Time Reversal in Ultrasound
Hassan Masoom
April 26, 2010
Contents
1 Introduction
3
2 Background
4
3 The Time Reversal Process
7
3.1
Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.2
The Stokes Experiment . . . . . . . . . . . . . . . . . . . . . . .
7
3.3
Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.4
The Time Reversal Process . . . . . . . . . . . . . . . . . . . . .
10
3.5
Multiple Scattering . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4 DORT and FDORT
16
4.1
DORT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
4.2
FDORT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
5 Simulations
23
5.1
Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
5.2
Two well Resolved Scatterers . . . . . . . . . . . . . . . . . . . .
25
1
5.3
Five not-so-well Resolved Scatterers . . . . . . . . . . . . . . . .
29
5.4
Scatterers of Equal Magnitudes . . . . . . . . . . . . . . . . . . .
32
5.5
The Addition of Speckle . . . . . . . . . . . . . . . . . . . . . . .
34
5.6
Less Speckle and Stronger Scatterers . . . . . . . . . . . . . . . .
39
6 Conclusion
41
2
Chapter 1
Introduction
Time delay focusing in ultrasound is the process of adjusting the time delays
of individual elements in an ultrasonic transducer. The time delays can be calculated a priori but inhomogeneities in the insonified medium and variations of
the elements can lead to poor spatial focusing. Adaptive focusing is the process
of dynamically changing the focusing properties of an ultrasound transducer
to focus on a specific location. The time reversal process aims to solve this
problem.
This paper discusses the time reversal process and the associated time reversal
operator as they are used to achieve dynamic focusing. First, some background
on ultrasound and another method used to achieve adaptive focusing is discussed. Then, the time reversal procedure is introduced and simulated in the
case of a single scatterer and multiple scatterers. Following this, the DORT and
FDORT procedures are discussed. In the last section, the FDORT algorithm is
simulated and some results are presented.
3
Chapter 2
Background
The basis behind ultrasound imaging is the reflection of ultrasound waves by
abrupt changes in density. This is more commonly known as scattering. The
ultrasound transducer transmits a pulse which is then reflected back after encountering a scatterer. The reflected wave is then detected by the ultrasound
receiver. Scatterers can be of various size and types. For example, the interface
between tissue and bone or a cluster of cancerous cells can both cause scattering.
The transmitted beam is created by several transducers by methods such as
time-delay beam-forming. This allows the transmitted waveform to constructively interfere at the specified location, thus focusing the energy from the transducer at this location. If there is a scatterer at this location, this will result a
strong back-scattered response. An example of this is given in Figure 2.1.
Given a focal location, if the speed of sound in the medium is assumed to
be constant, the propagation time of the wave from each transducer can be
calculated. These propagation times can be used to determine the time delay
required to focus on the given focal location.
The problem with this approach is that the speed of sound is assumed to be
constant. Inhomogeneities in the insonified medium can influence the speed
of the emitted wave thus decreasing the accuracy of focusing [1]. Therefore,
instead of constructive interference, destructive interference can take place at
4
Transducer
Scatterer
Element
Figure 2.1: Time delay focusing.
the intended focusing location.
Increased focusing benefits both ultrasonic imaging as well as therapeutic ultrasound. In imaging, increasing the accuracy of focusing improves resolution and
contrast. As well, improvements in therapeutic ultrasound can also be made.
For example, lithotripsy is the processes of using high intensity ultrasonic waves
to dissolve calcifications within the human body. This includes kidney stones
as well as stones in the gallbladder and liver. To achieve this, accurate focusing
of the calcification is required.
One approach to adaptive time delay focusing is cross-correlation (Figure 2.2).
In this approach, the delay which gives the maximum cross-correlation between
the receive signals is calculated and from this, the appropriate delays are then
determined [2]. The cross-correlation between two functions f and g is defined
as
Z
∞
f ∗ (τ ) g (t + τ ) dτ
(f ? g) (t) =
−∞
If f and g are the receive signals of two elements, the t which maximizes the
cross-correlation is used as the time delay between these two elements.
One issue with this approach is attenuation in the medium. Both frequency
dependent and independent attenuation [3] can cause the received signal to
vary in shape and size. Thus, cross correlation may not always find the correct
5
Delays
Correlation
Figure 2.2: The cross-correlation method.
delay for each element.
The time reversal process was developed to solve the problem of adaptive timedelay focusing. It is invariant of the transmission signals and so does not suffer
from the problems associated with cross-correlation. Furthermore, it makes
no assumptions on the medium nor does it make any assumptions on the the
properties of the ultrasound transducer elements so it is invariant to their inconsistencies.
6
Chapter 3
The Time Reversal Process
3.1
Time Reversal
The time reversal process relies on a fundamental symmetry that holds in all
areas of physics, except in the area of thermodynamics [4]. It states that the
reversal of the final conditions of a process will elicit the initial conditions. For
example, take the explosion of a brick. If one were to time reverse the explosion,
they would need to reverse the velocity of each of the individual particles of the
explosion. In this way, time reversal says that the original brick will be formed.
In other words, reversing the final conditions of the explosion will result in the
initial conditions, or in other words, the formation of the brick.
3.2
The Stokes Experiment
Although this is not practical with a brick, acoustic and electromagnetic waves
are easier to time reverse. In the mid 1800s, George Gabriel Stokes did a lot
of work that dealt with the wave theory of light. One of his experiments was
based on the time reversal process.
In the experiment, recreated in Figure 3.1, an incoming field E is incident on
7
Forwards
n1
n2
E
θ1
θ1
θ2
rE
Backwards
t’tE+r2E = E
tE
θ1
θ1
r’tE+rtE = 0 θ θ
2
2
rE
tE
Figure 3.1: The stokes relations.
a boundary between two media (with index of refractions n1 and n2 ) at an
angle θ1 . The result is two components, one reflected and one refracted. The
reflected component has magnitude rE and angle θ1 . The refracted component
has magnitude tE and angle θ2 . The stokes relations state that r + t = 1 so
that the intensity of the light is conserved.
In the next step of the experiment, the two resulting waves are reversed. The
reflected wave is thus incident at angle θ1 with amplitude rE and the refracted
wave is incident at θ2 with amplitude tE. The combination of these waves
results in the two waves shown in Figure 3.1. One of these waves, the reflected
wave in n1 has magnitude t0 tE + r2 E. With a little bit of math, it turns out
that t0 tE + r2 E = E. There is also, however, a wave that was reflected in
n2 . The magnitude of this wave is r0 tE + rtE and it turns out that this wave
entirely destructively interferes so it’s magnitude is 0. Thus, the process has
been successfully time reversed.
This example is highly simplistic but it can be expanded to more complex
scenarios with multiple scatterers. In these cases, the waves will reflect and
refract at many different angles. Thus, a transducer needs to sample the entire
field surrounding the insonified body. Moreover, to satisfy the Nyquist criteria,
the field must be sampled at every location no more than half a wavelength
apart. In two dimensions, this can be accomplished by a circular ring of elements
(Figure 3.2) and in three dimensions one needs a sphere of elements, often called
a time reversal cavity [5].
8
Figure 3.2: A ring transducer.
Although a transducer ring and transducer sphere are entirely realizable, they
are typically considered difficult to implement. Contemporary ultrasound transducers work from one direction of the object in pulse-echo mode. Furthermore,
to achieve acoustic impedance matching, there has to be no air between the
patient and transducers. In practice, the transducer is held against the body
and gel is used to remove any air. For a 2D ring of transducers, the patient will
simply have their head out of the tank of water and their body immersed in it
and this has been accomplished in practice. To measure the response from a
full set of elements in the 3D time reversal cavity, however, would require the
patient to be fully immersed in the body water.
3.3
Reciprocity
Reciprocity states that the reception of a signal at a location r1 from a source
at location r0 is identical to the reception of a signal at r1 from a source at r0
[6]. In terms of the Green’s function, this can be written as
G (r0 , t0 |r, t) = G (r, t0 |r0 , t)
where G (r0 , t0 |r, t) represents the Green’s function when a wave originates at
r0 at a time t0 and is received at r at a time t. The impulse response over the
surface of element i from a location at r0 can then be written as
9
hri (r0 , t) =
Z
G (r0 , t0 |r, t) dr
Si
where Si represents the area over the surface of element i. Likewise, for an
element at location r0 , the impulse response from a location r can be written as
hti (r0 , t) =
Z
G (r, t0 |r0 , t) dr
Si
Reciprocity tells us that these two are equal.
3.4
The Time Reversal Process
The time reversal process is now described and simulated. For the simulation
(Figure 3.3), a transducer of 16 elements 0.25 mm in the lateral direction and
1 mm in the elevation direction are used. The origin of the system is assumed
to be the center of the transducer array. A single scatterer is located at a depth
of 3 mm and lateral distance 1.5 mm.
For the simulation, the field II simulation program was used. It was developed
by Dr. Jørgen Arendt Jensen of the Technical University of Denmark [7]. It is
a collection of Matlab procedures that simulate the propagation of ultrasound
waves [8].
Transducer (mm)
Transducer (mm)
4
z
4
3
2
1
y
−1
−2
0
lateral
x
elevation
0
axial
axial
z
3
2
0
2 −2
2
1
y
0
x
−1
20
−2 −2
elevation
0
lateral
Figure 3.3: Simulation setup.
10
2
On the left side of Figure 3.4 is a conceptual depiction of the time reversal
process. Assume that the number of elements is given by M . The right side of
Figure 3.4while on the right side is the results of the simulation. Each of the
graphs show normalized traces for each of the 16 elements.
In the first step of the algorithm, a pulse from a single element is sent (Figure
3.4a). In the simulation, element 9 sends a Gaussian modulated sine. Note
that it does not matter which element performs the initial transmission. As the
scatterer is considered a point scatterer, transmission by another element will
simply impose a time delay to the resulting receive waveforms.
In the next step, the echoed signals are received by all of the elements of the
array. At this point, each of the elements have received the impulse response
hi (r0 , t). In the simulation, this is the impulse response convolved by a Gaussian
modulated sine or
hi (r0 , t) ⊗ x (t)
where x (t) is the waveform sent in step 1. In the simulation, note that because
elements 14 and 15 are closest to the scatterer, they receive the signals first
(Figure 3.4b).
Time reversal then takes place in step 3. In Figure 3.4c, we can see that signals
are simply time reversed versions those in Figure 3.4b. The transmit signal of
element i is thus
hi (r0 , −t) ⊗ x (−t) .
The signals then propagate through the medium and converge onto the scatterer
location so that they constructively interfere. Thus, they maximize energy at
the location r0 . The signal at the point r0 is given by
M
X
hj (r0 , −t) ⊗ x (−t) ⊗ hj (r0 , t) .
j=1
In the last step, the signals are read back by the elements so that element i
11
r0
r0
r0
r0
Normalized Response
Step 1 − Transmit Traces
15
10
5
0
0
2
4
6
Time (s)
8
−7
x 10
(a) First Step - Initial Transmission
r0
r0
r0
r0
Normalized Response
Step 2 − Receive Traces
15
10
5
0
4.5
5
5.5
6
Time (s)
6.5
7
7.5
−6
x 10
(b) Second Step - Reception
r0
r0
r0
r0
Normalized Response
Step 3 − Time Reversed Traces
15
10
5
0
0
1
2
3
Time (s)
−6
x 10
(c) Third Step - Time Reversal
Step 4 − Received Traces
Normalized Response
r0
r0
r0
r0
15
10
5
0
6
7
(d) Fourth Step - Second Reception
Figure 3.4: The time reversal process.
12
8
Time (s)
9
10
−6
x 10
receives


M
X

hj (r0 , −t) ⊗ x (−t) ⊗ hj (r0 , t) ⊗ hi (r0 , t) .
j=1
No assumptions were made of the transducer elements, nor of the medium.
The time reversal process is able to automatically focus on the location of the
scatterer, even through an inhomogeneous medium. It is also worth noting that
as long as energy is transmitted onto a scatterer, the resulting waveform exhibits
some form of resonance so that further iterations of the time reversal process
all focus energy directly on the scatterer.
3.5
Multiple Scattering
In the case of multiple scatterers, the time reversal process focuses energy on the
strongest scatterer. This can be seen in Figure 3.5. As with the time reversal
process on one scatterer, this was simulated with two scatterers. In this case,
one scatterer was placed at a depth of 30 mm and 6 mm in the positive lateral
direction and a second was placed at a depth of 30 mm and 6 mm in the negative
lateral direction. The scatterer in the negative lateral direction has a greater
scattering amplitude than the one in the positive lateral direction. The result
of the simulation can be found in Figure 3.6.
In the first iteration of the simulation, the responses from both scatterers are
clearly visible. After more iterations of the time reversal process, the response
from the weaker of the two scatterers is almost negligible. Thus, the time
reversal process has focused on the stronger of the two scatterers. Similarly,
whenever there are multiple scatterers, the time reversal process focuses on
the strongest of the scatterers. Moreover, the larger the contrast between the
strongest and second strongest scatterer, the quicker the algorithm focuses on
the stronger scatterer.
To explain this result, we can consider the insonification of each scatterer individually. The time reversal process for scatterer P focuses energy at the location
of P . The scattered pressure at a distance r and angle θ from a rigid sphere
13
(a)
(b)
(c)
(d)
Figure 3.5: Focusing on two scatterers with the time reversal process.
Iteration 2
15
10
5
0
4
4.2
Time (s)
4.4
4.6
−5
x 10
Normalized Response
Normalized Response
Iteration 1
15
10
5
0
4.2
4.4
(a)
10
5
4.6
4.8
Time (s)
5
5.2
−5
x 10
Iteration 4
5
5.2
5.4
−5
x 10
Normalized Response
Normalized Response
Iteration 3
4.4
4.8
Time (s)
(b)
15
0
4.2
4.6
(c)
15
10
5
0
5
5.5
Time (s)
(d)
Figure 3.6: Simulation of focusing on two scatterers.
14
6
−5
x 10
can be approximated by
ps (r, θ) ≈ −pim
k 2 a3 −jkr
3
e
1 − cos θ
3r
2
where pim is the incident pressure, k is the wave number and a is the radius of
the sphere [3]. The strength of a scatterer and it’s radius are correlated [9]. In
the first iteration of the algorithm, the same amount of energy is transmitted to
each of the two scatterers. The stronger scatterer will elicit a stronger response
and so on the second iteration, more energy will be sent to the stronger of the
two scatterers. Because of this, the difference between the returned energy in
the second iteration will be even greater. Further iterations will increase the
difference in received energy between the two scatterers. Eventually, only the
response of the stronger of the two scatterers will remain.
As was shown, the time reversal process, was able to focus on a single scatterer
without any prior information. Moreover, when there are multiple scatterers,
the time reversal process focuses on the strongest scatterer. However, there are
cases where it is beneficial to focus on other scatterers, namely scatterers which
to not have the largest magnitude. The DORT and FDORT algorithms use
something known as the time reversal operator or solve this problem.
15
Chapter 4
DORT and FDORT
We have seen that the time reversal process can focus on a single scatterer
and that it can selectively focus on the strongest of a collection of scatterers.
A question arises as to whether or not the time reversal process can focus on
weaker scatterers. The DORT and FDORT algorithms were created to solve
this problem [10].
The DORT and the FDORT are two algorithms developed by Dr. Mathias
Fink’s group. The DORT algorithm is the precursor to the FDORT algorithm
so it will be discussed first. Most of this is taken from the seminal work on the
DORT [10].
4.1
DORT
Consider a scenario with two transducers (e with M elements and r with N
elements) and two scatterers (P and Q) as depicted in Figure 4.1. Furthermore,
let knm (t) represent the impulse response when a signal is sent from element
m on transducer e to element n on transducer r. The received signal rn (t) on
element n is then given by
16
e
r
HTxT
HRx
HTx
HRxT
K
KH
D
Figure 4.1: The scenario for DORT.
rn (t) =
M
X
knm (t) ⊗ em (t)
m=1
where em (t) is the transmit waveform by element m on transducer e.
In the frequency domain, the convolution becomes a multiplication and so we
have
Rn (ω) =
M
X
Knm (ω) Em (ω)
m=1
where Rn (ω) is the frequency response of rn (r) and similarly for Knm (ω) and
Em (ω) . The frequency response matrix K (ω) can be written as

K11 (ω)
K12 (ω)
···

 K21 (ω) K22 (ω) · · ·

K (ω) = 
..
..
..

.
.
.

KN 1 (ω) KN 2 (ω) · · ·
K1M (ω)

K2M (ω) 

.
..

.

KN M (ω)
and has dimensions N × M .
If the excitation vector is defined as
T
E (ω) = [E1 (ω) , E2 (ω) , . . . , EM (ω)]
17

the received signal then becomes
R (ω) = K (ω) E (ω) .
In the following, the frequency term ω is omitted for clarity.
Because of reciprocity, if we were to transmit from transducer r and receive
at transducer e, the frequency response matrix would simply be KT where the
superscript T denotes the transpose. In other words, because of reciprocity, a
transmission from element m on e to element n on r is the same as a transmission
by n and reception by m. Therefore, KT represents the frequency response in
the reverse direction.
In the first iteration of the time reversal process, the received signal at r is given
by
R = KE.
In the second iteration of the process, the received signal is time reversed and
sent back to e. Time reversal in the frequency domain is simply the complex
conjugate. The received signal at e is then given by KT KE. Removing the
originally transmitted signal E, the conjugate of this is then the time reversal
operator KH K where the superscript H denotes conjugate transpose or the
hermitian operator.
As KH K is hermitian symmetric,
T
H T
H
T
KH K = KH K
= KT K = K K = KH K
it follows that its eigenvalues are orthogonal and its eigenvalues are real and
positive. Moreover, the eigenvalues of KH K are related to the strength of the
scatterers and the eigenvectors give us the focusing parameters needed to focus
on the scatterers.
The operation of the time reversal operator is not entirely intuitive. Thinking back to the previous section on the time reversal procedure, the resonance
18
property can be used to describe the eigenmodes of the time reversal operator.
Previously, it was noted that iterations of the time reversal process result in
the same transmit signal modulated by the impulse response of the system. As
previously mentioned, the received signal after a single iteration of the time
reversal process (from transducer e to r and back again) is given by
R0 = KT KE.
The second iteration of the time reversal process will result in a received signal
of
2
R00 = KT KKT KE = KT K E.
Likewise, on the nth iteration, we have
n
Rn = KT K E
as long as E focuses energy on the scatterer. From this, one can see the relan
tionship between KT K E and the definition of eigenvalues and eigenvectors
Ax = λx. Thus, the eigenvectors describe a way for us to focus energy onto the
scatterers and the eigenvalues are related to the strength of the scatterers.
To gain a better understanding of the significance of the eigenvalues, the following derivation is made. As in Figure 4.1, consider two scatterers P and Q.
Further, let HTx (P) by Green’s function from e to P and HRx (P) be the
Green’s function from r to P and similarly define HTx (Q) and HRx (Q) for
Q. Moreover, the propagation between the mth element of e and P is given
by HTx (P)m . Also, let D(P) represent the reflectivity of scatterer P . The
propagation of the wave from m to n can then be given as
Knm = HTx (P)m D(P)HRx (P)n + HTx (Q)m D(Q)HRx (Q)n .
Further, the transfer matrix K can be defined as
19
K = HRx T DHTx
where
D=
"
D (P)
0
0
D (Q)
#
and
"
HRx =
HRx (P)1
HRx (P)2
···
HRx (P)N
HRx (Q)1
HRx (Q)2
···
HRx (Q)N
#
and HT x is defined in a similar way. The time reversal operator then becomes
KKH = HRx T DHTx HTx H DHRx .
This can also be seen by tracing the paths in Figure 4.1. If the paths were
traced starting from the scatterer, the time reversal operator becomes
Tscat = DHTx HTx H DHRx HRx T .
If we take a closer look at HRx HRx T , we will find that
"
T
HRx HRx =
2
kHRx (P)k
#
hHRx (Q), HRx (P)i
hHRx (P), HRx (Q)i
2
kHRx (Q)k
Thus for the time reversal operator to be diagonalizable, we have to satisfy
hHRx (Q), HRx (P)i =
0
hHRx (P), HRx (Q)i =
0.
20
Physically, this means that we need to transmit to the scatterer P without
sending any energy to Q and vice-versa. In other words, the scatterers have to
be adequately spaced so as to not interfere with each other.
Furthermore, the eigenvalues can be found to be
hHRx (P), HRx (Q)i = 0.
Therefore, the eigenvalues of the DORT algorithm correlate to the strength of
the scatterers and the corresponding eigenvalues are the focusing parameters
required to focus on those scatterers. In this way, we are able to theoretically
focus on any scatterer we wish.
One of the main limitations of the DORT algorithm is its susceptibility to noise
and speckle [11]. In the presence of noise, the eigenvectors also become noisy and
are no longer able to accurately focus on scatterers. To alleviate this problem,
the FDORT algorithm was created.
4.2
FDORT
Instead of using a single element for the initial transmission as in DORT, to
create the time reversal operator, FDORT instead uses a series of focused pulses.
In FDORT, the same array is used for both transmission and reception. If L
different locations are focused on using our M element transducer e, let Bl
represent the M × 1 vector describing the lth transmitted pulse at location l.
Further, let B be the L × M matrix which describes focusing to each of the
L locations by our M element array. In this way, a generalized transfer Kf oc
matrix can be defined as
Kf oc = KBT .
Using our previous result for the transfer matrix, we have
21
Kf oc = HRx T DHRx BT .
Note that because we are using a single transducer for transmission and reception, we see the appearance of both HRx and HRx T . If we set HTx = HRx BT ,
we arrive at the previous result. Thus, FDORT can be used to determine the
focusing locations of a number of scatterers and less susceptible to noise.
The DORT and FDORT algorithms give us a way to focus on scatterers which
are not the strongest. In the next section, they will be simulated to show their
efficacy.
22
Chapter 5
Simulations
The DORT and FDORT algorithms were described in the previous section.
They use the time reversal operator KH K to calculate the focusing parameters
for scatterers in the medium. The eigenvalues of the time reversal operator
correspond to the magnitudes of the scatterers and the eigenvectors contain the
focusing parameters for focusing on the scatterers. Therefore, whereas the time
reversal process was only able to focus on dominant scatterers, the DORT and
FDORT algorithms are claimed to be able to focus on both strong and weak
scatterers.
In the following, we will use MATLAB to simulate the FDORT algorithm to
test its efficacy. The DORT algorithm was also implemented but the results for
it are not shown. As mentioned before, the DORT algorithm does not work in
the presence of noise. It is essentially superceeded by the FDORT algorithm as
the FDORT is more robust to noise. As before, Field II was used to simulate
the algorithms.
5.1
Simulation Setup
Two different transducers were used for the simulations. For the images of the
scattered field, the imaging array is a 256 element array of which, at any given
23
time, 64 elements are active. For these images, the response from 50 equally
spaced lines from −20 mm to 20 mm in the lateral direction perpendicular to
the array are measured. The resulting image is an interpolation of the lines and
is log compressed.
The transducer array for the FDORT is a 64 element array where all 64 elements
of the array are active. Besides the number of elements in the array, all other
properties of the arrays are the same. This includes, among other things, the
excitation waveform and dimensions of the individual elements. In both cases,
the same transducer is used for both transmission and reception. The images
are shown log compressed for display purposes.
The transmit pulse used is a Gaussian modulated sine wave with a center frequency of 3 MHz. The speed of sound was set to 1540 m/s. This gives us a
wavelength of about 0.25 mm which was used as the width of each element,
to prevent spatial aliasing. With the exception of speckle, the scatterers were
placed on the lateral-axial plane so that their distance in the elevation direction
was 0. Considering Figure 5.1, the elevation direction would be into and out of
the page.
For these simulations, the scatterers were always placed at a depth of 60 mm.
The FDORT algorithm requires that there be a set of predetermined focus
locations. For all of the simulations, 100 equally spaced locations between
(−20, 0, 60) mm and (20, 0, 60) mm were used where the position is defined as
(lateral,elevation,axial). In other words, 100 locations at a depth of 60 mm and
equally spaced between −20 mm and 20 mm) in the lateral direction were used.
The algorithm works as follows. Using time delay weighting, each of the N
locations is focused on. The time delays for each element are calculated based
upon the distance of the element from the target focus location and the speed
of sound. They are chosen such that they should converge on the focal location.
The responses on each of the M elements is recorded and their Fourier transform
is computed. Recall that K is a function of frequency, ω. The frequency component corresponding to the center frequency is extracted the resulting vector
then forms one row of our K matrix. More specifically, the response for focal
location n at element m is Knm . The time reversal operator, from before, is
defined as KH K. Because the dimensions of K are N × M , the time reversal
24
operator therefore has dimensions M × M . The eigenvectors are then of length
M and each correspond to the focusing parameter for on of the M elements.
To test the efficacy of the focusing parameter, the eigenvector for each of the
dominant eigenvalues is used to insonify the field. The maximum pressure field
is then measured to show how must energy is focused at each spacial location.
A good focusing mechanism will focus most of the energy onto a specific point
while poor focusing will result in a more diffuse response. Both of these cases
will be presented in the following.
5.2
Two well Resolved Scatterers
For the first simulation, the field consists solely of two scatterers, with locations
(−10, 0, 60) mm and (5, 0, 60) mm. The weights (in arbitrary units) of the two
scatterers are 20 and 30, respectively. An image of the phantom can be seen
in Figure 5.1. Note that the scatterer on the right is the stronger of the two
scatterers and that it’s response is stronger. Thus, we expect that this scatterer
will have a larger eigenvalue than the other.
Phantom 1
35
120
40
Axial distance [mm]
45
100
50
55
80
60
60
65
70
40
75
80
20
85
90
−20
−10
0
10
Lateral distance [mm]
Figure 5.1
25
20
Phantom 1 − Eigenvalues
10
10
5
Magnitude
10
0
10
−5
10
−10
10
−15
10
0
10
20
30
40
50
60
Eigenvalues
Figure 5.2
A plot of the eigenvalues is given in Figure 5.2. From this, it is clear that there
are two eigenvalues that are much greater than the rest (note that the magnitude
axis is in log scale). According to the FDORT algorithm, the eigenvectors which
correspond to the two largest eigenvalues should return the focusing parameters
needed to focus on each of the two scatterers.
The eigenvectors have a phase and magnitude component. The phase is used as
the delay for the element and the magnitude is used to modulate the amplitude
of the signal sent by the individual element. Combined, these should provide
time-delay focusing parameters for each of the M elements to focus on each of
the scatterers. The phase components corresponding to the largest eigenvector
are plotted in Figure 5.3.
The elements in close proximity to element 50 have the largest delay which
means that these elements should be closest to the scatterer. Referring back to
Figure 5.1, we can confirm that the largest scatterer is indeed in the positive
lateral direction. A quick calculation shows that element 51 should be closest
to the strongest scatterer, confirming our observation.
To test the focusing properties, the pressure responses were simulated. They
are shown in Figure 5.4. Given the excitation of the transducer, Field II has
the option to measure the field at a specific spatial location. The maximum
26
Phantom 1 − Eigenvector 1
0
−2
Phase (rad)
−4
−6
−8
−10
−12
−14
−16
0
10
20
30
40
50
60
Element
Figure 5.3: The phase component of the eigenvector corresponding to the
strongest eigenvalue of Phantom 1.
pressure at each spatial location recorded and displayed in the plot.
The pressure field when using the first eigenvector is shown in Figure 5.4a. As
we can see, the pressure response is maximized at the location of the dominant
scatterer. The pressure response for the second scatterer is shown in 5.4b.
The maximum pressure response for this case is at the location of the second
scatterer. The focusing properties of the third eigenvector, corresponding to an
eigenvalue of low magnitude, is shown in 5.4c. As we can see from this plot,
the pressure response is more or less diffuse across the imaged area. Subsequent
eigenvectors give similarly diffuse responses.
The FDORT algorithm has successfully given us focusing parameters for the two
scatterers in the field. In both cases, the eigenvectors allow us to individually
focus on one of the two scatterers without sending energy to the other. In this
simple example, the scatterers are spaced far apart and have relatively different
reflectivities.
27
Phantom 1 − Eigenvector 2
40
40
45
45
50
50
Axial distance [mm]
35
55
60
65
70
55
60
65
70
75
75
80
80
85
85
90
−20
−10
0
10
90
−20
20
Lateral distance [mm]
−10
0
(b)
Phantom 1 − Eigenvector 3
35
120
40
45
100
50
55
80
60
60
65
70
40
75
80
20
85
90
−20
−10
10
Lateral distance [mm]
(a)
Axial distance [mm]
Axial distance [mm]
Phantom 1 − Eigenvector 1
35
0
10
Lateral distance [mm]
(c)
Figure 5.4
28
20
20
5.3
Five not-so-well Resolved Scatterers
In the second simulation, the scatterers are spaced closer together and the differences in magnitude is decreased. Moreover, the second simulation includes
five scatterers rather than 2.
They are equally spaced at (−10, 0, 60) mm,
(−5, 0, 60) mm, (0, 0, 60) mm, (5, 0, 60) mm and (10, 0, 60) mm. Their reflectivities are 10, 12, 14, 16 and 18, respectively. Note that they are of increasing
magnitude from the left to the right.
The image of the second phantom is shown in Figure 5.5. From this, we can
verify that the scatterers are of increasing magnitude from left to right and that
the largest scatterer corresponds to the location (10, 0, 60) mm. From this image
is becoming clear that the scatterers are relatively close so that their responses
are starting to overlap. This should make it hard for the FDORT algorithm to
focus on each scatterer individually.
The corresponding eigenvalues are shown in Figure 5.6. There are five distinctly
large eigenvalues and the fifth largest eigenvalue is about 103 time as great as
the sixth largest. Note that the gap between the significant eigenvalues and the
Phantom 2
35
120
40
Axial distance [mm]
45
100
50
55
80
60
60
65
70
40
75
80
20
85
90
−20
−10
0
10
Lateral distance [mm]
Figure 5.5
29
20
Phantom 2 − Eigenvalues
10
10
5
Magnitude
10
0
10
−5
10
−10
10
−15
10
0
10
20
30
40
50
60
Eigenvalues
Figure 5.6
rest is smaller than in the first example.
The focusing properties of each of the eigenvectors is given in Figures 5.7 and 5.8.
In Figure 5.7a the eigenvector corresponding to the largest eigenvalue focuses
energy on the strongest scatterer but we can clearly see that it also focuses
energy on the scatterer adjacent to it, directly to its left (Figure 5.7a). This
is expected if the two scatterers are not resolvable as described in the previous
section.
The eigenvectors of the four subsequent eigenvalues all exhibit the same focusing
phenomenon. Each of them focuses some of their energy on to the scatterers
adjacent to them. The scatterers in the middle focus some energy onto scatterers
on either side of them and the scatterers at either end focus energy to the single
scatterer adjacent to it. Lastly, the sixth eigenvector does not show any focusing
properties as it corresponds to a significantly smaller eigenvalue.
Comparing the image of the pressure field generated by the first and fifth eigenvectors exhibits some interesting results. The amount of energy focused on the
second scatterer by the first eigenvector (which focuses on the first scatterer)
is greater than the amount of energy focused on fourth scatterer by the fifth
eigenvector (which focuses on the fifth scatterer). The scatterers are placed at
equal distances from each other. The right-most scatterer has a magnitude of
30
Phantom 2 − Eigenvector 2
35
40
40
45
45
50
50
Axial distance [mm]
Axial distance [mm]
Phantom 2 − Eigenvector 1
35
55
60
65
70
55
60
65
70
75
75
80
80
85
85
90
−20
−10
0
10
90
−20
20
Lateral distance [mm]
−10
(a)
20
Phantom 2 − Eigenvector 4
35
40
40
45
45
50
50
Axial distance [mm]
Axial distance [mm]
Phantom 2 − Eigenvector 3
55
60
65
70
55
60
65
70
75
75
80
80
85
85
−10
10
(b)
35
90
−20
0
Lateral distance [mm]
0
10
90
−20
20
Lateral distance [mm]
−10
0
10
Lateral distance [mm]
(c)
(d)
Figure 5.7
31
20
Phantom 2 − Eigenvector 6
35
40
40
45
45
50
50
Axial distance [mm]
Axial distance [mm]
Phantom 2 − Eigenvector 5
35
55
60
65
70
55
60
65
70
75
75
80
80
85
85
90
−20
−10
0
10
90
−20
20
Lateral distance [mm]
−10
0
10
20
Lateral distance [mm]
(a)
(b)
Figure 5.8
18 while it’s neighbor has a magnitude of 16. The left-most scatterer has a
magnitude of 10 and it’s neighbor has a magnitude of 12. Relative to their individual magnitudes, the difference between the reflectivity of left-most scatterer
and its neighbor is greater than the reflectivity of the right-most scatterer and
its neighbor. Therefore, it appears that the FDORT algorithm also depends on
the relative magnitudes of the scatterers.
5.4
Scatterers of Equal Magnitudes
To test if the magnitudes of the scatters affects the focusing ability of the
FDORT algorithm, a simulation with two scatterers of equal reflectivites is
performed. One scatterer is placed at (−5, 0, 60) mm and the second scatterer
is placed at (5, 0, 60) mm. Both have a magnitude of 10. They are shown in
Figure 5.9 and the corresponding eigenvalues are shown in Figure 5.10. As expected, there are two eigenvalues of comparable magnitude and which are much
larger than the rest.
32
Phantom 3
35
120
40
Axial distance [mm]
45
100
50
55
80
60
60
65
70
40
75
80
20
85
90
−20
−10
0
10
20
Lateral distance [mm]
Figure 5.9
Phantom 3 − Eigenvalues
10
10
5
Magnitude
10
0
10
−5
10
−10
10
−15
10
0
10
20
30
40
Eigenvalues
Figure 5.10
33
50
60
The pressure field resulting from using the eigenvectors (corresponding to the
three most significant eigenvalues) for focusing is shown in Figure 5.11. Something interesting happens in this simulation scenario. The FDORT algorithm is
not able to resolve the two scatterers even though they are spaced relatively far
apart. Because it cannot tell them apart, the algorithm focuses on both scatterers. Moreover, both of the eigenvectors with significant eigenvalues insonify
both of the scatterers.
Intuitively, and as derived in the previous section, this result makes sense because the eigenvectors and eigenvalues are based on the distance and magnitude
of the scatterers. If the two are the same, FDORT is not able to resolve the two
scatterers.
This result is interesting because the transducer is able to simultaneously focus
on two spatially distinct targets. This remarkable result may have interesting
applications.
5.5
The Addition of Speckle
Speckle is the result of scattering from sub-wavelength scatterers. It is not noise
in that it is not random. When measured several times, under similar condition,
the speckle pattern will be identical. Speckle is sometimes used in flow rate
calculations because the speckle pattern through vessels will be changing while
the speckle pattern through the stationary tissue will be constant. Because the
speckle pattern of moving objects varies with time, it can be averaged out of
the signal, providing some noise cancellation.
In the next simulation, we use the five scatterers from the second simulation but with the addition of speckle. As before, the locations of the five
scatterers are (−10, 0, 60) mm, (−5, 0, 60) mm, (0, 0, 60) mm, (5, 0, 60) mm and
(10, 0, 60) mm. Their reflectivities are 10, 12, 14, 16 and 18, respectively. As
before, they are of increasing magnitude from left to right and have relatively
similar magnitudes.
Speckle can be simulated by the presence of many scatterers of low reflectivity.
In practice, the density of scatterers required to simulate speckle ranges from
34
Phantom 3 − Eigenvector 2
40
40
45
45
50
50
Axial distance [mm]
35
55
60
65
70
55
60
65
70
75
75
80
80
85
85
90
−20
−10
0
10
90
−20
20
−10
0
(a)
(b)
Phantom 3 − Eigenvector 3
35
120
40
45
100
50
55
80
60
60
65
70
40
75
80
20
85
90
−20
−10
10
Lateral distance [mm]
Lateral distance [mm]
Axial distance [mm]
Axial distance [mm]
Phantom 3 − Eigenvector 1
35
0
10
Lateral distance [mm]
(c)
Figure 5.11
35
20
20
Phantom 5
35
120
40
Axial distance [mm]
45
100
50
55
80
60
60
65
70
40
75
80
20
85
90
−20
−10
0
10
20
Lateral distance [mm]
Figure 5.12
10 to 100 scatterers per cubic millimeter [12]. To speed up the simulation,
however, an average density of about 3.3 scatterers per cubic millimeter was
used for these simulations. More specifically, 100, 000 scatterers were randomly
placed in a volume of 50 × 10 × 60 mm. Their locations were chosen randomly
with a uniform distribution over the entire volume. The magnitudes of the
scatterers were were also randomly chosen but with a Gaussian distribution of
zero mean and unit variance.
The resulting ultrasound image can be seen in Figure 5.12. Note that the five
target scatterers are visible but they are now slightly harder to make out due to
the responses from other scatterers in the vicinity. Although still distinguishable
from the background, the left-most scatterer is now approaching the magnitude
of the speckle. Time gain compensation was not used for these images and so
it is clear that attenuation has reduced the magnitude of the signal at greater
depths.
For this simulation, there is a variation of the FDORT algorithm which was not
previously discussed. In it, the impulse responses for each of the scatterers is
time gated to a window that surrounds the scatterer. This is a trick introduced
36
Phantom 5 − Eigenvalues
6
10
5
10
Magnitude
4
10
3
10
2
10
1
10
0
10
0
10
20
30
40
50
60
Eigenvalues
Figure 5.13
in [11] to only use information close to the focal location. In this way, we
alleviate the influence from surrounding scatterers. Using the distance from the
element to the focal location, we create a window around the focal depth. The
resulting response is time gated using this window and the frequency response
of the time gated signal is used to form the K matrix. From there, the time
reversal operator is created in the same way.
The eigenvalues for this simulation are given in Figure 5.13. It is evident that
there is little difference between the five largest eigenvalues and the sixth largest
one. Thus, we don’t expect the eigenvectors to provide adequate focusing parameters.
The resulting pressure field from the eigenvectors is shown in Figure 5.14. As can
be seen from the images, even the focusing parameters from the the strongest
eigenvalue do not focus the energy at a specific target. All of the focusing
parameters return diffuse pressure responses.
The reason FDORT did so poorly may be because the magnitudes of the scatterers were similar and that they were spatially close. Moreover, perhaps the
magnitudes of the scatterers were not large enough to counter the combined
effect of the speckle. To test this these theories, a final simulation is performed
with stronger scatterers.
37
Phantom 5 − Eigenvector 2
35
40
40
45
45
50
50
Axial distance [mm]
Axial distance [mm]
Phantom 5 − Eigenvector 1
35
55
60
65
70
55
60
65
70
75
75
80
80
85
85
90
−20
−10
0
10
90
−20
20
Lateral distance [mm]
−10
(a)
20
Phantom 5 − Eigenvector 4
35
40
40
45
45
50
50
Axial distance [mm]
Axial distance [mm]
Phantom 5 − Eigenvector 3
55
60
65
70
60
70
80
85
85
90
−20
20
80
65
80
10
100
60
75
0
120
55
75
−10
10
(b)
35
90
−20
0
Lateral distance [mm]
Lateral distance [mm]
40
20
−10
0
10
Lateral distance [mm]
(c)
(d)
Figure 5.14
38
20
5.6
Less Speckle and Stronger Scatterers
In the next simulation, two scatterers far apart from each other are used. They
are placed at locations (−10, 0, 60) mm and (5, 0, 60) mm and have reflectivities
of 30 and 60, respectively. An image of the scattered field is given in Figure 5.15.
In this image, the scatterers are clearly distinguishable form the background
speckle. It is of note that logarithmic compression is performed on the pressure
field before it is displayed in the image. As such, the magnitudes vary more
than they appear.
The corresponding eigenvalues are shown in Figure 5.16. Although more significant than before, the single dominant eigenvalue is still close to second largest
eigenvalue. The focusing properties of the eigenvectors are shown in Figure 5.17.
It is evident that the eigenvectors do not focus the pressure at either of the two
scatterers. In the presence of speckle, it seems that the FDORT algorithm is
not able to focus on either of the two scatterers.
Phantom 4
35
120
40
Axial distance [mm]
45
100
50
55
80
60
60
65
70
40
75
80
20
85
90
−20
−10
0
10
Lateral distance [mm]
Figure 5.15
39
20
Phantom 4 − Eigenvalues
6
10
5
10
Magnitude
4
10
3
10
2
10
1
10
0
10
0
10
20
30
40
50
60
Eigenvalues
Figure 5.16
Phantom 4 − Eigenvector 2
35
40
40
45
45
50
50
Axial distance [mm]
Axial distance [mm]
Phantom 4 − Eigenvector 1
35
55
60
65
70
55
60
65
70
75
75
80
80
85
85
90
−20
−10
0
10
90
−20
20
Lateral distance [mm]
−10
0
10
Lateral distance [mm]
(a)
(b)
Figure 5.17
40
20
Chapter 6
Conclusion
In this report, the time reversal process and the time reversal operator were
described. Furthermore, both were simulated to understand their functionality.
The time reversal operator proved to automatically focus on the strongest of
two scatterers. To overcome the limitation of focusing on only the strongest
scatterer, the time reversal operator was derived and then discussed. It was
then simulated under various conditions. It was found that in the presence of
speckle and with scatterers that are spatially close, the DORT and FDORT
algorithm are not able to determine the focusing parameters for the individual
scatterers. Also of note is that the algorithms do not work with many scatterers
or regions of scattering rather than point source scatterers. This is an inherent
limitation of the two algorithms.
Given these shortcomings, the time reversal process and the time reversal operator provide a means for adaptively focusing on strongly reflecting targets.
They do not assume anything of the transducer elements and require little to
no initial setup.
41
References
[1] P. N. Marcus and E. L. Carstensen. Problems with absorption measurements of inhomgeneous solids. J. Acoust. Soc. Am., pages 1334–1335, 1975.
[2] S. W. Flax and M. O’Donnell. Phase aberration correction using signals
from point reflectors and diffuse scatters: basic principles. IEEE Trans.
Ultrason. Ferroelec. Freq. Contr, pages 758–67, 1988.
[3] R. S. C. Cobbold. Foundations of Biomedical Ultrasound. Oxford University Press, 1st edition, September 2006.
[4] M. Fink. Time reversed acoustics. Physics Today, pages 64–40, 1997.
[5] D. Cassereau, F. Wu, and M. Fink. Limits of self-focusing using closed
time-reversal cavities and mirrors— theory and experiment. Proc. IEEE
Ultrason. Symp., pages 1613–1618, 1990.
[6] P. M. Morse and K. U. Ingard. Theoretical Acoustics. McGraw-Hill, New
York, 1st edition, 1968.
[7] Jørgen Arendt Jensen.
Field ii simulation program.
http://server.
elektro.dtu.dk/personal/jaj/field/?main.html.
[8] Jørgen Arendt Jensen. Imaging of Complex Media with Acoustic and Seismic Waves, chapter Ultrasound Imaging and its Modeling. Springer Verlag,
2002.
[9] C. Prada, F. Wu, and M. Fink. The iterative time reversal mirror: A
solution to self-focusing in the pulse echo mode. J. Acoust. Soc. Am.,
90:1119–1129, 1991.
42
[10] C. Prada and M. Fink. Eigenmodes of the time reversal operator: a solution
to selective focusing in multiple-target media. Wave Motion, 20:151–163,
1994.
[11] J.-L. Robert, M. Burcher, C. Cohen-Bacrie, and M. Fink. Time reversal operator decomposition with focused transmission and robustness to speckle
noise: Application to microcalcification detection. J. Acoust. Soc. Am.,
119:3848–3859, 2006.
[12] Pinar Crombie. Fundamental Studies on Contrast Resolution of Ultrasound
B-Mode Images. PhD thesis, University of Toronto, 1999.
43