3.5 NUFFT 3-D Beamforming
3.5.3 Accuracy Evaluation
Unlike the frequency-domain direct method of (3.7) and (3.6), which is an exact method, the proposed method is an approximated one because the realization of NUFFT is an approximated method. In this sub-chapter, the beamforming errors caused by NUFFT are evaluated and proven to be negligible in the application of 3-D underwater acoustical imaging in both the far and near fields.
Fig. 3.7 a A spiral 2-D array with 256 elements. b A random 2-D array with 512 elements
In addition, although the CZT 3-D beamforming [3–5] cannot be directly applied to arbitrary arrays with off-the-grid elements, it is possible to be applied by approx-imating the array element positions with their closest on-the-grid positions. The accuracy loss of the approximated CZT is also discussed and compared with that of NUFFT method to verify the advantage of NUFFT.
The computational accuracy of NUFFT is determined by the two parameters:σ and J , as is mentioned above. To evaluate the influence of the two parameters on the accuracy of beamforming, NUFFT is realized based on the toolbox given by Fessler [27], whereσ and J can be set to different values.
Two arbitrary arrays: a spiral array and a random array, as depicted in Fig.3.7a and b are employed for computing the errors caused by NUFFT. The diameter of the two arrays is 60λ0, whereλ0is the wavelength of the carrier frequency f0. For a fully sampled uniform 2-D circular array with the diameter 60λ0, the total number of elements will be 11,305. The spiral 2-D array is composed of 256 omnidirectional punctiform elements, whose divergence angle, the key parameter of the spiral array, is 92.05°, as given in [28]. The random 2-D array with uniform distribution [29,30]
is composed of 512 elements. The transmitter located in the center of the array is omnidirectional. A 3-D scene with two reflecting scatterers is considered, the posi-tions of which are located at (ρ, θa, θe) (20 m, −15◦, −15◦) and (20 m, 0◦, 0◦), as described by the geometry of Fig.3.1a, whereρ denotes the distance from the center of the 2-D array to the scatterer. The scatterers are all taken as points and the medium is supposed to be isotropic and ideal without any frequency absorption. The trans-mitting signal is a Gaussian-envelope pulse with central frequency f0being 500 kHz and the 3 dB bandwidth being 180 kHz. This situation is obviously corresponding to the far-field condition. The received signals of the sensors are simulated by using the method adopted by Palmese and Trucco [31]. The details about the simulation technique will be given in Chap.5.
The relative errorεmax of the whole 3-D image obtained by using the NUFFT 3-D beamforming method can be computed [18] by
3.5 NUFFT 3-D Beamforming 41 different values ofσ and J
J
where ˆsm(n) and sm(n) represent the mth beam signal obtained by the NUFFT 3-D beamforming method and the frequency-domain direct method, respectively, and
“maxn,m( )” represents the maximum of the expression in the bracket for all the values of n and m. The procedure that computes the relative errors is depicted in Fig.3.8.
The relative errors for different values ofσ and J are computed and depicted in Fig.3.9. From Fig.3.9, it can be found that for both arrays, the relative errors are all above−32 dB when σ 1 and the relative errors will become less than or equal to
−87 dB when σ 2, with J being assigned different values between 4 and 6.
In order to reflect the accuracy of the proposed method intuitively, Fig.3.10(for the spiral array) and3.11(for the random array) show the 2-D slices at the focusing distance of the whole 3-D images obtained by the frequency-domain direct method and the NUFFT 3-D beamforming ( J 4 and σ 2). It can be found that there
Fig. 3.10 The 2-D slices at the focusing distance of the 3-D images obtained by the frequency-domain direct method and the NUFFT 3-D beamforming ( J 4 and σ 2) for the spiral array.
a for the frequency-domain direct method; b for the NUFFT 3-D beamforming
Fig. 3.11 The 2-D slices at the focusing distance of the 3-D images obtained by the frequency-domain direct method and the NUFFT 3-D beamforming ( J 4 and σ 2) for the random array.
a for the frequency-domain direct method; b for NUFFT 3-D beamforming
is nearly no difference between the 2-D slices obtained by the two methods for the different arrays. The relative errors of the side lobe level of the 2-D slices are all lower than−80 dB. The relative errors of the main lobe width are also lower than −80 dB.
The output beam signals in the direction (−15◦, −15◦) and (15◦, 15◦) computed by the frequency-domain direct method and the NUFFT 3-D beamforming ( J 4 and σ 2) are given in Fig.3.12. It is found that the two beam signals almost overlap with each other. The figures demonstrate that the proposed method based on NUFFT has a good accuracy in the far field.
To verify the high accuracy of the NUFFT 3-D beamforming further, the accuracy loss of the approximated CZT beamforming is also discussed and compared with that of the NUFFT method. The approximated CZT beamforming is realized by approximating the element positions of the spiral and random arrays with their closest on-the-grid positions and then applying the CZT [11,18]. The output time-domain beam signals of the approximated CZT method applied to the two arrays are shown
3.5 NUFFT 3-D Beamforming 43
Fig. 3.12 Output beam signals of the frequency-domain direct method (DM), the approximated CZT method (ACZT) and the NUFFT 3-D beamforming method ( J 4 and σ 2) for the spiral and random arrays. a Beam (−15◦, −15◦) for the spiral array; b Beam (15◦, 15◦) for the spiral array; c Beam (−15◦, −15◦) for the random array; d Beam (15◦, 15◦) for the random array
Table 3.1 The relative errors of the whole 3-D image caused by the proposed method based on NUFFT ( J 4 and σ 2) and the approximated CZT method (ACZT) applied to the spiral and random arrays
Array pattern Relative error for ACZT (dB) Relative error for NUFFT (dB)
Spiral −21.7145 −87.3355
Random −23.6289 −87.2251
in Fig.3.12. It can be seen that in the direction (15◦, 15◦) where exist no scatterers, the beam signals of the approximated CZT method deviate obviously from that of the frequency-domain direct method, while the beam signals of the proposed method still overlap with that of the frequency-domain direct method. The relative errors of the whole 3D image caused by the approximated CZT and the proposed methods are given in Table3.1. It can be found that the relative errors caused by the approximated CZT method are much higher than those caused by the NUFFT 3-D beamforming (Fig.3.11).
Table 3.2 The relative errors of the whole 3-D image in the near-field caused by the proposed method based on NUFFT ( J 4 and σ 2) and the Fresnel approximation direct method (FADM) applied to the spiral and random arrays
Array pattern Relative error for FADM (dB) Relative error for NUFFT (dB)
Spiral −30.2154 −30.2150
Random −25.2324 −25.2320
The above analyses are under the far-field condition, which might not be enough to verify the accuracy of the proposed method. In the following, the relative errors are discussed in the near field for the spiral and random arrays. In the near field, the errors of the proposed method are caused by NUFFT and the Fresnel approximation.
To evaluate the errors caused by the Fresnel approximation, the direct computation of (3.15) is conducted, which is called as the Fresnel approximation direction method (FADM). The direction computation of (3.7), namely the frequency-domain direct method which is accurate both in the near and far fields, is applied to provide reference results.
A field 3-D scene with two reflecting scatters is considered. The two near-field scatterers are located at (ρ, θa, θe) (1 m, −5◦, −5◦) and (1 m, 0◦, 0◦) respec-tively. Figure3.13shows that in the direction (10◦, 10◦) where exist no scatters, the beam signals of the proposed method based on NUFFT ( J 4 and σ 2) devi-ate obviously from those of the frequency-domain direct method but almost overlap with those of the FADM. Table3.2illustrates that the relative errors caused by the FADM and the proposed method based on NUFFT ( J 4 and σ 2) are nearly the same for both the spiral and random arrays. It can be concluded from Fig.3.13 and Table3.2that the error caused by NUFFT ( J 4 and σ 2) are negligible in the near field, compared with that caused by the Fresnel approximation.
On the basis of all the analyses, it comes to a conclusion thatσ 2 can be adopted to realize the NUFFT 3-D beamforming, and the relative errors are so small that they can be neglected in the application of underwater real-time 3-D acoustical imaging.