Learned Block Iterative Shrinkage Thresholding Algorithm for Photothermal Super Resolution Imaging

TL;DR

The benefits of using a learned block iterative shrinkage thresholding algorithm that is able to learn the choice of regularization parameters are shown and the use of the learned block-sparse optimization approach provides smaller normalized mean square errors for a small fixed number of iterations than without learning.

Abstract

Block-sparse regularization is already well-known in active thermal imaging and is used for multiple measurement based inverse problems. The main bottleneck of this method is the choice of regularization parameters which differs for each experiment. To avoid time-consuming manually selected regularization parameter, we propose a learned block-sparse optimization approach using an iterative algorithm unfolded into a deep neural network. More precisely, we show the benefits of using a learned block iterative shrinkage thresholding algorithm that is able to learn the choice of regularization parameters. In addition, this algorithm enables the determination of a suitable weight matrix to solve the underlying inverse problem. Therefore, in this paper we present the algorithm and compare it with state of the art block iterative shrinkage thresholding using synthetically generated test data and experimental test data from active thermography for defect reconstruction. Our results show that the use of the learned block-sparse optimization approach provides smaller normalized mean square errors for a small fixed number of iterations than without learning. Thus, this new approach allows to improve the convergence speed and only needs a few iterations to generate accurate defect reconstruction in photothermal super resolution imaging.

Figures (9)

• Figure 1: Exemplary specimen with defects shown as blackened stripes. A laser line array with twelve laser lines is used. The illumination pattern differs from measurement to measurement in the illuminated position and the number of laser lines (indicated by red area) which are switched on (randomly chosen). The dashed frame around the pattern indicates the covered illuminated area if all laser lines would be switched on.
• Figure 2: Numerical studies: Datasets used for training - case 1. All shown values and variables are unitless. Further, ${n\in \{1,\,\dots,\,N_r\cdot N_{\text{meas}}\}}$, ${d \in \{1,\,\dots,\,N_d\}}$. (a) A randomly normal Gaussian distributed measurement matrix A, (b) random uniform generated block-sparse matrix $X$, (c): result from the product of (a) and (b), (d): part of (a), whereby matrix $A^1$ is from ${A = [A^1,\,\dots,\,A^{32}]}$ with ${N_{\text{meas}} = 32}$, (e): part of (b) with matrix $X^1$ from ${X = [X^1,\,\dots,\,X^{32}]}$, (f): an exemplary block-sparse vector extracted from (b) for batch number $5$.
• Figure 4: The gain in dB using untied learning instead of tied learning is shown for different measurement numbers. The red dotted line indicates a gain of ${0\,\text{dB}}$ meaning that no improvement has been achieved using untied instead of tied learning. Positive gains indicate a degradation of the NMSE applying untied instead of tied learning.
• Figure 5: Exemplary synthetic training and synthetic test data for case 2 (convolution) as well as NMSE performance studies. (a): shape of used $\Phi_{\text{reduc}}$ in training, (b): comparison of the curve shapes for exemplary training datasets with e.g. ${x_{1,\,\text{train}}^{:,1} \in \mathbb{R}^{N_r}}$, (c): comparison of the curve shapes for exemplary test datasets, (d): reconstruction of $x_{\text{test}}$ using convolution-based tied LBISTA network ($6$ iterations/layers) vs BISTA ($1000$ iterations), (e): NMSE over iterations for BISTA with $95\,\%$ confidence interval, (f): NMSE over iterations for tied LBISTA with $95\,\%$ confidence interval
• Figure 6: Test data from photothermal structured illumination measurements. To obtain this image, averaging over the vertically arranged pixels as well as applying the maximum thermogram method (see Ref. ndteolensr) to eliminate the time dimension is necessary. At position = ${41\,\text{mm}}$ a blue marker line can be observed which refers to the marker line on the investigated surface of the specimen (see Fig. \ref{['erklaerbild']}).