Development of advanced regularization methods to improve photoacoustic tomography

Abstract

Photoacoustic tomography (PAT) is a scalable imaging modality having huge potential for imaging biological samples at very high depth to resolution ratio, thereby playing pivotal role in the areas of neuroscience, cardiovascular research, tumor biology and evolution research. The crucial step in PAT is the image reconstruction or the solving the inverse problem. The reconstruction can be performed by using analytical and model-based methods. The reconstruction schemes like backprojection, filtered backprojection, time reversal, delay and sum, or Fourier-based inversion have shown potential in providing qualitative reconstructions with an advantage of having lower computational complexity, but fails in irregular geometries and limited data scenarios. Model based reconstruction involves inverting a model-matrix that is generated either using impulse response or discretizing the solution of wave equation. Inversion in limited data scenarios is difficult due to ill-conditioned nature of the problem. Therefore typically prior statistics about the image is applied in form of regularization during the inversion. The prior works have attempted to choose the regularization in an automated fashion by minimizing some error metric like residual. In contrary, other schemes were proposed to mitigate the effects of regularization by using deconvolution approach using model-resolution matrix. Another perspective of regularization lies in its ability to define the resolution characteristic in the imaging domain. The resolution characteristics are heavily influenced by factors like ultrasound transducer sensitivity field, depth dependent fluence, bandwidth of the detector, and detector position etc. This thesis work attempts to develop advanced regularization methods that were based on numerical models as well as semi-norm of the data-fidelity terms The first half of thesis proposes two regularization schemes, developed with the standard Tikhonov framework, that are spatially varying to address problems pertaining to robustness to noise characteristics in the data and non-uniform resolution arising due to limited tomographic measurement positions. Model information is utilized to perform a model-resolution based spatially varying regularization having potential to mitigate resolution concerns arising due to limited detection positions. Secondly, fidelity embedded regularization, based on orthonormality between the columns of system matrix, is studied to perform robust reconstruction without necessarily requiring the noise statistics in the acquired data. The reconstruction schemes were compared with Tikhonov and total-variation based methods using numerical simulation and in-vivo mice data. The performance of the proposed spatially varying regularization schemes were superior (with upto 2 dB SNR improvements) than the Tikhonov/total-variation based regularization. The second half of this thesis work is based on singular value decomposition (SVD) which is widely used in regularization methods to know about the filtering applied to its spectral (eigen) values of the system. The state of the art methods like Tikhonov, total variation and sparse recovery based schemes assume equal weight to all the singular values (in the data fidelity term) irrespective of the amount of noise in the data. A fractional framework was developed, wherein the singular values are weight using a fractional power. The fractional power controls the amount of damping or smoothness in the reconstructed solution. The fractional framework was implemented for Tikhonov, `1-norm and total-variation a-priori constraints. In this work, automated way of choosing the fractional power was developed. Both theoretically and with numerical experiments it was shown that the fractional power is inversely related to the data noise level for fractional Tikhonov scheme. The fractional framework was on-par/outperforms the standard reconstructions i.e. Tikhonov, `1-norm and total-variation on numerical simulations, experimental phantoms and in-vivo mice data using figure of merits like contrast to noise ratio (CNR) and Pearson correlation (PC).