Heat diffusion blurs photothermal images with increasing depth

TL;DR

Photothermal imaging faces a depth-dependent blur because heat diffusion acts as a Wiener process that both spreads the mean temperature distribution and introduces fluctuations that erase information about the initial state. The authors connect entropy production to information loss through a fluctuation–dissipation framework, deriving a k-space cutoff $k_{ ext{cut}}$ and a corresponding spatial resolution $oldsymbol{ extdelta}_r(t)\,oldsymbol{=oldsymbol{ extpi}}\,igl(oldsymbol{ extalpha} t/oldsymbol{ extln}( ext{SNR})igr)^{1/2}$; they also formulate the thermal PSF via a complex wavenumber $oldsymbol{ extsigma}(oldsymbol{ extomega})=igl(1+iigr)rac{1}{oldsymbol{ extmu}(oldsymbol{ extomega})}$. To overcome this limit, they propose and evaluate strategies including increasing SNR, virtual-wave reconstruction with SAFT, and regularized sparse/positivity-constrained inversion via ADMM, as well as end-to-end and hybrid deep-learning approaches; experiments with graphite bars show that integrating priors and learning-based methods yields notable resolution gains. The framework provides a principled path to higher-resolution subsurface imaging in photothermal modalities and highlights the trade-offs between bandwidth, regularization, and prior information. The work thereby links diffusion physics, information theory, and modern reconstruction techniques to extend the capabilities of photothermal and related imaging methods.

Abstract

In this tutorial, we aim to directly recreate some of our "aha" moments when exploring the impact of heat diffusion on the spatial resolution limit of photothermal imaging. Our objective is also to communicate how this physical limit can nevertheless be overcome and include some concrete technological applications. Describing diffusion as a random walk, one insight is that such a stochastic process involves not only a Gaussian spread of the mean values in space, with the variance proportional to the diffusion time, but also temporal and spatial fluctuations around these mean values. All these fluctuations strongly influence the image reconstruction immediately after the short heating pulse. The Gaussian spread of the mean values in space increases the entropy, while the fluctuations lead to a loss of information that blurs the reconstruction of the initial temperature distribution and can be described mathematically by a spatial convolution with a Gaussian thermal point-spread-function (PSF). The information loss turns out to be equal to the mean entropy increase and limits the spatial resolution proportional to the depth of the imaged subsurface structures. This principal resolution limit can only be overcome by including additional information such as sparsity or positivity. Prior information can be also included by using a deep neural network with a finite degrees of freedom and trained on a specific class of image examples for image reconstruction.

Figures (12)

• Figure 1: One realization of the Wiener process with the particle starting at $x=0$ (a) and 500 realizations using reflecting boundaries at a distance of $\pm$20 (b).
• Figure 2: Left: Occupation number representation of the Wiener process for 5000 particles starting at $x=0$ at $t=20$ (a) and $t=100$ (b). For comparison the Gaussian distribution with a variance proportional to time is shown. For $t=100$ the reflecting boundaries at a distance of $\pm$20 can be already recognized by occupation numbers significantly increasing compared to the Gaussian distribution. Right: Galton board to demonstrate the occupation number representation. Licensed as Creative Commons BY-SA 4.0: Matemateca (IME/USP)/Rodrigo Tetsuo Argenton.
• Figure 3: Occupation number representation of the Wiener process for $N=5000$ particles starting at $x=0$ at $t=1000$ using 40 cells. At that time the influence of the reflecting boundaries at a distance of $\pm$20 dominates. For comparison the uniform distribution with the mean value 125 (5000 particles divided by 40 cells) and the erroroverline at $\pm$ the standard deviation is shown. The standard deviation is the square root of the variance given in Eq. (\ref{['Eq:Occupation']}) using $p_i=1/40$ for the uniform distribution.
• Figure 4: Occupation number representation of the Wiener process for for $N_\text{equi} =10 000, N_\text{cell} =40$ and $N_0=1000$ at a time $t=20$. For comparison the mean value ± the standard deviation (square root of variance) from Eq. (\ref{['Eq:Wiener']}) is shown, where the variance in a good approximation is constant at $N_\text{equi}/N_\text{cell}$.
• Figure 5: Quantum entanglement explains how probability comes into play. A global state of a large isolated system, called universe, which is in a quantum pure state, can be separated into a small diffusive system entangled with a larger environment. The reduced state of the diffusive system is given by tracing out the environment, and is almost indistinguishable from the thermodynamic state given by statistical mechanics.