Fractional Regularization to Improve Photoacoustic Tomographic Image Reconstruction

Photoacoustic tomography involves reconstructing the initial pressure rise distribution from the measured acoustic boundary data. The recovery of the initial pressure rise distribution tends to be an ill-posed problem in the presence of noise and when limited independent data is available, necessitating regularization. The standard regularization schemes include Tikhonov, ℓ
₁
-norm, and total-variation. These regularization schemes weigh the singular values equally irrespective of the noise level present in the data. This paper introduces a fractional framework to weigh the singular values with respect to a fractional power. This fractional framework was implemented for Tikhonov, ℓ
₁
-norm, and total-variation regularization schemes. Moreover, an automated method for choosing the fractional power was also proposed. It was shown theoretically and with numerical experiments that the fractional power is inversely related to the data noise level for fractional Tikhonov scheme. The fractional framework outperforms the standard regularization schemes, Tikhonov, ℓ
₁
-norm, and total-variation by 54% in numerical simulations, experimental phantoms, and in vivo rat data in terms of observed contrast/signal-to-noise-ratio of the reconstructed images.