A new algebraic reconstruction technique (ART), nonlinear auto-adjusting iterative reconstruction technique (NAIRT), is proposed and applied to reconstruct a section of an actual thermal air flow field. With numerical simulation, NAIRT was tested to reconstruct a complicated field to demonstrate its superior reconstructive capability. In contrast, three typical ARTs, the basic ART, simultaneous ART (SART), and a modified SART (MSART), were simulated to demonstrate the reconstructive capability improvement attained through the use of the proposed NAIRT. The calculated results were discussed with mean square error (MSE) and peak error (PE). A thermal air flow field was produced with an alcohol burner and was detected by a laser beam. With laser beam projections, a cross-section of the field was reconstructed by NAIRT. As a result, the reconstructive capability was improved much by NAIRT. The MSE decreased by 95.5%, and PE by 97.2% from that of the basic ART. Only NAIRT converged without filters while its reconstructive accuracy improved. By increasing the projections from 42 to 84, the accuracy of NAIRT without filters was improved significantly. NAIRT could effectively reconstruct the section of the thermal field. The proposed NAIRT needed no filter for its convergence and it had the highest reconstructive accuracy and simplest iterative expression of those analyzed.
The random-direction partial derivatives in optical deflection equations were transformed into numerical differences in
reference frames for tomography. The nonlinear deflection equations were transformed into linear tomography ones. A
detecting ray will turn its propagating direction when it runs through a heterogeneous refractive index field. Its deflecting
angle a is the function of refractive index n by n's first order partial derivative. So, the optical deflection equation
involves nonlinear first order partial derivative. This kind of detecting ray equations can't be resolved by tomography
algorithm directly. At first, the nonlinear partial derivative should be transformed into numerical difference. Here, a
practical transforming algorithm was put forward. The diagnosed field was divided into tiny foursquare grids. Each grid
and its refractive index were approximated to a correct cone with an irregular bottom. With the approximation, the space
partial increment calculation was much simplified at any grid, in any direction and to any detecting ray. It was assumed
that the refractive index distribution should be coplanar in the area between three grid centers of the three close-adjacent
grids. With the assumption, the refractive index partial increment could be calculated with a numerical difference
function of close-adjacent grid refractive indexes. With the approximation and assumption, the partial derivative was
transformed into numerical difference. As the result, partial derivative related to any detecting ray could be transformed
into numerical difference. Nonlinear deflection equations could be transformed into linear difference ones. So, the
deflected angles can directly be applied to reconstruction as projections.
It was demonstrated that the lowest sampling frequency fmin_sample doesn't exist in mathematics. With mathematical
analysis, the principles of sampling and reconstructing a continuous signal were strictly calculated. We found that the
spectra of sampling data at critical sampling frequency fcritical_sample should overlap at the highest frequency fmax of the
continuous signal. The fcritical_sample was defined as double of the fmax, viz. fcritical_sample=2fmax. As we know, the
reconstructed signal will be distorted with this kind of overlapped spectra. Here, we will further illustrate the theoretical
results. Aided with Fast Fourier Transform(FFT), the critical sampling and the process reconstructing continuous-time
signal from it were discussed by spectroscopy. A symmetrical frequency-limited spectrum F(ω) was constructed with
three modified rise-cosine pulses. Its corresponding time-domain signal f(t) was worked out theoretically. f(t) was
sampled with δT(t). By modifying T, the critical sampling signal was obtained. With FFT, the spectrum Fd(ω)of the
sampling signal was figured out. The calculated Fd(ω) was compared with the constructed F(ω), and was analyzed for
observing frequency alias. A cycle of Fd(ω) for restoring the continuous signal could be obtained when Fd(ω) was
filtered by an ideal low-passed filter. With FFT, a continuous signal was reconstructed from it. As the results, the spectra
of sampling data at the fcritical_sampleoverlapped at the fmax. The reconstructed signal distorted obviously. So, the lowest
sampling frequency fmin_sample doesn't exist. The sampling theorem couldn't include equal sign. It is unscientific to say
that the fmin_sample equal to double of the fmax.
A new deflection tomography algorithm was suggested and tested with a simulated flow field. The programs
calculating deflection projection and inverse projection were worked out based on optical refraction principle and
mathematical, physical significance of tomography. With our home-made Simple Self-correlative Algebraic
Reconstruction Technique (SSART), a new deflection tomography algorithm was programmed and named auto-adapted
deflection tomography system. A section of a complex flow field was simulated with Gauss and rectangle window
functions. One positive and one negative Gauss peaks were constructed on the section in order to make the model
contain double-polar components. One square tower was constructed on the section in order to make the model contain
much more high-frequency waves. The deflection projections were figured out according to the tomography algorithm
with direction interval one degree. So, we got 180 direction projections in total. 12 projections were selected out by
direction step 15 degree, and 84 projections were picked out by direction interval 2 or 3 degree. The section was
reconstructed with the projections by SSART. The reconstructed results were compared with the model. The
reconstructive effect was estimated with Mean-square error(MSE) and Peak error(PE). As the result, the system could
reconstruct the simulated field accurately. With twelve projections, MSE was about from 0.0001 to 0.0006 at the end of
300 cycle iterations, and PE was about from 0.007 to 0.020. With selected 84 projections, the MSE was 0.00000999, and
PE was 0.00013142. So, the auto-adapted deflection tomography system can accurately reconstruct complex flow fields.
A new deflection tomography system was put forward and named Auto-adapted Iterative Deflection Tomography
(AIDT). It was tested to approach accurate reconstruction of a high frequency field containing shock waves with
sufficient projections. A high frequency field was constructed with Gauss and rectangle window functions. It was
simulated to project in all directions with angle resolving rate per degree. Two cross projections was selected to simulate
reconstructing the model field by AIDT. Then, the number of projections was increased step by step in order to approach
the accurate values of the model. All reconstructed results were studied. As a result, all high frequency sections are
distorted much. With a few projections, such as two projections, the reconstructed results lost the basic characteristics of
the model. With projections increasing, the distortions get lighter and lighter. Near the critical projections, such as
twenty-four projections, the reconstructed field becomes similar to the model except some high frequency sections.
When eighty-four projections are employed, the reconstructed result is in accord with the model. In this case, the Mean
Square Error(MSE) is 0.00003609. But up to one hundred and eighty projections, the reconstructed result almost
stopped improving. In this case, the MSE is 0.00003231. As we know, these are most accurate reconstructions to now.
AIDT can accurately reconstruct high frequency fields with shock waves.
A new iterative reconstruction algorithm is developed and applied to moiré deflection tomography for flow field measurements. The algorithm is derived from the basic deflection formula and based on a modified algebraic reconstruction technique. The precision and convergence of the algorithm are analyzed through a numerical simulation. To capture multidirectional projection data, a rotatable deflectometric system is developed. The efficacy of the new algorithm is assessed by reconstructing an asymmetric temperature field. Furthermore, the algorithm is employed to investigate the image of a cross section of a free jet containing steep gradients and to reconstruct the density field of the rocket exhausted jet.