## Experimental and image-inversion optimization aspects of thermal-wave diffraction tomographic microscopy

Optics Express, Vol. 7, Issue 13, pp. 519-532 (2000)

http://dx.doi.org/10.1364/OE.7.000519

Acrobat PDF (374 KB)

### 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

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]

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]

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]

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]

## 2. Theoretical and Numerical Model

**r**and

**r**

_{0}are the coordinates of the observation point and the source point, with respect to the origin,

*T*is the scattering field,

_{s}*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:

*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 R

_{0}. 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. A. Mandelis, “Theory of Photothermal Wave Diffraction and Interference in Condensed Phases,” J. Opt. Soc. Am. A **6**, 298 (1989). [CrossRef]

*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.

**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],

**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**-

**b**‖

_{2}. 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]. 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}-

**b**‖

_{2}and the side constraint ‖

**x**

_{σ}‖

_{2}. 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]

**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. 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]

*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]

## 3. Experimental Method

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]

**13**, 1413 (1997). [CrossRef]

**13**, 1413 (1997). [CrossRef]

## 4. Tomographic Experiments

*α*=1.1×10

^{-5}m

^{2}/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]

*α*=1.55×10

^{-5}m

^{2}/s.

*f*=

*11*Hz with a thermal diffusion length, µ=(α/π

*f*)

^{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]

*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.

*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]

*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.

*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 (x

_{c}=0.15 mm, y

_{c}=0.25 mm). The 0.2-mm deep defect was centered at (x

_{c}=1.5 mm, y

_{c}=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, x

_{f}=0, 0.05, 0.1, 0.15, 0.2, 0.25 mm, and 13 detector positions.

## 5. Numerical Reconstructions

**13**, 1413 (1997). [CrossRef]

*x*

_{1}=0.5, 1.0, 1.5, 2 and 2.5 mm. The experimental amplitude and phase data of the respective laser positions were used as input parameters for the inversion technique. The frequency used in this experiment was

*f*=

*11*Hz with a thermal diffusion length of 0.56 mm. Figure 8 reconstructs the object function

*F*(

*x*,

*y*), with a contour of the reconstruction function at the bottom. The solid line on the contour of Figure 8 indicates the exact location and size of the defect, which was 0.6 mm in diameter, centered at (x

_{c}=1.5 mm, y

_{c}=1.6 mm). The defect is shown at the correct

*x*-position but it extends further to the back of the sample. This is an artifact of the reconstruction, which diminishes as a wider spatial-frequency spectrum range (laser positions) is used. Furthermore, the theoretical contrast was -0.5. On averaging, information was lost because a few reconstructions underestimated the defect contrast, and only limited information was available for reconstructing the defect, since detection occurred at the back surface of the sample only. Therefore, the contrast in Fig. 8 is underestimated. In conclusion, the use of more laser positions for averaging reconstructions, including reflection reconstructions, would ensure better positional accuracy and contrast for the object function. The optimal regularization parameter for the reconstruction was retrieved using the L-curve method. For each reconstruction performed at a specific laser position an L-curve was plotted for selecting the optimal solution corresponding to the corner regularization parameter, s. For one laser position the L-curve is shown in Fig. 9 and the optimal solution is σ=1×10

^{-6}. The average reconstruction was created by an average of all the optimal solutions. Figure 10 is the result of a reconstruction from the “vertical” part of the L-curve plot. The regularization parameter is σ=1×10

^{-8}. Perturbation noise dominates such a solution and the location of the defect is distorted. In Figure 11 an oversmoothed solution (“flat” part of the L-curve) is shown with the regularization parameter being σ=0.1. In this reconstruction the defect is overshadowed by the laser position x=1.5mm.

*ill-defined*problem provides a reasonable solution after being regularized. The regularization parameter is directly proportional to the ill-conditioning of the problem. The next reconstruction was of the reflection data from a 0.1-mm-deep defect at

*f*=80

*Hz*. Figure 12(a) is the average reconstruction of the aforementioned eleven laser positions. The optimal regularization parameter used for reconstruction was 10

^{-4}. All subsequent experimental reconstructions in reflection used this regularization parameter. On the other hand, the optimal value of the regularization parameter for transmission reconstructions (~10

^{-6}) was an indication that the transmission problem was less ill-conditioned than the reflection problem. In Fig. 12 the depth position of the defect at

*f*=80

*Hz*is correctly reproduced but the image is smaller than the true defect size shown by the solid line. Also, it is observed that the defect is somewhat asymmetric on the right hand side. This can be attributed to a slight inherent instrumental asymmetry of the photothermal radiometric microscope. The contrast is underestimated at -0.25. Some artifacts exist at the left and back surfaces of the defect. As the number of laser positions increases, these artifacts may decrease. Figure 12(b) shows the same reflection reconstruction at

*f*=

*300*Hz. Although the front of the defect is reconstructed well, the back boundary is shifted toward the front by about two thermal diffusion lengths (0.26 mm). This can be attributed to the fact that, in reflection, a scatterer can be seen no deeper than about two diffusion lengths. Deeper information is lost, resulting in severe distortions of deep-lying features. The reconstruction is, nevertheless, more symmetrical than that in Figure 12(a), as expected, since for a 0.3-mm scan the asymmetry of the instrument is much diminished.

**13**, 1413 (1997). [CrossRef]

*a*is the radius of the cylindrical defect,

*n*(eq. 2b) is the

_{δ}*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 λ

_{th}=2πµ). Even though there is no rigorous proof that the foregoing criterion is valid for non-propagating parabolic diffusion-wave fields [18], 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.

*f*=

*80*Hz.

**13**, 1413 (1997). [CrossRef]

## 6. Conclusions

*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.

*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]

## Acknowledgments

## References and Links

1. | G. Busse and K.F. Rank, “Stereoscopic Depth Analysis by Thermal Wave Transmission for Non-Destructive Evaluation,” Appl. Phys. Let. |

2. | A. Mandelis, “Theory of Photothermal Wave Diffraction Tomography via Spatial Laplace Spectral Decomposition,” J. Phys. A: Math. General |

3. | A. Mandelis, “Theory of Photothermal Wave Diffraction and Interference in Condensed Phases,” J. Opt. Soc. Am. A |

4. | M. Munidasa and A. Mandelis, “Photopyroelectric Thermal-Wave Tomography of Aluminum with Ray-Optic Reconstruction,” J. Opt. Soc. Am. A |

5. | O. Pade and A. Mandelis, “Computational Thermal-Wave Slice Tomography with Backpropagation and Transmission Reconstructions,” Rev. Sci. Instrum. |

6. | A. Mandelis, “Green’s Functions in Thermal Wave Physics: Cartesian Coordinate Representations,” J. Appl. Phys. , |

7. | L. Nicolaides and A. Mandelis, “Image-Enhanced Thermal-Wave Slice Diffraction Tomography with Numerically Simulated Reconstructions,” Inv. Prob. |

8. | L. Nicolaides, M. Munidasa, and A. Mandelis, “Thermal-Wave Infrared Radiometric Slice Diffraction Tomography with Back-Scattering and Transmission Reconstructions: Experimental,” Inv. Prob. |

9. | A. Mandelis, |

10. | A.N. Tikhonov, “On Stability Of Inverse Problems,” Dokl. Acad. Nauuk USSR , |

11. | B. Hofmann, |

12. | C. Hansen, “Analysis of Disrete Ill-Posed Problems by Means of the L-Curve,” SIAM Review , |

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. | A. Mandelis and M. Mieszkowski, “Thermal Wave Sub-Surface Defect Imaging and Tomography Apparatus,” U.S. Patent Number |

15. | L. Qian and P. Li, “Photothermal radiometry measurement of thermal diffusivity,” Appl. Opt. |

16. | G. Busse, D. Wu, and W. Karpen, “Thermal Wave Imaging with Phase Sensitive Modulated Thermography,” J. Appl. Phys. |

17. | A.C. Kak and M. Slaney, |

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 , |

**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

- G. Busse and K.F. Rank, "Stereoscopic Depth Analysis by Thermal Wave Transmission for Non-Destructive Evaluation," Appl. Phys. Let. 42, 366 (1983). [CrossRef]
- A. Mandelis, "Theory of Photothermal Wave Diffraction Tomography via Spatial Laplace Spectral Decomposition," J. Phys. A: Math. General 24, 2485 (1991). [CrossRef]
- A. Mandelis, "Theory of Photothermal Wave Diffraction and Interference in Condensed Phases," J. Opt. Soc. Am. A 6, 298 (1989). [CrossRef]
- M. Munidasa and A. Mandelis, "Photopyroelectric Thermal-Wave Tomography of Aluminum with Ray-Optic Reconstruction," J. Opt. Soc. Am. A 8,1851 (1991). [CrossRef]
- O. Pade and A. Mandelis, "Computational Thermal-Wave Slice Tomography with Backpropagation and Transmission Reconstructions," Rev. Sci. Instrum. 64, 3548 (1993). [CrossRef]
- A. Mandelis, "Green's Functions in Thermal Wave Physics: Cartesian Coordinate Representations," J. Appl. Phys. 78, 647 (1995). [CrossRef]
- L. Nicolaides and A. Mandelis, "Image-Enhanced Thermal-Wave Slice Diffraction Tomography with Numerically Simulated Reconstructions," Inv. Prob. 13, 1393 (1997). [CrossRef]
- 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]
- A. Mandelis, Diffusion Wave Fields: Green Functions and Mathematical Methods (Springer, New York, in press).
- A.N. Tikhonov, "On Stability Of Inverse Problems," Dokl. Acad. Nauuk USSR, 39, 195 (1943).
- B. Hofmann, Regularization for Applied and Ill-Posed Problems, (Teubner, 1986).
- C. Hansen, "Analysis of Disrete Ill-Posed Problems by Means of the L-Curve," SIAM Review, 34, 561 (1992). [CrossRef]
- 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]
- 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.
- L. Qian and P. Li, "Photothermal radiometry measurement of thermal diffusivity," Appl. Opt. 29, 4241 (1990). [CrossRef] [PubMed]
- G. Busse, D. Wu and W. Karpen, "Thermal Wave Imaging with Phase Sensitive Modulated Thermography," J. Appl. Phys. 71, 3962 (1992). [CrossRef]
- A.C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, (IEEE Press, New York, 1988).
- A. Mandelis, "Diffusion waves and their uses," Physics Today, 29, August 2000.
- E. Miller, L. Nicolaides and A. Mandelis, "Nonlinear Inverse Scattering Methods for Thermal-Wave Slice Tomography: A Wavelet Domain Approach," J. Opt. Soc. Am. A, 15, 1545 (1998). [CrossRef]

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