Stabilizing Laplacian Inversion in Fokker-Planck Image Retrieval using the Transport-of-Intensity Equation

Abstract: X-ray attenuation, phase, and dark-field images provide complementary information. Different experimental techniques can capture these contrast mechanisms, and the corresponding images can be retrieved using various theoretical algorithms. Our previous works developed the Multimodal Intrinsic Speckle-Tracking (MIST) algorithm, which is suitable for multimodal image retrieval from speckle-based X-ray imaging (SBXI) data. MIST is based on the X-ray Fokker-Planck equation, requiring the inversion of derivative operators that are often numerically unstable. These instabilities can be addressed by employing regularization techniques, such as Tikhonov regularization. The regularization output is highly sensitive to the choice of the Tikhonov regularization parameter, making it crucial to select this value carefully and optimally. Here, we present an automated iterative algorithm to optimize the regularization of the inverse Laplacian operator in our most recently published MIST variant, addressing the operator's instability near the Fourier-space origin. Our algorithm leverages the inherent stability of the phase solution obtained from the transport-of-intensity equation for SBXI, using it as a reliable ground truth for the more complex Fokker-Planck-based algorithms that incorporate the dark-field signal. We applied the algorithm to an SBXI dataset collected using synchrotron light of a four-rod sample. The four-rod sample's phase and dark-field images were optimally retrieved using our developed algorithm, eliminating the tedious and subjective task of selecting a suitable Tikhonov regularization parameter. The developed regularization-optimization algorithm makes MIST more user-friendly by eliminating the need for manual parameter selection. We anticipate that our optimization algorithm can also be applied to other image retrieval approaches derived from the Fokker-Planck equation.