Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Some concept of Optical Measurement Techniques in Thermal Sciences are Absorption Techniques, Alternative Approaches, Calibration Details, Computerized Tomography, Convolution Backprojection. Main points of this lecture are: Convolution Backprojection, Tomography, Interferometry, Computerized Tomography, Three-Dimensional Temperature, Temperature Field, Interferomeric Projections, Transmission, Tomography, Ray Configuration
Typology: Slides
1 / 14
This page cannot be seen from the preview
Don't miss anything!
Simple ART Gordon ART Gilbert ART Anderson ART
The three-dimensional temperature field can be reconstructed from its interferomeric projections using principles of tomography. Tomography is the process of recovery of a function from a set of its line integrals evaluated along some well-defined directions. In interferometry, the source of light (the laser) and the detector (CCD camera) lie on a straight line with the test cell in between. Further a parallel beam of light is used. This configuration is called transmission tomography and the ray configuration as the parallel beam geometry [28]. Tomographic algorithms used in interferometry reconstruct two- dimensional fields from their one-dimensional projections. Reconstruction is then applied sequentially from one plane to the next until the third dimension is filled.
Tomography can be classified into: (a) transform (b) series expansion, and (c) optimization methods. Transform methods generally require a large number of projections for a meaningful answer [92]. In practice, projections can be recorded either by rotating the experimental setup or the source-detector combination. In interferometry, the latter is particularly difficult and more so with the Mach–Zehnder configuration. With the first option, It is not possible to record a large number of projections, partly owing to inconvenience and partly due to time and cost. Hence, as a rule, a large number of projections cannot be acquired with interferometry and one must look for methods that converge with just a few projections. Limited-view tomography is best accomplished using the series expansion method [29]. As limited-view tomography does not have a unique solution, the algorithms are expected to be sensitive to the initial guess of the field the start the iterations. Optimization-based algorithms are known to be independent of initial guess, but the choice of the optimization functional plays an important role in the result obtained. Depending on the mathematical definition used, the entropy extremization route may yield good results, while the energy minimization principle may be suitable in other applications. For the algebraic techniques considered in the present study, an unbiased initial guess such as a constant profile was seen to be good enough to predict the correct temperature field. A complete random number guess can also be viewed as an unbiased initial guess. Tomography begin an inverse technique, was seen to preserve (and amplify under certain conditions) the noise in the initial data. The dominant trend in the field variable was seen to be however captured during tomography inversion.
The first integral in the form given above is divergent with respect to the spatial frequency practical implementation of the formula replaces , where W is a window function that vanishes outside the interval The cut-off frequency can be shown to be inversely related to the ray-spacing for a consistent numerical calculation of the integral. When the filter is purely of the band-pass type , the Radon formula can be cast as a convolution integral:
(19)
where
and
The inner integral over s is one dimensional convolution and the outer integral, an averaging operation over is called back projection .This implementation of the convolution backprojection algorithm is commonly used in medical imaging. Applications of the CBP algorithm to flow and heat transfer problems have been reviewed by Munshi [94].
Series expansion methods are perhaps the most appropriate tomographic technique for interferometry since they work limited projection data. These methods are iterative in nature and consist necessarily of four major steps, namely:
initial assumption of the field to be reconstructed over a grid, calculation of the correction for each pixel, application of the correction, and test for convergence,
The central idea behind the calculation of the correction (step2) is the following. With the assumed field, one can explicitly compute projection values by numerical integration. The difference between the computed projection and experimentally recorded projection data is a measure of the error in the assumed solution this error can be redistributed to the pixels so that error is reduced to zero. Repetition of these steps is expected to converge to a meaningful solution. The series expansion techniques differ only in the manner in which the errors are redistributed over the grid. The word convergence in step 4 is used in an engineering sense as a stopping criterion for the iterations, and not in the strict mathematical sense, where a formal proof is needed to show convergence of the numerical solution to the exact solution The iterative methods require the discretization of the plane to be reconstructed by a rectangular grid (Figure 4.66). The length of the intercept of the i-th ray with the j–th cell in given projection is known as the weight function w (^) ij. It can be shown that
(20)
where refers to the projection data. The number of unknowns N in most cases is much larger than the number of unknowns M. This discretization produces a matrix equation
(21)
The problem of reconstruction thus is a problem of inversion of a rectangular matrix. Iterative techniques that are used in the tomography can be viewed as developing a generalized inverse of the matrix This matrix in a typical laser tomography problem has large dimensions for the differentially heated fluid layer, the greatest matrix size encountered was This is a sparse matrix with many of its elements being zero. General purpose matrix libraries cannot be used to invert such matrices since they are highly ill-conditioned and rectangular in structure. The tomographic algorithms can be seen as a systematic route towards a meaningful inversion of the matrix equation.
Various ART algorithms are available in the literature owing their origin to Kacz-marz [95] and Tanabe [96]. They differ from each other in the way the correction is applied. Those presented below have been tested successfully by the author and his coworkers in the context of interferometry.
Simple ART
This algorithm has been suggested by Mayinger [6]. The corrections are applied through a weight factor. Computed as an average correction along a ray. The deference between the calculated projections with the measured projection data gives the total correction to be applied for a particular ray. The average correction is then the contribution to each cell falling in the path of the ray. This is computed by dividing the total correction obtained with the length of the ray. The calculated projection are computed once for a particular angle. Though are field values are continuously updated the calculated projection values remain unchanged until the completion of all the rays for a given angle. This algorithm will be referred to as ART1 in future discussions.
Let be the projection due to the^ -th ray in the^ direction of projection and^ be the initial guess of the field value. Numerically the projection using the current field values is defined as:
(22)
The individual steps in the algorithm are listed below. Calculate the total value of weight function along each ray as:
Start: 1 For each projection angle Start:2 For each ray Start:3 For each cell
Close: Close: Close:
Start:4 start iterations Start:5 For each projection angle Start:6 For each ray
Compute the numerical projection (Equation 21)
Close: Start:7 For each ray
Calculate the correction as:
Calculate the average of correction as:
Close: Start:8 For each ray Start: 9 For each cell If is non –zero then
where is a relaxation factor. Close: Close: Close: Check for convergence as: If
(where is the prescribed convergence, say STOP: Else: continue close:
Gilbert ART
Gilbert [98] has developed independently a form ART known as SIRT (Simultaneous Iterative Reconstruction Algorithm). In SIRT, the elements of the field function are modified after all the corrections corresponding to individual pixels have been calculated. This will be referred to as ART3. The numerically generated projection is computed once for all the angles and gets updated only after the completion of calculations through all the rays. For each ray from all angles, all the cells are examined to look for those rays which pass through a particular cell. For each cell, the rays which pass through it will contribute a correction that is decided by the weight factor. The algebraic average of all these corrections is implemented on the cell. This procedure will be called ART3. Its individual steps are: Calculate the total value of weight function along each ray as: Start:1 For each projection angle Start:2 For each ray Start:3 For each cell
Close: 3 Close: 2 Close: 1 Start: 4 Start iterations Start: 5 For each projection angle Start: 6 For each ray Compute the numerical projection (Eqution21) Calculate the correction as:
Close: 6 Close: 5 Start:7 For each cell
Identify all the rays passing through given cell (j)let be the total number of rays passing through the th cell)and corresponding and
Apply correction as:
Where is relaxation factor. Close: 7 Cheek for convergence as: If
Anderson ART
Anderson and Kak [99] have proposed a variation to the ART algorithm. This algorithm is abbreviated as SART (Simultaneous Algebraic Reconstruction Technique). The method of implementing the correction is similar to ART1. The only difference this algorithm has from ART 1is in the calculation of correction for each cell The weight factor used here is the exact intersection of a ray with the concerned cell. In contrast, ART1 uses the average correction for all the cells. This algorithm will be referred to as ART4. The individual steps are: Calculate the total value of weight function along each ray as: start:1 For each projection angle start: 2 For each ray start: 3 For each cell
close: 3 close: 2 close: 1 start: 4 start iterations start: 5 For each projection angle start: 6 For each ray Compute the numerical projection (Equation21) close: 6 start: 7 For each ray Calculate the correction as:
start:8 For each cell If non-zero then:
where is a relaxation factor. close: 8 close: 7 close: 5 Check for convergence as: If
Else: continue close: 4
