GUJARAT TECHNOLOGICAL UNIVERSITY

8763005231525457825523152Chandkheda, Ahmadabad Affiliated

A. D. PATEL INSTITUTE OF TECHNOLOGY

A UDP PROJECT REPORT ON

3-D RECONSTRUCTION OF FLAME USING 2-D IMAGES

BE-IV, SEMESTER – VII (MECHANICAL ENGINEERING)

DR. MITESH I. SHAHPROF. MAHARSHI THAKKAR

(Faculty Guide)(Faculty Guide)

DR. VISHAL N. SINGH

Head of Department

Sr. No. Name of student Enrollment No.

1. Bhumi Dhameliya 150010119012

2. Jay Patel 150010119070

3. Nandini Rajguru 150010119089

4. Preet Tejwani 150010119112

ACADEMIC YEAR (2018-19)

ACKNOWLEDGEMENT

.

We would like to express our heartfelt gratitude to our mentor Dr. Mitesh Shah, co-mentor Prof. Maharshi Thakkar as well as our Head of Department, Dr. Vishal Singh to give us the golden opportunity to pursue this wonderful project on the topic, “3-D Reconstruction Of Flame using 2-D Images”, which also helped us to do a lot of research and came to know many new things, we are really thankful to them. Secondly, we would also like to thank our parents and friends who helped us throughout the project.

CONTENTS

INTRODUCTION

About 2-D and 3-D images

Tomography

Engineering Applications of Tomography

Motivation

LITERATURE REVIEW

Brief of Various Research Paper referred

Review of Different Iterative Techniques

Outcomes of Literature Review

Reconstruction Methodology and Implementation Strategy

Mathematical Representation of image of 3-D object as projection

Weight Function Calculation

Methodology for MENT

Monte Carlo Sampling Method

Algorithm for generating theoretical Projection

COMPARISON WITH BENCHMARK PROBLEM

Standard Cosine Phantom Object

Reconstruction Of Cosine Phantom Using MENT Algorithm

CONCLUSION

References

Appendix

ABOUT PROJECT REPORT

The project report consists of five chapters. The first chapter of introduction gives the description of the difference between 2-D and 3-D images; what is tomography and its various applications that used in medical and engineering field. The introduction chapter ends on a note of motivation of why we chose to pursue this topic for the final year project.

The second chapter is regarding literature review which is sub-divided into three topics i.e., the brief about various research papers referred during this academic semester; various methods that are used for reconstruction of objects and conclusion drawn after performing the literature review.

The third chapter is about reconstruction methodology which consists of mathematical representation of image of 3-d object as projection, how weight function is calculated, the methodology for operating Maximization of Entropy Technique (MENT), about Monte Carlo Sampling Method and the algorithm that is used for generating theoretical Projection.

The fourth chapter comprises of 3-D cosine Phantom object that was theoretically generated in MATLAB, along with understanding the procedure that has been used in order to generate experimental results for reconstruction of flame object whose results are yet to be obtained. The fifth chapter of conclusion explains the future scope.

Lastly, References followed by Appendix are also provided that consists of Periodic Progress Reports (PPR), Patent Search & Analysis Report (PSAR) and Design Engineering Canvases.

1. INTRODUCTION

About 2-D and 3-D images

The two dimensions pictures depict length and width, where the objects on the picture are flat. However, both images given below are 2-D, but 3-D picture contain yet another dimension i.e., depth. This type is the most realistic one, as the depiction of objects or environments resembles the way we see them through our own eyes.

12382501025842

(B)

Fig 1.1.1 (A) 2-D Image (B) 3-D Image

Examples

Table 1.1.1 Summary of difference between 2-D and 3-D

Source: DifferenceBetween.net

Tomography

‘Tomography’ consists of two Greek words ‘tomo’ which means ‘slicing or sectioning’ and ‘grapho’ which means ‘to draw’. Tomography means ‘To reconstruct 3D shape of objects from its different 2D images captured at different angles’. Tomography is imaging by sections or sectioning, through the use of any kind of penetrating wave.

General applications of tomography

1. CT scanner2. MRI scanner3. Head scanner4. Medical NDT

Tomography is also used in radiology, geophysics, atmospheric science, material science, astrophysics, quantum information and other areas of science.

How a CT system works?

A motorized table moves the patient (Figure 1.3.1) through a circular opening in the CT imaging system.

As the patient passes through the CT imaging system,

4743450131103a source of x rays rotates around the inside of the circular opening. A single rotation takes about 1 second. The x-ray source produces a narrow, fan-shaped beam of x rays used to irradiate a section of the patient's body. The thickness of the fan beam can be as small as 1 millimeter or as large as 10 millimeters. In typical examinations there are several phases where is each made up of 10 to 50 rotations of the x-ray tube around the patient in coordination with the table moving through the circular opening.

Detectors on the exit side of the patient record the x rays exiting the section of the patient's body being irradiated as an x-ray "snapshot" at one position (angle) of the source of x rays. Many different "snapshots" (angles) are collected during one complete rotation.

The data are sent to a computer to reconstruct all of the individual "snapshots" into a cross-sectional image (slice) of the internal organs and tissues for each complete rotation of the source of x rays.

Fig 1.2.2: Patient in CT Imaging System Source: www.fda.gov

Engineering applications of tomography

1. Heat transfer and fluid applications2. Flame and combustion analysis

3. Assembling components4. Non-Destructive testing (NDT)

1.3.1 Role of tomography in mechanical field

There are a lot many roles of tomography in mechanical field like analysis of heat transfer and fluid flow, but here we will focus our knowledge on applications like:

Non-destructive evaluation (NDE)

It is a technique to detect the defects without affecting the intended applications or damaging the structure, which is mainly used for damage assessment and control of quality. Suppose during welding a part, weld defects such as slag inclusion, lack of penetration and burn through can cause a serious disaster during if it remains undetected during the time of welding.

Image reconstruction is a methodology through which the image of an interior part of an object is reconstructed from the projection data of an object. Through Computed Tomography, we can detect internal defects without damaging the structure by reconstructing a 2D or 3D image of the cross-section of the defective region even when the projected data are quite noisy.

1857375100570

Fig 1.3.1 (A) Non-destructive testing for a given object Source: Quality Magazine

Flame tomography

Flame tomography is the study of flame structure and its various parameters by reconstructing the 3D image by slicing or sectioning. Here, after taking the 2D image, slicing of each plane is done and then all the planes are stacked in order to reconstruct 3D image.

By reconstructing the image, we can get the correct field intensity as well as temperature distributions at each and every required point. With the use of flame tomography, we can also get the correct shape of the flame at a particular instant.

Fig 1.3.1 (B): 3-DReconstruction of flame by slicing method Source: Digital Imaging based 3-D visualization and characterization of coal fired flames by G. Gilabert

1.4 Motivation

In medical applications like CT scanner , Head scanner and MRI scanner images of person’s body are taken at different angles and then reconstructed in 3-Dimension, based on which doctors will analyze it and give proper treatment accordingly. Similarly, we can apply this idea for doing analysis in heat transfer, fluids and in combustion process where number of projections is limited.

By taking 2-D images of flame at different angles with the help of CCD cameras and then reconstructing it into 3-D using appropriate algorithm, we can get the 3-D shape of flame, which will help us to understand and improve the combustion phenomena i.e. what will be the shape of flame and according to that how will be the temperature distribution in the combustion chamber. It will be useful in research applications like how the design of nozzle and burner should be modified to get optimum flame shape and temperature distribution.

We can also analyze dynamic and turbulent behaviors of combustion flame, soot particles fraction, broadband emission and beam steering which will help us to improve the efficiency of a particular system like Boilers, I.C. engines, Nuclear reactors etc. Hence, these gives us the motivation to work under the domain ‘Flame Tomography’ and get the correct shape of a flame and accurate results of flame parameters at a given point.

2. LITERATURE SURVEY

As per the brief discussion about flame tomography in previous chapter, here some brief of the research papers related to this project is presented, that we have tried to understand during this academic tenure to fully comprehend the process which could help us to obtain more knowledge and deep understanding about the topic, which is described below in section 2.1.

Also due to the limitations of data, images and favorable environment, there are various techniques that have been devised here in order to effectively reconstruct objects using different algorithms. With various numbers of projection and projection angles, five distinct methods have been taken into consideration in section 2.2, where each projection of the object is deciphered as a path integral of the light-sensitive property of the object in a particular direction.

Brief of various research paper referred

Influence analysis of radiative properties and flame temperature reconstruction based on optical tomography by Yuan Yuan, Tian-Jiao Li, Yan-Long Zhu, Yong Shuai, He- Ping Tan

Optical computed tomography is a new technology based on the technology of computer information process, and its most outstanding feature is that it can precisely measure a certain level of transient physical distribution without disturbing the distribution of the measured field. A combination of radiative spectroscopy and optical tomography can be used to reconstruct the structure of a flame in a section manner to achieve the goal of detecting its 3D interior structure.

As light field imaging technology is used, the influence of radiative properties on flame distribution at different positions is hard to evaluate. Therefore, this study simulates the light field imaging of section flames to analyze the influence of radiative properties on flame distribution. A tomographic image is reconstructed from three-dimensional light field imaging, which is evaluated as similar to the section flame images. Then the temperature reconstructed from the tomographic image is compared to the known temperature, with a small relative error, under 10%.

In optical section imaging method, flame imaging in different attenuation media is simulated, and a temperature reconstruction light field imaging system based on tomography algorithm is established. The analysis showed that the attenuation has a significant impact on the flame section imaging process. As attenuation coefficients increase, the section flame images become darker. Reconstructed images are obtained by deconvolving the CCD imaging of 3D flame in the attenuation media (Ke = 10??1)by solving the PSF at a certain position.

The relative error of flame temperature reconstruction is within 10% in the range of 1450–1800 K. For the flame centre with high temperature (;1600 K), the error is less than 3.5%. Larger errors are found in the low temperature range (;1600 K) like flame edges. The reason is that the low sensitivity of the model to detect the low temperature, the accuracy of the reconstruction temperature is limited.

Simulation of Rayleigh-Benard convection using a lattice Boltzmann method by

Xiaowen Shan

Lattice Boltzmann equation (LBE) method has been developed as a computational fluid dynamics (CFD) method. The LBE method improves the idea by following only the 3ensemble-averaged distribution functions, thereby eliminating the time consuming statistical average step in the original LGA (Lattice Gas Automata method which originated form a Boolean fluid model). More importantly, as fluid motion is simulated at the level of the distribution functions, the microscopic physics of the fluid particles can be incorporated easily as in other particle methods. Many complex fluid phenomena due to inter-particle interactions, such as capillary phenomena, multiple phase flows, and nonlinear diffusion, can be simulated naturally.

The density field of the second component, which evolves according to the advection- diffusion equation of a passive scalar, is used to simulate the temperature field. A body force proportional to the temperature is applied, and the system satisfies the Boussinesq equation except for a slight compressibility. The critical Rayleigh number for the onset of the Rayleigh-Benard convection agrees with the theoretical prediction. As the Rayleigh number is increased higher, the steady two-dimensional convection rolls become unstable. The wavy instability and a periodic motion observed, as well as the Nusselt number as a function of the Rayleigh number, are in good agreement with experimental observations and theoretical predictions. The LBE model is found to be efficient, accurate, and numerically stable for the simulation of fluid flows with heat and mass transfer.

By simulating the temperature field using an additional component, they were able to avoid the numerical instability plaguing the thermal LBE models. The algorithm is simple, and the requirement on computational resources is twice of that for a non-thermal LBE code. The results agree very well with theoretical predictions and experimental observations both at near-critical and moderate Rayleigh numbers.

Three-dimensional visualization and quantitative characterization of gaseous Flames by H C Bheemul, G Lu and Y Yan

Model Reconstruction Based on Contour Extraction involves monitoring and characterization of combustion flames that has become increasingly important to combustion engineers for improving the understanding and on-line optimization of combustion conditions. Hence, optimized operating conditions in such systems are required to enhance furnace safety, improve combustion efficiency and reduce pollutant emissions. Industrial trials of such systems have also been undertaken recently and results obtained have demonstrated their operability and effectiveness.

In the system, three cameras had been placed equidistant and equiangular from each other around the flame which were being monitored and captured the two-dimensional images of the flame simultaneously from the three different directions. Dedicated computing algorithms have been developed to reconstruct three-dimensional models of the flame using its contours extracted from the two-dimensional images. A set of geometric parameters, including volume, surface area, orientation, length and circularity, had been defined to characterize the flame from the model generated. The accuracy and spatial resolution of the system have been evaluated using purpose-designed templates.

A series of experiments were conducted on a gas-fired combustion rig to evaluate the performance of the system. The results obtained from a series of experiments have demonstrated that the system is capable of measuring three-dimensional geometric parameters of the flame. It can therefore be deduced that, as expected, the quantitative characteristics of the flames are dependent on the operational conditions of the combustion rig. Further investigations can be undertaken to quantify luminous and thermodynamics parameters of a flame (brightness, uniformity, luminosity distribution, temperature distribution, soot concentration and flicker).

Three – Dimensional Computerized Tomographic Reconstruction of Instantaneous Distribution of Chemiluminescence of a Turbulent Premixed Flame by Yojiro ISHINO and Norio OHIWA

Maximum Likelihood Expectation Maximization (MLEM) Algorithm: Advance Computerized Tomography (CT) reconstruction technique is used for measuring an instantaneous three dimensional distribution of chemiluminescence of a turbulent pre- mixed flame is accomplished. In this technique, first instantaneous two-dimensional (‘projection-images’) of an objective flame are simultaneously taken from forty horizontal directions with a forty-lens camera. Next, four hundred horizontal CT images that were reconstructed from the ‘projection – images’ by this MLEM method, are vertically accumulated, resulting in an instantaneous three-dimensional distribution of flame-chemiluminescence.

The results for a propane-air fuel rich-premixed turbulent flame showed that the flame front is observed to be a thin wrinkled luminous region of 0.6 mm in thickness and that the cusps observed in the horizontal cross-sections correspond to ridges of the three dimensional flame fronts. It was found that the luminosity distribution is quenched along the ridges by Lewis Number effect. Inferring heat release rate or fuel consumption rate from chemiluminescence has long been a common and largely accepted practice.

The emission light has been assumed to be parallel ray for simplification of calculation. Although inclusion of absorption term was possible in MLEM method but, absorption was not considered in this investigation because of self-absorption at bands of CH, HCO and C2 is very weak in small flames. Half width at half maximum (HWHM) width of luminous region is measured at 5 pixels (approx) that is 0.6 mm in width. Luminous region of turbulent pre- mixed flames with weak turbulent intensity was considered to be thinner than pixel size 0.12mm. This difference resulted from

Long exposure time for fast movement

Inaccuracy of CT reconstruction which concerns number of iteration, etc.,

No compensation in emission intensity data processing (especially, assumption of linear proportion between digitalized pixel value and emission intensity)

Optical error (aberrations of lens, halation on film etc.,)

Various types of display of three-dimensional emission intensity distribution data set were performed, for example, bird’s eye views, a vertical sectional view, and horizontal sectional views. These displays helped to recognize the detail and total shape of objective flame.

Three-dimensional temperature field measurement of flame using a single light field camera by Jun Sun, Chuanlong Xu,* Biao Zhang, Md. Moinul Hossain, Shimin Wang, Hong Qi, and Heping Tan

Least Square QR factorization (LSQR) Algorithm: The combustion apparatuses in industries are continuously facing new challenges in order to increase the combustion efficiency, reliability and flexibility, and to reduce their environmental impact. The flame temperature has a direct influence on flame properties such as flame height and width, species and soot concentrations, and hence it is the one of the most important characteristic parameter of the flame closely linked to the performance of the combustion apparatus. Therefore it is desirable to determine the 3-D temperature field of flame for in-depth understanding of the combustion mechanism, and subsequent optimization of combustion process and pollutant formation process. Laser-based techniques are active measurement methods, which employ the measured scattering, absorption and fluorescent signals caused by the laser crossing the flame to derive the temperature.

However, due to the complexity and high cost of the laser-based diagnostic systems, these techniques are generally unsuitable for the applications in hostile industrial environments. In radiative imaging technique, the visible radiation information is usually applied to measure the temperature fields of flames. This technique doesn’t require imposing external signal and hence they are simple in system setup compared with laser-based diagnostic system.

17796562154551A novel method has been proposed for reconstructing three-dimensional (3-D) temperature field of a flame based on a single light field camera. A radiative imaging of a single light field camera is also modeled for the flame. In the model, the principal ray represents the beam projected onto the pixel of the CCD sensor. The radiation direction of the ray from the flame outside the camera is obtained according to thin lens equation based on geometrical optics. The intensities of the principal rays recorded by the pixels on the CCD sensor have been mathematically modeled based on radiative transfer equation. The temperature distribution of the flame was then reconstructed by solving the mathematical model through the use of least square QR-factorization algorithm (LSQR). The numerical simulations and experiments were carried out to investigate the validity of the method that was proposed. The results presented in the study showed that the method is capable of reconstructing the 3-D temperature field of a flame.

Fig 2.1.5 Schematic diagram of the experimental setup.Source: Jun Sun et al.

The simulation results indicated that the relative error of the flame temperature was not greater than 0.5% for the proposed method. Preliminary experiments were also carried out to reconstruct the 3-D temperature field of the ethylene diffusion flame on a purpose-built experimental setup. The results obtained from the experiments indicated that the LSQR method is capable of reconstructing 3-D flame temperature field.

Tomographic Reconstruction of the Luminosity Distribution of a Combustion Flame by Guillermo Gilabert, Gang Lu, Jiaqing Shao, Yong Yan

Filtered Back-Projection algorithm (FBP): Computing algorithms for the tomographic reconstruction of the luminosity distribution of a combustion flame from a limited number of projections has been described using filtered back-projection algorithm. The reconstruction results can be used to determine the 3D geometrical, luminous and fluid- dynamic (temperature and oscillation frequency) parameters of a flame. The process of capturing the light from a combustion flame onto an imaging sensor was found to be physically equivalent to a Radon transformation where a 2D flame cross-section underwent transformation to produce a one-dimensional (1D) section projection.

Computing algorithms for the tomographic reconstruction of the luminosity distribution of a combustion flame from its two-dimensional images captured simultaneously by CCD cameras. Two reconstruction algorithms, filtered back-projection and algebraic reconstruction, were described. A direct comparison between the two approaches was made in respect of their reconstruction effectiveness through computer simulation for a varying number of projections. Images of candle and gaseous flames were used to test the effectiveness of the algorithms developed. The outcome of the investigation led to the establishment of an improved method for the reconstruction of the luminosity distribution of flame cross-sections and ultimately the quantitative characterization of the inter-structures of a combustion flame. The FBP is not sensitive to small angles between projections, In contrast, with ART; the FBP proves itself to be unable to establish the shape and presents a highly fuzzy inner structure of the object for such a limited number of projections.

Estimation of radiative properties and temperature distributions in coal-fired boiler furnaces by a portable image processing system by Wenhao Li, Chun Lou , Yipeng Sun, Huaichun Zhou

Combustion measurement in Coal-Fired Boiler Furnaces: An experimental investigation on the estimation of radiative properties and temperature distributions in a 670 t/h coal-fired boiler furnace by a portable imaging processing system was performed. The portable system had been calibrated by a blackbody furnace. Flame temperatures and emissivities were measured by the portable system and equivalent blackbody temperatures were deduced. Comparing the equivalent blackbody temperatures measured by the portable system and the infrared pyrometer, the relative difference was less than 4%.

The reconstructed pseudo-instantaneous 2-D temperature distributions in two cross-sections could disclose the combustion status inside the furnace. The measured radiative properties of particles in the furnace proved significant scattering in coal-fired boiler furnaces and provided useful information for the calculation of radiative heat transfer and numerical

simulation of combustion in coal-fired boiler furnaces. The preliminary experimental results showed that this technology can be helpful for the combustion diagnosis in coal-fired boiler furnaces.

Both temperature distributions and radiative properties of the particulate medium play key roles in combustion. The temperature distributions in coal-fired boiler furnaces are closely related to pollutant emissions, e.g., NOx, and combustion efficiency. Various laser diagnostic measurement techniques have been applied in combustion system . However, impressive large dimensions of coal-fired furnaces cause significant attenuation for any laser. Besides that, there are many physical constraints such as noise, vibrations, and limited optical access in industrial-scale furnaces. Laser diagnostic techniques are seldom used in experimental investigations in coal-fired furnaces. Radiative properties of particles, such as emissivity, absorption and scattering coefficients, are important parameters in the numerical simulation of combustion and calculation of radiative heat transfer in coal-fired furnaces.

The reconstructed pseudo-instantaneous 2-D temperature distributions in the two cross- sections under the furnace load of 170MW and 140MW have been shown in figure. It can be seen that the 2-D temperature distributions in the horizontal cross-sections of the furnace appear to be a single-peaked shape with temperatures higher in the center and lower near the water-wall surfaces in section A and B. It is due to the four-cornered, tangential combustion. Besides, the 2-D temperature distributions in 170MW are higher than those in 140 MW. This is because more fuels are burned in higher furnace load and it will release more heat in the furnace, which will increase the temperature in the furnace.

(A)

(B)

Fig 2.1.7(A) Schematic of the furnace and the viewing ports, through which flame images were captured by the portable system by turn.(B) Pseudo-instantaneous 2-D temperature distributions in section A and B of the 670 t/h boiler furnace in two furnace loads of 170MW and 140 MW, respectively (K). Source: Wenhao Li et al.

Moreover, the temperatures in the upper burner zone (section B) are lower than those in lower burner zone furnace (section A), which is caused by the combustion of blast gas fed into the furnace from the lower burners. The input parameters include the absorption coefficient, the scattering albedo, and the emissivity of particles. Any uncertainty of these parameters will cause some errors in the simulation results. The measured results of the portable system can provide useful input information for combustion simulation.

The reconstructed pseudo-instantaneous 2-D temperature distributions in the two cross- sections investigated could disclose the combustion status inside the furnace. The measured radiative properties of particles in the furnace showed that there is significant scattering in coal-fired boiler furnaces and it can provide useful information for the calculation of radiative heat transfer and numerical simulation of combustion in coal-fired boiler furnaces. The preliminary experimental results showed that the portable image processing system is flexible and useful for engineers conducting combustion diagnosis in coal-fired boiler furnaces.

Review of different iterative Techniques

The various algorithms used are namely additive Algebraic Reconstruction Technique (ART), Maximization Entropy Technique (MENT), Minimum Energy Method (MEM) etc. The techniques have been discussed in detail below, as follows:

Algebraic Reconstruction Technique (ART)

This reconstruction method is based on series expansions which made its first appearance in the scientific literature and in the computerized tomography (CT) scanner industry in around1970. ART is an example which is frequently used. All algorithms that fall into the ART family are categorized into two major groups, namely additive ART and multiplicative ART (MART). These algorithms are further classified into sub-groups based on how different parts of the algorithms are implemented. The major steps of an ART algorithm is the integration procedure which is used to obtain approximate projections, the calculation of weighting functions, the structure of correction terms and the procedures used to correct the field value. The iterative reconstruction methods are based on the discretization of the cross- sectional plane by a square grid.

Simple ART

This method was suggested by Mayinger F., which is the simplest possible iterative construction algorithm that resembles in many ways to the algebraic reconstruction algorithm. With ??? as the projection due to ????ray with angle of irradiation ? andfi’ as the initial guess of the field value, computation of the approximation projection????using the test field has been carried out.

For each angle of radiation ?

?

???? = ? ???,???

?=1

, where ??? = 1, 2, 3…….M?Eq (1)

Where,?? denotes the ith ray of an irradiation with angle ?, and 1 ????M?. The subsequent steps are as follows:

For each ray??, calculation for the correction is given by:???? = ???? ????

So, average value of correction is ???

????

?

?? =

??

Applying a correction for each cell j of the test field is given asf ????= f ???? + ????

????

Procedure is repeated for all angles of irradiation and that completes the kth global iteration.

?×

Iteration is done until ??+1 ? ??

?

100

?e, where is the stopping criterion.

Table 2.2.1.1 Errors and convergence of Simple Art

Gordon ART

Source: Performance of iterative tomographic algorithms

The ART algorithm that was originally proposed for CT applications by Gordon et al is taken into consideration. In this method corrections are applied to all the cells through which the ith ray passes, before calculating the correction for next ray. Hence the number of rays per angle of irradiation is not important. As per the weight function equation, the approximate projection data is computed as

?

??? = ? ?????

?=1

, where i = 1, 2 …MEq (2)

Where M is the grand total number of rays and N is the number of cells. The remaining part of the algorithm can be stated as follows:

For each kth iteration

After calculating the correction for each ray i, computation for correction coefficient

?=1

is given by ?? = ??

???2

Correction is applied a to each cell j of the test field through which the present ray passes as

f ???? f ???? + ? ??? ???

?= ?

??

As the steps are repeated for all rays, it completes the kth iteration.

?×

Iteration is done until ??+1 ? ??

?

100

?e.

Approximate projection is calculated again using Eq (2) and iteration is carried out until the stopping rule is satisfied.

Gilbert ART

Gilbert independently developed a form of an ART, also called as Simultaneous Iterative Reconstruction Technique (SIRT). In SIRT, the elements of the field function are modified after all the correction values corresponding to individual rays have been calculated. The algorithm is similar to ART but the correction is applied as given below:

After calculating the correction for each ray i, computation for correction coefficient

?=?

is given by ?? = ??

???2

All the rays (Ncj) that pass through a given cell are identified along with its corresponding wij and ???.

The algebraic sum of all possible correction terms is applied to each cell j given as below which completes the kth iteration.

f ???? f ???? + ?Nc??

? ??? ???

?= ?

??=0

??

?×

Iteration is performed until ??+1 ? ??

?

100

?e.

A new value of the approximate projection is calculated using Eq (2) until the stopping criterion is satisfied.

Anderson ART

This method was proposed by Anderson and Kak, a new algorithm also called as Simultaneous ART (SART) which combines the ART and SIRT algorithms. It is found to be very efficient and superior in implementation. The method of applying a correction is similar to simple ART but the structure is similar to ART. The algorithm is as follows:

For each angle of radiation ?,

After calculating the correction for each ray ??, the correction coefficient ??? is computed as:

?=?

??? = ??

???,?2

Above step is repeated for all rays that belong to irradiation with angle?.

Apply a correction to each cell j of the test field as

f ???? = f ???? + ? ??? ????

…..Where, ? is the relaxation factor

??

???

Above steps are repeated for all rays of the irradiation with angle ?. The new value of the approximate projection using Eq (2) is calculated. The procedure is repeated to all angles of irradiation which will complete the kth global iteration.

?×

Iteration is performed until ??+1 ? ??

?

100

?e. Iterations are continued until the

stopping rule is satisfied.

Multiplicative ART (MART)

The correction strategies presented in above are called additive ART (or simply ART). When the correction is multiplicative, the ART is called multiplicative ART (MART). MART with three different types of correction formulae has been implemented. The initial approximate projection is computed using Equation (2). The MART algorithms considered are as follows:

The approximate projection ??? for each ray i is calculated. All the rays passing through a given cell(the total number of rays per cell being Ncj)and corresponding i, wij,?? and ???.

The product of all possible correction terms are computed for each cell j. This can be accomplished in three different ways:

MART 1: f ???? = f ???? × ?1 ? ? (1 ? ?? )

??

???

???

MART 2: f ???? = f ???? × ?1 ? ? ? (1 ? ?? )

??

???

??

???

? ???

MART 3: f ????f ???? × ? ??

?= ?

???

???

?×

Iteration is carried until E = ??+1 ? ??

?

100 ?e

The approximate projection is updated using Equation (2). Iterations are continued until E ? e.

955516285154Table 2.2.1.5 Performance of MART

Source: Performance of iterative tomographic algorithms

Table 2.2.1 Performance of Various ART Methods

Source: Performance of iterative tomographic algorithms

Minimum Energy Method (MEM)

Gull and Newton have suggested four functions which can be extremized to solve reconstruction. Entropy and energy functions are attractive and natural in engineering. The minimum energy method as implemented is described below:

Maximize(– ??

? 2)Subject to ?

?

= ?

?=1

Wij

??

?=1

?

i

Compared to MENT, the energy minimization is simple to formulate. However, Gull and Newton have concluded that this method produces a field which is negatively correlated and hence biased.

880370234230Table 2.2.2 Tomography of Rayleigh-Benard convection: performance of minimum energy

Source: Performance Evaluation of iterative tomographic algorithms

Maximum Entropy Technique (MENT)

Entropy optimization refers to maximizing the function F(x, y, z) given as: F(x, y, z) = ? ? f(?, ?, ?)lnf(?, ?, ?)d?d?d?

In the reconstruction process, following function is maximized using the Lagrange multiplier

method;

F ? = ?? ??(?) ?

?? W? ? ?

???

???ijk ?kj

The above equation is used to find the maxima of the function for all values of fi. The expression for fi is obtained as:

??

= exp {?1 ? ??

?(?)

??

?=1

??

Wijk}

?=1

?=1

Substituting??in the first equation of this page we have,

?kj

?

= ?

?=1

exp {?1 ? ??

?(?)

??

?=1

??

Wijk}

973908288337Table 2.2.3 (1)Tomography of Rayleigh-Benard convection: performance of MENT

Source: Performance Evaluation of iterative tomographic algorithms

2.3 Outcomes of Literature review

The conclusions that we observed after studying various methods of reconstruction techniques using different algorithms is mentioned below, in order to present why and which method do we intend to select for our further project exploration :

MENT and MART 3 performs’ better than all other algorithms, even with noisy data.

Not using 0 and 90° projections in any data set enhances errors and decelerates error decay rates.

Simple ART shows systematic behavior with respect to number of projections, view angle, and noise level. It works with a high value of relaxation. However, its performance is poor when compared to MART.

The minimum energy method predicts a field with high error.

MART and MENT show less sensitivity to noise in input data compared to ART. Noise amplifications are reduced for all algorithms if fewer views and large view angle are considered.

The MENT has following advantages over other methods:

Unlike MART, MENT does not require the evaluation of logarithms and exponentials.

The storage requirements are also lower for MENT.

It yields the image with the lowest information content consistent with the available data.

It is efficient even when the available projection data are incomplete or degraded by noise errors.

MENT has a faster rate of convergence even with noisy data compared to multiplicative ART (MART) used with accurate data.

Henceforth, MENT algorithm should be preferred in our project over other algorithms.

DESIGN METHODOLOGY AND IMPLEMENTATION STRATEGY

As per the discussion in previous chapter, due to advantages of MENT algorithm over other algorithms, in reconstruction of 3D flame shape it should be preferred. Here in this chapter the detailed methodology is discussed for achieving our final object, which includes mathematical representation of image, weight function calculation, MENT algorithm and Monte Carlo sampling. Based on that for implementation purpose the algorithm has been developed which is as follows:

Mathematical representation of images of 3D objects as projections:

A flame can be assumed as a 3D distribution of luminous intensity in a volume, based on this assumption, we choose a distribution of scalar property (fi) representing flame intensity, inside a cubical volume. The cubical volume is discretized into small voxels as shown in Fig. 3.1.

1714499194702

Fig. 3.1. Visualization of a beam passing through a grid Source: Subbarao et al.

The voxels which have a non-zero value of fi represent the portion of the cubical region where the flame intensity is non-zero. A beam of light, assumed to be of constant rectangular cross-section, is shown to intersect the volume and meet on a projection plane. The light beam, during its intersection, may intercept a fraction of volume of many voxels. The projection of each light beam is, therefore, a summation of the scalar field in each voxel multiplied by the weight factor of that voxel.

The image of an object (in the present case, flames) can be viewed as a two dimensional distribution of intensity on a grid. Our aim, while using MENT method, is to utilize these images and reconstruct the original field.

Some basic terms associated with the process of viewing and obtaining information from such a grid are as follows:

ANGLE OF PROJECTION (?, ?):? represents the angle measured from a reference line in the horizontal plane, while?is the angle measured in the vertical plane. Thus the angle of projection helps give the direction from which the projection of the flame is being taken.

ANGLE OF VIEW: The range of projection angles (maximum projection angle-minimum projection angle) possible in a given reconstruction problem is defined as angle of view. Theoretically, the angle of view must be equal to 180? for any reconstruction problem, but in fluid flow and heat transfer problems, because of the size and construction of the test object, it is not possible to cover a complete angle of view. This requires a good algorithm that works with a limited angle of view.

Weight Function (wj,k,i ) and It’s Calculation

As shown in the fig 3.2, consider the jth beam passing through object which is discretized in no. of pixels. Now it may be possible that the beam will pass from only some of the pixels. It is also possible that in some pixels beam will completely pass and in some pixels only fraction of beam will pass. So for calculating that fraction weight function will be useful.

‘The fraction of the area intercepted by a beam with the particular cell in a given projection is called weight function.’

A weighting factor defined by,

w

Fig. 3.2 weight function calculation

Aj,k,i

=

…..(1)

j,k,i

dx.dz

Where, Aj,k,i is the area of intercept for the ith cell of the jth beam in the kth projection, dx and dz

are the dimension of each cell along the x and z directions. Therefore wj,k,i is zero if jth beam does not intersect with the ith cell. If fi is the field value in the ith cell, the projection value of jth beam in the kth projection is given as:

?=1

?jk= ??

wj,k,i

??

…..(2)

This discretization produces a matrix equation:

Wj,k,i{?? } = {?jk}(3)

The problem of tomographic reconstruction thus reduces to inversion of this matrix. Iterative techniques can be thought of as generating a solution of eq. (3) through a generalized equation of matrix Wj,k,i. In the present work, maximum entropy tomography (MENT) algorithm is used to carry out the inverse of the matrix.

Methodology for Maximization of Entropy (MENT)

Principle of maximum entropy: The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge is one with largest entropy, in the context of precisely stated prior data. In some problems where, due to very incomplete data and a very complex process giving rise to the limited data, the probabilistic model has been chosen instead of deterministic model. In this case there are large numbers of models that can fit the data. So the problem arises to choose the best model with the limited data. So according to the solution given by Boltzmann from all the possible models the one which has largest entropy is the best model.

The way maximum entropy tomography (MENT) algorithm has been implemented is as follows:

Consider a continuous function f(x, y, z) which satisfies the condition

f(x, y, z) ? 0(4)

Entropy optimization refers to maximizing the function F(x, y, z) given as:

F(x, y, z) = ? ? ?(?, ?, ?)ln?(?, ?, ?)d?d?d?(5)

Where, f lnf is the entropy of f. Now, if fi is the discrete form of f(x,y,z)then eq. 5 can be written as,

Fi= ?

?

?=1

??ln(fi)(6)

In image reconstruction the collected data and any other priori information comprises the constraint over which entropy is maximized. The aim is to maximize,

? ?

?

?=1

??lnfi(7)

Subject to the projection data in the form of

?=1

?jk= ??

wjki

??

k = 1,2.N and fi ? 0

Satisfying all the constraint represented in the form of projection data for all the beams in all the projections. To incorporate these conditions into one equation introducing an auxiliary function,

F? = ? ?? ? ln fi? ??

??(?) ?

?? W

? ? ?

…..(8)

??=1 ?

?=1

?=1

?,??

jki?jk

Where, i is the cell number, k is the projection number, j is the beam number in the kth projection,

wkijis the weighting factor corresponding to the ith cell, jth beam in kth projection,

fi is the field value for cell i,

?kj is the Lagrangian multiplier and

?jk is the projection value given by Eq 2.

N is the total number of projections, n(k) is the number of beams in the kth projection. M is the total number of cells, and

Akij is the area intercept of the jth beam in the kth projection with the ith cell.

As the auxiliary function is a function of fi and for maximization, it has to be differentiated with respect to fi and equating to zero

-1 – lnfi – ??

??(?) ?w=0

?=1

?=1??jki

– (1 + ln fi) – ??

??(?) ?w

=0…..(9)

?=1

?=1

??

jki

(1 + ln fi) = – ??

??(?) ?w

?=1

?=1??

jki

lnfi = – 1– ??

??(?) ?w

?=1

?=1

??

jki

orfi = ?

?

? 1– ?

?=1

?=1

??(?) ???w

jki(10)

Substituting fi in projection data of Eq (2);

?=1

F? (???) = ?jk = ????,?,? ?

?

? 1– ?

?=1

??(?) ???w

?=1

jki(11)

?11?12?13

?21?22?23 ?jk = ?31?32?33

?1N?1?1N

??2N?1?2N ?3N?1?3N

…..(12)

???

?n1?n2?n3??nN?1?nN

?=1

Where;?11 = ???11 ?

?

? 1– ?

?=1

?=1

??(?) ???w

ijk

?=1

?21 = ???21 ?

?

? 1– ?

?=1

?=1

??(?) ???w

jki

?=1

?n1 = ????1 ?

?

? 1– ?

?=1

?=1

??(?) ???w

kij

Similarly all other elements of Eq.(13) can be determined. For all k’s and all j’s this produces a square matrix with equal no of unknowns Lagrangian multipliers and non-linear simultaneous equations. An iterative inversion is done using Newton’s Raphson Method. The procedure is as follows: each row of the above mentioned matrix represents the contribution of a given beam corresponding to which there is a Lagrangian Multiplier. Initially these Lagrangian multipliers are initialized to some value, say zero, these subsequent iteration values are calculated using:

F ( ?)

?k

? k+1=? k –?? x relaxation(13)

??

??

??

F??( ? k )

These iterations are carried till all the Lagrangian multipliers have converged.

Monte Carlo Sampling Method

The Monte Carlo method, also called Monte Carlo analysis, is a means of statistical evaluation of mathematical functions using random samples. This requires a good source of random numbers . The method is useful for obtaining numerical solutions to problems which are too complicated to solve analytically.

Monte Carlo simulation is also referred to as probability simulation. There is always some error involved with this scheme, but the larger the number of random samples taken, the more accurate the result.

In slice by slice reconstruction, estimating the weight function (wij) requires the calculation of the intersected area between a square pixel (i) and a rectangular ray (j), which has a simple analytical solution. However, in Direct-3D reconstruction, estimating the weight function requires the calculation of the volume of overlap between a cuboidal beam and a cubical voxel. Monte Carlo (MC) Sampling can also be used in such a problem to find the volume of intersection.

In the MC sampling method, points are taken randomly from a domain, such that the region which is being measured is a subset of this domain. The voxel will be taken as the domain and points will be drawn at random from this three dimensional volume. These points will be then checked to see if they lie inside the cuboidal beam or not. If Pin denoted the number of points which were found to lie inside the cuboidal beam and PT denoted the total number of points, the volume of the intersecting region is calculated as:

Vcal = Pin ×Vvoxel

PT

Subbarao et al. concluded that MC sampling was an acceptable method for weight function calculation.

Algorithm for Generating The Theoretical Projections

Algorithm for generating theoretical projections and weight function is described below as a part of implementation strategy. It should be noted that the projection value will be computed as sum of individual cells intersecting with the beam.

1.) Use the input parameters as beam thickness, beam spacing, grid size and field value function.

2.) Specify the angle of projection (theta) for which projection is to be carried out.

3.) Rotate the coordinate frame at angle (theta-pi/2) so that it becomes normal to the beam and the projections could be obtained. Also find minimum and maximum value of x and y coordinates in that rotated frame.

4.) Taking first beam, find it’s coordinates in original frame.

5.) Find the bottom most row number and topmost row number which the beam attains in the body.

6.) Taking 1st row, compute the leftmost and rightmost column which the beam intersects corresponding to it.

7.) Take left bottom most cell in that row from which beam is passing and calculate coordinates of intersecting points at which the left most and right most ray of beam is intersecting with that pixel edges(i.e. A,B,C,D,E,F as shown in fig 3.3).

8.) Now calculate intercepted area of the pixel through which the beam passes.

9.) Multiplying this intercept area with the field intensity of that pixel will give the value of projection for that pixel.

10.) Similarly, one by one, take adjacent cells in that particular row until it attains the right most column of beam and repeat the steps 7 to 9 for all cells and add all the projections for cells.

11.) Repeat the steps 6 to 10 for each and every row until the top most row through which the beam passes.

12.) Similarly, take all the beams one by one and repeat all the steps from 4 to 11.

13.) After summation of the projection values of each and every beam intersecting with each pixel, total projection value for that particular projection angle will thus be obtained.

COMPARISON WITH BENCHMARK PROBLEM

As per the discussion in previous chapter of literature review and design methodology, we decided to use MENT algorithm for reconstruction of flame. So first it is necessary to verify the reliability and accuracy of the MENT algorithm and for doing that we have taken the standard cosine phantom object, whose standard output can be known from it’s defining equation. For same we will create the MENT algorithm and reconstruct it. These both standard and reconstructed results will be compared for verifying the accuracy of MENT algorithm.

Standard Cosine Phantom Object

In the present study, a 3D intensity distribution has been generated using a cosine phantom object with a linear property variation in z-direction and whose shape is similar to flame shape, which is formulated as follows:

f (x,y) = 0.25 {1-cos 2? (1.25 x ( X + 0.4 ))0.8 } {1-cos 2 ? x (1.25 ( y + 0.4 ))0.67 }

where,-0.4 ; x ; 0.4,

-0.4 ; y ;0.4 .

The parameter that is taken to reconstruct the intensity field is as follows: where the size of the field is 40 x 40 grids. The necessary codes for the required result were written in MATLAB.

Result:

3429002845194105275240958The plot of the cosine phantom object performed in MATLAB is as follows:

Fig 4.1 schematic of the cosine phantom objectFig 4.2 Top view of the cosine phantom object

Reconstruction Of Cosine Phantom Using MENT Algorithm

Input parameters to be entered:

Side of square = a

Size of the grid = del_x

Percentage error you want = p_error

Take new projection and enter projection angle ?.

? is converted into radians by multiplying pi/180 Beam thickness = t

Where, t = del_x , when, sin?; 0.001 Otherwise, t = del_x*sin?

Finding minimum and maximum values of x1, x2, y1, y2

x1 is minimum value on x-axis and y1 is minimum value on y-axis x2 is maximum value on x-axis and y2 is maximum value on y-axis, When coordinate axis have been rotated by (?-pi/2)

???? ???? 0

So for the rotation in the clockwise direction R? = ????? ???? 0

0 0 1

Initially, X1, X2, Y1, Y2=0

For point 1(a, 0) =; Angle = ( ? -pi/2)with C.W. rotation

4068586-9542?1? = cos(? ? pi/2)sin(? ? pi/2)?

?1??sin(? ? pi/2)cos(? ? pi/2)0

?

Therefore, ?? = ????(? ? ??/?) and

?

?? = ?????(? ? ??/?)

Now, for point 2(a, a)

?2? = cos(? ? pi/2)sin(? ? pi/2)?

?2??sin(? ? pi/2)cos(? ? pi/2)?

Hence, ??? = a ???(? ? ??/?) + a sin(? ? ??/?)

??? = ?????(? ? ??/?)+ acos(? ? ??/?)

For point 3 (a, 0)

?3? = cos(? ? pi/2)sin(? ? pi/2)0

?3??sin(? ? pi/2)cos(? ? pi/2)?

So, ??? = a ???(? ? ??/?) and ??? = ? ???(? ? ??/?)

Calculation for a particular beam

n is the beam number

a_new is the x-coordinate of the centre of the beam in the rotated frame

So take, a_new = x1 for taking 1st beam in the calculation and repeat the same calculation, Until a_new; (x2 +t/2).

Now as the rotated frame is rotated back to original frame, the new transformed points for (a_new-t/2, y1) and (a_new+t/2, y1) are (ax1, ay1) and (ax2, ay2) respectively.

??1 = cos(? ? pi/2)?sin(? ? pi/2) ???? ? ?/2(1)

??1

sin(? ? pi/2)cos(? ? pi/2)

?1

??2 = cos(? ? pi/2)?sin(? ? pi/2) ???? + ?/2(2)

??2

sin(? ? pi/2)cos(? ? pi/2)

?1

Therefore solving matrices eqn. (1) and (2) we get,

38397151032011ax1 = (a_new – t/2) *???(? ? ??/?) – y1 *???(? ? ??/?); ay1 = (a_new – t/2) * ???(? ? ??/?) + y1 *???(? ? ??/?); ax2 = (a_new + t/2) * ???(? ? ??/?) – y1 *???(? ? ??/?); ay2 = (a_new + t/2)* ???(? ? ??/?) + y1 * ???(? ? ??/?);

The projection value is sum of

individual cells intersecting with the beam.

Let, valuen = array that stores the projection value corresponding to the nth beam.

Here, n has been initialized as 0 So, valuen = 0;

Computation of bottommost and Topmost row number which the beam attains in the body

If,cos (theta) ;0.001, ayy1=0 and ayy2 = floor (a/del_x)*del_x; Else,

Value of ayy1 is selected as minimum of 4 eqns. given below,

For

???1???1 = tan?;???1???1 = tan?; ???1???2 = tan?; ???1???2 = tan?;

0???1

????1

0???2

????2

4189920183836And value of ayy2 is selected as maximum of 4 eqns. given below,

???2???1 = tan?;???2???1 = tan?;

0???1????1

???2???2 = tan?; ???2???2 = tan?;

0???2????2

Hence, final value of ayy1 is taken as maximum of 0 and ayy1, thereafter,

ayy1= floor (ayy1/del_x)*del_x

Final value of ayy2 is taken as minimum of a and ayy2, thereafter,

ayy2= floor (ayy2/del_x)*del_x

Now let, ayyy = ayy1 and do calculation as described below until ayyy;= (ayy2+0.000001)

That is calculation is done row by row.

3845898368650Computation for leftmost and rightmost column which the beam intersects corresponding to it

1312095223455

If sin (theta) ;0.001Then, axx1 = 0; and axx2 = floor (a/del_x)*del_x;

Else, value of axx1 is selected as minimum out of 4 equations that are given below:-

???1???1 = cot?;???1???1

= cot?;

of

???????1????+???_????1

???1???2 = cot?;???1???2

= cot?;

???????2????+???_????2

And value of axx2 is taken as maximum out the four equations given as below:-

???2???1 = cot?;???2???1

= cot?;

???????1????+???_????1

???2???2 = cot?;???2???2

= cot?;

???????2????+???_????2

So the final value of axx1 is taken as maximum of 0 and axx1, thereafter

axx1 = floor (axx1/del_x)*del_x

Similarly, final value for axx2 is taken as minimum of a and axx2, thereafter

axx2 = floor (axx2/del_x)*del_x

Now let, axxx = axx1and do calculation as described below until axxx;= axx2+0.000001

That is calculation is done pixel by pixel in particular row.

4243011418753Take each cells in row simultaneously starting with left-bottom coordinates as (axxx, ayyy) and finds contribution corresponding to all in the projection value.

????1 ????1 =

cos(? ? 1.5707)sin(? ? 1.5707)

????

????

?sin(? ? 1.5707)cos(? ? 1.5707)

????2

And, ????2 =

cos(? ? 1.5707)sin(? ? 1.5707)

????

???? + ???_?

?sin(? ? 1.5707)cos(? ? 1.5707)

Now, check whether bottom left corner (axxx1), lies in the beam using condition, axxx1 ;= (a_new + t/2);; axxx1;= (a_new – t/2)

So, ptxi = axxx and ptyi = ayyy and i = i+1

Now for cell compute 8 different points on the pixel edges if the beam passing through it intersects.

3642242-84917POINT 1: Intersection point of leftmost ray and left edge of the cell

POINT 2: Intersection point of

the rightmost ray and left edge of the cell

POINT 3: Intersection point of

the leftmost ray and top edge of the cell

POINT 4: Intersection point of

the rightmost ray and the top edge of the cell

POINT 5: Intersection point of the leftmost ray and the right edge of the cell POINT 6: Intersection point of the rightmost ray and the right edge of the cell POINT 7: Intersection point of the leftmost ray and the bottom edge of the cell POINT 8: Intersection point of the rightmost ray and the bottom edge of the cell

…..if at all they intersect.

Now beginning with POINT 1

4036578555155Condition for intersection of coordinate for left ray and left edge is checked Ifaxxx1 ;a_new-t/2

;; axxx2 ;a_new-t/2 ||

axxx1 ;a_new-t/2 ;; axxx2;a_new-t/2

Then, Ptxi = axxx; and

ptyi = ay1 + (axxx – ax1)*tan?; i = i+1;

Now beginning with POINT 2

397973661616Condition for intersection of coordinate for rightmost ray and left edge is checked

If axxx1;a_new + t/2 ;; axxx2;a_new +t/2

|| axxx1;a_new + t/2 ;; axxx2 ;a_new+t/2 Then, ptxi = axxx;

42590571171688ptyi = ay2 + (axxx-ax2)*tan? And i=i+1;

Now let axxx1=axxx2;

axxx2= (axxx + del_x)*cos (?-1.5707)

+ (ayyy + del_x)*sin (?-1.5707)

4133072-305077Now it will check whether the top left corner of the cell lies in the beam

If axxx1;=a_new+t/2 ;; axxx1;= a_new-t/2 Ptxi = axxx; ptyi = ayyy;

i=i+1;

Now beginning with POINT 3

4251949421239Condition for intersection of coordinate for leftmost ray and top edge is checked If axxx1 ; (a_new-t/2) ;; axxx2 ;a_new-t/2

|| axxx1; (a_new – t/2) ;; axxx2; (a_new –t/2) Ptyi = ayyy +del_x;

Ptxi = ax1 + (ayyy+del_x-ay1)*cot ? i=i+1;

Now beginning with POINT 4

Condition for intersection of coordinate for rightmost ray and top edge is checked If axxx1;a_new + t/2 ;; axxx2;a_new +t/2

|| axxx1;a_new +t/2 ;; axxx2 ; (a_new +t/2); ptyi = ayyy + del_x;

ptxi = ax2 + (ayyy + del_x – ay2)*cot ? i=i+1;

now let axxx1=axxx2;

axxx2 = (axxx + del_x)*cos (?-1.5707)

+ ayyy*sin (?-1.5707);

Now it will check whether the top right corner of the cell lies in the beam or not.

If axxx1;= (a_new + t/2) ;; axxx1 ; = (a_new –t/2) Then, ptxi = axxx + del_x; ptyi = ayyy + del_x;

i = i+1;

Now beginning with POINT 5

4182888279862Condition for intersection of coordinate for leftmost ray and right edge is checked if axxx1 ; (a_new –t/2) ;; axxx2 ; (a_new –t/2)

|| axxx1 ; (a_new – t/2) ;; axxx2 ; (a_new –t/2) Ptxi = axxx + del_x;

Ptyi = ay1 + (axxx + del_x – ax1)*tan ? i = i +1;

Now beginning with POINT 6

4281482384229Condition for intersection of coordinate for rightmost ray and right edge is checked If axxx1 ; (a_new –t/2) ;; axxx2 ; (a_new + t/2)

|| axxx1 ; (a_new +t/2) ;; axxx2 ; (a_new+t/2) Ptxi = axxx + del_x;

Ptyi = ay2 + (axxx + del_x – ax2)*tan ? And i = i + 1;

3194704232217Now let, axxx1 = axxx2;

axxx2 = axxx * cos (? – 1.5707) + ayyy * sin (? – 1.5707);

4477936-174358Now it will check if the bottom right corner of the cell lies in the beam or not

If axxx1 ;= a_new +t/2 ;; axxx1 ;= a_new – t/2 Ptxi = axxx + del_x; ptyi = ayyy;

i = i+1;

Now beginning with POINT 7

4241284336480Condition for intersection of coordinate for leftmost ray and bottom edge is checked If axxx1 ; (a_new –t/2) ;; axxx2 ; (a_new –t/2)

|| axxx1 ; (a_new –t/2) ;; axxx2 ; (a_new – t/2) Then, ptyi = ayyy;

ptxi = ax1 + (ayyy –ay1)* cot ? i = i+1;

Now beginning with POINT 8

4286570435520Condition for intersection of coordinates for rightmost ray and bottom edge is checked. If axxx1 ; (a_new +t/2) ;; axxx2 ; (a_new +t/2)

|| axxx1 ; (a_new + t/2) ;; axxx2 ; (a_new +t/2) Then, ptyi = ayyy; ptxi = ax2 + (ayyy – ay2)*cot ? i = i + 1;

902465566485Using all this intersection points, intercept area can be found.

Intercept Area = ?ABC + ?ACD + ?ADE + ?AEF

Multiplying this intercept area with field intensity of this pixel will give the projection value corresponding to that pixel.

Summation of projection value for each cell corresponding to each beam will be the final value of projection value for that projection angle.

Currently we have performed up to step 9 successfully in MATLAB and we are yet to obtain our final results.

CONCLUSION

In various medical applications like CT scanner, Head scanner and MRI scanner, images of person’s body can be taken at different angles for 3D reconstruction purpose. There are numerous techniques approached by various researchers in order to get the most accurate results like ART, MART, MENT, MEM, LSQR etc. However, due to limited number of views in heat transfer, fluid science and in combustion processes, out of numerous techniques we found MENT methodology to be most effective due to various advantages that dominated other approaches.

Furthermore in this semester, we have understood the MENT methodology in detail and for verifying its result, we took a cosine phantom object and plotted it in MATLAB. For reconstructing it using MENT, we have also made theoretical algorithm in this semester and we will compare theoretical results with the reconstruction results in MATLAB.

Hence, in the next semester we aim to create an experimental setup, thereby capturing 2-D images of flame at different angles with the help of CCD cameras; we will reconstruct it into 3-D using our algorithm. This will help us to understand 3-D flame shape and improve the combustion phenomena. It will also act as a useful tool in research applications like designing of nozzle and burner, by modifying it, in order to get optimum flame shape and temperature distribution.

In future, we could also analyze dynamic and turbulent behaviors of combustion flame, soot particles fraction, broadband emission and beam steering that will help to improve the efficiency of particular systems like Boilers, I.C. engines, Nuclear reactors etc.

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APPENDIX

Periodic Progress Reports

Patent Search Analysis Report

Design Engineering Canvases