Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 795092, 9 pages

http://dx.doi.org/10.1155/2015/795092

## Time Reversal Reconstruction Algorithm Based on PSO Optimized SVM Interpolation for Photoacoustic Imaging

Department of Control Science and Engineering, School of Astronautics, Harbin Institute of Technology, Room H618, Main Building, No. 2 West Wenhua Road, Weihai, Shandong 264209, China

Received 9 November 2014; Revised 25 March 2015; Accepted 22 April 2015

Academic Editor: Javier Plaza

Copyright © 2015 Mingjian Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Photoacoustic imaging is an innovative imaging technique to image biomedical tissues. The time reversal reconstruction algorithm in which a numerical model of the acoustic forward problem is run backwards in time is widely used. In the paper, a time reversal reconstruction algorithm based on particle swarm optimization (PSO) optimized support vector machine (SVM) interpolation method is proposed for photoacoustics imaging. Numerical results show that the reconstructed images of the proposed algorithm are more accurate than those of the nearest neighbor interpolation, linear interpolation, and cubic convolution interpolation based time reversal algorithm, which can provide higher imaging quality by using significantly fewer measurement positions or scanning times.

#### 1. Introduction

Recently, photoacoustic tomography (PAT) has emerged as a powerful imaging technology for many biomedical applications [1, 2]. PAT is able to provide functional imaging of physiological parameters. Photoacoustic imaging combines electromagnetic and ultrasonic wave synergistically, providing relatively deep speckle-free imaging with high electromagnetic contrast at high ultrasonic resolution. In PAT, the developing reconstruction methods have been an important task [3, 4]. Various reconstruction methods, such as filtered backprojection, Fourier transform, alternative algorithm, time reversal, inversion of the linear Radon transform, and delay and sum beamforming, have been developed under different assumptions and approximation [5–9]. Besides, different shaped measurement surfaces will lead to different shaped final artifacts.

In a word, the image reconstruction progress can be regarded as running a numerical model of forward problem backwards in time domain, at each time step, in which the measured time-varying pressure signals are enforced (in time reversed order) as a Dirichlet boundary condition at their recorded positions that is called time reversal reconstruction method. It has been described as the “least restrictive” imaging algorithm on the basis that it relies on fewer assumptions than many other imaging reconstruction algorithms [10, 11]. The idea of time reversal to PAT was suggested by Fink and Prada [12] and later used for PAT by Xu and Wang [13], and the time reversal method was applied to deriving exact and approximate reconstruction algorithms for PAT in an arbitrary configuration and heterogeneous acoustic media.

In conventional time reversal imaging reconstruction, the recorded pressure time series are enforced in time reversed order as a Dirichlet boundary condition as the position of detectors on the measurement surface. If a sparse array of the detectors points is used to collect the measurement rather than a continuous surface, the enforced time reversed boundary condition will necessarily be discontinuous. This can cause significant blurring in the reconstructed images. To solve the problem, Treeby and Cox [14] improved time reversal image reconstruction by using interpolated sensor data. In the course, the interaction can be avoided by interpolating the recorded data onto a continuous rather than discrete measurement surface within the -space grid used for the reconstruction. The edges of the reconstructed image are considerably sharper and the magnitude has also been improved. After that, Cox and Treeby [15] used the enforced time reversal boundary condition to trap artifacts in the final image, and by truncating the data, or introducing an adaptive threshold boundary condition, this artifact trapping can be mitigated to some extent.

In this paper, what we are concerned with in the method is the artifacts elimination and interaction of image reconstruction when using “time reversal” algorithm. An optimized hybrid interpolation algorithm has been used to create a continuous surface that is spatially equivalent to the original Cartesian measurement surface via interpolation. It is demonstrated that the proposed algorithm can reduce artifact and improve the acoustic magnitude and the signal-to-noise ratio through partial correction for the discontinuous aperture.

#### 2. Theory and Methods

##### 2.1. Acoustic Propagation Theory in PAT

For PAT, in a lossless acoustic medium, the photoacoustic wave equation can be reformulated as an initial value problem. The photoacoustic forward problem may be written aswhere the initial conditions are given bywhere is the acoustic pressure at time and point inside the imaging region , is the acoustic particle velocity, is the sound speed, and , are the acoustic and ambient densities, respectively. Using this framework, it is straightforward to modify the adiabatic equation of state to account for acoustic absorption or nonlinear effects.

##### 2.2. Time Reversal Image Reconstruction

In PAT, the photoacoustic image reconstruction problem is to estimate the initial pressure distribution inside the imaging region given measurement of on arbitrary measurement surface . In time reversal imaging, this estimate is achieved by using the recorded measurements of the acoustic pressure over an arbitrary surface for some time to . In this case, the initial pressure in (2) is set to zero, givingFrom the equation, the pressure in the imaging field from to can be obtained; at the same time, the reverse-time variable from to also can be computed. In a practical sense, the reconstruction is performed by using the acoustic pressure time histories measured on for to in time reversed order as an enforced (time-varying) Dirichlet boundary condition on within a numerical acoustic propagation model. Here, and both and are in silico equivalents to the real world and .

During the time reversal reconstruction, if and vary spatially, the time reversed waves will no longer interfere precisely to reproduce the original wave field, which will lead to the production of additional waves. The extra waves are called vestigial waves, which are the time reversed vestiges of the scattered waves. Vestigial waves are artifact-producing waves, in the sense that any of them remaining in the computational domain would constitute artifacts in the final PAT image. To improve the reconstruction result of the discontinuous aperture, the artifacts and interaction can be largely removed by interpolating the recorded data onto a continuous measurement surface within the -space grid used for the reconstruction. As an optimized hybrid interpolation algorithm, PSO optimized SVM interpolation algorithm can provide higher convergence rate and optimization precision, which has been used to create a continuous surface that is spatially equivalent to the original Cartesian measurement surface via interpolation to solve the problem.

##### 2.3. PSO Optimized SVM Interpolation Algorithm

The essence of the support vector machine (SVM) training parameter selection is a process of optimization search [16]. In the process of interpolation, the index of the training model of the support vector machine, such as the fitting precision and the generalization ability, is directly related to the selection of the nuclear function, nuclear parameter, the penalty coefficient, and other parameters in the parameter optimization algorithm [17].

The particle swarm optimization (PSO) algorithm is a kind of bionic optimization algorithm, which is derived from the approximation behavior simulation of birds and fish population. It is an optimization tool based on iteration, searching for the optimal solution through the collaboration of the individual particles. The algorithm has a strong ability in global optimization. Using the particle swarm optimization algorithm to solve the optimization problem is equal to the search for the space location of a bird. The birds in the space are called “particles.” Each particle adjusts its flight path randomly based on the flight experience of its own so as to close to the optimal point finally. Different particle has different location and speed and individual fitness corresponding to the flight objective function. The flight path is adjusted by tracking two “extreme values.” One of the extreme values is called individual extremum which indicates the optimal solution of the particle itself. The other one is the global extremum, which indicates the optimal solution of the whole swarm. When the two extreme values are found, the particles update their speed and locations as follows:where is the inertia weight; is the current evolutionary iteration time; , ; and are random numbers distributed in the interval ; is the swarm size; and are the accelerating factors; is the -dimension component of the location of the th particle in the iteration; is the -dimension component of the velocity of the th particle in the iteration; is the individual extreme value; is the -dimension component of the best location of the th particle; is the global extreme value; is the -dimension component of the best location of the swarm.

Using the particle swarm optimization algorithm to solve the optimization problem is mainly discussing the optimization problem of the penalty factor , insensitive parameter , and the nuclear parameter . The progress of PSO-SVM method is shown in Figure 1. In general, the parameters of , , and are related with learning samples and practical problems. The penalty factor controls the degree of punishment when samples are misclassified and achieves a compromise between training error and model complexity. The larger is, the higher fitting precision is required. It makes training so difficult and also takes a longer time. But when is smaller, it will lower the fitting precision. Too big or too small value of will deteriorate the generalization capability of the system; the former is called “overlearning,” and the latter is called “less learning.” The insensitive parameter reflects the regression model’s sensitivity of the noise included in input variable. The larger is, the lower model fitting accuracy and complexity is also prone to overfit. The nuclear parameter represents the mean square error of Gaussian function and is the width of function in independent variables direction. When is smaller, the kernel function has better fitting performance, but it will deteriorate the generalization ability. So, using the particle swarm optimization algorithm to solve the optimization problem is mainly discussing the optimization problem of the penalty factor , insensitive parameter , and the nuclear parameter .