1. Introduction

Identification of bacteria is the procedure carried out every day in all microbiological laboratories around the world [1–4]. There are many techniques for identifying the bacterial species with high reliability, but most of them are expensive (e.g. DNA based methods), need additional knowledge (e.g. real time PCR (Polymerase Chain Reaction), where we need to know precisely what we are looking for) or are time consuming (e.g. classical biochemical methods). The optical methods for bacteria identification possess many advantages as non-contact and non-destructive measurement’s character and no need for advanced and time-consuming sample preparation. Moreover, they can be reliable, fast, not expensive and what is extremely important, they do not require any knowledge in advance [5–10]. The optical examination of biological samples has non-contact and non-destructive character, therefore in case of results that need confirmation, the sample can be verified by other methods, thus offering significant advantage in comparison with biochemical and molecular methods. In our previous works, it was demonstrated that the analysis of forward light scattering on bacterial colonies can be used for identification of different bacteria species [5–7, 10, 11]. Based on these results, our group developed novel method for bacteria identification based on analysis of the light diffraction on bacterial colonies in an optical system with converging spherical wave illumination, which was originally proposed in [5]. The studies conducted so far led to the development of the recognition process that allowed identification of the bacteria with high accuracy (over 98%) [6, 7]. The fundamental concept of the proposed method is based on the already verified assumption that in case of bacterial colonies growing on the solid nutrient medium, the variation of the optical properties (as the refractive indices and transmission coefficients) and morphology properties (as the profile, size and shape the colony) depending on the spatial orientation and metabolism of the bacteria cells in the colony, are responsible for generation of Fresnel diffraction patterns that are unique for each bacteria species and strains. The practical implementation of the method in microbiological diagnosis is as so far limited by the understanding the specific light transformation on different bacteria species and strains and also by the providing accurate and efficient methodology of diffraction patterns processing, followed by the quantitative analysis. To overcome these limitations, is the main issues of our current scientific interest, focused on the optimization of the proposed recognition method.

As the number of bacterial classes (strains) under study that make up our data set is smaller than the target data set (e.g. number of bacterial strains routinely identified in the microbiological laboratory), slight deterioration of the results after application of the recognition process to target (larger) data set, is expected. Therefore, our concern is to improve the recognition process (already described in [7]), thus to make it suitable for analysis of significantly larger data sets. The method can be improved in three ways: optimization of the optical system, optimization of the statistical analysis process or both, simultaneously.

Previously performed examination by means of digital holographic microscope (DHM) have shown that the bacterial colonies exhibit light focusing properties analogical to classical optical lenses, responsible for additional phase modulation of the incoming optical field [15]. Moreover, various bacteria produce colonies of different profiles that are responsible for different light focusing properties. This affects the convergence of diffracted optical field and the spatial dimension of recorded diffraction patterns, what can be responsible for losing some features important for the bacteria classification process. Therefore, choosing the optimal recording plane location enables to record the diffraction pattern possessing the most significant features, necessary for a correct identification. To achieve this goal, the analysis of the bacteria identification of the diffraction patterns recorded in different locations, was performed. Moreover, we have extracted and then selected new features based mostly on the central statistical moments, best describing morphological and textural properties of the bacterial colonies Fresnel diffraction patterns. Since the data dimension (data matrix size is number of registered bacterial colonies patterns by number of numerical features extracted from the patterns) did not allow for the classification due to larger number of features than number of observations (patterns) for each class (species/strain), we added the group of features ranking to choose only the most useful features for classification purposes [12]. Additionally, the method for the feature selection and modified partitioning of the data for the cross-validation by application of stratified sampling that allow splitting of the data into homogeneous subsets, was introduced (for details see Fig. 3).

In the laboratory work it is extremely important to have reproducible results. During the previous study the distance between the location of the diffraction patterns observation plane and the colony was selected to adjust the Fresnel pattern to fit into the image frame. However, it is necessary to determine the optimal location of the Fresnel diffraction recording plane, so thus to receive required number of features that can be used for the most accurate bacteria species/strains classification. As our work leads to automation of the diffraction patterns registration procedure, we decided to fix the location of the registration plane of the optical system at the distance, for which identification of the patterns would give best results. The analyzed parameter is further called as distance parameter or fixed distance parameter.

In the present examination three different bacteria species and one with two strains, were examined: Escherichia coli, Staphylococcus aureus, Proteus mirabilis, Salmonella Enteritidis and Salmonella typhimurium. The examination of two strains of Salmonella (Enteritidis and typhimurium), the most significant foodborne pathogens causing among others food contamination, has shown that the proposed method is capable to identify various strains of the same species.

2. Light focusing properties of bacterial colony affecting the location of the Fresnel diffraction patterns and the accuracy of the proposed identification method

Previously performed experimental examination of the Escherichia coli colonies has shown that analyzed biological object exhibits the light focusing properties similar to the conventional aspherical microlenses [11]. However, contrary to the classical lens, the bacterial colony is semi-transparent flat-convex object and the light transmission through the colony will be limited, therefore besides the introduced phase shift, the additional amplitude modulation of incoming wave, is observed. Therefore, the bacterial colony can be treated as amplitude and phase light modulator. Bacterial colonies have convex shapes, whereas the profile can be approximated by spherical, aspherical or Gaussian function depending on the bacteria species or strains (see Fig. 1).

 

Fig. 1 Exemplary colonies of the Escherichia coli (a) and Pseudomonas vulgaris (b) recorded by scanning confocal microscope Olympus LEXT 3D Measuring Laser Microscope (laser 405 nm, objective: 5X, 2X) showing different profiles of bacterial colonies (The picture was recorded by I. Buzalewicz by courtesy of Prof. Małgorzata Kujawińska from the Institute of Micromechanics and Photonics at Warsaw University of Technology).

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The amplitude transmittance of bacterial colony can be described by the following expression:

The proposed optical technique for bacteria identification is based on the analysis of the bacterial colonies Fresnel diffraction patterns recorded in optical system with the converging spherical wave illumination was broadly described in [5]. On the Fig. 2, the bacterial colony is illuminated by spherical wave converging towards the plane (x_F; y_F), which is the focal plane of the transforming lens. Due to the light focusing properties of the colony indicated above, bacterial colony acts as classical lens with positive optical power and introduces the additional phase modulation of the incoming wave, causing converging of the incoming wave towards the plane (x_F1;y_F1). The transforming lens and bacterial colony acts as the system of two lenses, however, because of the roughness of analyzed bacterial colony surface, as described in [15], the bacterial colony can be considered also as an optical diffuser, which is causing the spread of focal point and spatial light intensity variations in effective focal plane of the system composed of two lenses (transforming lens and bacterial colony). The colony profile and refraction index heterogeneity influences the transmission and phase properties responsible for the light diffraction. However, the convergence of the diffracted beam affects the spatial size of the Fresnel diffraction patterns in the observation space, which is limited to the space between the colony plane (x_S;y_S) and the effective focal plane (x_F1;y_F1) of the system composed of the transforming lens and the bacterial colony.

 

Fig. 2 The light transformation on bacterial colony in the proposed optical system: (a) additional phase modulation of bacterial colony, (b) exemplary Fresnel diffraction patterns of Escherichia coli colony depending on the location of the recording plane

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In consequence, for different recording planes the Fresnel diffraction patterns of bacterial colonies will have different spatial intensity distributions (see Fig. 2(b)). Therefore, the choice of the appropriate location of the recording plane is the key issue of the proposed technique, limiting accuracy and efficiency of bacteria identification. It is necessary to choose the optimal location, for which recorded Fresnel patterns will exhibit the most significant features enabling proper bacteria species/strains classification.

3. Materials and methods

3.1 Preparation of bacterial colonies samples

In this work five bacteria species, were examined: Escherichia coli (PCM O119), Staphylococcus aureus (PCM 2267), Proteus mirabilis (PCM 547), Salmonella Enteritidis (ATCC 13076) and Salmonella typhimurium (ATCC 14028). 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 first incubated for 18 hours at the temperature of 37°C. Bacteria suspensions in respective 10⁻⁵ and 10⁻⁶ dilutions were seeded on the surface of the solid nutrient medium in Petri dishes with Columbia agar (Oxoid), so as to obtain 12-20 colonies per plate, and were again incubated at 37°C for the next 18 hours.

3.2. The optical system for bacteria identification and measurement procedure

The set of Fresnel diffraction patterns of bacterial colonies were recorded in the optical system with converging spherical wave illumination in seven different locations of the recording planes. The optical system extensively described in [5] consists of the laser diode module (635 nm, 1 mW, collimated Thorlabs), beam expander (Edmund Optics), transforming lens (achromatic doublet, focal distance: 48.6 cm, clear aperture: 6.35cm, Edmund Optics), camera (EO-1312, Edmund Optics) and XYZ sample positioning stage with Petri dish. The main purpose of the experiment was to optimize the distance from the bacterial colonies to registration plane of the optical system with respect to the identification possibilities limited by the changes of the diffraction patterns size caused by different light focusing properties of different bacteria species/strains colonies. In contrary to the previous studies, where this distance were fixed and equaled approx. 2 cm, here the seven locations of the recording plane were chosen at intervals of 1 cm from the shortest available (0.8 cm) to the distance (6.8 cm), for which the narrowest diffracted beam diameter was observed. In a result, the recorded diffraction patterns had various sizes. Each bacterial colony was recorded in all seven locations in order to determine the best distance parameter.

In further analysis of the influence of the Fresnel diffraction patterns recording planes location, only the data from patterns recorded with distance parameters 3 (2.8 cm) and 4 (3.8 cm) from the bacterial colony were considered. Shorter distances: parameter 1 - 0.8 cm and 2 - 1.8 cm were excluded from the analysis as the patterns in many cases went beyond the area of image and thus it was impossible to obtain complete data. In contrary, recording in longer distances: parameters 5 - 4.8 cm, 6 - 5.8 cm and 6 - 5.8 cm, resulted in deformations of the patterns as the recording plane location was approaching the effective focal plane of the optical system combined the transforming lens and the bacterial colony. The pattern deformations caused by convergence of the diffracted optical field were observed mainly as a change of the patterns size, shape and the uneven distribution of the naturally existing rings (see Fig. 6). Defining the proper distance parameter allowed to obtain additional classification feature - the radius of the diffraction pattern. Further analysis proved that the radius is also unique for the given bacteria and can be exploited as valuable classification feature, since it is correlated with the unique light focusing properties of bacterial colonies associated with the colony profile. This will be discussed more in detail in the results section of the paper.

The target data set will consist of many classes (dozens of important, existing and yet unknown pathogenic bacteria species and their strains). In the current studies about 50 diffraction patterns for each species, were recorded (see Table 1).

Table 1. Number of colonies registered for various bacteria species. Each colony was registered in 7 different locations of recording plane.

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3.3. Image processing algorithm

Previously conducted research has shown that Fresnel diffraction patterns of bacterial colonies exhibit unique spatial structures [10]. Beside different radius, the patterns contain set of the diffraction rings, which number and size depend on the bacteria species. Morphological properties of the bacterial colonies are reflected by the morphological and textural properties of their diffraction patterns. Therefore, we decided to examine morphological and textural properties of the diffraction patterns, that are easily interpretable.

In the present study, image analysis was performed according to the same workflow as previously [6, 7], however here, new features were extracted from diffraction patterns for optimized bacteria identification (see Fig. 3).

 

Fig. 3 Summary of the differences between the original and optimized method workflows. The differences are depicted in bold and on darker background for the optimized method workflow. Bacteria registration process was optimized. Feature extraction procedure has been extended with additional morphological and textural features. Feature selection routine has been enhanced by calculation of additional ranking (due to bigger data set) and introducing SNR (signal to noise ratio also known as Fisher divergence) along with ANOVA (analysis of variance) for ranking construction. Classification has been limited to QDA (Quadratic Discriminant Analysis) and SVM (Support Vector Machine) due to inefficiency of LDA (Linear Discriminant Analysis). Finally, classification performance assessment part was improved by application of stratified sampling procedure that allowed for sampling of homogeneous subsets.

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3.4 Preprocessing

The extraction of the new and interpretable features from the diffraction patterns was preceded by usage of the dedicated macro written in the ImageJ free software (http://rsb.info.nih.gov/ij/) with human interaction for distinguishing the center and edges of the diffraction patterns [13]. Marking of edges and center was followed by partitioning each of patterns into 10 disjoint rings of equal thickness. Partitioning into 10 rings was shown to be the best of fixed splits [7]. Then, the normalization process was performed under the assumption of the black background of each single registered pattern as they were recorded in the darkness. The normalization was performed with use of the standard algorithm for histogram stretching. The mean values of the intensities of pixels belonging to the background (beyond the pattern edges) were calculated separately for each pattern and the value was set as the left edge of the stretched histograms, while the value of the right edge of the histograms has not been changed [7, 14, 15].

3.5 Feature extraction

Within each of the 10 rings of every registered diffraction pattern numerical features based on statistical central moments were calculated depending on the rings pixels intensities. Statistical moments based numerical features are often used as texture features of images as they are easily interpretable [14, 15]. For each ring of each pattern we calculated values of the new features, along with previously obtained mean (first moment) and standard deviation (second moment), denoting brightness and roughness of the regions of interest, respectively. New features considered as statistical central moments are skewness (third moment) and kurtosis (fourth moment). Skewness is referring to the measure of the symmetry of the shape of the pixel intensities distribution within each ring. Greater value of the skewness means longer tail in the bright direction of the pixel intensities histogram. Kurtosis is a measure of the flatness or peakedness of a distribution of the pixel intensities within each ring. We also calculated relative smoothness, which is a measure based on the standard deviation given with formula $\text{1}-\text{1}/\left(\text{1}+s{d}^2\right)$for each ring, and thus standard deviation and relative smoothness are dependent features. Uniformity was also calculated as${\displaystyle \sum p}{\left(x_i\right)}^{\text{2}}$, where p(x) is a probability mass function of the pixel intensities within the rings and i takes values in the range i ∈ [0,255]. Lastly, we calculated entropy, which quantifies expected value of information carried by each of the pattern rings and is given by the formula: $-{\displaystyle \sum p}\left(x_i\right){\text{log}}_{\text{2}}p\left(x_i\right)$. The formula describes Shannon entropy [14, 15]. All of the features are labeled feature-name.x, where x means ring number starting from the pattern center. Additionally, the Fresnel diffraction pattern radius was calculated, as it is an important predictor.

3.6 Feature selection

To decide, which features are the best for building the classification models, we used ANOVA analysis of variance [16] and Fisher divergence measure [17] to estimate the features separation possibilities measure. The measure denotes possibilities of the given feature to separate various classes (bacteria strains).

Fisher divergence is also called signal to noise ratio and thus further will be called SNR. Both tests were used to find features that differentiate bacteria species in the best way, with respect to distance parameter of the optical system. Explicit, SNR formula $\mu /\sigma$, where µ and σ denotes mean value and standard deviation respectively, was used to calculate separation measure value of the given feature by averaging separation values calculated in one-versus-all scheme. It means that for each feature under study we calculated five separation measures using SNR formula for two distance series of the data: $SNR=\frac{\text{1}}{n}{\displaystyle \sum \frac{\left({\mu }_i-{\mu }_j\right)}{{\sigma }_i+{\sigma }_j}}$, where n is the number of bacteria under study and in our case is equal five, i is the given bacteria species, while j represents complement of the bacteria set i in the data set. The procedure was performed for the two chosen distance parameters (3 and 4).

After feature extraction procedure, 71 numerical features for every registered pattern were obtained. However, for building classification models only those features that are good predictors, will be used. Some features have large variance and thus are poor predictors on their own. Impact of the variance of the data on the discriminant properties of the features is depicted in the Fig. 4. It shows that even good predictors are not satisfactory in all the cases, as the entropy.1 feature, which is one of the best predictors in our data set for distance 3 (Fig. 9). However, for the Escherichia coli and Salmonella Enteritidis the feature density overlap with each other due to large variance of both (shown with Staphylococcus aureus as an example). Simultaneously, the same feature can have very good prediction properties, if combined with some other feature or features. On the other hand, the QDA (Quadratic Discriminant Analysis) classification is not applicable to the whole data set, because of the QDA assumption of fulfilled condition m < p + 1, where m is the number of features building the model and p is the number of patterns in each of classes. It means that the number of features taken for the QDA classifier must not exceed 49 as this is the smallest number of patterns (registered for Escherichia coli class). This requirement is another reason for limiting the number of features. The final reason is possibility of interpretation - the smaller number of features, the easier to interpret the model.

 

Fig. 4 Exemplary density of the entropy.1 feature from distance 3 data subset for the species Escherichia coli, Staphylococcus aureus and Salmonella Enteritidis. Vertical lines correspond to mean value of the feature for given species. It can be seen that mean values of the entropy.1 feature differ among the classes, for Escherichia coli it takes values in the same range that for Salmonella Enteritidis, because of the high variance of the feature data, while for Staphylococcus aureus the small overlap with both Escherichia coli and Salmonella Enteritidis, is observed.

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To choose the best predictors from all of the available data, the prediction ranking for the groups of features (e.g. mean and entropy) without distinguishing between the rings, were firstly calculated. Secondly, ranking of features from best groups was prepared based on ANOVA and SNR, so the results could be compared with these from the previous study [10]. Both rankings were prepared with the distinction of the two distance parameter subsets. The ranking for groups of variables was created on the basis of QDA classification errors for models built with all the features included in the group. Group of features to build the models were selected according to the rule of combinations without repetitions for at most 4 groups. This gave 162 models in total (8 built upon 1 group, 28 built upon 2 groups, 56 built upon 3 groups and 70 built upon 4 groups). Each group occurred in 64 models. The ranking was constructed by summarizing error rates of models, in which given group occurred for each group of features and normalizing it by the number of occurrences. For the top 4 feature groups, ranking of the separation capability held by the features was constructed based on ANOVA and SNR.

3.7 Classification

In the study, only QDA and SVM (Support Vector Machine) as LDA (Linear Discriminant Analysis) proved to be not sufficient for the task of classifying bacteria species and strains. From the best predictors of each pattern, classification models of varying complexity, starting from one feature (first in the ranking) and in next steps adding following features accordingly to their order in the ranking, were built. The procedure was repeated for top 4 feature groups, as well as for mean and standard deviation as used during previous research for comparison in order to compare results between the previous one and current optimized analysis.

3.8 Classifier performance assessment

As we are dealing with five classes (bacteria strains), cross-validation (CV) was chosen as classifier performance assessment method. The task of the performance assessment method is to estimate the unknown classification error that will occur after using the classification model on other, independent data sets. CV accomplishes the task by splitting the data set into two disjoint subsets (learning and test sets. The model is built with use of given feature set (predictors) on the learning set, while its performance is tested on the test set. The procedure is repeated given number of times and upon the results the classification error is estimated. The procedure was performed 50 times in our analysis.

The partitioning of the data into 10 disjoint folds for the CV usage was performed according to the stratified sampling algorithm. Stratified sampling is a method of sampling from a data set that ensures that the distribution of the samples belonging to various classes between subsets is homogeneous.

Usually the classification analysis concerns two classes (e.g. case and control), but our data consist of five classes. Therefore, sensitivity and specificity cannot be calculated in the ordinary (two class) way. There are many approaches for calculating sensitivity and specificity measures in multi-class case. For the analysis, the one that is most widely accepted was chosen. It considers one of classes as case class and combines the rest of the classes as the control class. The rule is performed as long as all of the classes are the case class and is often called one-versus-all as it applies binary definition for the number of classes in the data set. After application of the rule for all of the classes the sensitivity and specificity values are averaged and give the final results for the experiment. The accuracy of the classification is given as 1-CV error.

4. Results and discussions

4.1 The recorded Fresnel diffraction patterns of bacterial colonies

The samples of various bacterial species and strains were examined in the optical system with converging spherical wave illumination. Exemplary Fresnel diffraction patterns of bacterial colony are shown on Fig. 5.

 

Fig. 5 Exemplary Fresnel patterns of bacterial colonies of various bacteria species recorded in the optical system at the same distance from the sample. Differences can be observed not only in the pattern itself, but also in the patterns size.

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Their visual inspection indicates that colonies of different bacteria species and strains generate the diffraction patterns with unique spatial intensity distribution, which can be used for their identification by advanced statistical methods.

4.2 Different locations of the observation plane

The theoretical considerations presented in Section 2 have shown that the light focusing properties of bacterial colonies resulting from their geometrical structure and index of refraction, can significantly affects the conditions of Fresnel diffraction patterns recording. Experimental results shown on Fig. 6 indicate that the Fresnel diffraction patterns are significantly affected by the light focusing properties of bacterial colony, which is manifested by the limitation of the diffraction patterns size with the increase of the registration plane distance from the colony; so-called distance parameter.

 

Fig. 6 The exemplary Fresnel patterns of single bacterial colony of Proteus mirabilis species recorded in the optical system in seven different locations

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Moreover, performed examinations have shown that the colonies of different bacteria species and strains exhibit also different light focusing properties, which significantly influence on the dimension and spatial intensity distribution of the Fresnel diffraction patterns (see Fig. 7). It correlates with the colony profile changes of different bacteria species and strains. The convergence of the diffracted beam changes depending on bacteria species/strains, and is limiting the observation space of the Fresnel patterns. The location of the bacterial colony Fresnel patterns registration plane is limited to the space between the bacterial colony plane and the effective focal plane of the system composed of both: transforming lens and colony. Obtained results have shown that the choice of an appropriate recording plane affects the accuracy of the proposed method of bacteria identification, because the Fresnel patterns even for the same bacteria species/strains depends significantly on the location of the registration plane. This is caused by phase modulation of the incoming illumination wave, which by transformation and diffraction on bacterial colony, is changing its convergence. Moreover, obtained results have shown that the shape of the Fresnel patterns in case of some bacteria species/strains near the focal plane are not symmetrical. This effect can be correlated with the deformation of the symmetry of the bacterial colony profile, which will cause that the diffracted/transformed on bacterial colonies waves will propagate in off-axis directions.

 

Fig. 7 The exemplary Fresnel diffraction patterns of bacterial colonies of different species and strains for different locations of the registration plane.

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Moreover, Fresnel diffraction patterns depicted by Fig. 8 confirm the predictions that during the bacteria colonies growth their light focusing properties are changing. It can be explained by the changes of the colony geometrical structure, particularly the colony profile, as it was already reported [11]. However, it is correlated with the bacteria species/strains. In case of Staphylococcus aureus colonies, for longer times of incubation the diameter and maximal thickness of the colony increases, what leads to the increase of the optical power, caused by the decrease of the colony focal distance. In another words, the growth of Staphylococus aureus leads to the decrease of colony radius of curvature. In consequence, the diffracted beam convergence is changing with the time of bacteria colony incubation (see Fig. 8) according to the bacteria species/strains and colony morphology. Moreover, the increase of the colony diameter leads to the decrease of the Fresnel patterns size due to the properties of Fourier transform. Therefore, it should be pointed out that the choice of an appropriate location of the diffraction patterns registration plane should be considered only for the given time of bacterial colony incubation. Moreover, from the optimization point of view, the longer time of the bacterial colony incubation will require the decrease of the distance of Fresnel patterns recording, what will lead to limitation of the observation plane extension.

 

Fig. 8 The exemplary changes of the Fresnel diffraction patterns of Staphylococcus aureus colonies for different locations of the registration plane and various incubation times.

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4.3 Feature selection

The features group ranking, depicted in Table 2, shows that for both distance data subsets (distances 3 and 4) the ranking pointed the same feature groups for the top 4 feature groups (mean, entropy, radius and standard deviation). Figure 9 and Fig. 10 present examples of the features separation possibilities measured by ANOVA. These rankings were prepared in the way that allow for comparison of the results obtained during this study and previous, presented in the [10]. The main goal of the comparison is to show that optimizing the analysis improved the identification by decreasing the error rate, while making the method more robust at the same time (fixed distance parameter).

Table 2. The groups of feature ranking based on the values of the summarized QDA models error rates for each model that contained given group. The error was normalized over number of models built with use of the given feature group. The models were built with use of at least 1 and at most 4 groups of features, all possible combinations of feature groups were included into the ranking. The models were built with use of all features of the given group.

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Fig. 9 ANOVA based features separation possibilities measure ranking for distance 3 (2.8 cm) data subset.

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Fig. 10 ANOVA based features separation possibilities measure ranking for distance 4 (3.8 cm) data subset.

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It can be observed that features separation measures of both subsets are different (Fig. 9 and Fig. 10). These findings, as well as results presented in Table 2 confirm the fact that data from the two subsets (distance parameter 3 and 4) have different identification possibilities. This suggests that the distance has a significant impact on the identification possibilities and thus it will be important to optimize the distance parameter for the future automation of the identification process.

4.4 Identification performance assessment

The comparison of the results of bacteria identification by the proposed method concerns two separate optimization processes performed during this study. Firstly, we compared previously reported analysis and classification features [7] with new optimized analysis and new features.

Secondly, two subsets of the data denoting registration with two different distance parameters of the optical system, are compared. All results are obtained with use of newly collected data with adjusted location of the registration plane and compared with results from previously conducted study. The identification results of the original analysis performed on the new data (two distance parameter data subsets) can be found in the Table 3. The results suggests, as it was expected that the comparison of the original analysis performed on the originally gathered data [7] and the new one with fixed distance parameter of the optical system, are different. We expected that original analysis run on the new data will give slightly worse results, but it is not so in every case. QDA classifier gave, as expected, worse results for both of new data subsets and both feature selection methods (error rate rose form 1.43% to 2.9%). However, SVM classifier error rate for patterns registered with distance 4 (3.8 cm) decreased from 2.83% for 16 features to 2.27% for only 12 features. This shows clearly that data (patterns) registered at the distance 4 have better possibilities of identifying bacteria for both feature selection methods, as not only error rate decreased, but the model is simpler and thus easier for interpretation purposes.

Table 3. Identification results for the previously reported analysis and three data subsets (distance parameter 3 and 4 and originally registered data). Two feature selection methods (ANOVA and SNR) were used for the features predictive properties rankings preparation for new data. The original analysis was intact. Number of best predictors used for the model building, identification error, multi-class sensitivity and specificity are depicted in the table. Most significant results are marked in bold.

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The identification results of the optimized analysis performed on the new data (two distance parameter data subsets) can be found in the Table 4. The optimized analysis gives better results for SVM classification than for QDA, while there is statistically non-significant difference between SNR and ANOVA feature selection methods on the same data. The results are consistent with those of the original analysis from Table 3. Exactly, as it is presented in Table 3, the distance parameter 4 (3.8 cm) data subset proved to have better bacteria identification capabilities with error as small as 1.34%, sensitivity equal 0.9759 and specificity 0.9903. Final results are better than those from the analysis on original data (without fixed distance parameter). Unfortunately, the complexity of the model increased from 13 features as previously to 27 features in the current study.

Table 4. Identification results for the optimized analysis and two data subsets (distance parameter 3 and 4). Two feature selection methods (ANOVA and SNR) were used for the features predictive properties ranking. Number of best predictors used for the model building, identification error, multi-class sensitivity and specificity are depicted in the table. Most significant results are marked in bold.

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5. Conclusions

Described here examination concerns bacteria identification procedure basing on the analysis of the light focusing properties of bacterial colonies and changes of the Fresnel diffraction patterns depending on the location of the registration plane. It was demonstrated that time-dependent light focusing properties of bacterial colonies are significantly affecting the Fresnel patterns by changes of the diffracted wave’s convergence depending on the bacteria species/strains. Therefore, the choice of the diffraction patterns registration plane location is a significant factor limiting the accuracy of the identification method. In order to achieve the best possible identification accuracy, Fresnel patterns exhibiting the most significant bacteria species/strains indicators, were exploited.

Further on, the optimization of the analysis resulted in building new statistical models of higher accuracy and more complexity. Proposed image processing algorithms and statistical methods, confirmed that the distance parameter of the optical system has significant impact on the identification possibilities of the proposed method. Optimization of distance parameter combined with optimization of analysis resulted in very small identification error 1.34% with high sensitivity 0.9759 and specificity 0.9903 measures. Obtained results are highly satisfying and allow further research on the automation of the method. Moreover, the analyzed data contained Fresnel diffraction patterns of two strains of dangerous pathogens: Salmonella bacteria (Enteritidis and Typhimurium), that are easily identified in our system. This proves that proposed optical method is suitable for bacteria identifying, not only on the level of bacterial species, but also on the level of bacteria strains.