Tomographic density imaging using modified DF–DBIM approach

Abstract

Ultrasonic computed tomography based on back scattering theory is the most powerful and accurate tool in ultrasound based imaging approaches because it is capable of providing quantitative information about the imaged target and detects very small targets. The duple-frequency distorted Born iterative method (DF–DBIM), which uses density information along with sound contrast for imaging, is a promising approach for imaging targets at the level of biological tissues. With two frequencies f1 (low) and f2 (high) through \({\mathbf{N}}_{{{\mathbf{f}}_{1}}}\) and \({\mathbf{N}}_{{{\mathbf{f}}_{2}}}\) iterations respectively, this method is used to estimate target density along with sound contrast. The implications of duple-frequency fusion for the image reconstruction quality of density information along with sound contrast based ultrasound tomography have been analyzed in this paper. In this paper, we concentrate on the selection of parameters that is supposed to be the best to improve the reconstruction quality of ultrasound tomography. When there are restraints imposed on simulated scenarios to have control of the computational cost, the iteration number \({\mathbf{N}}_{{{\mathbf{f}}_{1}}}\) is determined resulting in giving the best performance. The DF–DBIM is only effective if there are a moderate number of iterations, transmitters and receivers. In case that the number of transducers is either too large or too small, a result of reconstruction which is better than that of the single frequency approach is not produced by the implementation of DF–DBIM. A fixed sum \({\mathbf{N}}_{{{\mathbf{iter}} }}\) of \({\mathbf{N}}_{{{\mathbf{f}}_{1}}}\) and \({\mathbf{N}}_{{{\mathbf{f}}_{2}}}\) was given, the investigation of simulation results shows that the best value of \({\mathbf{N}}_{{{\mathbf{f}}_{1}}}\) is \(\left[{\frac{{{\mathbf{N}}_{{{\mathbf{iter}}}}}}{2} - 1} \right]\). The error, when applying this way of choosing the parameters, will be normalized with the reduction of 56.11%, compared to use single frequency as used in the conventional DBIM method. The target density along with sound contrast is used to image targets in this paper. It is a fact that low-frequency offers fine convergence, and high-frequency offers fine spatial resolution. Wherefore, this technique can effectively expand DBIM’s applicability to the problem of biological tissue reconstruction. Thanks to the usage of empirical data, this work will be further developed prior to its application in reality.

Background

The human body is a heterogeneous medium, consisting of many different organs and structures. When the ultrasonic beam travels to the boundary of two environments with different acoustic impedances, one part goes in the original direction and continues into the next while the other part is reflected. The reflected intensity depends on the difference in impedance between the two media [1]. The sound impedance (Z) is a physical quantity that indicates the ability of the environment to resist the penetration of ultrasound, which depends on the density (ρ) and sound velocity (v) of the medium (Z = ρ.v). In addition, when the plane which separates the two media is non-planar, ultrasonic scattering occurs besides the reflected and penetrating phenomena. Ultrasonic scattering is common when ultrasound wave encounters small structures whose diameters are equivalent to the wavelength of the incident wave. Back scattering is considered to be one of the most modern and accurate ultrasound methods [2]. However, most of backscattering formulas discard density changes to reproduce sound contrast and sound attenuation [2].

In biomedical imaging applications, the quality of image reconstruction using backscatter theory can be affected when ignoring the density information due to formulation simplification. Experimental results in [3, 4] have pointed out that the relative density variation ρ in tissue is comparable in magnitude to relative sound speed variation. In addition, the reconstructed image based on density information may contain useful information or as an image contrast source. This is essential in quantitative ultrasound imaging. At present, ultrasound computed tomography has been being applied for early detection and diagnosis of breast cancer. Notwithstanding, the information provided by current density imaging is not clearly understood. Although the density and speed of sound have high correlation in the benign tissue [5], the actual values of density are not known for many disease states. In [6], authors pointed out that the density usually increases with the increase of the speed of sound. Nevertheless, this does not happen when there are large amount of fibrous tissue mixed with fat tissues. As a matter of fact, there are some valuable studies in the works [7, 8] which indicate that density information can play a momentous role in scattering from tissues. Therefore, extra information or sound contrast in detecting cancer can be provided by specifying the density distribution of interest targets. Thus, theoretically, the use of ultrasonic computed tomography considering density variation associated with acoustic contrast may be useful for quantitative ultrasound imaging by providing three-dimensional impedance maps of tissues.

Born approximation is normally used in ultrasound tomography. There are two well-known methods Born Iterative Method (BIM) and Distorted Born Iterative Method (DBIM) for diffraction tomography [9]. One disadvantage of Born approximation is with strong scattering medium where it is invalid [10]. This problem can be solved by using a duple-frequency, frequencies f1 and f2, to reconstruct the target through \({\text{N}}_{{{\text{f}}_{1}}}\) and \({\text{N}}_{{{\text{f}}_{2}}}\) iterations. At low frequency f1, although the convergence of the algorithm to a contrast level is close to the real one, it offers a low spatial resolution. At high frequency f2, spatial resolution is improved while still preserving the convergence. In [11], the authors combined more than two frequencies to obtain the reconstructed results which are better than that of single frequency or duple-frequency. However, if more than two frequencies are combined, a large amount of data will be acquired and they will require a large memory for storage [11, 12]. In [11, 12], the authors did not concern to how to obtain the best-reconstructed result while switching among frequencies. In our previous work [13], two frequencies, f1 (low) and f2 (high), were used to estimate the sound contrast in \({\text{N}}_{{{\text{f}}_{1}}}\) and \({\text{N}}_{{{\text{f}}_{2}}}\) iterations respectively. Given a fixed sum \({\text{N}}_{\text{iter }}\) of \({\text{N}}_{{{\text{f}}_{1}}}\) and \({\text{N}}_{{{\text{f}}_{2}}}\), we proved that the best performance was archived if \({\text{N}}_{{{\text{f}}_{1}}}\) is \(\frac{{{\text{N}}_{\text{iter}}}}{2}\). However, this selection is only for sound contrast imaging without density consideration. Thus, this paper aims to reconstruct a large target and density along with sound contrast using the modified DF–DBIM approach shown in [13]. In this paper, a duple-frequency method is applied with low frequency f1 = 0.5 MHz and high frequency f2 = 1 MHz in the simulated scenarios. The simulation results obtained in this paper have shown that given a fixed sum \({\text{N}}_{\text{iter }}\) of \({\text{N}}_{{{\text{f}}_{1}}}\) and \({\text{N}}_{{{\text{f}}_{2}}}\), the best value of \({\text{N}}_{{{\text{f}}_{1}}}\) is \(\left[{\frac{{{\text{N}}_{\text{iter}}}}{2} - 1} \right]\). Besides, the normalized error (NE) will be reduced by 56.11%, compared to the normal single frequency DBIM method.

Methodology

The geometrical and acoustic configuration is shown in Fig. 1 where B1 is a homogenized background medium and B2 is an target medium. The measured data, the pressure signal from transmitter, is brought to DBIM to estimate the sound constrast and density. The alteration of the target density along with sound contrast will find any tissues if they exist.

Fig. 1
figure1

Geometrical and acoustical configuration

Assume that the wave numbers of B1 and B2 are k0 and k(r) respectively. We have the equation of the scheme as follows:

$${\text{p}}\left({\vec{r}} \right) = {\text{p}}^{\text{inc}} \left({\vec{r}} \right) + {\text{p}}^{\text{sc}} \left({\vec{r}} \right),$$
(1)

where \({\text{p}}\left({\vec{r}} \right)\) is overall signal, \({\text{p}}^{\text{sc}} \left({\vec{r}} \right)\) is scattered signal and \({\text{p}}^{\text{inc}} \left({\vec{r}} \right)\) is incident signal. Apply the Green function G0(·) to Eq. (1) to yield

$${\text{p}}\left( {{\vec{\text{r}}}} \right) = {\text{p}}^{\text{inc}} \left( {{\vec{\text{r}}}} \right) + \iint {{\text{X}}\left( {{\vec{\text{r}}}} \right){\text{p}}(\overrightarrow {{{\text{r}}^{\prime } }} ){\text{G}}_{0} \left( {k_{0} ,\left| {{\vec{\text{r}}} - \overrightarrow {{{\text{r}}^{\prime } }} } \right|} \right){\text{d}}\overrightarrow {{{\text{r}}^{\prime } }} },$$
(2)

where \({\text{X}}\left({\text{r}} \right)\) represents the target function required to be estimated. We can calculate the function by the following formula:

$${\text{X}}\left({\text{r}} \right) = \left\{{\begin{array}{*{20}l} {k\left(r \right)^{2} {-}k_{0}^{2} - \rho^{1/2} \left(r \right)\nabla^{2} \rho^{- 1/2} \left(r \right) = \omega^{2} \left({\frac{1}{{v^{2}}} - \frac{1}{{v_{0}^{2}}}} \right) - \rho^{1/2} \left(r \right)\nabla^{2} \rho^{- 1/2} \left(r \right)} \hfill & { if r \le R} \hfill \\ 0 \hfill & {if r > R} \hfill \\ \end{array}} \right.$$
(3)

where \(R\) is the radius of the target, \(\rho \left(r \right)\) is the target density, \(\omega\) is the frequency of the incident signal (\(\omega = 2\pi f\)), v0 is the sound speed difference of the background medium, and v is the target medium. When the background medium is homogeneous, G0 is the 0th Hankel function of the first kind

$$G_{0} \left( {k_{0} ,\left| {{\vec{\text{r}}} - \overrightarrow {{{\text{r}}^{\prime } }} } \right|} \right) = \frac{ - i}{4}H_{0}^{\left( 1 \right)} \left( {k_{0} \left| {{\vec{\text{r}}} - \overrightarrow {{{\text{r}}^{\prime } }} } \right|} \right) = \frac{ - i}{4}\sqrt {\frac{2}{{\pi k_{0} \left| {{\vec{\text{r}}} - \overrightarrow {{{\text{r}}^{\prime } }} } \right|}}} e^{{i\left( {k_{0} \left| {{\vec{\text{r}}} - \overrightarrow {{{\text{r}}^{\prime } }} } \right| - \pi /4} \right)}} .$$
(4)

The formula to express the overall pressure field in the observed mesh area (which is equal to N multiply N points) can be:

$$\bar{p} = \left({\bar{I} - \bar{C}.D\left({\bar{X}} \right)} \right)p^{inc},$$
(5)

where \(\bar{I}\) represents unit matrix,\(\bar{C}\) represents the Green matrix which shows the interactions among pixels and \(D\left(\cdot \right)\) returns a square diagonal matrix of the input vector. We have the formula expressing the scattered signal which is in the form of NtNr × 1 vector being as follows:

$$\bar{p}^{sc} = \bar{B}.D\left({\bar{X}} \right). \bar{p},$$
(6)

where \(\bar{B}\) represents all of pixels interaction to receivers shown through the Green matrix. As the result of rewriting Eqs. (5) and (6), we have [14]:

$$\Delta p^{sc} = \bar{B}.D\left({\bar{p}} \right).\Delta \bar{X} = \bar{M}.\Delta \bar{X},$$
(7)

such that \(\bar{M} = \bar{B}.D\left({\bar{p}} \right)\). Two parameters \(\bar{p}\) and \(\bar{X}\) need to be determined. The target function \(\bar{X}\) has \(N^{2}\) variables which are pixels in the meshing area and it can be calculated by:

$$\bar{X}^{n} = \bar{X}^{{\left({n - 1} \right)}} + \Delta \bar{X}^{{\left({n - 1} \right)}},$$
(8)

where \(n\) and \(n - 1\) are two consecutive discrete-time points. We can use Tikhonov’s regularization to estimate the value of \(\Delta \bar{X}\) [15]:

$$\Delta \bar{X} = { \arg }\mathop {\min}\limits_{{\Delta \bar{X}}} \left\| {\bar{p}^{sc}_{t} - \bar{M}_{t} \Delta \bar{X}_{2}^{2}} \right\| + \gamma \left\| {\Delta \bar{X}} \right\|_{2}^{2},$$
(9)

where \(\Delta \bar{p}^{sc}\) expresses the discrepancy between approximated and measured scattering signals and has the size of \(N_{t} N_{r} \times 1\); obtained results are arranged in matrix form \(\bar{M}_{t }\) of (\(N_{t} N_{r} \times N^{2}\)) elements; for the regularization factor \(\gamma\) which is required to be chosen carefully as the stability of the system can be affected mostly by it. The reconstructed image can be rough as the consequence of high values of \(\gamma\) to be chosen. However, if small values of \(\gamma\) are used, they can cause highly computational complexity. The regularization parameter \(\gamma\) stated in this paper is chosen as a function of the forward error. We can estimate the value of σ0 (where σ0 is the first singular value of the inverse solver matrix \(\bar{M}_{t }\)) and choose the parameter \(\gamma\) expressed in [16]. It is necessary to be notice that there will be changes of the inverse solver matrix \(\bar{M}_{t }\) in every iteration, as a result, \(\gamma\) is also altered. The computed value of \(\gamma\) in the first iteration of our numerical simulation is 1.2 × 10−12.

The reconstruction performance of imaging system is determined by using Relative Residual Error (\(RRE\)) as follows:

$$RRE = \mathop \sum \limits_{{\varvec{i} = 1}}^{\varvec{N}} \mathop \sum \limits_{{\varvec{j} = 1}}^{\varvec{N}} \frac{{\left| {\varvec{C}_{{\varvec{ij}}} - \hat{\varvec{C}}_{{\varvec{ij}}}} \right|}}{{\varvec{C}_{{\varvec{ij}}}}}$$
(10)

An outright description of an DF–DBIM is expressed through Algorithm 1.

figurea

After Algorithm 1, we will have Algorithm 2 which describes the procedure studying RRE by using different values of \({\text{N}}_{{{\text{f}}_{1}}}\):

figureb

Four simulated scenarios are performed in this paper. The corresponding parameters are shown in Table 1. By using Nt × Nr measurements, the variables in form of N × N in the target function will be estimated. Large numbers, small number of Nt and Nr will be considered through the first and second scenarios respectively while moderate numbers Nt and Nr will be considered under the third and the fourth scenarios.

Table 1 Simulation parameters of scenarios

The relationship of the number of measurements and variables of scenarios is stated in Table 2. It can be seen that in the first scenario, the number of variables is chosen to be smaller than the number of equations. In other words, this is the case where the number of transmitters and receivers are exceedingly determined. Therefore, the highly computational complexity of the system is caused as the consequence of the over-determination. The number of variables is chosen to be larger than the number of equations in the other three scenarios. In the scenarios, where there given small and moderate number of transmitters and receivers, a small change of the number of variables can cause a big change in RRE. Hence, this problem is suitable for practical implementation.

Table 2 The relationship between the number of measurements and variables of scenarios

Assume the incident signal to be a Bessel beam of zero order in Bessel function which is described as follows:

$$\bar{p}^{inc} = J_{0} \left({k_{0} \left| {r - r_{k}} \right|} \right),$$
(11)

For the 0th order Bessel functions \(J_{0},\; \left| {r - r_{k}} \right|\) is the distance between a transmitter and a kth pixel in the interested region. The excess phase \(\Delta \varphi\) (the phase alteration between the incident wave and the propagating wave through the target) is calculated by

$$\Delta \varphi = 2\omega \left({\frac{1}{v} - \frac{1}{{v_{0}}}} \right)R,$$
(12)

It is noticed that the artifact-free reconstruction will occur provided that \(\Delta \varphi < \pi\). In the DBIM, the choice of initialization frequency depends on the condition of Born’s approximation, i.e. choosing the f1 value such that \(\Delta \varphi < \pi\). For the first frequency f1, we will restore the initial target with the contrast value of \(v\)1. Then, by the selection of the frequency hopping value, from which we have the frequency f2, we will restore the target with the contrast value of \(v\)2 updated from the contrast \(v\)1. As we continue increasing the frequency, we will get closer to the real contrast of the interested target \(v\)* (\(v\)1 < \(v\)2 < ··· < \(v\)*). Increasing frequency gradually enables the optimization of the resolution. In principle, we can use frequency sweeping method to create images, but when using this method, for each frequency we will have a set of measured values, so the storage and calculation will be large. This is a barrier of the DBIM method that so far there are very few commercialized devices using this technology. Indeed, for one frequency, we must perform the transmitter and reception of signals to collect W set of measured values; for K frequencies, we must obtain K × W set of measured values. The measurement time increases K times and the stored data increases K times. Moreover, the implementation of multiple measurements will have errors in measurement and measuring multiple frequencies will have many artifacts which will affect the quality of the recovered image. Therefore, in this paper, we limit the use of two frequencies for image recovery. This study also improves upon previous studies [17, 18], by combining two frequencies in density imaging. By carefully choosing the frequency hopping value Nf1, we were able to avoid the cases showing poorer image quality than that using single frequency. Based on the above analysis, in terms of image quality (specifically the convergence rate), the MF-DBIM method will give better quality than the DF–DBIM method. However, in a certain amount of storage, computation and imaging time constraints, the use of the DF–DBIM method is also an effective choice if it is necessary to compromise the convergence rate and the calculation volume.

It can be seen from (12) that the sound contrast increases when the incident frequency decreases; the sound contrast decreases when the incident frequency increases. As consequence, in order to avoid artifacts, the small sound contrast needs a large frequency and vice versa. In this paper, we focus on reconstructing targets having large sound contrast which is 25%. The low frequency f1 = 0.5 MHz is used in the simulation scenario. Normally, the obtained results show poor image resolution with small frequencies though they achieve good convergence. However, with this value of f1, it allows both good convergence and acceptable resolution. The high frequency f2 = 1 MHz is then used to increase the image resolution. In [19], the small value of the sound contrast (from 0.06% to 6%) is selected. Nevertheless, the intent of our approach is to ponder the more complicated issue (i.e. included density information, large target and sound-contrast). The parameter of the excess phase used in this paper is 4.33π (Δφ = 4.33π). In spite of the fact that the condition of the Born approximation is not guaranteed, the target is still successfully retraced with some artifacts which appear near the target’s center. Therefore, fine solution of Born approximation is still ensured. In [20], the investigation was conducted by the authors for the phase range from 0.004π to 16π. It still generates the successful reconstruction result. Our study concentrates on describing how two frequencies f1 and f2 can bring the highest proficiency of the combined frequency problem and the compromise of the iteration steps. We have conducted a number of simpler scenarios with different sound contrasts such as 0.5% with satisfactory phase changes that brought much better results of the reconstructions.

Numerical simulation and results

Determining the finest value of \(N_{{f_{1}}}\)

The hypothetical target function \(X\left(r \right)\) is exhibited in Fig. 2. The axis Z indicates the percentage of target’s sound-contrast (i.e. 25% in our numerical simulation) while the axes X and Y indicate the meshing region in pixels. The normalized error (NE) between the obtained image and the initial image at every iteration is determined by using this formula:

$$\varepsilon = \frac{1}{V \times W}\sum\limits_{i = 1}^{V} {} \sum\limits_{j = 1}^{W} {} \frac{{\left| {m_{ij} - \hat{m}_{ij}} \right|}}{{\left| {m_{ij}} \right|}}$$
(13)

where m is a V × W initial image (i.e. hypothetical target function) and \(\hat{m}\) is the obtained image.

Fig. 2
figure2

The target function which is hypothetical with N = 20

The NEs of the proposed approach (DF–DBIM) after every iteration with various \({\text{N}}_{{{\text{f}}_{1}}}\) values in the first-to-fourth scenarios are shown from Figs. 3, 4, 5 to 6, respectively. We can find the errors of scenarios stated in Table 3 with every value of \({\text{N}}_{{{\text{f}}_{1}}}\) after the total number of iterations \({\text{N}}_{\text{iter}}\). The simulation results show that the number of \(N_{t}\) and \(N_{r}\) will affect the value of \({\text{N}}_{{{\text{f}}_{1}}}\). If Nt and Nr are small, \({\text{N}}_{{{\text{f}}_{1}}}\) is large and the result depends only on f1 (as shown in Fig. 4, \({\text{N}}_{{{\text{f}}_{1}}}\) = 5); when \(N_{t}\) and \(N_{r}\) are enormous, the consequence only relies on f2 (as shown in Fig. 3, \({\text{N}}_{{{\text{f}}_{1}}}\) = 1). Thus, we can see that the performance of reconstruction using duple-frequency method does not give better performance than using single-frequency. As the matter of fact, for Nt × Nr which is larger than N × N, we will have the number of variables smaller than the number of measurements and the algorithm convergence of DF–DBIM is ensured. In such situation, the better performance shall be offered provided that the higher frequency f2 is used. It is contrary to the case where Nt × Nr is smaller than N × N and the number of variables becomes large than the number of measurements. In this case, the convergence of DF–BDIM is achieved only by using the lower frequency.

Fig. 3
figure3

NEs of the proposed approach after iterations homologous with various \({\text{N}}_{{{\text{f}}_{1}}}\) values (Scenario 1)

Fig. 4
figure4

NEs of the proposed approach after iterations homologous with various \({\text{N}}_{{{\text{f}}_{1}}}\) values (Scenario 2)

Fig. 5
figure5

NEs of the proposed approach after iterations homologous with various \({\text{N}}_{{{\text{f}}_{1}}}\) values (Scenario 3)

Fig. 6
figure6

NEs of the proposed approach after iterations homologous with various \({\text{N}}_{{{\text{f}}_{1}}}\) values (Scenario 4)

Table 3 NEs of various scenarios to each value of \({\text{N}}_{{{\text{f}}_{1}}}\) after the overall number of 8 iterations

In the other scenarios such as third and fourth scenarios where Nt and Nr values are moderate, it is suggested that \({\text{N}}_{{{\text{f}}_{1}}}\) can be chosen to exploit both f1 and f2. Accordingly, the applying of DF–DBIM is significantly practical in the case of the moderate Nt and Nr,, which, can be seen better in Figs. 5 and 6 with \({\text{N}}_{{{\text{f}}_{1}}}\) = 3. Because of that, for deeper investigation, it is suggested to choose the scenario 3 where the number of measurements is 80 percentage of the number of variables.

Numerical simulation of the traditional (DBIM) and proposed (DF–DBIM) approaches

According to scenario 3, \({\text{N}}_{\text{iter }}\) is assigned 8; simultaneously, the best value of iteration correlated to the first frequency can be worked out by formula \({\text{N}}_{{{\text{f}}_{1}}} = \frac{{ {\text{N}}_{\text{iter}}}}{2} - 1\) (equal to 3). In fact, the findings of simulations referring to different input parameters (counting moderate values of \({\text{N}}_{\text{iter}}\)) have proved that \(\frac{{{\text{N}}_{\text{iter}}}}{2} - 1\) is the best value of iteration linked to the first frequency. This result is unsurprising on the grounds that when f1 (\({\text{N}}_{{{\text{f}}_{1}}}\) closed to \({\text{N}}_{\text{iter}}\) and \({\text{N}}_{{{\text{f}}_{2}}}\) closed to 1) is mainly applied, the algorithm converges quickly but shows a low precision; in contrast to the usage of f2 (with \({\text{N}}_{{{\text{f}}_{2}}}\) closed to \({\text{N}}_{\text{iter}}\) and \({\text{N}}_{{{\text{f}}_{1}}}\) closed to 1) which prompts the algorithm to give a high precision but low convergence speed. A trade-off between \({\text{N}}_{{{\text{f}}_{1}}}\) and \({\text{N}}_{{{\text{f}}_{2}}}\) corresponding to \({\text{N}}_{{{\text{f}}_{1}}} = \frac{{{\text{N}}_{\text{iter}}}}{2} - 1\) would be a sensible solution to \({\text{N}}_{{{\text{f}}_{1}}}\). The explicit disparateness between the obtained images using a fusion of frequencies in comparison with the images that use single frequency are exhibited in iterations from 4 to 8 iterations (shown in Figs. 7, 8, 9 and 10). The duple-frequency approach presents a more noticeable convergence rate in comparison with the one using single frequency. To recapitulate, the result recorded by the method of applying two frequencies is closer to the hypothetical target function than the one using single frequency.

Fig. 7
figure7

The obtained images of the individual approaches from iteration 1 to iteration 4 (Scenario 3)

Fig. 8
figure8

The obtained images of the individual approaches from iteration 5 to iteration 8 (Scenario 3)

Fig. 9
figure9

Reconstruction profiles of the different approaches through iterations 1 to 4 (Scenario 3)

Fig. 10
figure10

Reconstruction profiles of the different approaches through iterations 5 to 8 (Scenario 3)

The NEs of three various schemes (using f1, using f2, and fusing f1&f2) according to the third scenario are depicted in Fig. 11 so as to demonstrate the performance of the DF–DBIM approach. It is an observable fact that the duple-frequency approach deducts 56.1% of the NE compared with the conventional DBIM method which uses single frequency. In addition, the compound of two frequencies can make good use of both lower and higher frequencies. It brings out a better convergence rate, concurrently minimizes NEs. Figure 12 shows the radial profiles of reconstruction based on the third scenario, namely the hypothetical profile, the last obtained result of the f1-based DBIM, the last obtained result of the f2-based DBIM, the last obtained result of the f1&f2-based DBIM. According to the results shown in both figures Figs. 11 and 12, the proposed scheme (using f1&f2) produces the best performance (Fig. 13).

Fig. 11
figure11

The contrast of between the NEs of DF–DBIM and DBIM through \({\text{N}}_{\text{iter}}\) iterations (Scenario 3)

Fig. 12
figure12

Reconstruction profile of the proposed and traditional approaches after \({\text{N}}_{\text{iter}}\) iterations (Scenario 3)

Fig. 13
figure13

NE comparison of the DF–DBIM approach after \({\text{N}}_{\text{iter}}\) iterations with different values of the target density (Scenario 3)

In order to demonstrate the effectiveness of the proposed method, in addition to using one pair of frequencies, we also consider three other frequency pairs. We set the initial frequency f0 = 0.5 MHz. Scenario 3 is used again for simulation when applying different-frequency pairs instead of f1 = f0, f2 = 2f0, specifically as follows: (f1 = f0, f2 = 1.25f0); (f1 = f0, f2 = 1.5f0); (f1 = f0, f2 = 1.75f0). The numerical simulation results are shown in Table 4:

Table 4 Normalized error when using pairs of different frequencies

The numerical simulation results in Table 4 and Fig. 14 show that, when using three different-frequency pairs, the normalized error still reaches the smallest value when the number of loops of frequency f1 is 3, i.e. \({\mathbf{N}}_{{{\mathbf{f}}_{1}}} = \left[{\frac{{{\mathbf{N}}_{{{\mathbf{iter}}}}}}{2} - 1} \right]\). However, when the frequency hopping value \(\Delta f = f_{2} - f_{1}\) is small, the normalized error increases and vice versa. This is easy to understand that after recovering with frequency f1, the hopping value needs to be large enough to increase the convergence speed. With the large enough frequency f2, it is possible to update the target with the contrast as close to the real contrast as possible. As a result, the recovery quality will be better.

Fig. 14
figure14

Normalized error when using pairs of different frequencies

In this section, three off-centered targets were used to reconstruct using the modified DF–DBIM method. We set up three targets with different sizes and contrasts in the area of interest. In addition to the change in the simulation parameters of the above three targets, the other parameters are used as in Scenario 3 (the scenario is suitable for actual situation). Figure 15 shows the ideal target function, with three targets in the area of interest in the order from left to right respectively with contrast and diameter (21–5.71 mm); (25–5.0 mm); (23–4.44 mm). Figures 16, 17 and 18 show the normalized errors and the recovered results of approaches the DBIM-Using f1, the DBIM-Using f2 and the DBIM-Combining f1&f2.

Fig. 15
figure15

Ideal targets (3 objects need to be recovered)

Fig. 16
figure16

Normalized errors of approaches the DBIM-using f1, the DBIM-using f2 and the DBIM-Combining f1&f2

Fig. 17
figure17

The obtained images of approaches the DBIM-using f1, the DBIM-using f2 and the DBIM-combining f1&f2 from iteration 5 to iteration 8

Fig. 18
figure18

The obtained images of approaches the DBIM-using f1, the DBIM-Using f2 and the DBIM-combining f1&f2 from iteration 5 to iteration 8

Figure 19 illustrates the implementation of the DF–DBIM, commencing with the initialization of three factors \(\bar{O}_{n}\), \({\bar{\text{p}}}_{0}\), and n (\(\bar{O}_{n} = \bar{O}_{0}\); \({\bar{\text{p}}}_{0} = {\bar{\text{p}}}^{\text{inc}}\); n = 0) based on the presumption that there are number of pixels of the hypothetical target function (N) and overall number of iterations (\({\text{N}}_{\text{iter}}\)). Besides, chosen \(N_{t} N_{r}\) should be smaller than \(N^{2}\) (pixels of the hypothetical target function). After that, the aforementioned procedure for implementing DF–DBIM is classified into 2 stages. In the first stage, the low-frequency f1 is applied to the DBIM from iteration 1 to iteration [\(\frac{{{\text{N}}_{\text{iter}}}}{2} - 1\)]. The high-frequency f2 is used in the second stage and it is executed with the DBIM from iteration \(\frac{{{\text{N}}_{\text{iter}}}}{2}\) to iteration \({\text{N}}_{\text{iter}}\). After \({\text{N}}_{\text{iter}}\) iterations, the obtained output is the reconstructed function.

Fig. 19
figure19

Proposed flowchart of the DF–DBIM procedure

Conclusions

The implications of duple-frequency fusion for the image reconstruction quality of density information along with sound contrast based ultrasound tomography have been analyzed in this paper. As a matter of fact, the quality of reconstruction is under the influence of a large number of factors such as transducer number, interference, iteration number, frequency values, variable number, etc.. When there are restraints imposed on simulated scenarios to have control of the computational cost, the iteration number \({\text{N}}_{{{\text{f}}_{1}}}\) is determined resulting in giving the best performance. In addition, the DF–DBIM is only effective if there are a moderate number of iterations, transmitters and receivers. In case that the number of transducers is either too large or too small, a result of reconstruction which is better than that of the single frequency approach is not produced by the implementation of DF–DBIM. After observing the facts mentioned, we have selected several more pragmatic scenarios in order to analyze more profoundly. In pragmatic scenarios, the finest value of \({\text{N}}_{{{\text{f}}_{1}}}\) which is corresponding to [\(\frac{{{\text{N}}_{\text{iter}}}}{2} - 1\)] decreases 56.11% of the NE in comparison with the traditional approach using single frequency. Thanks to the usage of empirical data, this work will be further developed prior to its application in reality.

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Correspondence to Tran Duc Tan.

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Huy, T.Q., Cuc, N.T., Nguyen, V.D. et al. Tomographic density imaging using modified DF–DBIM approach. Biomed. Eng. Lett. 9, 449–465 (2019). https://doi.org/10.1007/s13534-019-00129-5

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Keywords

  • Ultrasound tomography
  • Distorted Born iterative method (DBIM)
  • Density imaging
  • Inverse scattering
  • Duple frequency (DF)