Automatic characterization of titanium dioxide nanotubes by image processing of scanning electron microscopic images

Introduction

The development of imaging tools gives rise to new opportunities in nanomaterial characterization. Several publications have recently appeared, dealing with image processing as a tool for material assessment. ^{1 –3} Actual methods of electron microscopy and other imaging devices in the computer tomography family enable image data to be taken in high resolution, in order to offer many new opportunities of material assessment. Persistent popularity of the scanning electron microscopic (SEM) images in the field of nanomaterial characterization is proved by the recent papers, ^{4 –8} Authors have implemented SEM images evaluation into routine process of nanomaterial characterization. The aforementioned papers are focused on the analysis of new materials. Both subjective and objective approaches have been applied to material description, but the subjective way prevails over the objective approach. Considering these facts, experimental image data may possibly be processed subjectively, however, it is necessary to use exact mathematical methods of image processing for objective evaluation.

This article is focused on the possibilities of utilizing adaptive methods to allow complete characterization of the titanium alloy digital image with respect to its diversity. This challenging task demands the application of various basic and advanced image-processing methods, including signal filtering, image segmentation, mathematical morphology and image classification. These methods have been described in detail in many publications. For this reason, the number of various scientific branches using this approach is rapidly increasing. ^9,10 The authors present a method for analysis of the nanostructure of graphene layers. They take advantage of low-pass filtering, the skeletization method from the mathematical morphology family and thresholding.

Images studied in the presented article were taken with a SEM. They represent a cluster of titanium dioxide (TiO₂) nanotubes developed on the surface of Ti-6Al-4V alloy by electrochemical oxidation in an ammonium sulphate and ammonium fluoride-based electrolyte. The electrochemical anodization consisted of a potential ramp from open circuit potential to the final potential with a 100 mV/s sweep rate, which was followed by potentiostatic exposition at the final potential for 2000 s. The nanotubular layer is firmly adhered to the surface. The adhesive strength of 42 MPa for the nanotubes on the Ti-6Al-4V alloy was published in the study by Fojt. ¹¹ Nanotubes should have an approximately circular cross section. They are separated from each other by the intertubular space. Nanotubes should be hollow along their entire length.

A specific tool for objective evaluation is necessary to be integrated into the process of a new material development. This method could assess and test different conditions’ settings to assure repeatability of the process. It is also necessary to suggest evaluation criteria. From the specific state of material formation, image classification into three basic classes is necessary. These are the intertubular space, the innertubular space and the nanotube wall.

Figures 1 and 2 present the selected image and its three-dimensional (3-D) cut, respectively. Non-stationary behaviour of studied material could be observed in the structure presented in the former. Generally, disturbances occurring in image data may be described as follows.

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Figure 1.

SEM image of nanostructured surface of Ti-6Al-4V alloy, resolution 1 px = 20 nm, image size 890 × 890 px. SEM: scanning electron microscopic; px: pixel.

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Figure 2.

3-D visualization of SEM grey-scale image of Ti-6Al-4V alloy nanostructured surface. The intensity corresponds to the grey level and x and y are special coordinates. 3-D: three-dimensional; SEM: scanning electron microscopic.

These features present a challenge to image evaluation, but, in addition, they can provide some suggestions for development of the evaluation procedure from a mathematical point of view.

Nanotubes with badly developed walls should not be extracted from the evaluation. Therefore, the evaluation process should deal with this problem by mathematical replenishment of the missing data. The process should also be resistant to shape abnormalities. All the various shapes should pass the evaluation to be included in the final result.

In totality, the new procedure was proposed with respect to all the above-mentioned facts connected with specific material features. The suggested procedure is intended to be used as a tool for testing the newly developed Ti-6Al-4V nanostructured material. In addition, it may serve as an inspiration for the quality assessment of other nanomaterials. Obviously, the procedure should be adapted to the actual conditions in all cases.

This section presents the current state of progress in imaging and image processing in connection with material engineering. The next chapter describes a suggested algorithm for image classification as well as the principles of the mathematical methods used. Their properties are illustrated on real images of the nanostructured material. The final chapters conclude the presented findings and discuss specific features of image data and the results obtained from the suggested procedure.

Methods for nanostructured material characterization

As was mentioned above, this article describes one of the up-to-date approaches serving for the automatic classification and evaluation of the titanium alloy nanostructured surface. The design of a robust algorithm is the main goal. It should be capable of classifying the image data of a wide range of quality. In addition, the algorithm should deal with a large amount of data. These limitations present an opportunity to focus on the possibilities of the utilization of image processing.

The processing of all images requires a specific strategy. The approach presented below may be used during the completion of this task. In general, the typical strategy of image processing consists of preprocessing, core processing and post-processing.

Image preprocessing may consist of noise reduction and intensity correction. The resulting image is input for the following step.

Figure 3 presents the selected result of the algorithm described above.

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Figure 3.

Classified nanostructured surface of Ti-6Al-4V alloy. Black: nanotube walls; Blue: innertubular space; Red: intertubular space; Green: unclassified pixels.

This article is not dedicated to a detailed description of mathematical methods but is focused on their practical application. Mathematical principles are presented in the specific range adapted to the needs of the image processing of TiO₂-nanostructured material.

Preprocessing

Very often, noise reduction is the basic technique which is applied as a first step in image processing. It usually makes the image data understandable or enables the application of other methods in subsequent steps.

There are many principles used for noise reduction. ¹² The methods used stem from a description of an image as a two-dimensional (2-D) discrete function I = I(x,y), where x and y are spatial coordinates x = [x ₁,x ₂,…x _M] and y = [y ₁,y ₂,…y _N], respectively, and M and N specifies image size in pixels. The value of I represents pixel intensity or grey level. Intensity usually has 256 (2 ⁸ ) or 65,636 (2 ¹⁶ ) levels. These levels may be normalized to the interval I ∈ (0, 1).

Image filtering is a common strategy for noise reduction. Image filtering methods can be differentiated between linear and non-linear, from one point of view. The averaging filters family may be mentioned as a representative of linear methods.

Spatial image filtering is practically applied by the use of 2-D discrete convolution according to equation (1):

g ( x , y ) = ∑ ( m , n ) ∈ O I ( x − m , y − n ) h ( m , n )

1

The suggested approach used the simplest finite response filter, the averaging filter, for a noise reduction. Averaging filter means a linear special filter smoothing an image by use of the average pixel intensity. It replaces the value of every pixel in an image by the average of the intensity level in a neighbourhood specified by the filter mask h. Equation (2) presents an example of the filter mask h with filter size equal to m = n = 3:

h = 1 9 [ 1 1 1 1 1 1 1 1 1 ]

2

As a result, the averaging filter produces an image with reduced sharp transitions. Nevertheless, averaging filters have an undesirable effect on image edges. In the context of noise reduction, it is necessary to eliminate all interfering effects. On the one hand, a larger filter could suitably be used for noise elimination. On the other hand, this approach could overly smooth the edges in the image, making subsequent processing a challenging task, especially for WT. Figure 4 describes the influence of image filtering on WT. WT ranks among the methods which are very noise-sensitive, therefore image preprocessing is unavoidable.

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Figure 4.

WT: Influence of image preprocessing using averaging filter on WT. (a) Original image cut. (b) Under-segmentation: WT applied to image treated with low-pass filtering, filter size (m, n) = (30, 30). (c) Over-segmentation: WT applied to image treated with low-pass filtering, filter size (m, n) = (3, 3). (d) Proper segmentation: WT applied to image treated with low-pass filtering, filter size (m, n) = (10, 10). WT: watershed transform.

Figure 4(a) presents the selected original image cut. Figure 4(b) and (c) shows the results of image segmentation by WT, preceded by image filtering. The filters applied were both too large and too small in size. The first option smooths the image until it makes the image impossible to be processed by WT. The segments founded in too much smoothed image mismatch the real pattern. This phenomenon is called under-segmentation. In contrast, application of a filter which is too small may cause over-segmentation due to the remaining noise in the image. Real segments are visible, but there are a lot of invalid segments, preventing a fine segmentation.

Image segmentation

The image was denoised in the previous step and continues as an input to the second step of the proposed method. The image segmentation step constitutes the main part of the material assessment. This part consists of:

A WT is used for image segmentation.

Adaptive thresholding. Inside each founded region, the individual threshold has been evaluated and grey-level image has been transformed into a black and white (BW) image.

Post-processing of the BW image is obtained. This covers mathematical morphology and object classification by means of cluster analysis.

The objective of this task is the separation of the innertubular and inter spaces from other parts of the image.

Watershed transform

WT belongs to the segmentation methods family. It arises from the mathematical morphology of grey-scale images, providing a simple yet powerful instrument for image segmentation. The principles of mathematical morphology will be discussed in section ‘Post-processing’.

In WT, the detection of segment boundaries is based on an assumption that the boundary is located in a place where the gradient of the image function is maximal. What is called the Beucher gradient was presented in the study by Beucher ¹³ as a simple approximation of the image I gradient evaluation, see equation (3). B signifies the structuring element (SE).

grad ( I ) = ( I ⊕ B ) − ( I ⊖ B )

3

A disadvantage of this approach is the possible inclination to over-segmentation, as was discussed in section ‘Preprocessing’ in connection with image denoising.

A result of the WT application is presented in Figure 4(d). The founded segments correspond to real data in most cases, so they can be used as an input to the subsequent adaptive thresholding part of the classification algorithm.

Adaptive thresholding

Because of the variability and simplicity of implementation, image thresholding ranks among the popular methods of grey-level image segmentation. The basic principle is presented by equation (4). The output of the thresholding operation is the binary image BW(x,y) where objects are characterized and denoted by 1:

W ( x , y ) = { 1 if I ( x , y ) > T 0 if I ( x , y ) ≤ T

4

Various factors, such as nonstationary noise, blurred edges between an object and a background and inadequate contrast, complicate the thresholding operation. Alternatively, a simple thresholding expressed by equation (4) may not be possible. Consequently, many modifications to the original simple method of threshold evaluation were brought out in the literaures. ^12,14,15

Considering the image data characteristics, adaptive thresholding was chosen for the processing of the studied images. It was assumed that it is possible to separate nanotubes in every obtained region and a different threshold could be established for every region. So the threshold value depends on the intensity level just inside the given region. This approach solves the problem of non-uniform grey-level intensity in different parts of the original image. The Otsu method ¹⁶ was considered as a suitable method in this study.

To sum up, the threshold determination inside each region is the goal of this step. Figure 5 illustrates the threshold levels set in the selected image cut. This example proves the original image had a different intensity in particular parts of the image.

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Figure 5.

Adaptive thresholding: Threshold levels in each region. Original image cut used corresponds to Figure 4.

Selected results of adaptive thresholding are presented in Figure 6(b).

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Figure 6.

Global and adaptive thresholding: Difference between application of simple global and adaptive thresholding on selected image cut. The Otsu method was used in both cases. Original image cut used corresponds to Figure 4. (a) Global thresholding. (b) Adaptive thresholding.

Post-processing

In this part, the segmented image will be refined. In addition, it may contain residual disturbances which resisted noise reduction and undesirable region boundary lines resulting from WT. Mathematical morphology methods are suggested for this task. This approach has been used in processing of nanomaterials in the studies by Pré ¹⁷ and Munoz-Mármol et al. ¹⁸

Morphological transform Ψ(X) is given by a relation between the image I and the set of points B which is called an SE. The SE usually includes representative point P. Selected examples of simple SEs are presented in Figure 7.

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Figure 7.

Examples of simple SEs. SEs: structuring elements.

Practical application of the morphological transform can be visualized as a shifting of the SE across the image. The result of this relation is recorded to the output image on the representative pixel location. Erosion and dilatation belong among the basic morphological operations.

The dilatation is defined as a vector product of two sets, the image I and the SE A, in equation (5). It is denoted by I ⊕ A:

I ⊕ A = { p ∈ Z 2 : p = i + a , i ∈ I , a ∈ A }

5

The preceding definition can be presented in a more intuitive form when the SE A is viewed as an analogy of the convolution mask. In practical significance, dilatation is used for the growing of the thickness of the objects in a binary image. The specific manner of this thickening is controlled by the shape of the SE used.

The erosion of the image I and the SE B, denoted by I ⊖ B, is given by equation (6).

I ⊖ B = { p ∈ E 2 : p + b ∈ I ∀ b ∈ B }

6

Using erosion, the element B has to be contained in I. If so, the value of the result is set at 1. In another case, the value is set at 0. In practice, erosion is used for making an object thinner as well as for the elimination of residual white pixels caused by previous image processing. Boundary object extraction is a popular application of erosion and dilatation. The object boundary can be simply found as the difference between original and eroded object. ¹²

The correct choice of shape as well as size of the SE is crucial for the successful application of mathematical morphology. The influence of various element shapes is presented in Figure 8. It is obvious that SE used in Figure 8(c) is too large and causes loss of image information. Whereas SE applied to Figure 8(d) seems to be suitable. Examples of the disk-shaped SE with various sizes are presented in Figure 7.OPEN IN VIEWER

Figure 8.

Mathematical morphology: Influence of SE selection on image closing. (a) Original image cut. (b) Corresponding BW image arising from previous processing — WT, and adaptive image thresholding with various SE use. (c) Image closing uses SE of type disk, size 5. (d) Image closing uses SE of type disk, size 1. SE: structuring element.

Mathematical morphology application was successful in the processing of nanotube microscopic images. Artefacts of residual noise and undesirable boundary markers were cleared away by use of mathematical morphology.

Object classification

As objects were found in the image (see Figure 8(d) in which white areas represent both innertubular and intertubular spaces), now it is necessary to distinguish into which group each object belongs. The adaptive thresholding step (section ‘Adaptive thresholding’) differs the image to two groups of objects. Namely, a correctly chosen threshold separated image data into two groups of objects, the nanotube wall and the background. Further separation of the background into innertubular and intertubular spaces is the objective of this part. By completing this task, the nanotube perimeter can also be evaluated.

Object classification was treated using the clustering analysis from the decision-theoretic methods family. This method is often used for sorting the objects into several groups (clusters) according to their characteristic features. A proper choice of these features is crucial for successful classification. Geometrical object properties, for example, area region, perimeter, diameter, eccentricity, convex hull, bounding box and others or their combination, can serve for this purpose in the considered case. A successful application of cluster analysis can be found in the study by Jain and Celebi et al. ^19,20

The classification criterion used was derived from the region area and its bounding box. Bounding box area S_B means the area of the smallest rectangle covering the whole region. While the inner nanotube part is a convex region with a shape similar to a circle, the intertubular space regions are strictly non-convex. Because of this, the ratio C of region area S_R and bounding box area S_B was used as the classification criterion C

C = S B S R

7

Following the clustering analysis application, the object classification was carried out. The classification stems from the classification criterion mentioned above. Each group of objects was labelled by the following principle. In the case of the assessed region belonging to the innertubular space object group, the value of the criterion C is expected to be close to 1. In contrast, regions belonging to the intertubular space object group have an indented shape, so their area of region and area of bounding box are expected to be totally different.

To sum up, criterion C serves as a characteristic feature expected to be capable of differentiating the innertubular space from the intertubular space by means of the K-means algorithm.

The K-means algorithm is a popular method in the field of image classification for its simplicity and robustness. Its task is sorting a vector of characteristic features F = [f ₁, f ₂,…f _Z] counting Z elements into k clusters. Elements of F can be multidimensional quantities, that is, there can be group of features describing image properties attending to the evaluation. Elements are clustered into k groups according to their deviation of the cluster mean value. The deviation is evaluated by the method of the least squares, see equation (8).

J = ∑ i = 1 k ∑ f j ∈ F ( f j − μ i ) 2

8

The resulting image classification is presented in Figure 9. The studied image is completely segmented into two groups based on their characteristics – the innertubular space, the intertubular space and nanotube walls. This segmentation will be used for the further evaluation of nanotube wall thickness.

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Figure 9.

Innertubular space and intertubular detection in selected image cut. (a) Original image cut. (b) Original image segmented by adaptive thresholding. Objects intended for classification are marked as white. (c) Classification result. Parts of the image which have not been considered in the classification process form a grey-level background. The innertubular space is marked as white, while the intertubular space is marked as black.

Wall thickness evaluation

The wall thickness evaluation is the last step in the nanomaterial characterization process. Two different approaches were compared for evaluation of the variable quality of image data. The first method is based on mathematical morphology. It may be used for image data in which many discrepancies are expected. The second method takes the advantage of spatial filtering by Laplacian of Gaussian (LoG). The suggested method is designed for better quality data. Obviously, wall thickness evaluation is highly dependent on image resolution. It is possible only to detect whole pixel values. This means that if 1 pixel corresponds to 20 nm, the wall thickness can only be scaled by 20 nm.

Evaluation based on mathematical morphology

The first method used for nanotube wall evaluation is based on the mathematical morphology. Morphological erosion according to equation (6) was used. Input to the erosion process was a binary image after adaptive thresholding (see Figure 6(b)). Objects used for the wall evaluation process are marked by the black colour. During the wall thickness evaluation procedure, the aforementioned objects were eroded in layer by layer of pixels. This erosion was repeated until the whole wall disappeared. For example, if the object assigned to the wall was eroded after five steps of erosion, 10 pixels would be removed, so wall thickness was 10 pixels. The described process is graphically illustrated in Figure 10. Pixel value was transferred to nanometres afterwards using the resolution value of the studied image.

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Figure 10.

Illustration of erosion as a tool for nanotube wall thickness evaluation. Each grey-level layer represents one-pixel layer eroded in one step of erosion. Dark grey pixels represent the first erosion step. Light grey pixels represent the last step when almost the whole object disappears.

The advantage of this approach is its robustness to different wall thicknesses. It is indifferent to disturbances caused by doubled walls, for example. The nanotube wall was measured along the whole perimeter in one step, so thickness measurement avoided corrupted parts.

Figure 10 illustrates the application of this method on the real image data. Each grey level represents one step of erosion. Erosion will continue until all the pixels disappear.

Evaluation of correct nanotube thickness is a crucial part. The count of founded regions in the image was suggested as a criterion for the selection of the erosion level suitable for nanotube wall evaluation. It is assumed that the count of founded regions in the image will rapidly increase when the thinnest part of the nanotube wall is literally cut apart by the process of erosion. This process is illustrated for the selected cut in Figure 11, where the count of founded regions in each step of the erosion process is presented. It is obvious that the count of regions in the image increases in step 4, so the corresponding wall thickness of the nanotube wall is 8 pixels. This value was transferred to nanometres afterwards using the resolution value of the studied image. The example presented in Figure 11 serves as a case study. Different image data are expected to have a variable development of count of regions in the image.

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Figure 11.

Nanotube wall evaluation: Count of regions in selected image cut in each step of erosion.

To sum up, the erosion method from the mathematical morphology family is a suitable method for evaluation of asymmetrical or inhomogeneous nanotubes. The suggested principle allows the evaluation of partly corrupted walls as well. The count of regions founded in the image at each step was observed and its maximal value was proposed as a key feature of nanotube wall thickness evaluation.

Evaluation based on spatial image filtering

This approach stems from image filtering, discussed in the preceding section ‘Preprocessing’. LoG function was chosen for nanotube wall thickness evaluation, see equation (9). Due to its shape, LoG filter is suitable for detecting fuzzy edges present in real images (see Figure 12). This specific type object detection is based on a convolution of the image and predefined LoG filter of a given width. The mathematical principle of convolution was described in section ‘Preprocessing’ ¹² :

LoG ( x ) = − 1 π σ 4 [ 1 − x 2 2 σ 2 ] e − x 2 2 σ 2

9

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Figure 12.

Intensity profile of nanotube wall compared to LoG filter, σ = 3. LoG: Laplacian of Gaussian.

Zero-crossing points of LoG are important features for nanotube wall thickness evaluation. The purposed evaluation approach presumes that nanotube walls can be detected by a feature specific for LoG filtering. The edges in the image are given as zero crossings of an image convolved with the LoG filter. Positions of these points can be obtained by a solution of equation (9) equal to 0. The zero-crossing points formulation is presented in equation (11):

LoG ( x ) = 0

10

x 1,2 = ± σ 2

11

It is necessary to extend LoG filter to 2-D for the purpose of image processing (see equation (12)). Vectors x and y represent spatial coordinates. An important parameter of this filter is standard deviation σ representing filter width.

LoG ( x , y ) = − 1 π σ 4 [ 1 − x 2 + y 2 2 σ 2 ] e − x 2 + y 2 2 σ 2

12

Nanotube wall measurement is based on the successful result of image filtering by LoG filter. As is obvious from Figure 12, the width of the LoG filter can be a clue leading to nanotube wall width evaluation. The main idea is based on the presumption that the known filter peak width w can match the wall width. The filter peak width w is defined as the distance between two zero-crossing points of LoG filter. The equation for peak width w evaluation is presented in equation (13):

w = 2 σ 2

13

On the whole, finding the proper value of standard deviation σ is the objective of nanotube wall width evaluation by LoG filter. Several σ values were tested on the selected image. Figure 13 shows the output of LoG filtering with various σ value settings.

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Figure 13.

Detection of nanotube wall for wall thickness evaluation by LoG filtering with various standard deviation settings. (a) σ = 1; (b) σ = 3; (c) σ = 5 and (d) σ = 7. LoG: Laplacian of Gaussian.

The best σ value was chosen by the number of falsely detected pixels (see Table 1). These pixels were wrongly detected by LoG and the filter declared them as pixels belonging to the nanotube wall. In fact, they belong to the innertubular space or the intertubular space. The number of falsely detected pixels should be minimal, so this assumption was used for the evaluation of the best filter setting (see Table 1).

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Table 1.

Overview of LoG filter results.^a

Characteristics evaluated from LoG application. Total pixels in the image: 65 636
Standard deviation σ	1 2	3 2	5 2	7 2
Total non-classified pixels	26 454	22 302	13 270	24 578
Total non-classified pixels’ (%)	40.37	34.03	20.25	37.50

LoG: Laplacian of Gaussian.

^aBold values represent the best standard deviation setting used for the nanotube wall thickness evaluation.

Results

The design and the development of the automatic evaluation of TiO₂ nanotubes is the goal of the presented paper. Nanostructured material was prepared by electrochemical oxidation carried out on Ti-6Al-4V alloy in an ammonium sulphate and ammonium fluoride-based electrolyte. Various types of exposures were chosen to achieve a range of dimensions of nanostructures. Studied image data were recorded the SEM with various scanning parameters. Simple, effective and undemanding processing methods were chosen due to the amount of processed data. The suggested strategy uses the mathematical background of image filtering, WT, adaptive thresholding, mathematical morphology and object classification.

The proposed algorithm’s functionality was tested on more than 50 images obtained during the new material development. The representative features of presented algorithm and the nanomaterial characteristics were observed during this testing.

Nanotube innertubular space and nanotube wall thickness have been during the study as well. Their values have been significant for observation of the material development process. They serve as a tool for evaluation of changes in material structure while the preparation process parameter has been changed. Table 2 presents the selected results of the of Ti-6Al-4V nanomaterial evaluation by the suggested algorithm.

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Table 2.

Selected results of nanostructured material evaluation by the suggested algorithm.^a

Nanotube				Algorithm setting
Sample no.	The area of the innertubular space (nm2)	The nanotube wall
		Thickness (nm)	Evaluation method	Filter size
T2-02-3	5157 ± 2828	8 ± 4	LoG	10
T2-04-3	4739 ± 2760	6 ± 2	LoG	10
T2-05-3	4242 ± 2195	5 ± 1	LoG	10
T2-06-3	4148 ± 2257	5 ± 2	LoG	10
T2-14-3	4133 ± 2438	6 ± 3	ME	15
T2-18-3	4040 ± 2199	6 ± 1	LoG	10
T2-19-3	986 ± 574	4 ± 1	ME	5
PH-036-3	2402 ± 1293	5 ± 2	LoG	10
PH-037-3	2349 ± 1233	4 ± 1	LoG	10
PH-038-4	1064 ± 628	4 ± 2	ME	10
PH-040-3	2692 ± 1307	4 ± 1	LoG	10
PH-041-3	1800 ± 942	3 ± 2	ME	10
PH-045-3	4533 ± 1934	7 ± 2	LoG	10
PH-046-3	4777 ± 2215	7 ± 2	LoG	10

LoG: Laplacian of Gaussian; SE: structuring element.

^aPresented results are obtained with fixed SE setting – shape: disk; size: 1.

Presented results have been chosen as a representative set for illustration of the experiment significant. It is obvious that the values of the innertubular space of all samples are burdened by a higher standard deviation. This phenomenon is caused by the variable size of the nanotubes in the real sample. This assumption is based on the histogram of the distribution of the innertubular space area in high-quality material which is presented in Figure 14.

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Figure 14.

Histogram of the area of the innertubular space distribution in selected sample of the well-developed material.

It is obvious that the area of innertubular space of the most of nanotubes lies in lower values. In contrast, the higher values of innertubular space, which are not as numerous as the lower values, influence the average value and the standard definition of the innertubular space. The similar situation rises for the nanotube wall thickness assessment. This histogram shape is typical for the high-quality material with the well-developed nanotubes. Contrary, the distribution of the area of the innertubular space for the low-level quality material is shown in Figure 15. The low values of the innertubular space area are significant for this type of the material. It was assumed that these values belong to small regions where the image classification failed.

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Figure 15.

Histogram of the area of the innertubular space distribution in selected sample of the low-level quality material.

For better illustration, both the low-level quality and the high-level quality samples are presented in Figure 16. Both samples have been classified by the proposed method and the percentage of the unclassified pixels was evaluated.

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Figure 16.

The percentage of unclassified pixels in material with various quality. (a) The low-level quality material. (b) The high-level quality material. Unclassified pixels are depictured by green colour. The percentage of unclassified pixels is denoted by P. Black – nanotube walls. Blue – innertubular space. Red – intertubular space. Green – unclassified pixels.

Considering the results presented, it was assumed that the percentage of the unclassified pixels in the image is suitable parameter for the assessment of the material corruption. According to this fact, the algorithm was tested for variable material quality. The following examples illustrate extreme possible case studies focused on material quality. The low level of quality of Ti-6Al-4V nanomaterial is represented in Figure 17. Almost every feature of nanotubes is not observable, apart from several well-developed nanotubes in the low part of the image. Obviously, the classification algorithm failed in the corrupted part of the studied sample, but well-rounded nanotubes were successfully found and marked. The number of unclassified pixels (green colour) is remarkable in upper part of image, so the corrupted parts have been determined as unclassified pixels. It should be denoted that well-developed nanotubes in the lower part in the image has been detected and classified successfully. This result proves the effectivity of the proposed algorithm. Further classification of the corruption level could be carried out by methods of texture analysis, ²¹ which is planned for our future work.

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Figure 17.

Example of classification of fully corrupted material.

The second case study was focused on local disturbances observable on a single nanotube. There are several nanotubes with corrupted walls in the lower part of the example in Figure 18. The algorithm did not recognize them as nanotubes, so they were interpreted as innertubular space or background. In contrast, well-developed nanotubes were successfully identified. This result confirms the algorithm ability to detect the nanotubes in between unstable neighbouring disturbances.

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Figure 18.

Example of disturbances on a wall of a single nanotube and their influence on classification.

The final example presented in Figure 19 depicts a well-developed material. It can be seen that most of the material was successfully classified. The number of unclassified pixels (green colour) is minimal in comparison with the sample of corrupted material.

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Figure 19.

Example of a well-developed material and its classification.

The method suggested was integrated into the evaluation process during the development of the TiO₂ nanostructured material, in collaboration with the Department of Metals and Corrosion Engineering. The method was applied for testing the experimental conditions’ setting. Several nanotube parameters were observed, but diameter was selected as the most representative. The influence of voltage, exposure time and surface treatment of the sample on nanotube diameter was observed during the material development process. The method proposed was also used for the evaluation of the reproducibility of nanotube preparation.

Figure 20 shows the dependence of voltage on nanotube diameter. It is obvious that the diameter of the nanotube increases with the increasing voltage applied during the experiment.

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Figure 20.

Example of the voltage – tube diameter dependence. The dot on the graph represents the mean value and a line segment signifies a variance.

Conclusion

The presented algorithm has proved as a suitable tool for observation of material parameters during development of the nanostructured material. Selected representative parameters have been assessed for all tested images, while their values serve for evaluation of the nanotube material process.

Together with the material development process, the algorithm itself has been tested on the real image data. It has been discovered that the success of the evaluation is highly influenced by quality of the material and also quality of imaging. While the images of high-quality material with low amount of corruption are processed, the automatic evaluation is successful. In case of the low quality of image data are processed, the subjective intervention to the algorithm parameters can be required. Namely, the proper method for the nanotube wall thickness evaluation should be selected.

This statement is supported by experimental results considering the various material quality. They show that the false-pixel parameter can serve as an indicator of material corruption. It is assumed that while the proper parts of the nanostructures can be detected in the material with well-developed nanotubes, the number of unclassified pixels is minimal. Contrary, the number of unclassified pixels in the corrupted material is significantly higher. In addition, experiments show that the algorithm proposed is able to detect the well-developed nanotube in area of significant corruption. As a results, it has been assumed that the percentage of pixels unclassified by the suggested algorithm can be used as a clue for the material quality assessment.

During the algorithm testing, it has been assumed that the denoising filter setting, choice of the SE and selection of the proper method for nanotube wall thickness have proven to be key parameters for the successful evaluation of nanotube image data.

It has been found out that the filter type and its size setting determined at the beginning of the testing has been suitable for all set of image data and these parameters could not be changed. In this case, the averaging filter application was successful but this filter can be replaced considering the character of image data. This filter should not be suitable in those cases when the image contains fine details, which should be preserved, or the edges in the image concentrate important information for successful characterization. The weighted averaging filter of the bilateral filter can be suitable choices in these situations. Filter size setting was constant for most of the image data. The small group of images required the increasing or decreasing the filter size value. The lower material quality or the lower quality of imaging was a collective feature of this group of images. In these cases, the filter size setting is suggested to be selected subjectively, afterwards. It should be noticed that the choice of the filter size influences the quality of image segmentation using WT and succeeding adaptive thresholding.

The SE type and its size belonged to the further important parameters of the presented algorithm. It has been observed that the selected SE type and size should not be changed during the evaluation of the group of samples. It can be assumed that the SE type and its size must be selected in the initial part of the algorithm preparation and the same values should be used during the evaluation of the whole set of image data. This assumption is applicable only in the case when the set of images have the same resolution. If not, the reassessment of the SE setting is recommended. The size and shape of SE has been selected empirically in this work because the image resolution has not been changing during the experiments. In the case when the image resolution may vary among the image data, the spatially variant mathematical morphology and the intrinsic SEs are recommended to apply, see the studies by Debayle and Pinoli and Charif-Chefchaouni and Schonfeld. ^22,23

The influence of the image data quality to the selection of the SE parameters has not been observed.

The nanotube wall determination has been suggested by two different approaches. These methods were selected and designed due to their difference in evaluation principle. This decision allowed wall width measurement in many types of images, in consideration of the quality of image data. The first method was based on mathematical morphology. The nanotube wall was measured by the ability of morphological erosion to artificially decrease the nanotube wall. The desired value was derived from the number of iterations leading to total collapse of the studied object. Its suitability for lower quality image data is the main advantage of this method. Its principle allows focusing on the narrow parts of nanotube walls, so the undesirable wall thickening caused by material corruption is ignored. The high dependence on the image resolution is disadvantage of this approach. Nevertheless, this method is able to evaluate the nanotube wall thickness with no difference in image data quality. The second method was based on LoG filtering. Its principle is based on the comparison of the filter shape with the shape of the nanotube wall. This method has possibility to detect the nanotube wall with higher accuracy than the preceding method. It should be noticed that this method fails in case that the lower quality image data are processed.