Photoacoustic tomography based on the application of virtual detectors

1. International INTERNATIONAL Journal of Electronics and Communication JOURNAL Engineering OF ELECTRONICS & Technology (IJECET), AND
ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 11, November (2014), pp. 17-24 © IAEME
COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET)
ISSN 0976 – 6464(Print)
ISSN 0976 – 6472(Online)
Volume 5, Issue 11, November (2014), pp. 17-24
© IAEME: http://www.iaeme.com/IJECET.asp
Journal Impact Factor (2014): 7.2836 (Calculated by GISI)
www.jifactor.com
17
IJECET
© I A E M E
PHOTOACOUSTIC TOMOGRAPHY BASED ON THE
APPLICATION OF VIRTUAL DETECTORS
Hu Danfeng
(School of Electronic and Information, Soochow University, Suzhou, Jiangsu, China)
ABSTRACT
Photoacoustic tomography (PAT) based on the filtered back projection (FBP) algorithm
needs full-scanned photoacoustic data of the sample to reconstruct good quality images.
Unfortunately, the number of ultrasonic sensor locations and the sampling angles are subject to the
restriction of hardware conditions and spatial size in the practical application. The consequence is
that the quality of reconstructed images becomes poor. Proposed the concept of virtual detectors
using interpolation and compressed sensing method to preprocess the photoacoustic data in order to
solve these problems. Results of numerical simulation with Matlab demonstrate that the peak
signal-to-noise ratio (PSNR) of the images has been significantly improved.
Keywords: Compressed sensing, Interpolation method, MATLAB, Photoacoustic tomography,
Virtual detector.
INTRODUCTION
Photoacoustic tomography (PAT) is an emerging noninvasive biomedical imaging technique
combining the advantages of both optical and ultrasonic imaging [1]. It is based on the photoacoustic
effect which was discovered by Bell in 1880[2]. In this effect, when a photon is absorbed by the
object and then converted into heat, the temperature of the object will increase rapidly, which causes
thermal expansion, thereby generating wideband ultrasonic waves transmitted to the boundaries.
With the ultrasound signals acquired by one or some ultrasonic transducers, the initial pressure
distribution in the tissue can be reconstructed.

2. International Journal of Electronics and Communication Engineering Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 11, November (2014), pp. 17-24 © IAEME
¶ b ¶
( , ) ( , ) ( ) ( )
p r t p r t A r I t
c t C t
2
¶
0 0
pb = −
18
In many PAT applications, a lot of positions’ ultrasonic signal around the sample need to be
acquired in order to reconstruct good quality images, especially when using conventional
back-projection reconstruction algorithms [3-5]. These algorithms are also built on the basis of full
scan detection. It can cause cost rising, along with the increase of data acquisition and processing
time. What’s more, in actual cases, the number and locations of the sampling points are subjected to
the hardware conditions and spatial location restrictions. So the problem is how to guarantee the
quality of reconstructed image with the incomplete measurement data.
In order to solve the problem mentioned above, some methods have been proposed. Provost
et al. demonstrated that the photoacoustic tomographic image can be reconstructed in the framework
of CS (compressed sensing) theory with measurements acquired from a small number of angles [6].
Dong Liang et al. validated a CS-based reconstruction algorithm for random optical illuminated
photoacoustic imaging on numerical phantoms [7]; Phantom and in vivo studies by Zijian Guo et al.
showed that the CS method can effectively reduce the number of under-sampling artifacts in the PAT
reconstruction [8]. Jing Meng et al. reported the application and validation of CS for
frequency-domain photoacoustic tomography reconstruction [9]. Mingjian Sun et al. demonstrated
that the Bayesian compressive sensing method can achieve superior performance than other
state-of-the-art CS-reconstruction algorithms in PAT reconstruction [10]. In our paper, we proposed
using virtual detectors by interpolating and compressed sensing method to preprocess the
photoacoustic data.
2. METHODS
2.1 Photoacoustic Equation
According to the photoacoustic theory, the generation and propagation of photoacoustic
waves in a lossless medium can be described by the photoacoustic wave equation [11]:
2
2
1
2 2
p
Ñ − = −
¶ ¶
(1)
where p(r, t) is the acoustic pressure at position r and time t , c is the speed of acoustic wave in
the medium, b is the thermal expansion coefficient,
p C is the specific heat, A(r) is the absorbed
energy density, I (t) is the normalized temporal profile of the illuminating pulse.
For two-dimensional circular scanning, the problem of photoacoustic imaging is to recover
A(r) from (1) and the pressure 0 p(r , t) detected by transducers [12]:
j
4 0
0 0
/
1 ( , )
( )
2
p
t r r c
r C p r t
A r d
c t t j
= −
¶ (2)
Where 0 j is a certain radian of the detecting surface and 0 r is the corresponding position.

3. International Journal of Electronics and Communication Engineering Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 11, November (2014), pp. 17-24 © IAEME
l m
19
2.2 Virtual Detectors Applying
The filtered back-projection (FBP) method is widely used in the photoacoustic image
reconstruction. It needs intensive and full-scanned photoacoustic data to balance the number of
unknowns for achieving superior image quality. Virtual detectors applying can solve the problem
caused by insufficient real detectors.
2.1 Interpolation Method
The virtual detectors are assumed exist between the real ones. We use interpolation method to
calculate the signals acquired by virtual detectors. Suppose
( , ) l p r t
and ( , ) m p r t are two
photoacoustic signals at adjacent positions l and m , the signal ( , ) virtual p r t of virtual detector
between them is given by
( , ) ( , )
( , )
2
virtual
p r t p r t
p r t
+
= (3)
The detectors are doubled by using this method once. The issue caused by incomplete
detector angles is also dealt with in the same way.
2.2.2 CS Method
CS Method has been considered as a break-through innovation in information theory [9]. It
can be used for signal reconstruction with sparse sampling based on the sparsity of the signal itself or
its transform in another domain. In our paper, we proposed to increase the number of measurements
by virtual detectors in the framework of CS. That is to say, the incomplete data obtained under some
hardware and spatial location restrictions can be recovered just like some detectors are added into the
measuring positions. The key point of applying this method is trying to find a set of bases to span a
space so that most of the energies of the signal concentrate on a small number of coefficients in such
a space. The original signal can be recovered by padding zeros to the compressed signals and
performing inverse transform.
Suppose that x is the complete measurement data with real and virtual detectors, y is the
actual incomplete measurement data in PAT. The mission is try to recover x from y [12]. In order
to resolve such an underdetermined problem in the framework of CS, we try to find an appropriate
transform y so that x is sparse in the transformed domain, i.e., y : q =y x and
q M , where
l 0
M is the number of components in complete data x , q is a compressed version of x upon
performing transform y . In this way, recovering complete data x from incomplete data y can be
converted to recover it from its compressed version q if y has been found. According to the
principles of the CS theory, the transform y can be determined by solving the following
constrained optimization problem:
1
=
y − 1
q min
. .
l
q
s t y K
(4)
Since the 0 l norm is an NP-hard problem and can’t be solved efficiently, it has been
replaced with the 1 l norm in (4). Such a substitution is plausible because the solution to it is sparse
enough. Then we can get the solution.

4. International Journal of Electronics and Communication Engineering Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 11, November (2014), pp. 17-24 © IAEME
2.2.3 CS Method Base On Dictionary Learning
d = E − d q (6)
= − , j
E x d q
-5 -4 -3 -2 -1 0 1 2 3 4
20
q q
The algorithm mentioned above is based on the fixed base functions which are not optimal
[13]. For complex signal, fixed base is sometimes not enough to capture a variety of characteristics
of the signal, so we can’t guarantee that the signal is sparse enough in the transform domain. CS
method based on dictionary learning can be used to learn an over-complete dictionary from a set of
training signals.
Let the dictionary 1 D y − = . The purpose here is to find a D makes the measurement data x can
be approximately expressed with the least error. In order to achieve this, the compressed matrix and
the dictionary D are iteratively updated in turn.
With a given D, is updated:
= −
0
2
, 2
£ =
0
min
D
q
q
. . , 1, 2,...,
i l
x D
s t T i N
(5)
Where 0 T is sparse coefficient,
2
2
x − Dq is the reconstruction error.
With the updated , a better D is updated:
2
2
min
k
k
k k k T
d
Where k d is the th k column of D, j
k j T
¹
j k
T q is the th j column of T q .
After the D is found, the q is sparse enough now. The next step is the same as equation (4)
and y can be recovered.
3. RECONSTRUCTION RESULTS AND DISCUSSION
In order to evaluate the performance of our proposed method, we use the k-Wave Toolbox [14]
to generate photoacoustic data and filtered back-propagation (FBP) method is to reconstruct the
photoacoustic image. The phantom is shown in Fig. 1.
Different numbers of unfocused ultrasound detectors are uniformly distributed around the
phantom.
x -pos ition [mm]
y-position [mm]
-5
-4
-3
-2
-1
0
1
2
3
4
Fig. 1: The original phantom image

5. International Journal of Electronics and Communication Engineering Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 11, November (2014), pp. 17-24 © IAEME
21
(1) Reconstructed by FBP directly
x -pos it io n [m m ]
5
4
3
2
1
0
-1
-2
-3
-4
-5 -4 -3 -2 -1 0 1 2 3 4 5
y-position [mm]
-5
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
x -pos it io n [m m ]
5
4
3
2
1
0
-1
-2
-3
-4
-5 -4 -3 -2 -1 0 1 2 3 4 5
y-position [mm]
-5
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
Fig. 2: FBP with 100 detectors Fig. 3: FBP with 50 detectors
(2) Reconstructed by FBP with interpolation method
Fig. 4: Interpolation with 100 detectors Fig. 5: Interpolation with 50 detectors
(3) Reconstructed by FBP with CS method
x -po s it io n [m m ]
5
4
3
2
1
0
-1
-2
-3
-4
-5 0 5
y-position [mm]
-5
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
x -pos it io n [m m ]
5
4
3
2
1
0
-1
-2
-3
-4
-5 -4 -3 -2 -1 0 1 2 3 4 5
y-position [mm]
-5
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
Fig. 6: CS with 100 detectors Fig. 7: CS with 50 detectors

6. International Journal of Electronics and Communication Engineering Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 11, November (2014), pp. 17-24 © IAEME
(4) Reconstructed by FBP with dictionary based CS method
= × (8)
22
= − ¢
MSE f f
x -pos it io n [m m ]
5
4
3
2
1
0
-1
-2
-3
-4
-5 -4 -3 -2 -1 0 1 2 3 4 5
y-position [mm]
-5
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
Fig. 8: Dictionary based CS method Fig. 9: Dictionary based CS method
with 100 detectors with 50 detectors
In order to give a quantitative evaluation of the performance of different virtual detectors
generating methods, both MSE (mean squared error) and the PSNR (peak signal to noise rate
defined as follows are introduced:
2
M N 0 £ 0
£
1
( ) ij ij
i N j M
×
(7)
and
2 255
PSNR 10 lg
MSE
where M, N are, respectively, the numbers of rows and columns in the image to be reconstructed, ij f
is value of the pixel in the phantom to be reconstructed ij f ¢
is the corresponding value of the pixel
in the reconstructed image. Tables 1 and 2 give both the MSEs and the PSNRs for the reconstructed
results from different methods.
Table 1: MSE for different reconstruction methods
Sample Points
Direct
Reconstruction
Interpolation
Method
CS Method
Dictionary
based CS
Method
100 0.0178 0.0153 0.0132 0.0117
50 0.0344 0.0206 0.0187 0.0169
Table 2: PSNR for different reconstruction methods
Sample Points
Direct
Reconstruction
Interpolation
Method
CS Method
Dictionary
based CS
Method
100 65.62 66.28 68.81 69.85
50 62.76 64.99 65.24 66.72

7. International Journal of Electronics and Communication Engineering Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 11, November (2014), pp. 17-24 © IAEME
23
IV. CONCLUSION
We proposed several virtual detectors generating methods for photoacoustic image
reconstruction which try to recover larger number of data from a smaller number of measurements.
They are particularly suitable for the case when only very small number of measurements can be
performed due to some constraints. Simulated results demonstrate that our methods archive better
results than direct reconstruction.
V. ACKNOWLEDGEMENT
This work was financially supported by the Jiangsu college students practice and innovation
training plan (111028527), Science and technology project in Suzhou(SYG201113) and National
computer and information technology experiment teaching demonstration center of China.
REFERENCE
[1] L. V. Wang and S. Hu, Photoacoustic tomography: in vivo imaging from organelles to organs,
Science, Volume 335, Issue 6075, 2012, pp. 1458 – 1462.
[2] A. G. Bell. On the Production and Reproduction of Sound by Light [J]. American Journal of
Science, 1880, 5(118): 305-324.
[3] Kruger R.A, Kiser Jr W.L, Miller K.D, et al. Thermoacoustic CT: imaging principles [J]. Proc.
Of SPIE, 2000, 3916: 150-159.
[4] Minghua Xu, Lihong V. Wang. Time-domain reconstruction algorithms and numerical
simulation for thermoacoustic tomography in various geometries [J]. IEEE Trans on
biomedical engineering, 2003, 50(9): 1086-1099.
[5] Yi Wang, Da Xing, Yaguang Zeng, et al. Photoacoustic imaging with deconvolution
algorithm [J]. Phys. Med. Biol, 2004, 49(14): 3117-3124.
[6] J. Provost and F. Lesage, The application of compressed sensing for photo-acoustic
tomography, IEEE Transactions on Medical Imaging, vol. 28, no.4, pp. 585-594, Apr. 2009.
[7] D. Liang, H. F. Zhang, and L. Ying, Compressed-sensing photoacoustic imaging based on
random optical illumination, International Journal of Functional Informatics and Personalized
Medicine, vol.2, no. 4, pp. 394-406, Apr. 2009.
[8] Z. J. Guo, C. H. Li, L. Song, and L. V. Wang, Compressed sensing in photoacoustic
tomography in vivo, Journal of Biomedical Optics, vol. 15, no. 2, pp. 021311, Mar.-Apr.
2010.
[9] Jing Meng, Dong Liang, Compressed Sensing Photoacoustic Tomography in Vivo in Time
and Frequency Domains, Proceedings of the IEEE-EMBS International Conference on
Biomedical and Health Informatics, 2012, 717-720.
[10] Mingjian Sun, Naizhang Feng, Photoacoustic image reconstruction based on Bayesian
compressive sensing algorithm, CHINESE OPTICS LETTERS, COL 9(6), pp. 061002, 1-4,
2011.
[11] Xu M, Wang L V. Pulsed-microwave-induced thermoacoustic tomography: Filtered
back-projection a circular measurement configuration [J]. Medical Physics, 2002, 29(8):
1661-1669.

8. International Journal of Electronics and Communication Engineering Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online), Volume 5, Issue 11, November (2014), pp. 17-24 © IAEME
24
[12] Danfeng Hu, Jiajun Wang, Erxi Fang, Wei Zhou, Yue Zhou. The application of compressed
sensing method in photoacoustic image reconstruction. IEEE International Conference on
Information Science and Technology, 2014.
[13] M. Aharon, M. Elad, and A.Bruckstein, K-svd: An algorithm for designing over complete
dictionaries for sparse representation. IEEE Trans. on Signal Processing, 54(1), pp.4311-4322,
2006.
[14] Bradley E. Treeby, B. T. Cox. k-Wave: MATLAB toolbox for the simulation and
reconstruction of photoacoustic wave fields [J]. Journal of Biomedical Optics, 2010, 15(2):
1-12.
[15] P Mahalakshmi and M R Reddy, “Cochlear Implant Acoustic Simulation Model Based on
Critical Band Filters”, International Journal of Electronics and Communication Engineering
Technology (IJECET), Volume 3, Issue 2, 2012, pp. 116 - 129, ISSN Print: 0976- 6464,
ISSN Online: 0976 –6472.