Data

Below are a few algorithms and methods we picked to help simulate on MATLAB as well as their corresponding graphs and findings

Delay Multiply and Sum

To kick off our data we are going to be taking a look at the Delay Multiply and Sum algorithm. This was originally invented to improve on the soon-to-be outdated algorithm Delay and Sum. The Delay and Sum algorithm is based on wave propagation and is relatively simple to implement, however, is overall limited by the aperture size and operating frequency. In comes the Delay Multiply and Sum.

Studies show that using the algorithm above leads to a significantly higher contrast resolution in imaging, which leads to an increased dynamic range. This is in part due to the scan line signal Sij(t), as when they were uncoupled, the resulting images may be distorted. This was fixed as seen below when a signed geometric mean is applied to the coupled signals.

To code and plot the algorithms described above, we had to download the USTB Repository and Matlab Toolbox V2.1

Plot them Both

As we can clearly see in the labeled plots above, in both cases the DMAS plot came out with a lower amplitude based on the group of scatterers given, thus proving that it is able to create better imaging and higher contrast resolution.

Now that we have plotted both the DMAS and DAS images, we can compare their resolutions to see which is better

Hilbert Transform Method

In addition to the Delay, Multiply, and Sum Algorithm we are also taking a look at the Hilbert Transform Algorithm. This is most commonly referred to as the most important operator in terms of analysis. It comes up in several different contexts and can be used in a multitude of different ways such as instantaneous amplitude/frequency estimation, construction of causal filters for amplitude only, small-redundancy 2D directional wavelets, etc.

It can be most commonly viewed as below:

Which can also be altered to view as a convolution among other things. An important thing to note is that this transform can make use of and denote the Fast Fourier Transform based on the Convolution theorem to calculate the Hilbert Transform which can be expressed as

The Hilbert Transform overall makes the formation of an analytical signal much easier and is quite useful in bandpass signal processing, which in turn can be very useful for Communications as well as Ultrasound Signal processing and determining the best methods for recording waves

Plotted above is a normal Hilbert Transform in which it is related to the data by a 90-degree phase shift where the sines become cosines and cosines become sines

Now that we have plotted the Hilbert Transform we need to make a connection to Ultrasound Processing and how it could possibly help with imaging and higher resolution. One of the biggest problems when dealing with Ultrasound recordings is the power needed. Scans of a large magnitude with great detail need significantly more processing power of which is sometimes not available leaving lackluster scans as a result. In order to counteract this, we need to determine the best methods for improvement to quantify acoustic attenuation in Diagnostic Ultrasounds. This is where Power Spectral Density Estimation methods come into use. By comparing different Power Spectral Density estimation methods like the short-time Fourier Transform, Thomson's multitaper technique, or Welch's periodogram we can better choose a method that is the most efficient and doesn't consume as much power.

Sample Hilbert Transform sampled for 1 second at 10kHZ

The Graph above is plotted in between 0.01 seconds and 0.03 seconds with three different sinusoidals. After plotting this we can take the analytical line plot and put it into the power spectral density form by uncommenting out the three lines on the bottom.

Shown above are the Welch estimates of the power spectral density for the original signal and then the Hilbert one. By dividing the sequences into Hamming-Windowed, non-overlapping sections with a length of 256 we can clearly see the power per frequency rate. Now that we have the rate for one, let us take a look at another method in order to compare it with the welch method.

Hilbert Transform Fir Filter

FIR Frequency Response

Here we have both the 60th order Hibert Transform FIR Filter. From there we can extrapolate the imaginary part of the analytical signal as we see in the second photo. Finally, after plotting both of these graphs we can now plot the Power Spectral Density of the signal and compare it to the Hilbert counterpart.

From what we can gather both the Power Spectral Density Estimations are quite similar for both Welch's method and the FIR filter. Based upon these results you can safely say that either method can be good for estimation. Hopefully, with these results, we can determine how to better attain power levels and pick a method that can attain the most power per frequency used. The Fir Filter while similar to the Welch Method has the added advantage of a better frequency response as well as higher quality when it came to color filtering as it contributes to clutter rejection when taking a scan allowing lower velocity blood flow to be measured as well.