## Abstract

A novel method for evaluation of bacterial colonies number (Colony Forming Units - CFU), is described. Proposed algorithm, based on the Mellin transform, allows the CFU evaluation, invariant for the spatial orientation and scale changes. The proposed method involves image recording of bacteria grown in Petri dishes, calculation of the Fourier spectrum followed by coordinates transformation, and determination of the Mellin transform. It was proved that there is a high correlation between CFU and maxima of Mellin spectra. The method was practically implemented for evaluation of antibacterial activity of silver-based nanomaterials and the effect of an additional laser light irradiation.

©2010 Optical Society of America

## 1. Introduction

In a routine microbiological examination determination of the number of bacteria colonies (so called Colony Forming Units - CFU) is a standard procedure, which is mostly performed manually. Some attempts were made to improve quality and efficiency of this process, whereas various semi-automated and automated algorithms have been proposed [1,2]. Computer aided image processing techniques are often exploited [3–7], as well. Moreover, some dyes are applied to increase bacteria colonies visibility and in a consequence to facilitate counting procedures [8].

Recently published results demonstrated that the scattering patterns of bacteria colonies exhibit some specific features, which may be exploited for identification of various bacteria species [9–14]. Forward light scattering, generally associated with diffraction effects, may be analyzed, whereas the colonies topography, radial structure and composition are correlated with the amplitude and phase modulation of optical field [9]. It was shown that it is possible to detect and classify closely related pathogens. However, due to the lack of nominal bacteria models, it is difficult to classify the species basing only on the analysis of scattered light pattern. Therefore, the expected variations of cells (topography, shape, refractive index etc.) should be taken into consideration. To overcome this problem, computational predictions and multivariate statistical methods were proposed for classification and recognition based on angular light scatter [10,13]. It was also shown that the growth of bacteria colonies on agar results in changes of forward-scattering pattern [14].

Our approach presented here exploits correlation between Fourier spectrum, which can be considered as a far-field diffraction pattern, and the amount of analyzed objects – bacteria colonies. First, the specific features of diffraction patterns caused by increasing amount of bacteria colonies, were characterized. The theoretical analysis based on a scalar diffraction theory was performed. It was demonstrated that there is a high correlation between spectrum modulation background values and number of objects. Additionally, further study in order to ensure the scale and rotation invariance were performed by means of Mellin transform. In this way, it was possible to eliminate the influence of bacteria colonies random size fluctuations on the Fourier spectrum.

Although Mellin transform was already proposed for microbiological studies to identify the Vibrio cholerae O1 bacteria by image correlation, however a different computational algorithm was used [5,6]. The novelty of our approach expands the previous concept and rely on exploiting of Fourier and Mellin spectra analysis for evaluation of bacteria colonies concentration. Thus, it is possible to assess the efficiency of antibacterial activity of examined agents.

## 2. Influence of objects number on Fourier spectrum

Fourier spectra analysis has found many applications in various fields as e.g. in signal analysis, optics, statistical analysis or image processing [15–17]. One of the basic property of optical Fourier spectrum, it is a shift invariance. Regardless, where the object might be located in the input plane, its spectrum is always located symmetrically on the optical axis. In the case of multiple randomly located objects with the same size and shape, their Fourier spectrum contains contributions from each object, involving the phase factors from all possible objects locations. Therefore, the mutual configuration of objects in input plane, is reflected in random modulation of initial spectrum, which would be observed for a single object, as a function of transverse spatial coordinates. The nature of this modulation is associated with the objects number and their locations. This can be explained by means of scalar diffraction theory. Presented in this section theoretical consideration will show the correlation between modulation of Fourier spectrum and the object number, since these properties will be used in our approach to determine the number of bacteria colonies in further sections.

First, let us consider the light diffraction on multiple apertures of the same shape and predetermined periodic spatial configuration. If the aperture center is placed at the point (*x _{i}, y_{i}*) (

*i = 1, 2, …, n*, where

*n*is a number of apertures) and the object is illuminated by a monochromatic, coherent plane wave of an unit amplitude, propagating perpendicularly to the (

*x*,

*y*) plane, then the total optical field

*U*(

_{n}*x,y*) in object plane can be described by the convolution of the amplitude transmittance of single aperture with the localization function $\Gamma (\xi ,\eta )$:

According to the convolution theorem [16], the Fourier transform of the total optical field expressed by Eq. (1) can be described as:

We can see that localization function contains all information about the number of objects and their spatial configuration. Therefore, by means of Eq. (3), the Fourier transform of *n* objects $\Im \left\{{U}_{n}(x,y)\right\}$ contains this information, as well. According to the shift properties of Dirac delta function, the Fourier transform of the localization function Eq. (2) can be expressed by

Additionally, if we define $F{}_{0}({f}_{x},{f}_{y})$ as the Fourier transform of a single object, then according to the Eq. (3) the Fourier transform of objects array is described as:

Finally according to the Eq. (4) and Eq. (5), the Fourier spectrum is depicted by the following formula:

As one can see from Eq. (6), the Fourier spectrum $S({f}_{x},{f}_{y})$ of one single object ${\left|{F}_{0}({f}_{x},{f}_{y})\right|}^{2}$ is modified by some additional term proportional to sum of the objects number *n* and modulation factor *m(f _{x}, f_{y})*, which represents phase relationship associated with mutual spatial configuration of analyzed objects. The modulation factor can be expressed as a sum of $n(n-1)/2$ cosines. For asymmetrically located object it takes a form:

Further, the normalized Fourier spectrum ${S}_{output}({f}_{x},{f}_{y})=S({f}_{x},{f}_{y})/{\left|{F}_{0}({f}_{x},{f}_{y})\right|}^{2}$ will be analyzed:

where introduced factor*N*(so - called modulation background) is correlated with the number of objects

*n. S*can be considered as a modulation of initial Fourier spectrum of single object, where

_{output}(f_{x}, f_{y})*N*is a modulation associated with objects number and

*m(f*is correlated to the spatial configuration of objects. Figure 1 depicts schematically normalized Fourier spectrum along

_{x}, f_{y})*f*axis.

_{x}Factor *N* can be considered as a constant component of *S _{output}(f_{x}, f_{y}),* while the factor

*m(f*is an additional cosinusoidal modulation. It should be pointed out that determination of

_{x}, f_{y})*S*is possible only for set of objects with the same size and shape. In the case of an unknown number of objects in the input plane, the modulation background may be exploited for determination of the objects number. Furthermore, this approach may be used to perform comparative analysis of samples with various objects numbers, like e.g. bacteria colonies. It based on correlation between Fourier spectrum properties and number of analyzed objects.

_{output}(f_{x}, f_{y})## 3. Verification of the theoretical considerations by numerical simulations

In order to verify the theoretical consideration presented above, appropriate simulations in Matlab environment were performed. First, the case of square apertures 16 x 16 pixels was analyzed, whereas the apertures number varied from 1 to 30. The dimension of the spatial window was 512 x 512 pixels. Illumination conditions were the same for every apertures configuration. It can be seen that the higher intensity is observed for the higher object’s number, what is caused by the increase of the modulation background, which is correlated to the number of analyzed objects. The initial Fourier spectrum of single aperture is additionally modulated by the factor *m(f _{x}, f_{y})* associated with the mutual spatial configuration of these apertures.

As it was already mentioned, it is possible to identify the number of identical objects by analyzing *S _{output}(f_{x}, f_{y})* (Eq. (8)). As an example, the function

*S*for

_{output}(f_{x}, f_{y})*n*= 5 objects is presented on Fig. 2 . In order to estimate the modulation background value

*N,*we will define it as a mean value of maxima and minima of

*S*:

_{output}(f_{x}, f_{y})*N*can be considered as a constant component of the modulation.

This approach was next applied to determine the modulation background in the case of objects composed of square apertures, whereas the apertures number varied from 1 to 30. According to the Eq. (8), first the modulation term $N+m({f}_{x},{f}_{y})$ of *S _{output}(f_{x}, f_{y})* was calculated. Then, the modulation background

*N*was estimated by use of Eq. (9). The high correlation between the modulation background and the number of apertures was stated (see Fig. 3 ).

The function on Fig. 3 is not linear. Therefore, we will introduce an additional term $error(n)$ describing the distinction between objects number *n* and calculated *N* value by using proposed approach. This value is increasing, when the number of objects increases. The relationship between modulation background value *N* and number of objects *n* is expressed as follows:

Occurred difference *error(n)* between *N* and *n* described by Eq. (10) is caused by the limited bandwidth of Discrete Fourier Transform (DFT). The Fourier transform of the analyzed objects limited by the window can be considered as a convolution of the Fourier transforms of the objects and the window. It is known that convolution result is depending on the shape of both convoluted functions. When the window size is infinite, then the object space is unlimited and the Fourier transform of the window takes a form of a Dirac delta function. Therefore the final Fourier transform of the analyzed objects and the window is equal to the Fourier transform of the object itself according to the “sifting” properties of Dirac delta function. The finite size of the window causes smoothing and spreading of the original Fourier spectrum of the objects. When the size of the window decreases this effects are more significant. In our analysis the size of the window limiting the object space is constant. Therefore the increasing number of analyzed objects in input space decreases the area of the windows causing the spreading and smoothing of the analyzed objects Fourier spectrum. This effect affects the proposed approach by occurring the difference *error(n)* between the modulation background and the number of analyzed objects.

In order to determine, how sensitive is this approach to the spatial orientation of apertures in the object plane, the further analysis was performed on 15 random configurations of seven apertures. It was stated that the standard deviation of modulation background *N* for various configurations of apertures is equal to 0.036 [a.u.] Therefore, the proposed method might be used to evaluate objects number with the significant accuracy, providing that the objects have the same shape and size.

According to the properties of Fourier transform, change of object’s size in the spatial domain causes an inversely proportional change of Fourier spectrum in the spatial frequency domain [15,16,18]. Therefore, in the case of multiple objects of various sizes, the Fourier analysis cannot be used, since it does not assure the scale invariant approach. Some additional methods must be introduced, what will be proposed below.

## 4. Application of Mellin transform for scale invariant analysis

In spite of many advantages of Fourier transform as e.g. shift invariant, it’s application for defining the number of objects in input plane is limited to the objects of the same shape and size. Bacteria colonies have often circular shapes, however they may have various sizes. To omit the problem of size fluctuation, the Mellin transform is used in optical image processing [18]. Although not so widely used as the Fourier transform, it offers certain advantages in pattern recognition [19–23], atmospheric optics [24,25], determination of out-of place displacement in digital holography [26], in hydrology [27,28], for speech analysis and in the acoustics [29–33] and some biomedical applications [5,6,34–36].

The Mellin transform of an one-dimensional function $g(\xi )$ according to [18], is defined by the following expression:

where, in general,*s*is a complex variable. If the complex variable

*s*associated with imaginary axis equals $s=i2\pi f$, the Mellin transform can be reduced to the Fourier transform of an exponentially stretched function by the substitution of variables:$\xi ={e}^{-x}$. The main property of the Mellin transform, which is particularly important for our analysis, it is the scale and rotation invariance. It offers the possibility to determine objects number in spite of objects sizes fluctuation and rotation.

Practical realization of the Mellin transform can be performed by introducing modified coordinates system, where space variables are logarithmically stretched, according to the relation $x=-\mathrm{ln}\xi $. To perform the Mellin transform, the analyzed image is first geometrically transformed in order to convert the coordinates from a linear scale to a logarithmic scale. Several interpolation algorithms of discrete data were proposed, however the most widely used in optics, is the convolution of the input data and sinc function [37,38]. The optical Mellin transform can be also realized through Haar wavelet transformation [39]. Some hybrid algorithms were also proposed [20].

The Mellin transform provides scale and rotation invariant analysis, but if objects are randomly located in the input space, the localization of Mellin spectrum depends on transverse coordinates. In other words, the main advantages of Fourier spectrum – shift invariance and a location of the spectrum symmetrically to the optical axis, are completely lost. To omit this problem, the algorithm combining scale and rotation invariance of the Mellin transform with the shift invariance of the Fourier transform, as described by Casasent and Psaltis [22,23], will be exploited in further analysis (see Fig. 4 ).

The proposed algorithm involves the following steps: image recording, calculation of the Fourier spectrum, coordinates transformation, and determination of the Mellin transform. Polar transformation of the Fourier spectrum from spatial frequency coordinates ${\left|F({f}_{x},{f}_{y})\right|}^{2}$ to polar coordinates ${\left|F\left(\rho ,\theta \right)\right|}^{2}$ is performed by using appropriate substitution of variables $\theta ={\mathrm{tan}}^{-1}\left({f}_{y}/{f}_{x}\right)$ and $\rho =\sqrt{{f}_{x}^{2}+{f}_{y}^{2}}$. This step ensures the separation of rotation and scale changes. The main advantage of polar transformation of Fourier spectrum is limiting two-dimensional scaling of ${\left|F({f}_{x},{f}_{y})\right|}^{2}$ to one-dimensional scaling of ${\left|F\left(\rho ,\theta \right)\right|}^{2}$ along radial coordinate.

Let assume that the input image in the space domain is scaled by factor m, then according to the similarity theorem of the Fourier transform its Fourier spectrum is inversely scaled as well in the logarithmic scale, the scaling process can be reduced to the simple translation:

According to the Eq. (12), if the Fourier spectrum ${\left|F(\rho ,\theta )\right|}^{2}$ is scaled due to the object size change, after log-polar transformation it will be expressed by |F(ln*ρ* - ln*m*, *θ*)|^{2}, so the scale change is now represented by the shift along $\mathrm{ln}\rho $ axis (see Fig. 5
).

As one may see, the one-dimensional Fourier transform of log-polar transformed spectrum, which is equivalent of the Mellin transform along logarithmic coordinate, is scale invariant. This can be obtained, since the scale change is represented by the shift along logarithmic coordinate, on which the Fourier transform is invariant. Realization of one–dimensional Mellin transform by determination of the Fourier transform requires a logarithmic scaling of *ρ* coordinate, what causes log-polar transformation of the input Fourier spectrum. So, one-dimensional (1D) Mellin transform described by Eq. (11) (see Fig. 6
) can be expressed by:

To ensure the rotation invariant process, the additional one-dimensional Fourier transform along angular coordinate, should be performed in order to obtain complete two-dimensional Mellin transform. Therefore, determination of the two-dimensional Fourier transform of log-polar transformed Fourier spectrum leads to the scale and rotation invariant transformation.

It should be pointed out that presented in this section analysis does not consider the random rotation of the individual objects, since bacteria colonies have almost circular shapes. The above theoretical considerations were verified by computer simulation. The calculations were performed for various number of objects (squares with the size 16 x16 pixels) varying from 10 to 240, randomly located in the input plane (window: 512 x 512 pixels). The random sizes fluctuations were in range of 0.25-4.0 of the original size of the object. First, the Fourier transform of the input objects was calculated and the high pass frequency filter was used to eliminate the zero-order component of Fourier spectrum. Further, the Mellin transform followed by the log-polar transformation, was performed. Following, the two-dimensional Fourier transform was computed in order to obtain scale and rotation invariance. As it was demonstrated, Fourier and log-polar transformations, enable the position, scale and rotation invariant analysis. It was stated that the zero-order maximum of the Mellin spectrum (MMS) is correlated with the number of objects (see Fig. 7
). The correlation between MMS value and the number of objects n is obvious, when the Mellin transform of the normalized Fourier spectrum *S _{output}(f_{x}, f_{y})* (Eq. (8)) will be analyzed according to the presented above algorithm. After required log-polar transformation of the

*S*to

_{output}(f_{x}, f_{y})*S*${e}^{-\tilde{\rho}}$,θ), the Fourier transform Eq. (13), which in the following manner is equivalent to the Mellin transform can be described as

_{output}(The modulation background is considered here as a constant component, therefore its Fourier transform is equal to Nδ(0,0). From Eq. (14) it can be seen that in Mellin transform the information about number of analyzed object contained in the value of modulation background is expressed by the term Nδ(0,0), which is correlated with the zero-order maximum of the Mellin spectrum (MMS). Therefore the value of the MMS can be used to evaluated the number of analyzed object.

In the case of bacteria colonies, the Mellin spectrum will be influenced also by transmission properties of growth medium. Therefore, the comparative analysis is possible only for bacteria seeded in the media with the same physical/optical properties,

## 5. Practical implementation of Mellin transform for evaluation of the antibacterial activity of some agents

In order to validate the presented above idea, some experiments on bacteria colonies *in vitro*, were performed. The problem of growing bacteria resistance to various antibacterial agents and sterilization methods, is known worldwide. Many laboratories are working towards elaboration of new methods and materials for combating pathogens. Recently, antibacterial features of nanomaterials are examined and it was proved that e.g. silver based nanomaterials exhibit certain antimicrobial activity [40,41]. In our group, we have also demonstrated that silica nanospheres with immobilized silver and silver-gold nanoparticles have antibacterial properties [42].

#### 5.1 Material and methods

The antibacterial activity of Ag-doped silica nanoparticles against Escherichia coli species was examined. The cultures were obtained from the microbiological laboratory of the Department of Epizootiology and Veterinary Administration with Clinic of Infectious Diseases of the Wroclaw University of Environmental and Life Science.

Bacteria suspensions were incubated for 24 hours at the temperature of 37°C. After incubation, the colloidal solutions of Ag-doped silica nanoparticles were added to the suspensions and then, bacteria were seeded on the MacConkey agar and incubated for next 24 hours. Exemplary images (756 x 756 pixels) are demonstrated on Fig. 8 .

Three various concentrations of antibacterial Ag-doped silica nanoparticles were tested: 0.25:1 (0.25 ml of colloid on 1 ml bacteria suspension), 0.5:1, 1:1 and three samples of each analyzed concentration were prepared. As a control, the bacteria samples without any antibacterial agent, were used. Additionally, three samples of each concentration of antibacterial agent (0.25:1, 0.5:1, 1:1) were irradiated for 4 minutes by laser light (wavelength 410 nm, output power 50mW). Therefore eighteen samples were evaluated. The dependence between bacteria CFU (Colonies Forming Units) and parameters characterizing the Mellin spectra, was examined.

#### 5.2 Correlation between Mellin spectrum properties and CFU

First, bacteria colonies were recorded by means of CCD camera and the Fourier transform of the central circular image, reflecting the shape of Petri dish, was performed. The obtained Fourier spectrum was log-polar transformed (see Fig. 9 ) and the second 2D Fourier transform, which is an equivalent to the 2D Mellin transform, was calculated.

Following, the maxima of calculated Mellin spectra (MMS) were determined and compared with manually counted CFU for seven different samples of bacteria colonies to determine the calibration curve. Obtained results are presented on Fig. 10 . We see that for the higher CFU, the Mellin spectra maxima are higher, as well.

Presented dependence was interpolated by 2^{th} order polynomial expressed by the following formula:

*a*= 1.321 x 10

_{2}^{−14};

*a*= −1.0166 x 10

_{1}^{−5};

*a*= 2397.5. The R

_{0}^{2}value, describing how good is the fitting of provided experimental data by 2

^{th}order polynomial, is equal 0.9963.

## 6. Discussion

Practical implementation of Mellin transform for evaluation of bacteria colonies number showed that Mellin spectra characteristics are related to the objects number. From our knowledge it is the first automated image processing method used for evaluation of bacteria colonies number based on scale and rotation invariant analysis of optical transforms. The developed methodology was used for evaluation of efficiency of antibacterial activity against *Escherichia coli* of materials containing silver nanoparticles. By manual counting of colonies, we have stated that additional irradiation of samples with laser light enhanced the antibacterial effect. Analogical results were achieved from analysis of Mellin spectra after system calibration, whereas calibration means defining the dependence of colonies number CFU and maxima of Mellin spectra MMS. Although, the relative small number of samples were used for calibration (seven only) and the dependence was interpolated by 2^{th} order polynomial, the good agreement between calculations and manual counting for twelve samples was achieved (the differences range varied from 1 to 3% and standard deviation was equal 4.51). Based on analyzed references [1–8] it is not possible to present direct comparison of our results with other bacteria colonies counting methods, since different number of bacteria colonies, sizes and shapes of plates with the samples, were used in these examinations. Moreover, in the reported methods some additional preliminary samples preparation techniques were applied as for example addition the dye to the colonies [8], what in significant way can affect the accuracy of evaluation of bacteria colonies number, since it makes them clearly visible for automated counters.

It should be pointed out that during analysis of significant number of bacteria colonies, for example 596, the manual counting takes at least 4-5 minutes, when proposed method takes below 2 minutes, including the image acquisition (1 minute), image processing (20 seconds) and evaluation of bacteria colonies (15 seconds). Such data were obtained for 756 x 756 pixels images of bacteria colony samples on Petri dish and PC (1,8 GHz, Intel Core ^{TM} 2 duo, 1GB RAM). This time can be extended for the images recorded with higher resolution. The performance speed and accuracy of proposed algorithm can be increased by achieving higher contrast between bacteria colonies and agar background, for example by using dyes to color colony or by using appropriate image processing algorithm to obtain binary mask of examined samples on Petri dish. It should be mentioned as well, that any defect of medium, possible structural and optical non-homogeneities may affect the described above analysis. Proposed approach was considered the case of bacteria colonies with the same shape, therefore the analysis of samples containing bacteria colonies with different shapes by proposed method can lead to significant errors. It should be pointed out as well, that all bacteria sample images should be recorded in the same illumination conditions.

## 7. Conclusion

The Fourier spectra may be used for defining the number of objects in the object plane, providing that objects have the same shape and size. Log-polar transformation of initial Fourier power spectrum and performance of an additional Fourier transform as an equivalent to the Mellin transform, assures the spatial orientation and scale invariance. In this way, it was possible to perform quantitative analysis of bacteria colonies number CFU. Presented algorithm, can be used for comparative evaluation of changes in bacteria colonies number due to antimicrobial materials, sterilization, photosterilization etc.

## Acknowledgement

This work was partially financed by the research program “GRANT - Support of the Research Projects by Scholarships for Ph.D. Students” granted by EC and Regional Government (No GRANT/II/23/2009, GRANT/II/23/2009P). The partial support of the Polish Ministry of Science and Higher Education (No N N518 327335) is gratefully acknowledged.

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