C. ZHANG ET AL.

Copyright © 2013 SciRes. ENG

PAI. The image reconstruction problem is to determine

an estimate of A(r) from the know ledge of p(r, t). A va-

riety of image reconstruction algorithms [3-10] ha s been

developed for the inversion of (2). However, those algo-

rithms neglect the response characteristics of the trans-

ducer. Therefore they may produce significant image

blurring and low contrast in the reconstructed images.

2.2. Finite Bandwidth Effect of a Transducer

Most of PAI reconstruction algorithms are based on the

assumption that the bandwidth of ultrasonic transducers

which are employed to receive signals is infinite. How-

ever, actual ultrasound transducer s are all with the finite

bandwidth. The finite bandwidth effect of transducers

can be expressed as a convolution of received signals and

an impulse respon se. The actual projection data can be

expressed as:

00

( ,)( ,)(,)

ideal

prtprt hrt= ∗

, (3)

where the pideal(r0,t) is the idealized projection data based

on the assumption that the transducers have the infinite

bandwidth, p(r0,t) is the actual projection data obtained

by the transducers, h(r0,t) is the impulse response of the

transducer related to its bandwidth, * denotes a one-di-

mensional temporal convolution.

Figure 1 compares ultrasonic signals obtained by an

idealized transducer and a finite-bandwidth transducer.

Ultrasonic signals are excited by a point optical absorber.

Figure 1 illustrated that the finite-bandwidth effect may

cause the reduction of the signal density and the convo-

lutional noise. These are major reasons of image blurr ing

and low contrast in reconstructed images if image recon-

struction algorithms fail to take the finite-bandwidth ef-

fect into account.

Figure 1. A comparison between ultrasou nd signals obtaine d

by an idealized transducer and a finite-bandwidth trans-

ducer.

2.3. Compensating for the Finite Bandwidth

Effect

Based on (3), the purpos e of compensation for the finite

bandwidth effect is to obtain the pideal(r0,t). As the re-

ceived signals are non-stationary signals, we use the

short time Fourier transform (STFT) to transform the

data into the Fourier domain. Here we take the additive

noise into account. Equation (3) is transformed as:

00

(,)(,) (,)

ideal

PrP rHre

ω ωω

= ⋅+

(4)

where e is the additive noise, H(r, ω)is the bandwidth of

the transducer, P(r0, ω) and Pideal(r0, ω)is the Fourier

spectrum of the actual projection data and the idealized

data.

An optimal filter T can be derived to obtain received

signals wher e th e finite bandwidth effect is compensated.

(5)

In this study, this optimal filter T is designed to mi-

nimize the error between P(r0, ω) and Pideal(r0, ω). At the

same time, th e optimal filter should have good robustness

against the additive noise. We can easily get the band-

width characteristics of a transducer from its data sheet

or from the physical meansurement experiment. In the

Fourier domain, its distribution function is Gaussian. The

optimal filter T can be found:

, (6)

where

is a parameter determine the noise suppres-

sion effect and

a Gaussian kernel function which can

be expressed as:

. (7)

Here μ is the mean of the function, σ is the variance of

the function. Both of those parameters can be obtained

from the bandwidth characteristics of the transducer.

Once we obtained the actual projection data, we can

reconstruct the image with the compensated data.

3. Results and Discussion

Simulations are performed to ev alu ate the efficacy of this

method on the improvement of resolution and contrast of

the reconstructed image. The comparison are made

among this method and those reconstruction algorithms

assuming finite-bandwidth ultrasound transducers. All

simulations are done using the K-wave toolbox [13,14]

of MATLAB.

In this simulation, three optical absorbers with differ-

ent radius and optical absorption density are located in

the test image. Its original optical energy distribution is

showed in Figure 2. We set the radius of scanning circle

as 250 pixels and assume the speed of sound consistent