# Application of Inverse Abel Transform on flame image

## Abstract

The present work involves the development and implementation of a mathematical transformation called Inverse Abel Transform and its application to diagnostics of reacting flow systems. Chemiluminescence is the phenomena of emission of light during a chemical reaction. It takes place due to the formation of an intermediate excited state which further stabilizes and produces light. Chemiluminescence has a wide range of application like analysis of inorganic species in a liquid phase, combustion analysis (concentration of CH* and OH*), glow sticks, etc. In combustion systems particularly, the location of maximum concentration of CH* and OH* is useful as they are considered to be the heat release markers in premixed flames. The concentration field of a given species can be obtained by viewing the reacting zone through an appropriate optical bandpass filter. This filter allows only the wavelength of light emitted by the species of interest. However, such an image contains the concentration of the entire volume, as each pixel captures the integrated light from the entire volume passing through its line of sight. In order to resolve an axisymmetric integrated line of the sight data distribution to one at a given azimuthal plane, a mathematical transformation called Inverse Abel Transform is used. The IAT takes a 2D projection and reconstructs a slice of the cylindrically symmetric 3D distribution. IAT also plays an important role in analyzing the projection of plasma plumes and flames. The major difficulties in implementing IAT are the discontinuity and its dependence on the first derivative of the original function. The latter is particularly challenging when used with experimental data, which is discrete and can be noisy. In order to remove these difficulties, several methods are available. Presently, a direct discretization method, implemented in MATLAB, is used to calculate the IAT. This method addresses the singularity by modifying a term suitably to avoid division by zero. The implemented algorithm was tested with various trial functions and was seen to provide an acceptable match. Subsequently, image processing of a CH* chemiluminescence data from literature is performed to obtain the azimuthal distribution of CH* relative concentration.

**Keywords****:** Inverse Abel Transform, Chemiluminescence, image processing

## Abbreviations

IAT | Inverse Abel Transform |

AT | Abel Transform |

DD | Direct Discretization |

NO | Nestor Olsen |

GUI | Graphical User Interface |

GUIDE | Graphical User Interface Development Environment |

## INTRODUCTION

## Chemiluminescence Phenomena

## Inversion Function

Inverse Abel Transform (IAT) is a mathematical function which takes a 2D projection and reconstructs a slice of the cylindrically symmetric 3D distribution. IAT also plays an essential role in analyzing the projection of plasma plumes and flames.

When the source is assumed to be cylindrically symmetric and optically thin, the observed lateral data can be converted to radial distribution profile. The Abel transformation (AT) is given by:

The IAT of a system is given as:

Where F(x) = Abel transform

f(r) = circularly symmetric function

x = distance from origin

r = radius

L= Maximum limit of radius

The significant difficulties in implementing IAT are the discontinuity and its dependence on the first derivative of the original function. The latter is particularly challenging when used with experimental data, which is discrete and can be noisy. To remove these difficulties, several methods are available. Presently, a direct discretization method (DD) and Nestor Olsen (NO) method implemented in MATLAB is used to calculate the IAT.

## Difference between the Numerical and Analytic Approach

Analytical methods fit the experimental data to a function and allow direct integration of function and avoid the discretization problems [1-3]. Some functions, such as Gaussian function are used to serve the purpose.

Numerical methods convert the Abel Inversion equation (Eq.2) to a summation to allow processing on a discrete set of data from Ref. [4]. It is a fast way of solving the integral equation.

## METHODOLOGY

The IAT was operated on data points obtained from a flame image from Ref. [5]. The algorithm which was generated tested initially in MATLAB on data points obtained through a different trial function such as linear function, exponential function, and logarithmic function. It consists of two steps:

First, the discretized Abel Transform was performed on the trial function to generate the data points which treated as data points of a flame image.

Second, IAT was performed on previously obtained data points to get the original function.

The algorithm seems to provide satisfactory results. Further, the method was implemented on actual data of a flame image.

## Algorithm for Discretized Abel Transform

The following algorithm for Abel Transform was implemented on trial function (f_r = log(r)) for 51 axisymmetric data points, as shown in Fig 1.

## Algorithm for Inverse Abel Transform

There are two methods of implementing the IAT on the given data points. The following algorithms are to be applied on a 2n+1 data point discrete distribution I(x) with axial symmetry. The first method is known as direct discretization (DD) from Ref. [6], and the second method is Nestor – Olsen (NO) from Ref. [4].

The DD is based on a direct discretization of (Eq.2). This method addresses the singularity by modifying a term suitably to avoid division by zero. It is given as:

The Nestor Olsen method uses a numerical algorithm which is easy to compute, and the IAT is given as follows:

Where M(x) = measured lateral intensity

N = number of data points

a = increment in value of x

B (k, n) = -A (k, k) for n=k

B (k, n) = A (k, n-1) – A (k, n) for n>= k+1

Where k and n are the integer position indices for radial and lateral intensity.

## RESULTS AND DISCUSSION

## Validation of Algorithm

The algorithm generated to solve the Abel transformation numerically, and Inverse Abel transformation was performed on different trial function.

## Able Transformation

Abel Transform function was executed analytically and numerically, and the results were compared. Four functions were taken as a trial function.

The first function is a linear function (f(r) = r² + 2r) where r is the radius and its Abel transform was plotted in original and discretized form as shown in Fig 2. Analytically, Abel transform was represented by (O_AT), and numerically, it was described by (D_AT).

Second function is exponential function (f(r) = exp(r)) and Abel transform is plotted original and discretized as shown in Fig 3.

Third function is function f(r) = √r and Abel transform is plotted original and discretized as shown in Fig 4.

Fourth function is logarithmic function (f(r) = log(r)) and Abel transform is plotted original and discretized as shown in Fig 5.

## Inverse Abel Transform

Both methods of IAT was executed discretely and compared with the actual function values. The results were compared. IAT for trail function is shown in the figures below.

When the function is f(r) = (r² + 2r), IAT is shown in Fig 6. The original function is represented by (F_r), and IAT is represented by (F_r1) corresponding to DD method and by (F_r2) corresponding to NO method.

When the function is (f(r) = exp(r)), IAT is shown in Fig 7.

When the function is f(r) = √r, IAT is shown in Fig 8.

When function is f(r) = log(r), IAT is shown in Fig 9.

## Application to Chemiluminescence

Image processing of a CH* chemiluminescence data from literature Ref. [5] is performed to obtain the azimuthal distribution of CH* relative concentration. Implementation of IAT on the flame image was done, and it was observed that the region of maximum concentration of CH* occurs at the outer edges of the flame which is the region where intensity is maximum. On analysing the inverted image by two methods, it was observed that the number of different zones of the flame image obtained with the help of DD method is more as compared to the NO method.

## MATLAB-based GUI for Inverse Able Transform

GUI was generated in MATLAB using GUIDE. GUI is a user interface that includes graphical elements such as icons and buttons. The significant advantage of GUI is that they make operation more intuitive and easy to learn and use. The user does not require any knowledge of programming. The GUI was generated for finding the IAT, and it allows the user to choose an image and the method for finding the Inverse Abel Transform. Depending upon the method selected by the user either DD or NO, it will show the IAT of the previously selected image. At the interface, the user can see the actual chosen image, half of the image with colormap and image after Abel inversion. The GUI for the IAT of combustion flame image is shown in Fig 10.

## CONCLUSION

The present work involves the implementation of a mathematical transformation called Inverse Abel Transform for the purposes of analysing chemiluminescence data of a reacting flow field. Two numerical methods, namely direct and Nestor-Olsen methods, were implemented in MATLAB. The methods are first validated for known one-dimensional functions and finally applied to a two-dimensional chemiluminescence data. A MATLAB-based function, including one with a GUI, is developed for this purpose. The results show that the regions of maximum radical concentration is captured adequately.

## REFERENCES

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## ACKNOWLEDGEMENTS

The summer training opportunity I had with the Indian Institute of Science, Bangalore was a great chance for learning and professional. Therefore I consider myself a very lucky individual as I was provided with an opportunity to be a part of it. I am also grateful for having a chance to meet so many wonderful people and professional who led me through this summer training period.

I express my sincerest thanks to Dr. Saptarshi Basu, professor at mechanical engineering department, IISc Bangalore and Dr. Keerthi, a postdoctoral researcher at mechanical engineering department, IISc, Bangalore for taking part in the useful decision and giving necessary advice and guidance and arrange all facilitate to work more comfortable. I would also like to thank Abhishek and Ankit, Project Assistant at IISc Bangalore, who helped me in my training. I perceive this training as an excellent opportunity to research at the undergraduate level. I will continue to work on my gained skills and knowledge to obtain desired career objectives. Hope to continue cooperation with all of you in the future.

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