An adaptive multiscale inverse scattering approach to photothermal depth profilometry

Photothermal depth profilometry is formulated as a nonlinear inverse scattering problem. Starting with the one-dimensional heat diffusion equation, we derive a mathematical model relating arbitrary variation in the depth-dependent thermal conductivity to observed thermal wavefields at the surface of a material sample. The form of the model is particularly convenient for incorporation into a nonlinear optimization framework for is particularly convenient for incorporation into a nonlinear optimization framework for recovering the conductivity based on thermal wave data obtained at multiple frequencies. We develop an adaptive, multiscale algorithm for solving this highly ill-posed inverse problem. The algorithm is designed to produce an accurate, low-order representation of the thermal conductivity by automatically controlling the level of detail in the reconstruction. This control is designed to reflect both (1) the nature of the underlying physics, which says that scale should decrease with depth, and (2) the particular structure of the conductivity profile, which may require a sparse collection of fine-scale components to adequately represent significant features such as a layering structure. The approach is demonstrated in a variety of synthetic examples representative of nondestructive evaluation problems seen in the steel industry.

[1]  Curtis R. Vogel,et al.  Ieee Transactions on Image Processing Fast, Robust Total Variation{based Reconstruction of Noisy, Blurred Images , 2022 .

[2]  Per Christian Hansen,et al.  Analysis of Discrete Ill-Posed Problems by Means of the L-Curve , 1992, SIAM Rev..

[3]  Eric L. Miller,et al.  Wavelet‐based methods for the nonlinear inverse scattering problem using the extended Born approximation , 1996 .

[4]  Philip E. Gill,et al.  Practical optimization , 1981 .

[5]  Jan Thoen,et al.  Photoacoustic frequency‐domain depth profiling of continuously inhomogeneous condensed phases: Theory and simulations for the inverse problem , 1991 .

[6]  Jan Thoen,et al.  Quantitative photoacoustic depth profilometry of magnetic field‐induced thermal diffusivity inhomogeneity in the liquid crystal octylcyanobiphenyl , 1991 .

[7]  Weng Cho Chew,et al.  Comparison of the born iterative method and tarantola's method for an electromagnetic time‐domain inverse problem , 1991, Int. J. Imaging Syst. Technol..

[8]  T. Habashy,et al.  Rapid 2.5‐dimensional forward modeling and inversion via a new nonlinear scattering approximation , 1994 .

[9]  Alternating heat diffusion in thermophysical depth profiles: multilayer and continuous descriptions , 1998 .

[10]  Eric L. Miller,et al.  Nonlinear inverse scattering methods for thermal-wave slice tomography: a wavelet domain approach , 1998 .

[11]  Truong Q. Nguyen,et al.  Wavelets and filter banks , 1996 .

[12]  Gene H. Golub,et al.  Matrix computations , 1983 .

[13]  H. G. Walther,et al.  Theory of microstructural depth profiling by photothermal measurements , 1995 .

[14]  Eric L. Miller,et al.  Simultaneous multiple regularization parameter selection by means of the L-hypersurface with applications to linear inverse problems posed in the wavelet transform domain , 1998, Optics & Photonics.

[15]  T. Habashy,et al.  Beyond the Born and Rytov approximations: A nonlinear approach to electromagnetic scattering , 1993 .

[16]  W. Chew Waves and Fields in Inhomogeneous Media , 1990 .

[17]  Andreas Mandelis,et al.  Generalized methodology for thermal diffusivity depth profile reconstruction in semi‐infinite and finitely thick inhomogeneous solids , 1996 .

[18]  A. Mandelis Hamilton–Jacobi formulation and quantum theory of thermal wave propagation in the solid state , 1985 .

[19]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .