Reconstruction of Rising Bubble With Digital Image Processing Method

Yuchen Bian, Feng Dong, Hongyi Wang Tianjin Key Laboratory of Process Measurement and Control School of Electrical Engineering and Automation

Tianjin University Tianjin, China

e-mail: fdong@tju.edu.cn

Abstract—The description for rising bubble is important for the study of principle of the Gas/liquid two-phase flow. As to the rising bubble, reconstruction the shape of the bubble can make a strong foundation for further description. The bubble images were collected with two cameras; the images were preprocessed using digital images processing methods; parameters of the simulated ellipse were extracted with Hough transformation; and bubble simulated ellipsoid model were reconstructed based on these extracted parameters.

Keywords-digtal image processing; parameter extraction; bubble ellipsoid; reconstruction

I. INTRODUCTION In nature and industry, especially various kinds of energy

industry such as oil and chemistry, multi-phase flow occurs frequently. Two-phase flow refers to a special flow pattern with relationship of mixture hydromechanics, in which there must be two kinds of phase coexisting and explicit interface between them [1]. The most common pattern is Gas/liquid two-phase flow, where the most basic pattern is bubble flow. Bubble flow plays an important role in the study of mechanism in Gas/liquid two-phase flow and also of great value in bubble columns of industrial application.

With the development of science and technology, various kinds of new technology are used in the measurement of multi-phase flow parameters. As a new measurement technique, digital image processing has been widely used in the measurement of multi-phase flow parameters, such as bubble deformation, flow velocity [2] and gas fraction. At first, initial images were processed; and then interested targets in the images were measured and projection parameters of bubble were extracted [3]; at last, deformed bubble model were reconstructed using these parameters for further study of the mechanism of bubble flow. Based on digital image processing for single image of bubble flow, Ford discussed the deformation and motion of single bubble, but the dip angle of bubble was ignored, the horizontal width and vertical height of the bubble image were simply considered as the axes of bubble [4]. In the study of high gas fraction bubble flow, Murai reconstructed the gas fraction distribution using image processing [5].

In this paper, taking videos of rising bubble from two vertical directions using two cameras; processing the images with digital image processing methods to idealized binary images; extracting parameters from the projection ellipse of bubble images; matching the two groups of parameters to calculate the 3-D parameters and reconstruct the ellipsoid model of deformed bubble. It provided a feasible way for further study of the mechanism of bubble flow.

II. COLLECTION OF BUBBLE IMAGES It is better and more accurate to take videos of rising bubble

using two cameras from two vertical directions than using single camera. The two-phase flow experimental facility in this study is a trough made of organic glass with 200mm×200mm sectional area and 1.25m height, as illustrated in Fig. 1. The diameter of the pore is 2mm. And the velocity, frequency, size of bubble can be controlled by adjusting the control valve below the pore. The dual-camera image collection facility consists of two cameras from two vertical directions, and the focal distance is 8mm. The image resolution ratio of images taken by these cameras is 760 (horizontal) × 575 (vertical) pixels, and the frame rate is 30 frames per second. In these conditions, smearing phenomenon was decreased and high quality continual images were collected.

Figure 1. Experimental facility of bubble column

III. PREPROCESSING FOR 2-D BUBBLE IMAGES RGB images were taken by the cameras, but binary images

were required in the extraction of bubble parameters later on, so it’s necessary to transform the initial RGB images with noise to binary images that apply for parameters extraction,

This work is supported by the National Natural Science Foundation of China (No. 50776063) and the Natural Science Foundation of Tianjin (No. 08JCZDJC17700).

978-1-4244-7935-1/11/$26.00 ©2011 IEEE

(a) Grayscale images (b) Binary images

(a) Initial bubble image (b)Mean fiter

(c) Median filter (d) Wiener filter

.

(a) Binary images with “hole” (b) Filling result

that is processing process for bubble images. In this process, many kinds of digital images processing methods were used, such as type conversion of image, image noise filtering, image painting algorithms and edge detection.

A. Conversion of Image Type Images are stored as a matrix in which every element is the

pixel value. Common image types are RGB images, grayscale images and binary images. In GRB images, pixel color consists of three components: red, green, blue; grayscale images only have the strength information and there isn’t color information in the images; binary images only have black and white which are referred using 0 and 1 respectively.

Transformation from RGB to grayscale images is a process of equating the three components. It is verified that when satisfied (1), reasonable grayscale images can be transformed.

BGRVBGR gray 11.059.030.0 ++==== (1)

B. Image Noise Filtering Noises mix in the initial images due to being disturbed.

Noises may generate because of sensitive components, particle on the photo negative film, disturbance of transmission channel and quantizing noise. Common methods of wiping of noises are linear filtering, median filter and auto-adapted filtering. As common kinds of linear filter, mean filter applies to filtering the grain noise in images. Median filter operates easily and can protect the boundary in images, especially it is effective for filtering the salt and pepper noise, but sometimes, thin lines and small target regions might be lost. As a common kind of auto-adapted filter, wiener filter can filter noise auto-adaptively, because it can adjust the output based on local variance of the image. In some occasions, wiener filter is better than linear filter and it has better selectivity, which can hold the edge and the detail information more effectively [6]. The results with all kinds of filter were shown as Fig. 2.

Figure 2. Results with all kinds of filter

C. Image Binaryzation Image binaryzation refers to a process from grayscale

image to binary image. In the processing for bubble images, the key to binaryzation is selecting an appropriate threshold T (if the pixel value is bigger than T, the value of binary image equals 1, i.e. the color of white; if smaller, the value equals 0, i.e. the color of black), which can divide the bubble and background clearly. If the T was too big, the target might be estimated as the background; and if too small, the background might be estimated as the target. In this paper, the difference between the target and the background of the bubble images is obvious, so the images could be binaried using a direct threshold. Fig. 3(b) is the bubble binary image of Fig. 3 (a) the grayscale image.

Figure 3. Image binaryzation

D. Bubble “Hole” Filling Highlight region might generate due to reflection when

images collecting, as Fig. 2(a). So “hole” might generate after images filtering and binaryzation, as Fig. 3(b). The “hole” must be filled with image corrosion and dilation so that the edge detection could proceed effectively.

Figure 4. Bubble “Hole” Filling

E. Bubble Edge Detection In image, the edge refers to the end of a feature area and the

beginning of another feature area. Edge detection actually is to detect the position where the images features have changed. The edge detection can be achieved based on grayscale images using the operator. The common operators for edge detection include differentiating operator, Laplacian of Gaussian operator and Canny operator. In Fig. 5, we used Canny operator to detect the bubble edge based on grayscale image.

Although the bubble edge was detected using operator of edge detection, there were a lot of background points being

(a) Binary images with no “hole” (b) Bubble edge

(a) Bubble grayscale images (b) Based on Canny operator

(a) Straight line 1+−= xy (b) Cocurrent points in plane θρ −

concluded as the edge, as Fig. 5(b) and it’s harmful for bubble parameters extraction. So it’s better to use the binary images with no “hole” to detect the bubble edge, and the bubble edge was only kept without other extra information.

Figure 5. Bubble edge detection

Figure 6. Edge detection for binary image

IV. ELLIPSE PARAMETERS EXTRACTION WITH HOUGH TRANSFORMATION

In bubble flow, the rising bubble is usually simulated as an ellipsoid model, so the projection image of the bubble could be considered as an ellipse. After bubble edge detection, Hough transformation was used in ellipse parameters extraction. Hough transformation is widely used in geometry recognition and actually a method of conversion between graphics domains and the parameter domain. Hough transformation could detect the curves which could be expressed with analysis formula, such as straight line, circle, ellipse, and parabola and so on. There is only a little effect of the gap of curves and the shape rotation when using Hough transformation to detect. Even though the target was defective, covered or polluted, the detection result would be correct. Hough transformation could be used in the detection of ellipse in bubble images.

A. Hough Transformation Taking straight line recognition with Hough transformation

as an example, the fundamental principle of Hough transformation will be explained. In fact, recognition of straight line with Hough transformation is a method of transformation between x-y coordinate system and coefficient coordinate system (k-b coordinate system). Using the relationship of collinearity and intersection, Hough transformation transforms the question of recognition to the question of counting the number of collinear points.

The equation of a straight line could be expressed with bkxy += , the k and b are two parameters, the slope and the offset respectively. So there is a one-to-one relationship between a straight line in x-y coordinate system and a pair of parameters ( )bk, in k-b coordinate system. And in reverse, there is also a one-to-one relationship between a straight line yxkb +−= in k-b coordinate system and a pair of parameters ( )yx, in x-y coordinate system. This two way relation is the so-called Hough transformation.

In the applications, the bkxy += expression of a straight line could not express the kind of straight line that parallel to the axis y, i.e. cx = . As in (2), Polar equation of a straight line is used so that the transformation domain has meaning.

θθρ sincos yx += (2)

In (2), ρ is the distance from the origin of coordinate system and the straight line, and θ is the angle between axis x and the normal of straight line. So there is a one-to-one relationship between a point in x-y coordinate system and a curve in θρ − coordinate system, and easily known, the curves in θρ − coordinate system that correspond to the collinear points in x-y coordinate system must be concurrent lines. As shown in Fig. 7, the three points in (a) correspond to the three curves in (b) and the three curves are concurrent lines.

Figure 7. Ralationship between straight line in x-y plane and the cocurrent point in plane θρ −

It’s also easily known that a straight line in x-y coordinate system corresponds to a point in θρ − coordinate system. If the numbers of concurrent curves of every point in some given area of x-y coordinate system were counted and stormed as a matrix, the maximum values of this matrix might be correspond to those straight lines in x-y coordinate system. In application, the θρ − plane is equally spaced into small grids according to the precision and every grid corresponds to an element of the matrix. For every point in x-y coordinate system, curves are drawn in θρ − coordinate system, and if a grid were crossed by a curve, the corresponding element of the matrix increases by one. So the matrix gives an index to the numbers of collinear points in x-y coordinate system. For those grids which have big count, the corresponding points in x-y coordinate system might be collinear points. That is the way to recognize straight line using Hough transformation.

(a) 3-D ellipsoid in space (b) Ellipsoid parameters

Hough transformation could be used in the recognition of geometry shape which could be expressed with an analysis formula, such as circle, ellipse and parabola.

B. Ellipse Parameters Extraction The images of rising bubble can be simulated as two semi-

ellipsoids top and bottom, as Fig. 8. Its major axis is a and the minor axes are b and bβ . The parameter β enables us to express various bubble shapes such as a dimpled hemispheroidal-cap (-1< β <0), a hemispheroidal-cap ( β =0), a distorted spheroid (0< β <1, β > 1) and a spheroid ( β =1) [7].

Figure 8. Minic diagram of deformed bubble( β >-1)

Take the semi-ellipse above as an example, the angle between the major axis and the horizontal axis is defined asθ , and the length of semi-major axis and semi-minor axis is a and b respectively, the center coordinate of the ellipse is ( )00 , yx , so the equation of the ellipse is:

[ ]

[ ]1

cos)(sin)(

sin)(cos)(

2

200

2

200

=−+−−

+−+−

byyxx

ayyxx

θθ

θθ (3)

As in (3) at least 5 parameters need to be extracted to establish an ellipse: the center coordinate ( )00 , yx ; the angle between the major axis and the horizontal axisθ ; the semi-major axis a and semi-minor axis b. Extracting directly these 5 parameters would consume too much resource, so it’s necessary to reduce the dimension to simplify the calculation.

Scanning the bubble edge in line, row, and one by one to get the center coordinate ( )00 , yx ; and then calculating the distance each two edge point, so the longest distance is 2a, and the angle θ could be obtained based on the two edge points; detecting the ellipse on the top and bottom with the method of Hough transformation and the semi-minor axes 1b and 2b could be obtained; and at last, optimizing with images area among the several extracting results.

Fig. 9 is the simulated ellipse with Hough transformation. To observe clearly, the bubble edge binary image was displayed with inverse value, as shown in Fig. 9, it’s relatively accuracy to reconstruct the bubble shape using Hough transformation.

Figure 9. Reconstruction of deformed bubble simulate ellipse

V. RECONSTRUCTION OF 3-D BUBBLE ELLIPSOID As mentioned in preceding text, the rising bubble could be

simulated as two semiellipsoids top and bottom though bubble shape changing frequently. The matching method for 3-D bubble ellipsoid model parameters could obtain the 3-D parameters and reconstruct the ellipsoid model of rising bubble based on the two groups of parameters extracted from the two projection planes using Hough transformation.

A. Matching Method of 3-D Ellipsoid Model Parameters To determine an ellipsoid uniquely in space, at least 9

parameters need to know: the center coordinate ( )000 ,, zyx , three semi-axis a, b, c and three space angles γβα ,, . As shown in Fig. 10, three space angles γβα ,, refer to the angle between axis a and axis x, axis a and axis z, axis b and axis z respectively.

Figure 10. 3-D ellipsoid and ellipsoid parameters

The reconstruction method is absolutely same for the two semi-ellipsoids on the top and the bottom, so take the top ellipsoid as an example to show the reconstruction process.

In the two projection planes there are 10 2-D parameters (Table I): the first group of 5 parameters, i.e. in the first plane (xoz plane): ( ) 1110101 ,,,, bayx θ and the second group of 5 parameters (yoz plane): ( ) 2220202 ,,,, bayx θ . So the reconstruction process is the matching process from these ten 2-D parameters to the nine 3-D parameters of ellipsoid (Table II), i.e. ( ) γβα ,,,,,,,, 000 cbazyx , as shown in Fig. 11.

(a) Projection on xoz plane (b) Projection on yoz plane

(c) Reconstruction ellipsoid

Figure 11. 3-D ellipsoid reconstruction

TABLE I. 2-D PARAMETERS

2-D Parameters

( )0201 xx ( )0201 yy ( )21 θθ ( )21 aa ( )21 bb

xoz plane 11.0960 38.2576 24.2277 3.6872 2.2528

yoz plane 4.6369 37.8901 14.0362 3.4140 2.3186

TABLE II. 3-D PARAMETERS

3-D Parameters

( )mmcba ,, ( )°γβα ,, ( )mmzyx 000 ,,

data

3.6872, 3.4140, 2.2528

24.2277, 14.0362, 0.0000

11.0960, 4.6369, 38.0739

Take the first bubble image as an example, the reconstruction process is:

(1) From 1a , 1b and 2a , 2b to cba ,,

=

==

),min( 21

2

1

bbcabaa

(4)

(2) From 1θ , 2θ to γβα ,,

°=

==

902

1

γθβθα

(5)

(3) From ( )0101, yx and ( )0202 , yx to ( )000 ,, zyx

+=

==

2/)( 02010

020

010

yyzxyxx

(6)

B. Analysis of Reconstruction Method The first step: Take the two semi-major axes 1a and 2a of

the two projection planes as two semi-axes ba, of the ellipsoid directly. Especially when determining the third semi-axis of bubble ellipsoid, take the lower value of 1b and 2b as the semi-axis value because of partial matching in some cases. This matching method apply to the projection theory, it could be improved by adjusting the three space angles γβα ,, .

The second step: It is thought that the angle 1θ and 2θ of the two projection planes equal to the space angle βα , of the ellipsoid. And as the rising bubble shape usually is flat ellipsoid, the angle between axis b and axis z, i.e.γ , equals to

°90 .

The third step: Two components of the center coordinate x and y equal to the horizontal ordinate of the center in the two projection planes. There is a little difference between the vertical ordinates of the two 2-D parameters due to desynchronization when taking bubble videos, so take the mean of the two ordinates of the two projection as the z axis value of bubble ellipsoid.

In fact, the rotate angles have relationships with the length of axes of the rising bubble. And in this method, the length of axes and the rotate angles has no relationship, so this matching method applies to rising bubble with low velocity in static flow field, because the flow field has little effect on the rising process of the bubble. In a word, this matching method is conformity with not only the projection theory but also the rising bubble shape.

C. Reconstruction Result Take an example with a bubble in the rising bubble

sequence, used the matching method described above for the top and bottom semi-ellipsoid and the result was shown as Fig. 12.

Figure 12. Results of top and bottom semiellipsoid reconstruction

(a) Projection on xoz plane (b) Projection on yoz plane

VI. EXPERMENTAL VERIFICATION To confirm the accuracy of this method of ellipsoid

reconstruction, as shown in Fig. 13, images of a known ellipsoid were taken and the pattern of this ellipsoid was familiar with rising bubble in water. The known parameters and the conducted parameters of this ellipsoid were shown in Table III. After comparison, the accuracy of this matching method for rising bubble ellipsoid model could be verified.

Figure 13. Images of known ellipsoid

TABLE III. COMPARISON OF KNOWN AND CONDUCTED PARAMETERS

Parameters

a(mm) b(mm) c(mm) ( )°α ( )°β ( )°γ

known 29.76 18.62 18.32 6.5 92.3 5.8

conducted 29.26 18.53 16.64 7.3 82.1 0.0

VII. CONCLUSION The bubble images were collected with two cameras, after

preprocessing using digital images processing methods, 2-D parameters of ellipse were extracted with Hough transformation, and bubble simulated ellipsoid model were reconstructed based on these 2-D parameters. This matching method applies to rising bubble with low velocity in static flow field and it is conformity with not only the projection theory but also the rising bubble shape.

It‘s important to use this model for further study of principle of the Gas/liquid two-phase flow, such as the shape and volume of the bubble, rising velocity, gas fraction and so on.

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