Photoaccoustic Image Reconstruction

Photoacoustic tomography (PAT) [1, 4, 5, 6, 7, 9, 10] is a novel hybrid imaging modality that is widely used to image the intrinsic and extrinsic tissue chromophores by using a combination of optical absorption and acoustic wave propagation. Upon shining with a laser pulse, the substance under investigation absorbs optical energy and undergoes thermoelastic expansion; as a result, the spatial distribution of the concentration of the substance gets translated into the distribution of pressure-rise. This initial pressure rise travels outwards as ultrasound waves which are collected by ultrasound transducers placed at the boundary. From the ultrasound signal measured by the transducers as a function of time, a PAT reconstruction method recovers an estimate of the initial pressure-rise by solving the associated inverse problem. By reconstructing the maps of the initial pressure rise using the acoustic measurements, PAT has been able to image tissues at a relatively large depths with high resolution compared to purely optical modalities. The inverse problem is however challenging. It is challenging because the image has to be recovered for the entire cross-sectional plane, whereas the samples of the acoustics pressure are available only from the points lying in the periphery of the imaging specimen where the transducers are located.

The image reconstruction methods in PAT use the measurements of acoustic waves to give an estimate of the initial pressure rise inside the tissue. In our work, the focus is on the model-based PAT image reconstruction with regularization. One common feature we find in all model-based reconstruction methods is that they are based on general purpose regularization namely total variation regularization. There has not been any model-based PAT reconstruction method with a regularization form that is specifically tuned for the characteristics of PAT images. We propose a model-based PAT reconstruction method involving a novel form of regularization; the regularization is constructed to suit the physical structure of typical PAT images. We construct it by combining second-order derivatives and intensity into a non-convex form to exploit a structural property of PAT images that we observe: in PAT images, high intensities and high second-order derivatives are jointly sparse.

We write the proposed regularization as

where p0 denotes the vector containing the pixels of the initial pressure rise p₀(r) in scanned form, ( .)_r denotes the component of its vector argument, the weight α (0, 1) controls the relative penalization and D_2,i represents the matrix of 2^nd order derivative filters. Here, we choose q < 0.5 meaning that the resulting regularization is non-convex.

This regularization is combined with a data fidelity cost, and the required image is obtained as the minimizer of this cost. We modify cost by adding quadratic penalty term to enforce positivity as given below:

Here H is the model-matrix and the parameters λ and λ_p controls the relative penalization of the regularization term and the positivity term respectively. As this cost functional is non-convex, the efficiency of the minimization method is crucial in obtaining artefact-free reconstructions. We develop a custom minimization method for efficiently handling this non-convex minimization problem. We adapt the preconditioned gradient search for minimizing the cost function. Since the cost function can have several local minima, such a minimum is likely to contain artifacts and is also influenced by the tolerances used in the algorithm. To alleviate this problem, we adopt the well-known graduated non-convexity (GNC) approach [3].

Further, as non-convex minimization requires a large number of iterations and the PAT forward model in the data-fidelity term has to be applied in the iterations, we propose a computational structure for efficient implementation of the forward model with reduced memory requirements. The proposed algorithm was compared against the FISTA based method of Huang et al. [2] for various levels of reduction in the measured data (16, 32, 64, and 128 transducers) with various levels of measurement noise (20, 30, and 40 dB). We considered both real and simulated data sets for our experiments, and the proposed method yielded superior reconstruction quality in all cases. [8]

Figure 1: Reconstructed images from input datasets corresponding to 16 transducers for various noise levels. (A), (B), and (C): reconstructions obtained from the proposed method corresponding to SNRs 20 dB, 30 dB, and 40 dB. (D), (E), and (F): reconstructions obtained from the FISTA method corresponding to SNRs 20 dB, 30 dB, and 40 dB.

References

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