With all the simulations described in the previous section the amount of information created is overwhelming. A major challenge is to compress the information to easily evaluate our hypothesis. The plots, of the type in Figure 5.11 and 5.10, from each simulation displaying the mean and variance of the correlation values and the error values are placed in Appendix D. If we investigate the plots of the correlation and the error we see the trend that we pointed out in Section 5.4.1; that higher correlation gives less error. The parameters giving both the highest correlation and the lowest error for the Capon beamformer are L = 32 with K = 0 and L=64 withK=5. If we remember back to Section 3.4.3 this is actually the parameters that gives the Capon beamformer approximately equal speckle statistics as the DAS beamformer. We will discuss this result in detail in the next section.
The fact that these parameters generally seems to be the best for Capon, allows us to only choose these parameters for Capon when we do the final evaluation of the beamformers. In the later analysis we will find that Capon gets similar performance as DAS only with the parameters giving the best results for Capon, the analysis is therefore simpler if we only use the parameters giving the best performance for Capon. If we further investigate the plots in the appendix we will see that the longest window used for correlation, 50 samples, gives the best result. Based on these observations we can extract only these data and compare them for different compression simulations both with and without noise.
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Error from crosscorrelation estimation
Compression [times ∆]
Error [dB]
Compression [times ∆]
Error [dB]
DAS Capon:K=0,L=32 Capon:K=5,L=64
(a) Crosscorrelation: Top: dynamic transmit,bottom:fixed transmit.
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Compression [times ∆]
Error [dB]
Error from pulsed−Doppler estimation DAS
Compression [times ∆]
Error [dB]
DAS Capon:K=0,L=32 Capon:K=5,L=64
(b) Pulsed-Doppler: Top: dynamic transmit,bottom:fixed transmit.
Figure 6.4: The error from the crosscorrelation displacement estimation (a) with 50 samples window length, and the pulsed-Doppler displacement estimation in (b). The top plots are from dynamic transmit focus, the bottom plots from fixed transmit focus. These plots are from simulations without noise.
In Figure 6.4 (a) we have plotted the error from the crosscorrelation estimation with 50 sample window length, and in (b) the error from the pulsed-Doppler estimation. The top plots are for the images created with dynamic transmit, and the bottom plots are from the images created with fixed transmit. We see that the errors are close to equal for both the beamformers, but for∆compression the CaponL=32,K=0 beamformer is slightly better than DAS for crosscorrelation estimation when dynamic focus is used, while DAS is slight better for the crosscorrelation estimation for fixed focus. For the pulsed-Doppler estimation the results are very similar.
Error from crosscorrelation estimation
Compression [times ∆]
Error [dB]
(a) Crosscorrelation: Top: dynamic transmit,bottom:fixed transmit
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Compression [times ∆]
Error [dB]
Error from pulsed−Doppler estimation DAS
(b) Pulsed-Doppler: Top: dynamic transmit,bottom:fixed transmit
Figure 6.5: The error from the crosscorrelation displacement estimation (a) with 50 samples window length, and the pulsed-Doppler displacement estimation in (b). The top plots are from dynamic transmit focus, the bottom plots from fixed transmit focus. These plots are from simulations with noise as described in Section 6.1.3
In Figure 6.5 we have plotted the same simulations as in Figure 6.4, but
now we have plotted the error values from the calculations where noise has been added to the data. From the plots we see that the noise has created higher error in the estimations, but it has not influenced one beamformer more than the other - the performance of the two beamformers are still very similar.
If we continue our investigation of the plots in the appendix, one thing we will notice is that the Capon beamformer seems to perform relatively best, compared to the DAS beamformer, for the crosscorrelation displacement estimation at the shortest window length, 12 samples, when noise has been added to the simulations.
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Error from crosscorrelation estimation
Compression [times ∆]
Error [dB]
DAS Capon:K=0,L=32 Capon:K=5,L=64
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Compression [times ∆]
Error [dB]
DAS Capon:K=0,L=32 Capon:K=5,L=64
Figure 6.6: Error from crosscorrelation estimation with noise for 12 samples windowlength.Top:dynamic transmit,bottom:fixed transmit.
In Figure 6.6 we have plotted the error from the different compression simulations for precisely this observation. From the plot we see that indeed, the Capon beamformer gets less error than DAS - but it is just slightly and the error is, for most of the simulations, very similar.
To make sure our previous assumptions that the lateral oversampling factor ofq = 4 is sufficient to not lose information when imaging speckle with Capon beamforming, Section 3.4.3, and that decimating the signal by a factor 4, Section 3.3.3, did not interfere with the result we did a final simulation. In this simulation we used a lateral oversampling factor of q = 16 and did not decimate the signal, meaning that we had 4 times the amount of data both in axial and lateral dimension. However, this information should not interfere with the results - and it did not.
In Figure 6.7 we see that this simulation follows the previous pattern of Capon with L = 32 and Capon with L = 64,K = 5 giving similar results as DAS, and actually slightly better for the crosscorrelation displacement estimation withL = 32. The reason for not using more and different parameters for the Capon beamformer in this simulation is the overwhelming simulation time, approximately 56 hours (see Appendix A), we get with 2048 lines in one ultrasound image.
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Sum of squared error : Capon L=64
DAS
Sum of squared error : Capon L=32
DAS Capon:K=0,L=32 Capon:K=5,L=32
(a) Error values from crosscorrelation estimation
Sum of squared error
DAS Capon:K=0 Capon:K=5
(b) Error values from pulsed-Doppler estimation
Figure 6.7: The error from the crosscorrelation estimation (a) and the pulsed-Doppler estimation (b). This if from a simulation with no decimation and with lateral oversampling factor q=16.