1

Laboratory 1: Signal processing using ultrasonic imaging signals C. Coussot, Y. Qiu, M.F. Insana November 26, 2006 Objective: The objective of this laboratory is to familiarize readers with some of the elementary signal processing required to produce medical ultrasonic images. The three types of images discussed in this lab are Brightness Mode (B-Mode), Color Doppler Mode (C-Mode) and Duplex Doppler Mode (D-Mode) images. Readers will learn about signal acquisition and processing, observe the effects of signal filtering and thresholding, and explore the consequences of temporal under-sampling on diagnostic image quality.

Different use of ultrasonic images in medicine. From:medical.philips.com

2

Acquisition of ultrasonic B-mode images: As shown in Fig 1, an ultrasound transducer is pressed against the skin surface. At least one short-duration (~0.3 μs) broadband pulse is transmitted into the body for each line of sight in the image. Immediately afterwards, the transducer switches from transmitter to receiver to listen for returning echo signals. All backscattered pressure echoes are detected, focused, filtered, and amplified to give a radio-frequency echo signal for each line of sight. These signals are then demodulated, logarithmically compressed in amplitude, and converted into gray-level values for digital image display along that line of sight in the image. The beam is then electronically steered to another line of sight and the process is repeated to build up a gray-scale B-mode image. These images are presented for display often at video rates, 30 frames/second. If blood is flowing in the region of interest and the color-flow imaging option is selected, then an additional set of pulses is transmitted and echoes are received to estimate blood velocity, leading to lower frame rates (from 10 to 20 frames/seconds). Velocity estimates are overlaid onto the B-mode image as color pixels to produce a C-mode image.

Figure 1. B-mode echo acquisition and representative signals are illustrated. Color-Doppler methods described in Fig 2 are combined with B-mode methods to give the C-mode image above.

Ultrasonic transducer

Sound beam axis defines line of sight

Arterial walls

Radio-frequency echo signal

Compressed amplitude echo

signal

line of sight

Ultrasonic image (B-mode with color-flow

overlay)

3

Figure 2. Illustration of C-mode echo acquisition and signal processing for blood velocity estimation. The Doppler spectrum is illustrated in the lower right corner. A C-mode image is shown in Fig 1. It is a cross sectional view of a normal common carotid artery. Acquisition and processing of color-flow images. When an operator wishes to image blood flow in an artery, an additional set of pulse-echo signals are recorded. Threaded between B-mode acquisitions described in Fig 1 are the C-mode acquisitions for which several longer duration (~1 μs), narrow-band pulses are transmitted and received at each line of sight. As illustrated in Fig 2 for 4 pulses, up to 32 pulse-echo waveforms may be acquired to form an echo ensemble for velocity estimation. Following each pulse transmission, echoes are received as a function of time t and digitally recorded in our instrument at 4 x107 echo samples per second. Pulses that generate the ensemble echo waveforms are transmitted at the relatively slow rate of approximately 103 pulses per second along the time axis t’ in Fig 2. The frequency of pulse transmission is the pulse-repetition frequency (PRF) that is the Doppler sampling frequency Sf . To estimate blood velocity we analyze echoes as a function of t′ . Often t is referred to as the fast time axis used in all modes of imaging and t′ is referred to as the slow time axis used primarily for velocity estimation. Therefore, instead of looking along a line of sight in the image, velocity estimates fix the fast time value t and analyze the evolution of the signal over slow time t′ (along the red line in Fig 2). Fig 2 shows only 4 waveform

t′

t

Blood velocity

Df

Stationary echoes (clutter)

Noise

FT

Blood vessel

Transducer

}Echo

ensemble

Ultrasonic beam axis

}

Four echo samples for Doppler spectral analysis

t′

echo amplitude

t′

echo ampl

Nf0

1Df

Dnf

4

samples, but we can use up to 32. Taking the Fourier transform of the echo amplitude at a fixed time t and as a function of t′ we then decompose the signal into sine waves at n different temporal frequencies Dnf . The amplitudes of the sine waves are presented in a Doppler spectrum as a function of Doppler frequency Df . Frequency is then converted to velocity v with the Doppler equation:

)cos(2 0 α⋅⋅

=f

cfv D ,

where c is the average speed of sound in tissue (~1540 m/s), 0f is the carrier frequency of the narrow-band pulse transmitted (2.5 – 15 MHz), and α is the Doppler angle specifying the direction of the sound beam axis with respect to the direction of blood motion. We then compute a Doppler spectrum of red blood cell velocity for each location in the field of view. These spectra have three components: one is from stationary or slowly moving echoes (black spectrum in Fig 2), one is from moving red blood cell echoes (red spectrum in Fig 2), and one is noise. At each location, the spectrum is filtered to minimize stationary and noise components, the results are averaged to find the mean velocity (spectral centroid), and that value is color coded for mapping into a C-mode image (Fig 1). Details of velocity estimation may be found in standard texts [1-3].

5

Wall filtering. A significant problem is to design a filter that minimizes stationary echoes without also affecting the blood flow echoes. This process is called wall filtering because it aims to filter the motion of the vessel walls, muscle tissues, and other slow moving scatterers that contribute significantly to the echo signal energy. When stationary and blood echo spectral components are distinct, filtering is easy; however, this is rarely the situation.

Figure 3. Example of an in vivo Doppler spectrum from a normal adult carotid artery. Sampling echo waveforms. First, let’s introduce a little sampling theory in the context of this example (Ch 4, [5]). The Nyquist frequency Nf (see Fig 2) is defined as the highest frequency having non-zero amplitude. If signals are properly sampled, then the sampling frequency Sf is at least twice the Nyquist rate, viz.,

NS ff 2≥ . If that criterion is not met, then we say that signals are aliased and the spectrum is distorted from under sampling. Consider the reduced Doppler frequency axis, SD fff =′ , for the spectrum of Fig 3. Since the spectrum of a sampled signal is symmetric and periodic of period Sf , the reduced frequency axis presents all the information we need for this problem within the range 5.00 ≤′≤ f . If the amplitude of the Doppler spectrum is not approximately zero at 5.0=′f , aliasing is possible. In fact, the signal in Fig 3 may have some minimal aliasing. That spectrum has an ensemble size of 16 pulses; therefore we have 8 points in the

5.00 ≤′≤ f frequency range. Since we want to pass only blood echoes, we choose to apply a high-pass filter with an ideal corner frequency near ′ f = 0.05. We will use these signals and conventions in this lab.

D Sf f f′ =

6

Selecting filter parameters (Ch 7 [5]). The parameters available for filtering include (a) the type of filter, (b) the cutoff (or corner) frequency, and (c) the filter order. Filter type and order number determine the sharpness of the transition region (Fig 4) and the smoothness of the spectral response in the pass and stop bands. While the goal is to eliminate all echoes below the cutoff frequency and pass those above, the realities of building filters force us to accept approximations. To minimize the transition region (sharpen the frequency response), we select a high-order filter. Unfortunately, higher order filters can increase instrument cost and processing time, which may lower the frame rate for color-flow imaging. One goal of this lab is to understand the tradeoffs and consequences of selecting wall filters for peripheral Doppler applications. We use simple IIR digital filters approximated from continuous-time Butterworth and Chebyshev filters in this lab. These filters are briefly described with their advantages and drawbacks to guide our selection of filter parameters. In Fig 4, the Doppler spectrum of Fig 3 is plotted along with various Chebyshev filter response functions to illustrate the effects. For each high-pass wall filter in Fig 4, the cutoff frequency we selected is ′ f = 0.15 and the filter order number ranges between 1 and 4. High-order filters yield short transition regions that help us eliminate stationary echoes with minimal effect on blood echoes. Fig 4 suggests that we use a 4th order Chebyshev filter provided we are willing to accept oscillations in the pass band. Oscillations can distort average velocity estimates that appear in C-mode images as color noise.

Figure 4. Example in vivo carotid Doppler spectrum is shown along with the transfer functions of four high-pass Chebyshev filters each with a cutoff frequency of 0.15. Increasing the order of the filter shortens the transition region.

Corner frequency

Oscillations in the pass band as the order rises

Transition regions (sharper if the order is high)

D Sf f f′ =

Doppler spectrum

7

Figure 5 shows the frequency response of four Butterworth filters with the same parameters described in Fig 4. Despite the oscillations, Chebyshev filters are preferred because of the shorter transition regions. Even high-order Butterworth filters are slow to go from stop to pass bands. Therefore stationary echo clutter cannot be filtered without also filtering part of the blood echoes.

Figure 5. An in vivo carotid Doppler spectrum is shown along with the transfer functions of four high-pass Butterworth filters each with a cutoff frequency of 0.15. Increasing the order of the filter shortens the transition region. For a fixed order, Butterworth filters transition more slowly than Chebyshev filters yet they remain maximally smooth in the stop and pass bands. Note that filtering blood echoes is equivalent to multiplying the Doppler spectrum by the filter response function. The product gives the spectrum of the filtered waveforms. Such filtered spectra are displayed on page 11. Now that both filters have been introduced and you can see what their transfer functions look like at different orders, let’s examine options for selecting the cutoff frequency. If we could tolerate long time-duration (high-order) filter functions, we could arbitrarily shorten the transition region in the frequency domain. This situation would allow us to choose a cutoff frequency at ′ f = 0.05 and thus obtain excellent separation of stationary and blood echoes. (The noise is so broad band that complete separation of noise from blood echoes is impossible with frequency filters.)

Large transition regions for any order

Smooth transition, no oscillation

D Sf f f′ =

8

Realistically, we need to select a cutoff frequency somewhere between two undesirable extreme situations: one where the cutoff frequency is so low that it passes all the blood echoes but also much of the clutter spectrum, or so high that it stops all of the clutter signal and much of the blood echo signals. Figure 6 displays a second-order Chebyshev filter with four different corner frequencies. At ′ f =0.1, the filter passes too much of the clutter spectrum, and yet at ′ f =0.3 it eliminates too much of the blood spectrum.

Figure 6. In-vivo Doppler spectrum and the magnitude of the transfer function of second-order Chebyshev filters with various cutoff frequencies. Figures 7, 8 and 9 show C-mode images filtered with a second-order Chebyshev wall filter having corner frequencies ′ f = 0.1, 0.2, and 0.3, respectively. Color noise in figure 7 means that a corner frequency of 0.1 may be too low; erosion of color inside and near the arterial wall in figure 9 suggests 0.3 may be too large. Obviously, an inappropriate choice of wall filter or filter parameter can bias the velocity estimates in the color-flow image or fail to eliminate color noise, and therefore adversely affect diagnostic performance.

D Sf f f′ =

Doppler spectrum

9

Figure 7. In vivo color-flow image filtered with a Chebyshev wall filter of order 2 and a cutoff frequency of 0.1.

Figure 8. In vivo color-flow image filtered with a Chebyshev wall filter of order 2 and a cutoff frequency of 0.2.

10

Figure 9. In vivo color-flow image filtered with a Chebyshev wall filter of order 2 and a cutoff frequency of 0.3.

11

Clutter thresholds. Frequency filtering is not very effective at reducing echo noise because that noise is very broadband. Echo noise appears in color-flow images as color noise. To eliminate weak amplitude echo signals, we set a clutter threshold whereby echoes of amplitude less than the threshold value are eliminated. Varying the threshold value controls color-flow noise not only by eliminating low amplitude electronic noise but by suppressing clutter from stationary echoes that leak through the wall filter. Figure 10 shows a Doppler spectrum after the signals were filtered with a second-order Chebyshev wall filter. Three clutter threshold amplitudes are indicated by the three horizontal lines at 100, 150 and 200. As the corner (cutoff) frequency increases from 0.1 to 0.3 in the three plots, the maximum amplitude of the filtered echo spectrum decreases and the centroid (amplitude-weighted average Doppler frequency used in the Doppler equation) increases from 0.21 to 0.32. However, when the cutoff frequency is 0.3, very little of the blood echo spectral energy remains. Also at 0.3 cutoff frequency and 200 threshold, no signals are passed, which indicates no blood flow. In this laboratory, the clutter threshold is expressed as a percentage of the maximum value of Doppler spectral amplitude. For example, a threshold equal to 0.01 where the maximum amplitude is 10000 (common in the tissue) will eliminate echo amplitudes lower than 100 (10000*0.01). Since we don’t know the amplitudes at the centroids and the maximum spectral value in the image, we must use our knowledge of vascular anatomy to help us decide how to set this threshold. One of your tasks in this laboratory will be to identify the best clutter threshold value.

Figure 10. Filtered Doppler spectra at three different corner (cutoff) frequencies. Three clutter thresholds levels are indicated by the three horizontal lines.

Cutoff=0.1 maximum value=450 Clutter thresholds

centroid

Cutoff=0.2 maximum value=390

centroid centroid

Cutoff=0.3 maximum value=170 Below the higher threshold!

D Sf f f′ =

D Sf f f′ = D Sf f f′ =

12

In figure 8 above, we set the threshold very low (0.01) and found some (red) clutter noise above and to the left of the vessel. We know it is noise because the B-mode image shows no vessel in this region. Figure 11 and 12 display the same data except that thresholds were increased to 0.02 and 0.03, respectively. As you can see, the clutter noise is reduced but the slowest blood velocities – those near the vessel wall – are also reduced. Physiologically, we know that blood velocity is greatest near the center of the artery and least near the wall. Therefore, while the clutter threshold did eliminate valid signals, the penalty is less severe because of the anatomical context of the flow patterns. Coupling anatomy (in the B-Mode image) with color-flow is the value of C-Mode imaging over other types of Doppler displays.

Figure 11. Color-flow image filtered with a second-order Chebyshev filter and a cutoff frequency of 0.2. The clutter threshold is set to 0.02. The wall of the artery appears thicker than in Fig 8 because flow near the vessel wall is too slow to be displayed.

Figure 12. Color-flow image filtered with a second-order Chebyshev filter and a cutoff frequency of 0.2. The clutter threshold is raised to 0.03. The wall of the artery appears thicker than in Figs 8 or 11. However, the “red” clutter noise is further reduced.

13

More on sampling and aliasing. The rate at which we transmit pulses while recording an echo ensemble, i.e., the PRF, is the sampling rate Sf for signals in the Doppler spectrum. If the PRF is very low and we do not satisfy the sampling criterion given by NS ff 2≥ , we alias the recorded signals. Consequently, energy from the highest frequencies in the waveform appears at lower frequencies in the recording. Figure 13 displays two sinusoids of different frequencies whose sample points are the same. With no other indication than the sample points, viewer cannot tell if the sampled sinusoid is from the properly sampled blue waveform or the under sampled red waveform. Energy from both signals would appear at the frequency of the blue waveform. In this way, aliasing distorts the waveform.

Figure 13. Plot illustrating the aliasing phenomenon. An under sampled red sinusoid would be mistakenly reconstructed as the blue sinusoid. (from Wikipedia). At the bandwidth limit where the Doppler frequency equals the Nyquist frequency,

fD = fN , we have fN '= fN

fS

≤ 0.5 to satisfy the sampling theorem. So it is easy to see

that the sampling frequency Sf must exceed twice the Nyquist frequency, i.e.,

NS ff 2≥ , to avoid distortion. Most electronic noise is very broadband. Therefore, low-pass anti-aliasing filters applied before sampling are always recommended in addition to wall filters to minimize aliasing. Another way to view the problem of aliasing is to recall that frequency spectra are sums of periodic functions that repeat at integer multiples of Sf . Under-sampling Doppler signals causes the repeating spectra to overlap, which misplaces some of the signal amplitudes at the wrong frequency, just as we saw happen in Fig 13. Fig 14 illustrates the effects of varying the sampling rate on the Doppler spectrum for the same input signals. Fig 15 shows how the blood component of the Doppler spectrum from the carotid artery of a volunteer can appear as noise when the waveforms are under sampled by setting the PRF too low.

14

Figure 14. The Doppler spectrum at the top results from properly sampled echo signals. Spectra on the bottom are from under sampled signals. On the left, we see the naturally periodic component spectra overlap. These are summed in the process of spectral analysis to give the distorted observed spectrum on the lower right.

Figure 15. Under sampling in vivo echo signals gives this Doppler spectrum that is distorted because of aliasing.

No overlapping: when we sum those spectra we keep them unchanged. The final spectrum from 0 to fs/2 is the spectrum of the initial

signal.

Sampling frequency

Spectra summed to get the spectrum of the sampled signal.

The 2 spectra are overlapping. When we sum them, we get the spectrum at the right,

which is a distortion of the original.

Aliased spectrum

' D Sf f f=

' D Sf f f= ' D Sf f f=

D Sf f f′ =

15

We do not need to examine the Doppler spectrum to find out if our signal is under sampled. We can also look at the color-flow image. If aliasing places echoes at the wrong frequency, then, according to the Doppler equation, those misplaced echoes will not accurately represent blood velocity in the color-flow image. Once the PRF is chosen, we know the maximum “authorized” value for Df and therefore we also know the maximum velocity that can be accurately detected. As displayed in figure 16, aliased signals are displayed with colors indicating that blood is moving in the opposite direction. In Fig 16, the blood is moving away from the transducer. Therefore we expect the flow to appear in the blue spectrum. However under-sampling the signals causes aliasing that causes the fastest velocities in the center the vessel to wrap around the blue colors are appear in the red colors. Because we have an anatomical context, we can vary the PRF to see if the colors change. If the red regions remain despite increases in the PRF, then turbulent flow may be present, which might indicate a potentially life-threatening condition. A sonographer’s job is to eliminate potential artifacts, like aliasing, that mask themselves as flow pathologies.

Figure 16. Under-sampled C-mode image of the blood flow in an artery. The effects of alsiasing are seen as a reversal of flow.

High velocity flow displayed as if going forward

Low velocity flow going backward

The color bar indicates the authorized velocity values depending on the PRF. The forward flows

are displayed in red-yellow and the backward ones are in blue-green. If the signal is aliasing, high

backward velocities will be displayed in yellow-red as shown by the red arrows.

High velocity flow displayed incorrectly, as if blood is flowing toward the transducer

Low velocity flow (blue) away from transducer

The color bar indicates the authorized velocity values depending on the PRF. Flows toward the transducer (located at the top of the image) are displayed in red-yellow and those away from the transducer in blue-

green. Aliased away signals are incorrectly displayed in yellow-red indicating flow toward the transducer.

16

Tissue threshold. As you can imagine, computing the Fourier transform of the ensemble at each point of the image, filtering it and computing the centroid takes time. If the computational load is too high, we must slow the frame rate and that might be unacceptable for diagnostic reasons. Note that there are usually only a few interesting regions of flow in most vascular images. For example, in Fig 16, flow is included in less than 1/10th of the field of view. We can reduce the computational load and maximize frame rate by introducing a tissue threshold, where we identify those regions for which Doppler measurements is necessary. Careful, tissue thresholds are not the same as clutter thresholds discussed above. Stationary tissues surrounding vessels are usually more scattering than blood, so that bright regions in the B-Mode image are likely to be tissue not requiring Doppler estimation. Figure 17 displays the echo power (squared amplitude) of the B-Mode image before the threshold is applied. As you can see, the artery region is black indicating low-power points. Conversely there are many bright elements outside the vessel in the tissue. Figure 18 indicates pixels where velocities have been computed, i.e. points where the echo power was below the threshold. These points correspond primarily to pixels in the artery.

Figure 17. The echo power (squared amplitude) of the B-Mode image is computed and displayed. The high power regions are bright corresponding to the tissue regions.

High intensity elements in the tissue (displayed in pink)

Low intensity elements in the artery (displayed in black)

17

Figure 18. Once the threshold has been applied, a zero power is assigned to the elements whose power was over the threshold. They appear in black whereas the pixels whose power was below the tissue threshold are bright. Only the velocity of the elements in the artery and of some remaining regions will be computed, leading to fewer calculations and faster C-mode frame rates. The tissue thresholding works as follows. Compute the square of the B-mode image. Limit velocity estimation to pixels in regions of power less than the tissue threshold. Setting a tissue threshold at 0.98 tells the system to subject to velocity processing only those pixels with power (1 - 0.98) x 100 = 2% of the peak and less. This tissue threshold was applied in Fig 18. Increasing the threshold improves C-mode frame rate but reduces the number of pixels displaying color values. For example, a threshold at 0.996 processes only 0.4% of the image pixels for flow. The problem is to find an optimum threshold – one where all the blood-flow pixels are processes and yet where no time is wasted processing non-flow regions. Once the tissue threshold has been applied, one can filter the Doppler spectrum of the remaining pixels and apply the clutter threshold. Figure 19 displays three images processed at different tissue thresholds: 0.95, 0.98 and 0.996. At the lowest threshold setting (0.95, top left), we obtain a smooth flow profile in the vessel and yet some color noise in the upper left corner. At the highest threshold (0.996, bottom), the color noise is nearly gone although holes begin to appear in the arterial flow, which we know is not a physiological condition.

The elements whose echo power is over the threshold appear in black. Their velocity will not be computed.

The elements in the artery appear bright. Power in those pixels are below the tissue threshold, so their velocities are computed.

18

Figure 19. Ultrasonic images pre-filtered with different tissue threshold values. As the threshold increases, clutter noise in the tissues disappears but holes may appear in the artery flow since some high density moving points in the artery can be misclassified as clutter.

Threshold=0.95

Threshold=0.996

Threshold=0.98

Color noise in the tissue Less noise in the tissue

No noise in the tissue

Smooth flow in the artery

Smooth flow in the artery

Holes in the blood flow

19

D-mode images. We have been looking at color-flow images that combine the tissue anatomy with blood velocity distributions in space and time. For this information, we collapsed the Doppler spectrum to a single value that was color coded by estimating the centroid of filtered and thresholded echo spectra. However, there is additional information in the Doppler spectrum that may be preserved by displaying D-mode images. A D-mode image (also called spectral Doppler or duplex Doppler) is a spectrogram displaying velocity (or frequency, since they are proportional) on the vertical axis and time on the horizontal axis for a fixed location in the body. Spectra at any instant of time are shown in gray scale, as in Fig 20 below. Bright gray pixels indicate high amplitude spectral values at the velocity and time corresponding to the location of the pixel in the image. Often an operator begins a patient exam using color-flow imaging to locate the position of blood flow in the anatomy. A small window is placed in a B-mode region under analysis and D-mode is switched on to show how the spectrum varies with time. This information is valuable in complex pulsatile flow situations, e.g., near suspected arterial stenoses.

Figure 20. Spectral Doppler (D-mode image) of a carotid artery. The white horizontal line near the top indicates zero velocity. High velocity away from the transducer (large negative values) is obtained near systole in the cardiac cycle. Echo processing is similar to color-flow imaging in principle, although C-mode images use many shortcuts to maximize frame rate that will not work in D-mode. As we described above, wall filters reduce echoes that are both high amplitude and slow moving (stationary). In this way, blood echoes remain while tissue echoes are eliminated at low-velocities near the baseline.

20

The choice of wall filter and its parameters are based on the same tradeoffs as before so we refer you to pages 5-11 of this lab. Likewise, the phenomenon of aliasing also impacts the D-Mode image. We showed earlier that under sampled echo acquisitions are displayed as blood traveling in both directions within the vessel. In D-mode images, the signal amplitude also “loops” and flow appears with the opposite sign. Figure 21 gives examples of D-mode (spectral Doppler) images in which different filters have been applied. With no filter applied, the tissue spectrum overwhelms the flow spectrum. Filtering suppresses the low velocity (there are no more white pixels near the baseline) although some low velocity flow is filtered.

Figure 21. Spectral Doppler images of a carotid artery filtered in different ways. When not filtered, we only see the high intensity tissue echoes around the baseline.

Low velocity-high intensity stationary tissue

High velocity, low intensity blood flows are invisible with no filter. Because the tissue echoes are so much stronger, the weaker blood echoes do not register in the display.

Spectral Doppler not filtered

Spectral Doppler filtered with a first-order Butterworth. Cutoff=0.4

Spectral Doppler filtered with a second-order Chebyshev filter. Cutoff=0.4

Filtered stationary tissue (no “white” anymore)

Blood flow

Partially filtered tissue

Blood flow

21

Figure 22 illustrates the appearance of aliasing in D-Mode images. As shown for the color-mode images, blood appears to be going forward at some slow times t and backward at some others.

Figure 22.Aliasing in spectral Doppler images.. Now that you know the basics of filtering and sampling, please go to the website and apply your new knowledge. You will be asked to make filtering decisions, view the image sequences that result, and explain the appearance. Data were acquired using the Siemens Antares system pictured below. We are most interested in having you discuss the tradeoffs and compromises required for selecting imaging parameters that optimize diagnostic information content. Have fun.

Figure 23.Antares SONOLINE ultrasonic device.

High velocity blood flow away from the transducer is displayed flow toward the transducer because of aliasing

22

References: [1] J.A. Zagzebski Essentials of Ultrasound Physics, Mosby, St. Louis, 1996. [2] B.A.J. Angelsen, Ultrasound Imaging: Waves, Signals and Signal Processing, Vol 1, Emantec AS, Trondheim Norway, 2000 http://www.ultrasoundbook.com [3] J.A. Jensen, Estimation of Blood Velocities Using Ultrasound: A Signal Processing Approach, Cambridge University Press, New York, 1996. [4] A. Stoylen, Basic ultrasound for clinicians, http://folk.ntnu.no/stoylen/strainrate/Ultrasound [5] A.V. Oppenheim, R.W. Schafer, J.R. Buck, Discrete-Time Signal Processing, 2/e, Prentice-Hall, Upper Saddle River NJ, 1999.