Application of fractal analysis methods to images obtained by crystallization modified by an additive

2.1. Crystallograms obtained with milk

The mixture to be evaporated has the following composition: 6 ml of milk dilution (6.25 μl∙ml^-1) in an aqueous $S$ solution (55 μmol$\bullet$ml^-1). Three sorts of milk were analysed: a) biodynamically produced raw milk (Sennerei Bachtel-Damalis AG, CH 8342 Wernetshausen; denoted as BD milk); b) milk from organic production microfiltrate homogenized and pasteurized (Molkerei Biedermann CH 9220 Bischofszell; denoted as MHP milk); ultra-heat treated milk (Coop, CH Basel; denoted as UHT milk).The obtained results are shown on Fig. 1.

All the images have size 2016×1512 and were obtained in RGB colour model.

Fig. 1Crystallograms obtained with a) BD milk, b) MHP milk, c) with UHT milk

a)

b)

c)

2.2. Crystallograms obtained with beans

The crystallograms obtained with bean leave extracts are shown on Fig. 2. They were obtained by evaporating 6 ml of an aqueous solution of copper chloride (24.5 μmol$\bullet$ml^-1) containing beans leave extract (130 µl$\bullet$ml^-1).

Bush bean seeds (cv. SAXA from Bingenheimer Saatgut AG, Kronstraße 24, D-61209 Echzell) are sown on the 18th of May, in 4 planters containing a soil mixture available of the shelf (Ökohum GmbH, Ebertingen – D – 88518; terriccio universale bio; characteristics: N₂: 65 mg/L (40-90); P₂O₅: 100 mg/L (60-140); K₂0: 760 mg/L; pH (CaCl₂): 6.3 (5.9-6.7; salts (KCl) 1.4 g/L (0.85-1.95 g/L; organic substances 20 % (15-25 %).

Fig. 2Crystallograms obtained with bean extracts. Beans were cultivated as mentioned above and sprayed with distilled water (B1) or with the suspension of silica (B2), or with two 501 preparations C10 and C11 (B3 and B4 respectively)

a) B1

b) B2

c) B3

c) B4

At 2-nod stage (cotyledons felt down, 2 opposite leaves, and 1 trifoliate leave), the beans were sprayed with approximately 20 ml water or suspension of silica or of 501 preparations (C10 and C11; [11]) after dynamization.

Additive preparation: in 5 days after spraying, 1 of the opposite leaves and the trifoliate leave from the next nod were harvested from different plants to make up 21 g; the leaves were crushed and suspended with 600 ml of distilled water and filtered after agitation for 10 min; the filtrate being additive ($A$).

All the images have size 1750×1500 and were obtained in RGB. By visual perception the images are rather similar. Hence, they may have close structures and thereby close numerical characteristics.

3.1. Fractal signature method

The method described is used to obtain fractal dimension of the image by calculating the approximate area of a special gray level surface for a given image. The method (blanket technic) was described in [12], and used for solving various application problems in [13-15]. We applied this method in different variants to analyse biomedical preparation images [16, 17]. Here we present brief description of the method.

Let $F=\{X_{ij},i=\mathrm{0,1},\dots ,K,j=\mathrm{0,1},\dots ,L\}$ be a gray level image and $X_{ij}$ be the intensity of the $(i,j)$th pixel. For the points with coordinates $(x,y)$, where $i<x<i+1$, $j<y<+1$ we assume that $F\left(x,y\right)=X_{ij}$. In a certain measure range the surface of function $F$ can be viewed as fractal. In image processing this surface is a nonempty bounded set in $R^3$. Surface area $A_{\mathrm{\delta }}$ may be calculated using the volume of a special $\delta$-parallel body (“blanket”) with thickness $\text{2}\delta$.

For $\delta =$ 1, 2, …, the blanket surfaces $u_{\delta }$ (upper) and $b_{\delta }$ (bottom) are defined iteratively as follows:

We assume that $u_0\left(i,j\right)=b_0\left(i,j\right)=X_{ij}.$ Volume ${Vol}_{\delta }$ of the blanket is:

The formulas for the calculation of the surface area are:

Fractal dimension $D$ of the image is calculated as:

Note that Eq. (3) is more preferable for pure fractal objects, whereas Eq. (4) is used for both fractal and non-fractal surfaces. The application of Eq. (4) is necessary if ${Vol}_{\delta }$ depends on all lesser scale features: the subtracting ${Vol}_{\delta -1}$ isolates the features that change from scale $\delta -1$ to $\delta$. In our calculations we use Eq. (4).

One of useful modification of the method is to divide the image on $N$ boxes and for each box $i$ calculate area $A_{\delta }\left(i\right)$ of the corresponding surface by the described method. Thus, to obtain area $A_{\delta }$ of the whole surface we sum up $A_{\delta }\left(i\right)$ by $i$. If we repeat this procedure for several values of the box size we obtain the dependence of the surface area on the size of a partition box, i.e. a vector characteristic. The values $A_{\delta }\left(i\right)$ may be also applied to obtain a segmentation of the image.

It should be noted that we operate with integer numbers in calculations, and values $A_{\delta }$ may be large enough: from 10³ to 10⁵. Because of this, on graphs values are given in the normalized form, so the maximal result be less than 10³. The box size is measured in pixels.

3.2. Calculation of multifractal spectrum by using density function

We calculate the characteristic of every pixel which the authors in [18] proposed to call density function. Then we combine all the pixels with close values of density function into the subset, which results in the partition of the image on the subsets. They are called level sets. For each level set we calculate its fractal dimension.

Let $\mu$ be a measure defined by pixel intensities. In our experiments for any $A\in R^2$ we assume that $\mu \left(A\right)$ is equal to the sum of intensities of pixels from $A$. For $x\in R^2$ we denote $B(x,r)$ as the square of length $r$ with its center $x$. For a given digital image let $\mu \left(B\left(x,r\right)\right)=k{r}^{d\left(x\right)}\left(x\right)$, where $d\left(x\right)$ is called local density function of $x$, and $k$ is some constant. Taking several values for $r$ we have:

The density function measures the non-uniformity of the intensity distribution in square$B(x,r)$. The set of all points $x$ with local density $\alpha$ is level set $E_{\alpha }=\left\{x\in R^2\mathrm{}:d\left(x\right)=\alpha \right\}$. In practice, not to increase the number of level sets, nonintersecting sets $E\left(\alpha ,\epsilon \right)$ are considered, such that $E\left(\alpha ,\epsilon \right)=\left\{x\in R^2:d\left(x\right)\in [\alpha ,\alpha +\epsilon )\right\}.$

For each set $E_{\alpha }$we calculate its fractal (capacity) dimension $f\left(E_{\alpha }\right)$ (denoted by $f\left(\alpha \right)$ for brevity) using well-known Least Square Method. These dimensions form multifractal spectrum$.$ As it follows from Eq. (5), the values of $d\left(x\right)$ are real numbers. The values of fractal dimensions for the subsets in plane are in interval [0, 2]. On graphs obtained in examples numerical values on $X$ axis left ends of intervals $[\alpha ,\alpha +\epsilon )$ are marked

It should be noted that for each image we obtain an interval $\left[{\alpha }_{min},{\alpha }_{max}\right]$ for values of density function, and these intervals are different for different images: the greater interval the more complex structure we reveal. That means that graphs of multifractal spectra have different domains of definition, and it is difficult to bring them to one start point.

3.3. Multifractal spectrum versus generalized statistical sum

Consider set $M$ and its finite partition $U$ consisting from $N$ boxes. Let box size $l$ and box measures $\left\{p_i\right\}$ be given. In [19] the authors offer to calculate the dimension of measure support $M$ of $\left\{p_i\right\}$ as:

For the fact that measure ${\{p}_i\}$ depends on the box size, and $l~1/N$, Eq. (6) is equivalent to:

The last formula is the information dimension of the subset on which measure $\left\{p_i\left(l\right)\right\}$ is defined, in other words we calculate the information dimension of measure $\left\{p_i\left(l\right)\right\}$support.

The authors also described the method for the calculation of multifractal spectrum which is based on the usage of direct multifractal transform of a given initial normed measure and computing the information dimensions of the subsets that are the supports of the initial measure and its multifractal transforms.

This method was described more fully in [20], and here we give only the short description. In what follows we assume that $p_i\approx l^{{\alpha }_i}$, then ${\alpha }_i\approx \ln p_i\left(l\right)/\ln l$. Consider the generalized statistical sum $\phi \left(q\right)=\sum _{i=1}^N{p}_i^q\left(l\right)$, where $q$ is a real number. Given initial distribution $\left\{p_i\right\}$ we construct the sequence of measures $\mu \left(q,l\right)=\left\{{\mu }_i\left(q,l\right)\right\}$ obtained by direct multifractal transform as:

For each measure $\mu \left(q,l\right)$ the information dimension of its support can be calculated by Eq. (7) and set $f\left(q\right)$ (dimensions of $\mu \left(q,l\right)$ supports) is obtained as follows:

We also calculate the average value of exponents ${\alpha }_i$ with respect to measure $\mu \left(q,l\right)$ as:

Hence Eqs. (8) and (9) define the set of dimensions (parametrized multifractal spectrum) $f\left(q\right)$ and the set of the average values of exponents $\alpha \left(q\right)$ as the functions of parameter $q$. Excluding $q$ one may obtain the relation between $\alpha$ and $f$. In other words the last formulas give a parametrized representation of multifractal spectrum.

4.1. Experiments with milk

1. Application of fractal signature method.

Initial images were obtained in RGB colour model. In experiments we used representations in grayscale and HSV. The transition from RGB to grayscale as a rule results in averaging of an image structure and the loss of some peculiarities, whereas the analysis on H-component of HSV allows us to reveal the texture of an image and show the difference between images.

If the milk images studied are represented in grayscale, graphs do not show clear separation between the images (Fig. 3(a)). At the same time, calculating surface area using H-component we obtain a numerical characteristic that differs UHT milk from BD and MHP (Fig. 3(b)). The result coincides with visual perception. Hence the choice of the colour model may influence results considerably.

As was mentioned in section 3.1, on graphs the box size is given in pixels, and the values of the surface area are given in normalized form, such that maximal value be less than 10³.

Fig. 3Calculation of surface area for different milk crystalograms: a) in grayscale, b) using H-component

a)

b)

2. Density function method.

By applying the second (density function) method we calculate multifractal spectra for the given images. The calculations were performed in grayscale.

In all these experiments $\epsilon =$ 0.4. It means that for the obtained diapason of $\alpha \in [{\alpha }_1,{\alpha }_2]$ level sets $E\left(\alpha ,\epsilon \right)=\left\{x\in R^2:d\left(x\right)\in \left[\alpha ,\mathrm{\alpha }+\epsilon \right)\right\}$ are calculated with $\epsilon =$ 0.4 beginning from ${\alpha }_1$. As it was mentioned above, different images have different diapasons of density function values, which reflects the peculiarities of image textures.

Fig. 4Multifractal spectra calculated for different milk crystallograms. The graph for UHT milk, c) differs from two others a), b)

a) BD milk

b) MHP milk

c) UHT milk

The following graphs show multifractal spectra for 3 sorts of milk. On $X$ axis the values of density function are shown, the values of fractal dimensions of the corresponding level sets are on $Y$ axis. As the level sets are represented as binary images, we calculate their capacity dimensions. To make the reading of pictures clear, on Fig. 4(a) and Fig. 4(b) values on $X$ axis are shown with gaps 0.8.

3. Generalized statistical sum.

For each image we obtain two spectra as the functions of real parameter $q$. The calculations were performed using H-component, $q\in$ [7, 0] by step 0.5. The values of $q$ are on $X$ axis, the graphs singularity power $a\left(q\right)$ and multifractal spectra $f\left(q\right)$ are shown for BD (a), MHP (b) and UHT (c) milk on Fig. 5. All pairs of the graphs are different, which points to the fact that the structures of the images given are distinct.

Fig. 5The graphs of singularity and multifractal spectra for 3 images of milk: a) BD milk, b) MHP milk and c) UHT milk

a) BD milk

b) MHP milk

c) UHT milk

4.2. Experiments with beans

1. Fractal signature method.

These experiments were performed in grayscale. Fractal signature method applied to the analysis of the images gives the following graphs presenting the dependence of the surface area on a box size.

Fig. 6The dependence of the surface area on box size: graphs for images B3 and B4 differ from B1, B2 and from each other. Graphs for B1 and B2 are close

2. Density function method.

In all the experiments $\epsilon =$ 0.25. The graphs below illustrate the difference between intervals for density function.

3. Multifractal spectrum versus generalized statistical sum.

For each image we obtain two spectra as the functions of real parameter $q$. The calculations were performed using H component, $q\in$[–7, 0] by step 0.5.

The graphs for images B1-B4 are shown on Fig. 8.

We see that all the pairs of graphs are different, and this method allows us to reveal small differences in the structure of the images.

Fig. 7The graphs of multifractal spectra for images B1, B2, B3 and B4

a) B1

b) B2

c) B3

d) B4

Fig. 8The graphs of singularity and multifractal spectra for images B1-B4, q∈ [–7, 0]

a) B1

b) B2

c) B3

d) B4