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Optics Express

  • Editor: J. H. Eberly
  • Vol. 7, Iss. 13 — Dec. 18, 2000
  • pp: 519–532
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Experimental and image-inversion optimization aspects of thermal-wave diffraction tomographic microscopy

Lena Nicolaides and Andreas Mandelis  »View Author Affiliations


Optics Express, Vol. 7, Issue 13, pp. 519-532 (2000)
http://dx.doi.org/10.1364/OE.7.000519


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Abstract

Thermal-wave Slice Diffraction Tomography (TSDT) is a photothermal imaging technique for non-destructive detection of subsurface cross-sectional defects in opaque solids in the very-near-surface region (µm-mm). Conventional reconstructions of the well-posed propagating wave-field tomographies cannot be applied to the ill-posed thermal wave problem. Photothermal tomographic microscopy is used to collect experimental data that are numerically inverted with the Tikhonov regularization method to produce thermal diffusivity cross-sectional images in materials. Multiplicity of solutions, which is inherent to ill-posed problems, is resolved by adopting the L-curve method for optimization. For tomographic imaging of sub-surface defects, a new high-resolution radiometric setup is constructed, which reduces the broadening of images associated with previous low-resolution setups.

© Optical Society of America

1. Introduction

Thermal-wave inverse problems are ill-posed problems, meaning they either have no solution, no unique solution or unstable solution. The main difficulty with such problems is that a regularization method is needed to isolate a unique solution. The ill-posed problem examined in this work is associated with Thermal-wave Slice Diffraction Tomography (TSDT), a relatively new cross-sectional photothermal imaging method for the detection of sub-surface defects in materials through mapping of the cross-sectional slice thermal diffusivity. Images obtained from conventional thermal wave imaging are two dimensional “projections” of subsurface features in the sense that the image is formed by mapping the sample surface temperature in a two dimensional raster scan without regard to the actual depth position of scatterers. Cross-sectional photothermal tomographic imaging refers to a line scan that obtains information about a sample cross-section normal to its surface plane. Although some early work has been done in obtaining elementary depth information on sub-surface features (defects) [1

1. G. Busse and K.F. Rank, “Stereoscopic Depth Analysis by Thermal Wave Transmission for Non-Destructive Evaluation,” Appl. Phys. Let. 42, 366 (1983). [CrossRef]

], it was not until the 1990s that TSDT was introduced in this Laboratory in a concerted effort to develop a viable tomographic imaging technology using thermal waves [2

2. A. Mandelis, “Theory of Photothermal Wave Diffraction Tomography via Spatial Laplace Spectral Decomposition,” J. Phys. A: Math. General 24, 2485 (1991). [CrossRef]

8

8. L. Nicolaides, M. Munidasa, and A. Mandelis, “Thermal-Wave Infrared Radiometric Slice Diffraction Tomography with Back-Scattering and Transmission Reconstructions: Experimental,” Inv. Prob. 13, 1413 (1997). [CrossRef]

]. Multiplicity of solutions was found to be an inherent feature to the (ill-posed) thermal-wave tomographic reconstruction problem. As a result, a “brute-force” approach [7

7. L. Nicolaides and A. Mandelis, “Image-Enhanced Thermal-Wave Slice Diffraction Tomography with Numerically Simulated Reconstructions,” Inv. Prob. 13, 1393 (1997). [CrossRef]

,8

8. L. Nicolaides, M. Munidasa, and A. Mandelis, “Thermal-Wave Infrared Radiometric Slice Diffraction Tomography with Back-Scattering and Transmission Reconstructions: Experimental,” Inv. Prob. 13, 1413 (1997). [CrossRef]

] was used to obtain a proper (reasonably stable) solution. However, in order to pinpoint a reasonable solution, it is important to adapt a rigorous optimization approach. This is especially important, if TSDT is to be successfully implemented for practical and industrial applications. In this work, a suitable optimization algorithm in TSDT is developed in conjunction with a high-resolution photothermal radiometric experimental tomographic setup. The purpose of such a high-resolution setup is to overcome the problem of image broadening associated with earlier low-resolution experimental set-ups [8

8. L. Nicolaides, M. Munidasa, and A. Mandelis, “Thermal-Wave Infrared Radiometric Slice Diffraction Tomography with Back-Scattering and Transmission Reconstructions: Experimental,” Inv. Prob. 13, 1413 (1997). [CrossRef]

]. The optimization algorithm is necessary for speeding up the image reconstruction procedure under optimal sensitivity and noise conditions.

2. Theoretical and Numerical Model

A forward theory of the thermal wave propagation process has been developed utilizing thermal wave diffraction as an integral solution of the pseudo-wave Helmholtz equation [2

2. A. Mandelis, “Theory of Photothermal Wave Diffraction Tomography via Spatial Laplace Spectral Decomposition,” J. Phys. A: Math. General 24, 2485 (1991). [CrossRef]

,3

3. A. Mandelis, “Theory of Photothermal Wave Diffraction and Interference in Condensed Phases,” J. Opt. Soc. Am. A 6, 298 (1989). [CrossRef]

]. The theoretical model that represents the behavior of a three dimensional thermal wave [6

6. A. Mandelis, “Green’s Functions in Thermal Wave Physics: Cartesian Coordinate Representations,” J. Appl. Phys. , 78 (2), 647 (1995). [CrossRef]

,9

9. A. Mandelis, Diffusion Wave Fields: Green Functions and Mathematical Methods (Springer, New York, in press).

] is correlated with the Born approximation utilizing the Tikhonov regularization method to invert almost singular matrices resulting from the ill-posedness of the inverse thermal-wave problem [7

7. L. Nicolaides and A. Mandelis, “Image-Enhanced Thermal-Wave Slice Diffraction Tomography with Numerically Simulated Reconstructions,” Inv. Prob. 13, 1393 (1997). [CrossRef]

]. The Born approximation results in the following integral equation [7

7. L. Nicolaides and A. Mandelis, “Image-Enhanced Thermal-Wave Slice Diffraction Tomography with Numerically Simulated Reconstructions,” Inv. Prob. 13, 1393 (1997). [CrossRef]

]

Ts(r)=V0G0(rr0)Ti(r0)F(r0)dV0
(1)

where r and r 0 are the coordinates of the observation point and the source point, with respect to the origin, Ts is the scattering field, G 0 is the appropriate Green function. F(r) is the object function and it represents the inhomogeneities of scattering object region R. The object function F(r) is zero at every point outside region R and has a non-zero value that represents the ratio of thermal diffusivities inside region R. If the inhomogeneous region R is removed from boundary S, then F(r) would be zero everywhere and thus the medium inside the volume enclosed by the boundary S would be totally homogeneous. Therefore, F(r) is defined as follows:

F(r)={q02[n2(r)1];rR0;rR
(2a)

with

n(r)=α0α(r)
(2b)

and

q0=(1+i)(ω2α0)12
(2c)

where q 0 is the complex wavenumber and n(r) is a measure of the variation of the thermal diffusivity values in the scattering object R from that of the surrounding (reference) region R0. The ratio in equation (2b) has been symbolized by n(r) deliberately, to suggest the analogy of this parameter to the effects of variations in the refractive index in conventional optical propagating fields [2

2. A. Mandelis, “Theory of Photothermal Wave Diffraction Tomography via Spatial Laplace Spectral Decomposition,” J. Phys. A: Math. General 24, 2485 (1991). [CrossRef]

,3

3. A. Mandelis, “Theory of Photothermal Wave Diffraction and Interference in Condensed Phases,” J. Opt. Soc. Am. A 6, 298 (1989). [CrossRef]

]. The object function reconstructed is the non-dimensional function n 2(r)-1, referred to as the contrast of the image. The background thermal diffusivity, α, thermal conductivity, k, modulation frequency, f, and laser beam size, w, are the necessary input parameters for calculating the homogeneous thermal-wave field.

Equation (1) is defined by a system of linear equation as Ax=b, where A is an ill-conditioned matrix that represents the discretized version of the intergral equation (1), b is the experimental known vector and x is a sought solution. To solve the ill-posed problem, Tikhonov’s regularization method is used and the regularized solution, x σ, as proposed by Tikhonov is [10

10. A.N. Tikhonov, “On Stability Of Inverse Problems,” Dokl. Acad. Nauuk USSR , 39(5), 195 (1943).

],

xσ=min{Axb2+σL(xx0)2}
(3)

where x 0 is an initial estimate of the solution, and matrix L is either the identity matrix I or a discrete approximation to a derivative operator. The regularization parameter, σ, controls the weight given to minimization of the side constraint, ‖L(x-x 0)‖2, relative to minimization of the residual norm, ‖Ax-b2. For this work, no particular knowledge about the desired solution is available, so x 0=0 is used; also matrix L is set as the identity matrix, I. In an ill-posed problem, small perturbations in the experimental data cause large perturbations in the solution [11

11. B. Hofmann, Regularization for Applied and Ill-Posed Problems, (Teubner, 1986).

]. Therefore, a regularization method must be implemented to stabilize the problem. The fundamental idea of Tikhonov regularization is to introduce a trade-off between the size of the residual norm ‖Ax σ-b2 and the side constraint ‖x σ2. By choosing a suitable regularization parameter, σ, a satisfactory solution can be found for which the two constraints are balanced [12

12. C. Hansen, “Analysis of Disrete Ill-Posed Problems by Means of the L-Curve,” SIAM Review , 34, 561 (1992). [CrossRef]

]. Too much regularization leaves out information actually available in b while too little regularization produces a solution dominated by errors. One method for choosing the optimal regularization parameter is the L-curve method [12

12. C. Hansen, “Analysis of Disrete Ill-Posed Problems by Means of the L-Curve,” SIAM Review , 34, 561 (1992). [CrossRef]

,13

13. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487 (1993). [CrossRef]

]. This is an a posteriori method based on plotting the side constraint of the solution versus the residual norm for a particular regularization parameter. A schematic of the L-curve is shown in Figure 1 and, as can be seen, the name “L-curve” is derived from the fact that the curve is L-shaped. The “corner” of the curve corresponds to a regularization parameter that is optimal [12

12. C. Hansen, “Analysis of Disrete Ill-Posed Problems by Means of the L-Curve,” SIAM Review , 34, 561 (1992). [CrossRef]

]. The “flat” part of the L-curve is dominated by regularization errors occurring from oversmoothing and the “vertical” part is dominated by perturbation errors occurring from undersmoothing. Thus the corner of the L-curve corresponds to a solution in which there is a fair balance between the regularization and perturbation errors.

Fig. 1. L-curve method: the corner of the curve corresponds to optimum regularization.

3. Experimental Method

The experimental technique needed to acquire the scanned data for thermal-wave tomograms is described briefly. The data are obtained at one cross-section from different laser and detector positions. Historically, the first detection method used to obtain tomographic slice data was photopyroelectric detection [4

4. M. Munidasa and A. Mandelis, “Photopyroelectric Thermal-Wave Tomography of Aluminum with Ray-Optic Reconstruction,” J. Opt. Soc. Am. A 8,1851 (1991). [CrossRef]

,14

14. A. Mandelis and M. Mieszkowski, “Thermal Wave Sub-Surface Defect Imaging and Tomography Apparatus,” U.S. Patent Number 4, 950, 897; Date: August 21, 1990.

], which was limited to transmission measurements. A photothermal infrared radiometric detection method then followed. However, a disadvantage was that a low-spatial-resolution setup was used, which resulted in image broadening [8

8. L. Nicolaides, M. Munidasa, and A. Mandelis, “Thermal-Wave Infrared Radiometric Slice Diffraction Tomography with Back-Scattering and Transmission Reconstructions: Experimental,” Inv. Prob. 13, 1413 (1997). [CrossRef]

]. In the low-spatial-resolution set-up the beam spot size was focused to about 50µm and the dimensions of the detector active area were 50µm×50µm. The major disadvantage of the system was that the collected infrared radiation with paraboloidal mirrors, was focused to a spot size much larger (~70µm) than the detector active area. This not only led to a significant loss of signal but also made the detector size the limiting factor of the observed image area. When a detector with an active area of 1mm×1mm was used [8

8. L. Nicolaides, M. Munidasa, and A. Mandelis, “Thermal-Wave Infrared Radiometric Slice Diffraction Tomography with Back-Scattering and Transmission Reconstructions: Experimental,” Inv. Prob. 13, 1413 (1997). [CrossRef]

], there was no signal loss, but a broader image was seen. This broadening, due to the detection size, was improved by constructing a new high-resolution experimental configuration for photothermal radiometric tomographic application. In the new set-up the infrared radiation is collected by two diffraction-limited reflecting objectives as oppose to the paraboloidal mirrors. Figure 2 illustrates the experimental configuration. A 514.5-nm cw Innova Ar+ laser from Coherent is modulated and then focused onto a sample to a spot size of approximately 27 µm. The modulation is performed by an external acousto-optic modulator (AOM) (ISOMET 1201E-1). The blackbody radiation from the optically excited sample is collected and collimated by the two axially aligned reflecting objectives. It is then focused onto a liquid-nitrogen-cooled photoconductive HgCdTe (Mercury-Cadmium-Telluride) detector (EG & G Judson Model J15D12-M204-S050U). The HgCdTe detector has an active square size of 50 µm×50 µm and a bandwidth of 2–12µm. The detector is operated at a cryogenic temperature of 77 K since its efficiency increases with decreasing temperature. An antireflection-coated germanium window with a transmission bandwidth of 2–14 µm is mounted in front of the detector to block any visible radiation from the pump laser. Before being sent to the digital lock-in amplifier (Stanford Research System Model SR850), the photothermal-radiometric signal is amplified by a preamplifier (EG&G Judson Model PA-300) specifically designed for operation with the HgCdTe detector. The lock-in amplifier, which is interfaced with a personal computer, receives the preamplifier output and rejects all stray signals except those that have the same modulation frequency as the excitation beam. This process of data acquisition, storage, and scanning is fully automated. The instrumentation has the ability to perform in either reflection or transmission mode by directing the laser beam to the front or rear surface of the sample using removable mirrors.

Fig. 2. High-resolution tomographic radiometric microscope experimental set-up.

In the experimental set-up, both the modulated heating source and the detector are localized, and therefore can be scanned independently. For each laser position, data are collected at several detector positions along a straight line. This defines the cross-sectional plane (tomographic scan) to be imaged, e.g. line PQ in Figure 3. The scan for the same cross-section is repeated for different laser positions. The experimental information is then used to reconstruct the respective cross sectional slice, using the method outlined in section 2. In practice, moving the sample becomes equivalent to moving the detector. For example, the transmission tomographic scan is performed by first positioning the laser and then moving the sample and laser together over the detector. Figure 4 illustrates a tomographic scan in transmission with three detection points and three laser positions, which ensures that the same cross-section is maintained at all times. When detector and laser are aligned with each other the signal is highest. For laser position #1, the maximum occurs at point 3; for laser position #2, the maximum occurs at point 2; and for laser position #3, the maximum occurs at point 1. These three scans (or more) can be used to reconstruct the experimental tomographic scan. In reflection mode the tomographic scan is achieved by attaching a 45° mirror to the first reflective objective, Fig. 2. The dead optical field that the reflective objective produces limits the size of the mirror. For this reason the dimensions of the mirror are 3 mm 2 mm, limiting the tomographic scan to 1.5 mm (3 mm/2). Experimentally, this limit was found to be 0.7 mm and thus tomographically only a maximum cross-section of 0.7 mm can be investigated at a time with the current set-up.

Fig. 3. Line scan for cross-sectional imaging, of sample thickness d with a round subsurface defect.
Fig. 4. Schematic of experimental method of thermal-wave tomographic scans for 3 laser and 3 detection positions.

4. Tomographic Experiments

Steel is a good thermal emitter and can be reasonably assumed homogeneous for testing the tomographic method. Two samples of a mild steel alloy with subsurface defects were used for the tomographic experiments. Sample 1 contained a subsurface hole of diameter 0.6 mm located 0.1 mm below the surface and had a thickness of 2 mm. The thermal diffusivity of the sample was measured in transmission and was found to be α=1.1×10-5m2/s, in agreement with literature reports [15

15. L. Qian and P. Li, “Photothermal radiometry measurement of thermal diffusivity,” Appl. Opt. 29, 4241 (1990). [CrossRef] [PubMed]

]. Since the experimental setup in reflection was limited by the tomographic range, a second steel sample (similar geometry as Sample 1) was customized to investigate the reflection experiments. Sample 2 had two 0.3-mm diameter defects, 0.1 mm and 0.2 mm deep in the material, respectively. The thermal diffusivity of this steel material was measured in transmission and was found to be α=1.55×10-5m2/s.

Fig. 5. Amplitude and phase in transmission of the line scan, f=11 Hz.

First, Sample 1 was tested in transmission at a modulation frequency, f=11 Hz with a thermal diffusion length, µ=(α/πf)1/2=0.56 mm. The tomographic scan was taken along a constant cross section. The defect imaged lay 0.1 mm from the front surface and was 0.6 mm in diameter. Figure 5 shows a conventional line scan along the imaged line. A line scan is achieved by scanning the laser and the detector together. This provides information about the defect’s x-location. Both the amplitude and phase of the scan exhibit a minimum at the defect location. This behavior is due to the fact that the defect (air) is a poor thermal conductor and thus acts as a thermal impedance to heat propagation in transmission. In general, the amplitude of the signal is influenced by surface blemishes and reflectance, whereas the phase is largely unaffected and truly represents subsurface defects [16

16. G. Busse, D. Wu, and W. Karpen, “Thermal Wave Imaging with Phase Sensitive Modulated Thermography,” J. Appl. Phys. 71, 3962 (1992). [CrossRef]

]. From the line scan data the defect is found to be at the center of the scan (x=1.5 mm). Figure 6 represents amplitude and phase images of the transmission tomographic scan with five laser positions along the cross-section (3 mm) at 49 detector points. The five laser positions used were x f=0.5, 1, 1.5, 2, 2.5 mm. From the tomographic scan, the information given from a line scan can also be obtained: In the former scan, the maximum signal always occurs when the laser and detector are aligned with each other. Therefore, the locus of maxima at all laser positions can produce the line scan. This is the diagonal line (x=y) in the experimental data as seen in Figure 6, where a minimum is observed for both amplitude and phase. These are equivalent to the line scan of Fig. 5.

Fig. 6. Transmission tomographic spatial-frequency scan of 0.1-mm deep defect at f=11 Hz. Amplitude and phase.

The homogeneous field of the samples, i.e the background field with no defect, was also investigated for both transmission and reflection modes. Figure 7 shows the field in reflection for f=80 Hz. The response is shown on a two-dimensional graph. Each laser position corresponds to a maximum in the experimental data. The theoretical fields, equations (20a) and (20b) in Ref [7

7. L. Nicolaides and A. Mandelis, “Image-Enhanced Thermal-Wave Slice Diffraction Tomography with Numerically Simulated Reconstructions,” Inv. Prob. 13, 1393 (1997). [CrossRef]

] for reflection and transmission, respectively, are also shown at each laser position for f=80 Hz. The agreement between theory and experiment at all frequencies and experimental modes is excellent although there is a slight instrumental asymmetry on the right-hand side of the experimental data.

Reflection tomographic scans at modulation frequencies of 80 Hz and 300 Hz were performed subsequently for two defects, 0.1-mm and 0.2-mm deep. The x-location of the defects was always at the center of the scan. The tomographic scans in reflection at 80 Hz were performed for 11 laser positions, x f=0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5 mm, and 21 detector positions. The length of this scan was 0.5 mm and the thermal diffusion length at 80 Hz was 0.25 mm. The 0.1-mm deep defect was centered at (xc=0.15 mm, yc=0.25 mm). The 0.2-mm deep defect was centered at (xc=1.5 mm, yc=0.35 mm). Similar tomographic experiments were performed at the same locations using 300 Hz. The thermal diffusion length at this frequency was 0.13 mm. The length of this scan was confined to 0.3 mm, since there was no signal at greater scanning distances. The reflection tomographic scans were thus performed for 7 laser positions, xf=0, 0.05, 0.1, 0.15, 0.2, 0.25 mm, and 13 detector positions.

Fig. 7. Homogeneous field of sample #2 at f=80 Hz. Experiment (square); Theory (eq. (20a) in Ref.[6]) (solid).

5. Numerical Reconstructions

Fig. 8. TSDT transmission reconstruction of Figure 6, averaged over 5 laser positions. True defect shown by solid line; average regularization parameter σ~1×10-6.
Fig. 9. L-curve of x=1.5mm laser position reconstruction of Fig. 8; optimal regularization parameter σ~1×10-6.
Fig. 10. Reconstruction from the vertical part of the L-curve (Fig. 9). Regularization σ=1×10-8.
Fig. 11. Reconstruction from the flat part of the L-curve (Fig. 9). Regularization σ=0.1.

The Born approximation seems to be adequate for the materials and defect geometries utilized in this investigation. Assuming tubular (cylindrical) defects, such as drilled holes in aluminum, Kak and Slaney [17

17. A.C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, (IEEE Press, New York, 1988).

] have given a mathematical condition for the validity of the first Born approximation for general propagating hyperbolic wave fields that obey the conventional Helmholtz wave equation. This condition can be expressed as [8

8. L. Nicolaides, M. Munidasa, and A. Mandelis, “Thermal-Wave Infrared Radiometric Slice Diffraction Tomography with Back-Scattering and Transmission Reconstructions: Experimental,” Inv. Prob. 13, 1413 (1997). [CrossRef]

,17

17. A.C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, (IEEE Press, New York, 1988).

],

anδ<λ4
(4)

where a is the radius of the cylindrical defect, nδ (eq. 2b) is the change in the refractive index between the surrounding medium and the defect, which in this case is the square root of the ratio of background to defect thermal diffusivity; λ is the probe field wavelength (in the thermal-wave case λth=2πµ). Even though there is no rigorous proof that the foregoing criterion is valid for non-propagating parabolic diffusion-wave fields [18

18. A. Mandelis, “Diffusion waves and their uses,” Physics Today, 29, August 2000.

], for all the cases examined the criterion was calculated in Table 1. It can be seen the criterion was valid for all the cases examined in this work. One issue that arises from this Table is that the criterion for reconstruction #2 is identical to that for reconstruction #4 and the criterion for reconstruction #3 is identical to #5. These cases have the same size defect located at different depths. With thermal waves the criterion of relation (4) is not strictly true, since the depth of the defect is as important as its size. In the future perhaps a better criterion can be formulated on the validity of the thermal wave Born approximation, which would include the depth of the defect.

Based on these observations, a “guide” can be formulated for optimal thermal-wave tomographic reconstructions. Reflection reconstructions can better resolve defects close to the surface. The probing frequency, and thus the thermal diffusion length, must be chosen with care. Large defects compared to the thermal diffusion length do not reconstruct as well as when the defect diameter is on the order of one thermal diffusion length. If the depth of the defect is greater than the thermal diffusion length, the defect will not be resolved depthwise. Therefore, for an optimum reconstruction the depth and size of the defect must be of the same order of one thermal diffusion length. This is consistent with the fact that in the cases examined the optimal reconstruction was that of a 0.2 mm deep defect at f=80 Hz.

Table 1. Born approximation validity for all thermal-wave reconstructions.

table-icon
View This Table
Fig. 12. Average reconstruction of 0.1-mm deep defect with 0.15-mm diameter (front is the bottom and back is the top of image): a) f=80 Hz, b) f=300 Hz. Average reconstruction of 0.2-mm deep defect with 0.15-mm diameter: c) f=80 Hz, d) f=300 Hz.

6. Conclusions

In this work, a robust computational method for TSDT was developed. Most importantly, the L-curve method was used to resolve the difficulty of multiplicity of solutions, thereby obtaining an optimal solution to the inverse problem. Furthermore, a new high-resolution photothermal microscope, suitable for tomographic scans and limited by the beam spot size (w=27 µm), was constructed. Machined defects were made in mild steel samples to test the performance of TSDT. It was concluded that when the defect depth and size were of the order of one thermal diffusion length, optimal reflection reconstructions were obtained.

Photothermal radiometric tomography was shown to be a method of collecting cross-sectional data, which are then numerically processed with the use of Tikhonov regularization to produce a tomogram. The implications of this work, besides establishing a reliable experimental foundation of TSDT, are quite broad in that it has the potential to address several other fields of diffusion-wave tomographic science where strong dispersive attenuation presents a reconstruction problem. These fields include medical diffuse photon-density-wave tomography, optical coherence imaging, attenuated acoustical imaging, and geophysical tomography. There are aspects that require further research in order to make TSDT a practical diffraction tomographic technique: Developments are presently being pursued to improve both the computational and experimental methodologies. There are several other methods besides Tikhonov regularization that can be investigated for solving the ill-conditioned thermal wave problem. Such methods of regularization are the truncated Singular-Value Decomposition (SVD) and the maximum-entropy, which are the subject of many publications [11

11. B. Hofmann, Regularization for Applied and Ill-Posed Problems, (Teubner, 1986).

]. A wavelet-based method developed by Miller et al. [19

19. E. Miller, L. Nicolaides, and A. Mandelis, “Nonlinear Inverse Scattering Methods for Thermal-Wave Slice Tomography: AWavelet Domain Approach,” J. Opt. Soc. Am. A , 15, 1545 (1998). [CrossRef]

] appears to be very promising for data inversion and cross-sectional thermal-diffusivity tomographic imaging.

Acknowledgments

This work has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), through a research grant.

References and Links

1.

G. Busse and K.F. Rank, “Stereoscopic Depth Analysis by Thermal Wave Transmission for Non-Destructive Evaluation,” Appl. Phys. Let. 42, 366 (1983). [CrossRef]

2.

A. Mandelis, “Theory of Photothermal Wave Diffraction Tomography via Spatial Laplace Spectral Decomposition,” J. Phys. A: Math. General 24, 2485 (1991). [CrossRef]

3.

A. Mandelis, “Theory of Photothermal Wave Diffraction and Interference in Condensed Phases,” J. Opt. Soc. Am. A 6, 298 (1989). [CrossRef]

4.

M. Munidasa and A. Mandelis, “Photopyroelectric Thermal-Wave Tomography of Aluminum with Ray-Optic Reconstruction,” J. Opt. Soc. Am. A 8,1851 (1991). [CrossRef]

5.

O. Pade and A. Mandelis, “Computational Thermal-Wave Slice Tomography with Backpropagation and Transmission Reconstructions,” Rev. Sci. Instrum. 64, 3548 (1993). [CrossRef]

6.

A. Mandelis, “Green’s Functions in Thermal Wave Physics: Cartesian Coordinate Representations,” J. Appl. Phys. , 78 (2), 647 (1995). [CrossRef]

7.

L. Nicolaides and A. Mandelis, “Image-Enhanced Thermal-Wave Slice Diffraction Tomography with Numerically Simulated Reconstructions,” Inv. Prob. 13, 1393 (1997). [CrossRef]

8.

L. Nicolaides, M. Munidasa, and A. Mandelis, “Thermal-Wave Infrared Radiometric Slice Diffraction Tomography with Back-Scattering and Transmission Reconstructions: Experimental,” Inv. Prob. 13, 1413 (1997). [CrossRef]

9.

A. Mandelis, Diffusion Wave Fields: Green Functions and Mathematical Methods (Springer, New York, in press).

10.

A.N. Tikhonov, “On Stability Of Inverse Problems,” Dokl. Acad. Nauuk USSR , 39(5), 195 (1943).

11.

B. Hofmann, Regularization for Applied and Ill-Posed Problems, (Teubner, 1986).

12.

C. Hansen, “Analysis of Disrete Ill-Posed Problems by Means of the L-Curve,” SIAM Review , 34, 561 (1992). [CrossRef]

13.

C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487 (1993). [CrossRef]

14.

A. Mandelis and M. Mieszkowski, “Thermal Wave Sub-Surface Defect Imaging and Tomography Apparatus,” U.S. Patent Number 4, 950, 897; Date: August 21, 1990.

15.

L. Qian and P. Li, “Photothermal radiometry measurement of thermal diffusivity,” Appl. Opt. 29, 4241 (1990). [CrossRef] [PubMed]

16.

G. Busse, D. Wu, and W. Karpen, “Thermal Wave Imaging with Phase Sensitive Modulated Thermography,” J. Appl. Phys. 71, 3962 (1992). [CrossRef]

17.

A.C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, (IEEE Press, New York, 1988).

18.

A. Mandelis, “Diffusion waves and their uses,” Physics Today, 29, August 2000.

19.

E. Miller, L. Nicolaides, and A. Mandelis, “Nonlinear Inverse Scattering Methods for Thermal-Wave Slice Tomography: AWavelet Domain Approach,” J. Opt. Soc. Am. A , 15, 1545 (1998). [CrossRef]

OCIS Codes
(100.6950) Image processing : Tomographic image processing
(290.3200) Scattering : Inverse scattering

ToC Category:
Focus Issue: Diffuse optical tomography

History
Original Manuscript: October 27, 2000
Published: December 18, 2000

Citation
Lena Nicolaides and Andreas Mandelis, "Experimental and image-inversion optimization aspects of thermal-wave diffraction tomographic microscopy," Opt. Express 7, 519-532 (2000)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-13-519


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References

  1. G. Busse and K.F. Rank, "Stereoscopic Depth Analysis by Thermal Wave Transmission for Non-Destructive Evaluation," Appl. Phys. Let. 42, 366 (1983). [CrossRef]
  2. A. Mandelis, "Theory of Photothermal Wave Diffraction Tomography via Spatial Laplace Spectral Decomposition," J. Phys. A: Math. General 24, 2485 (1991). [CrossRef]
  3. A. Mandelis, "Theory of Photothermal Wave Diffraction and Interference in Condensed Phases," J. Opt. Soc. Am. A 6, 298 (1989). [CrossRef]
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