Computationally efficient error estimate for evaluation of regularization in photoacoustic tomography

Abstract

The model-based image reconstruction techniques for photoacoustic (PA) tomography require an explicit regularization. An error estimate (?2) minimization-based approach was proposed and developed for the determination of a regularization parameter for PA imaging. The regularization was used within Lanczos bidiagonalization framework, which provides the advantage of dimensionality reduction for a large system of equations. It was shown that the proposed method is computationally faster than the state-of-the-art techniques and provides similar performance in terms of quantitative accuracy in reconstructed images. It was also shown that the error estimate (?2) can also be utilized in determining a suitable regularization parameter for other popular techniques such as Tikhonov, exponential, and nonsmooth (?1 and total variation norm based) regularization methods.

Accelerated image reconstruction using extrapolated Tikhonov filtering for photoacoustic tomography

“…The standard (zeroth order) choice for L is identity matrix ( I ); thus, the solution becomesThese regularization methods involve matrix–matrix multiplications as well as solving large system of equations, which is computationally expensive. Therefore, the Tikhonov regularization was implemented in a Lanczos bidiagonalization framework, to reduce the computational complexity . The Lanczos bidiagonalization of the system matrix A can be written aswhere M q = [ m 1 , m 2 ,…, m q ] and R q = [ r 1 , r 2 ,…, r q ] are the left and right orthogonal Lanczos matrices of dimensions m × ( q + 1) and n 2 × q respectively.…”

“…Using the above Lanczos bidiagonalization, Eq. reduces toThis method requires a suitable number of Lanczos iterations and value of the regularization parameter to be chosen and we achieve these using error estimate‐based method, which was proven to be computationally efficient . It is briefly reviewed here for completeness.…”

“…One way to obtain the solution at λ = 0 is to perform an extrapolation given x at multiple values of λ (assuming that the solution is a function of λ , which is true for any regularization scheme). Thus, the extrapolation method avoids the difficulty of evaluating the regularization parameter in an automated fashion, which is computationally demanding …”

“…An important step in Tikhonov regularization is to select an appropriate regularization parameter. It is well known that the reconstruction result is largely biased toward regularization parameter . Moreover, the regularization parameter always filters some of the natural characteristics of the expected image.…”

“…Recently, a least‐squares approach was developed and was shown to be an effective alternative compared to the L‐curve and GCV methods for automatic selection of regularization parameter . Subsequent to this work, an error estimate‐based method was proposed and shown to be computationally efficient compared to the least‐squares method in determining this regularization parameter automatically . Despite all these developments, automated evaluation of regularization parameter is still computationally demanding and the metric for an automated choice of regularization parameter may not provide the desired reconstructed image characteristics as regularization inherently blurs the reconstructed image .…”