with those provided by the commonly used Otsu’s algorithm [
A multiresolution model consists of generating different versions of a given image by decreasing the initial resolution, which also means decreasing the initial size. This is achieved by a downsampling operator which must be associated with an appropriate filtering to avoid aliasing phenomena (downsampling theorem). Multiresolution approaches have been investigated for different purposes such as image segmentation and image compression [
One usual property of images is that neighboring pixels are highly correlated. This property is inefficient to represent the complete image directly in terms of its pixel values, because most of the encoded information would be redundant. Burt and Adelson designed a technique, named Laplacian pyramid, for removing image correlation which combines characteristics of predictive and transform methods [
This pyramidal representation is useful for two important classes of computer graphics problems. The first class is composed of those tasks that involve analysis of existing images, such as merging images or interpolating to fill in missing data smoothly, become much more intuitive when we can manipulate easily visible local image features at several spatial resolutions. And the second, when we are synthesizing images, the pyramid becomes a multiresolution sketch pad. We can fill in the local spatial information at increasingly fine detail by specifying successive levels of a pyramid.
The first time that pyramidal structures were applied to multiresolution decompositions was at [
Pyramidal methods for multiresolution image analysis have been used since the 1970s. Early work in multiresolution image description was primarily motivated by a desire to reduce the computational cost of methods for image description and image matching. Later, multiresolution processing was generalized to computing multiple copies of an image by repeatedly summing nonoverlapping blocks of pixels and resampling until the image is reduced to a small number of pixels. Such a structure became known as a multiresolution pyramid [
Interest in multiresolution techniques for signal processing and analysis is increasing steadily [ The pyramid consists of a finite number of levels such that the information content decreases towards higher levels. Each step towards a higher level is constituted by an information-reducing analysis operator, whereas each step towards a lower level is modeled by an information-preserving synthesis operator. One basic assumption is necessary: synthesis followed by analysis yields the identity operator, meaning that no information is lost by these two consecutive steps.
The techniques based on fractals show promising results in the field of image understanding and visualization of high complexity data.
The high complexity of some images demands new techniques for understanding and analyzing them. The similarity of fractals and real world objects has been observed and studied from the very beginning. The fractal geometry became a tool for computer graphics and data visualization in the simulation of the real world. In order to perform visual analysis and comparisons between natural and synthetic scenes several techniques have been developed. After a period of qualitative experiments, fractal geometry began to be used for objective and accurate purposes: modeling images, evaluating their characteristics, analyzing their textures, and so forth.
Nowadays, there are a lot of fields where fractals appear [
Fractal geometry theory deals with the behaviors of sets of points
Mandelbrot defined a fractal as a shape made of parts similar to the whole in some way [
A fractal object is self-similar or self-affine at any scale. If the similarity is not described by deterministic laws stochastic resemblance criteria can be found. Such an object is said to be statistical self-similar. The natural fractal objects are statistically self-similar. A statistically self-similar fractal is by definition isotropic. To have a more precise, quantitative description of the fractal behavior of a set, a measure and a dimension are introduced. The rigorous mathematical description is done by Hausdorff’s measure and dimension [
Let
Now, if in (
There is an interesting property of the Hausdorff measure: If the Hausdorff dimension of the set
Hausdorff dimension of the set
Then, the value of the parameter
A set is said to be fractal if its Hausdorff dimension strictly exceeds its topological dimension,
Numerical evaluation of Hausdorff dimension is difficult because of the necessity to evaluate the infimum of the measure over all the coverings belongings to the set of interest. That is the reason to look for another definition for the dimension of a set. The box counting dimension allows the evaluation of the dimension of sets of points spread in an
Given a set of points
Depending on the geometry of the box and the modality to cover the set, several box counting dimensions can be defined using closed balls, cubes, and so on [
The equivalence of these definitions was proved. Also it was proved that these dimensions are inferior bounded by the Hausdorff dimension [
Fractal geometry provides a mathematical model for many complex objects found in nature, such as coastlines, mountains, and clouds [
Image segmentation is the process of partitioning an image into several regions, in order to be easier to analyze and work with.
In image segmentation the level to which the subdivision of an image into its constituent regions or objects is carried depending on the problem being solved. In other words, when the object of focus is separated, image segmentation should stop [
We study the simplest problem, dividing the image into just only two parts, foreground and background, or object pixels and background pixels. The intensity values, continuity or discontinuity, color, texture, and other image characteristics are the origin of the different image segmentation techniques. Reference [
So, some of the most important groups in image segmentation techniques are the threshold-based, the histogram-based, the edge-based, and the region based.
The threshold-based methods are based on pixels intensity values. The main goal here is to decide a threshold value
The edge-based methods show boundaries in the image, determining different regions where we have to decide if they are foreground or background. The boundaries are calculated analyzing high contrasts in intensity, color, or texture. On the other hand, an opposed point of view are the region-based methods divide the image into regions, searching for same textures, colors, or intensity values.
In soil science the porosity of a porous medium is defined by the ratio of the void area and the total bulk area. Therefore, porosity is a fraction whose numerical value is between 0 and 1, typically ranging from 0.005 to 0.015 for solid granite to 0.2 to 0.35 for sand. It may also be represented in percent terms by multiplying the number by 100. Porosity is a dimensionless quantity and can be reported either as a decimal fraction or as a percentage.
The total porosity of a porous medium is the ratio of the pore volume to the total volume of a representative sample of the medium. Assuming that the soil system is composed of three phases—solid, liquid (water), and gas (air)—where
Table
Range of porosity values.
Unconsolidated deposits | Porosity | Rocks | Porosity |
---|---|---|---|
Gravel | 0.25–0.40 | Fractured basalt | 0.05–0.50 |
Sand | 0.25–0.50 | Karst limestone | 0.05–0.50 |
Silt | 0.35–0.50 | Sandstone | 0.05–0.30 |
Clay | 0.40–0.70 | Limestone, dolomite | 0.00–0.20 |
Shale | 0.00–0.10 | ||
Fractured crystalline rock | 0.00–0.10 | ||
Dense crystalline rock | 0.00–0.05 |
Our work applies image segmentation techniques to calculate the porosity of soil images. Also, we have compared our results with the Otsu image segmentation algorithm.
The Laplacian pyramid representation expresses the original image as a sum of spatially band-passed images, while retaining local spatial information in each band. The Gaussian pyramid is created by low-pass-filtering an image
Representation of the one-dimensional Gaussian pyramid process.
Representation of the two-dimensional Gaussian pyramid process.
This process is repeated to form a Gaussian pyramid
Fractal dimension is a useful feature for texture segmentation, shape classification, and graphic analysis in many fields. The box-counting approach is one of the frequently used techniques to estimate the fractal dimension of an image.
There are several methods available to estimate the dimension of fractal sets. The Hausdorff dimension is the principal definition of fractal dimension. However, there are other definitions, like box-counting or box dimension, that is popular due to its relative ease of mathematical calculation and empirical estimation. The main idea to most definitions of fractal dimension is the idea of measurement at scale
These formulae are appealing for computational or experimental purposes, since
In order to determine the normality interval we use the Kolmogorov-Smirnov normality test [
For a data set
For
Let
The most common image segmentation methods are the histogram thresholding based, since thresholding is easy, fast, and economical in computation. For performing the image segmentation we need to calculate a threshold which will separate the objects and the background in our image. Since soil images are relatively simple when we just pay attention to void and bulk, so we are going to apply the global threshold technique, instead of more advanced variations (band thresholding, local thresholding, and multithresholding). The global thresholding technique consists of selecting one threshold value and applying it to the whole image.
The resultant image is a binary image where pixels that correspond to objects and background have values of 255 and 0, respectively. Quick and simple calculation is the main advantage of global thresholding.
Otsu’s method searches for the threshold that minimizes the intraclass variance (or within class variance)
Otsu [
The class probabilities
The class probabilities and class means can be computed iteratively. This idea yields an effective algorithm.
Otsu’s algorithm assumes just only two sets of pixel intensities, the foreground and the background, or void and bulk for soil images. The main idea of the Otsu’s method is to minimize the weighted sum of within-class variances of the foreground and background pixels to establish an optimum threshold. It can be formulated as
Let the pixels of a given picture be represented in 256 gray levels:
The threshold at level
Our 1D filters are defined by the weighting function
Note that the functions double in width with each level. The convolution acts as a low-pass filter with the band limit reduced correspondingly by one octave with each level. Because of this resemblance to the Gaussian density function we refer to the pyramid of low-pass images as the Gaussian pyramid. Just as the value of each node in the Gaussian pyramid could have been obtained directly by convolving a Gaussian-like equivalent weighting function with the original image, each value of this bandpass pyramid could be obtained by convolving a difference of two Gaussians with the original image. These functions closely resemble the Laplacian operators commonly used in image processing. For this reason the bandpass pyramid is known as a Laplacian pyramid. An important property of the Laplacian pyramid is that it is a complete image representation: the steps used to construct the pyramid may be reversed to recover the original image exactly. The top pyramidal level,
The weighting function
If the size is 5, we have the filter
Convolution is a basic operation of most signal analysis systems. When the convolution and decimation operators are applied repeatedly
Figure
First, second, and fourth iteration of the filter 1D (
Sixth iteration of 1D Laplacian pyramid filters,
We have tested different values
The first two filters are Gaussian-like, and the last two are fractal-like. It is possible to confirm these early conclusions. We have successfully applied normality tests to verify the normality of the filters obtained with the lowest
We can see the results of fractal dimension (FD) of filters 1D whose values are shown in Table
Fractal dimension of filters.
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The generation of bidimensional filters
When we calculate the bidimensional filters, we obtain filters like Figure
Bidimensional filters corresponding to
There is a relevant result when we study the normality of filters: the Gaussian function is separable if variables are independent. Burt and Adelson show that if we choose
Once we have a chosen value for
Because the symmetry property of the filter
Conditions to calculate
Specifically, the numerical simulation with
Gaussian filter adjustment.
Density function of normal distribution (black) and weighting function for
Figure
After these results, we have generated the Gaussian and Laplacian pyramids corresponding to one Gaussian value for
Gaussian and Laplacian pyramids of the soil image ((a)
Fractal dimensions and threshold values.
Fractal dimension of filters 1D
Threshold values from Laplacian pyramid filters and Otsu’s technique
Soil image and several segmentations.
Original
Otsu
When we have applied our method to segment images with different values for the parameter
The application of our methods with different values
We have presented threshold values obtained from Laplacian pyramid and the comparison with Otsu’s values. On the other hand, if we compare the pore size frequency distribution obtained by Otsu’s method and the threshold obtained based on Laplacian filter structure some difference is observed, as we can see in Figure
Pore size distribution.
The field of fractals has been developed as an interdisciplinary area between branches of mathematics and physics and found applications in different sciences and engineering fields. In geo-information interpretation the applications developed from simple verifications of the fractal behavior of natural land structures, simulations of artificial landscapes, and classification based on the evaluation of the fractal dimension to advanced remotely sensed image analysis, scene understanding, and accurate geometric and radiometric modeling of land and land cover structures.
Referring to the computational effort, fractal analysis generally asks high complexity algorithms. Both wavelets and hierarchic representation allow now the implementation of fast algorithms or parallel ones. As a consequence a development of new experiments and operational applications is expected.
We have seen that the different choice of the parameter
The different shape of filters, Gaussian/fractal, has perceptible effects when we generate new levels of the Gaussian and Laplacian pyramids, getting blurred or accentuated new images, at every new level. Filters generated with lowest
Also, there is a different behaviour of the energy of the different levels of the Laplacian pyramid if we choose different
These filters can be applied to image segmentation of soil images, with a simple computation and good results, quite similar or even better to some famous techniques such as Otsu’s method.
Moreover, results concerning porosity are similar but there are differences in pore size distribution that could improve percolation simulations. The implementation of this method in three dimensions is straightforward.
Future work could add other image segmentation techniques and neural networks methods to select the optimal threshold values from information and characteristics of the image.
The authors declare that there is no conflict of interests regarding the publication of this paper.