Computational and Mathematical Methods in Medicine

Volume 2016, Article ID 9732142, 8 pages

http://dx.doi.org/10.1155/2016/9732142

## Regularized Iterative Weighted Filtered Back-Projection for Few-View Data Photoacoustic Imaging

^{1}Department of Mathematics Science, Liaocheng University, Liaocheng 252000, China^{2}Life Sciences Research Center, School of Life Sciences and Technology, Xidian University, Xi’an 710071, China

Received 17 February 2016; Revised 16 June 2016; Accepted 19 June 2016

Academic Editor: Giancarlo Ferrigno

Copyright © 2016 Xueyan Liu and Dong Peng. 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 emerging noninvasive imaging technique with great potential for a wide range of biomedical imaging applications. However, with few-view data the filtered back-projection method will create streak artifacts. In this study, the regularized iterative weighted filtered back-projection method was applied to our photoacoustic imaging of the optical absorption in phantom from few-view data. This method is based on iterative application of a nonexact 2DFBP. By adding a regularization operation in the iterative loop, the streak artifacts have been reduced to a great extent and the convergence properties of the iterative scheme have been improved. Results of numerical simulations demonstrated that the proposed method was superior to the iterative FBP method in terms of both accuracy and robustness to noise. The quantitative image evaluation studies have shown that the proposed method outperforms conventional iterative methods.

#### 1. Introduction

Photoacoustic imaging (PAI) combining good acoustic resolution with high optical contrast in a single modality has great potential for tremendous clinical applications [1]. It is promising in many aspects, for example, the detection of breast cancer, skin cancer, and osteoarthritis in humans [2–4]. In the past decades, many algorithms have been proposed for image reconstruction when the ultrasonic transducer collects signals from a full view [5–7]. A limiting factor for these algorithms is a large number of measurements made with transducers. In addition, in many potential applications of PAI, such as ophthalmic imaging and breast imaging, the object is only accessible from limited angles. A practical need exists for reconstruction from few-view data, as this can greatly reduce the required scanning time and the number of ultrasound sensors [8–11].

Analytic algorithms like filtered back-projection (FBP) and time-reversal based reconstruction attain very fast reconstruction performance [5, 12]. However, these algorithms have an inherent limitation of requiring large number of data points around the target object for accurately estimating the optical absorption. And implementations of such formulae may cause streak-type artifacts and negative values in the reconstructed image. To overcome these limitations, iterative image reconstruction algorithms have been proposed to improve the reconstructed image quality [13–16], which can mitigate artifacts from incomplete few-view data and permit reductions in data-acquisition times. And iterative methods can be significantly accelerated with GPU-based reconstructions [17]. Because of this, the development of iterative image reconstruction algorithms for PAI is an important research topic of current interest. By minimizing the least-square error between the measured signals and the signals predicted by the exact photoacoustic propagation model, the model-based photoacoustic inversion method has been proved to be stable and accurate. However, its reconstruction is computationally burdensome which limits its application in the practical PAI [16–18]. Finally, iterative weighted algorithms can effectively mitigate image artifacts due to limited-view acoustic data [19–21]. All of these methods provide the opportunity for accurate image reconstruction from few-view data.

In this study, inspired by the iterative weighted approaches in CT [20, 22, 23], we derived a regularized iterative weighted FBP (RIWFBP) method to improve the convergence properties of the iterative loop and improve image quality in few-view PAI. During the reconstruction, we firstly use the effective few-view scanning angle improved FBP method to reconstruct an initial image of the optical absorption [24]. In each iteration step, the difference between the collected signals and the calculated signals was used to update the correction image, and a regularization operation that improved the convergence properties of the iterative loop was added. Numerical simulation and experimental results reveal the good performance of the RIWFBP method.

This paper is organized as follows. In Section 2, the iterative improved FBP method and the regularized enhanced iterative scheme are reviewed briefly. In Section 3, besides using numerical phantoms, we also conducted experimental measurements and applied our reconstruction method to the obtained data. Finally, the conclusions are drawn in Section 4.

#### 2. Methods

##### 2.1. Iterative Improved Filtered Back-Projection Method

According to the forward problem for an acoustically homogeneous model present in [8], the acoustic wave pressure at a detector position and time over a circle in the 2D imaging is related to the spatial distribution of electromagnetic absorption ,Here is the coefficient of volumetric thermal expansion; is the isobaric specific heat; is the speed of sound; and is the temporal profile. For 2D imaging, the approximate inverse solution for the circular-scan geometry can be represented by [25]To numerically model the above forward and inverse problems, we used vector to represent and vector to represent . Then the forward problem can be described as , and the reconstruction formula can be written as [5], where is the back-projection operator [8]. In real biological tissue imaging only the noisy signals can be detected from few-view data. is an ill-conditioned matrix; thus we cannot obtain an exact image.

Iterative improved FBP (IFBP) methods have been used to reduce artifacts due to an insufficient data and streaks due to missing angles. The update step of IFBP is then given by [13]In this way, a sequence of image vectors is produced.

##### 2.2. Regularized Iterative Weighted Filtered Back-Projection Method

In this section, we will present the RIWFBP method for the few-view PAI imaging. This contribution is an extension of theory and experiments on iterative weighted FBP (IWFBP) presented in [26]. By using the effective scanning angle the algorithm for full-view data can be approximately extended to the few-view case. The reconstructed intensity error problem induced by few-view scanning can be improved [24]where is the effective scanning angle and and are, respectively, the minimum and maximum angle of the signal acquisition position. Based on (4), we used the RIWFBP method to compensate for the nonexactness of FBP.

During the reconstruction, we firstly used weighted FBP method to reconstruct an initial distribution of the absorbed energy density. Then we applied the RIWFBP method to update the distribution of the absorbed energy density. To compensate for the difference between the reconstruction and the actual image , a weighted parameter [26],was used to correct the differences between the measured signals and computed signals . We obtained the error correction image from the differences between and at each iteration step. The recursion expression is as follows:In practice, it might be wise to employ only a fraction of the full values of the correction image . Inspired by the idea of regularization [22, 23], the RIWFBP method is designed to reconstruct the absorbed energy deposition by adding a regularization operation in the iterative loop of IFBP. This is accomplished through the quadratic regularization for least-squares minimization of the following functional:where is a parameter determining the amount of regularization and are the inverse distances between the pixels and in a 3^{2} neighborhood. The last term is obviously a penalty term. The minimization of (7) can be realized by differentiating with respect to and setting each of the resulting expressions to zero, leading to the following system of equations:where [22] and is the standard basis for . By using the steepest descent method the solution in an iterative form was given asFinally, the last term in (9) has been added to the update step in (6), resulting inThe convergence of the iteration has been explained by Sunnegårdh and Danielsson [22]. In this paper, the quality of the reconstructed image is measured via the normalized mean absolute error (NMAE), which is most sensitive to distortion artifacts defined asAfter a few iterations, artifacts are suppressed while the edge and detailed information is preserved well. When a desired minimum NMAE has been achieved the iterative process will stop, and then the results will be output.

#### 3. Results

##### 3.1. Reconstructions from Simulated Few-View Data

Computer simulations were conducted to demonstrate the effectiveness of the proposed method. The imaged source with a size of 256 × 256 pixels, as shown in Figure 1(a), was approximately located within a thin slab. The photoacoustic signals were calculated according to (1), and Gaussian noises were also added to simulated signals. In the experiments, all reconstruction methods were implemented in MATLAB (MathWorks, Natick, MA).