Title: Ultrasound Imaging: Lecture 2
1Ultrasound Imaging Lecture 2
Jan 14, 2009
Interactions of ultrasound with tissue
Beams and Arrays
 Steering
 Focusing
 Apodization
 Design rules
 Absorption
 Reflection
 Scatter
 Speed of sound
Image formation
 Signal modeling
 Signal Processing
 Statistics
2Anatomy of an ultrasound beam
 Near field or Fresnel zone
 Far field or Fraunhofer zone
 Neartofar field transition, L
L
Lateral distance (mm)
Depth (mm)
3Anatomy of an ultrasound beam
 Lateral Resolution (FWHM)
FWHM
Lateral distance (mm)
Depth (mm)
4Anatomy of an ultrasound beam
DOF
Lateral distance (mm)
Depth (mm)
5Array Geometries
 Schematic of a linear phased array
 Definition of azimuth, elevation
 Scanning angle shown, q, in negative scan
direction.
6Some Basic Geometry
 Delay determination
 simple path length difference
 reference point phase center
 apply Law of Cosines
 approximate for ASIC implementation
 In some cases, split delay into 2 parts
 beam steering
 dynamic focusing
7Far field beam steering
 For beam steering
 far field calculation particularly easy
 often implemented as a fixed delay
8Beamformation Focusing
 Basic focusing type beamformation
 Symmetrical delays about phase center.
9Beamformation Beam steering
 Beam steering with linear phased arrays.
 Asymmetrical delays, long delay lines
10Anatomy of an ultrasound beam
11Grating Lobes
How many elements? What Spacing?
Grating Lobe
 Linear array
 32 element array
 3 MHz
 pitch l 0.4 mm
 l 0.51 mm
 L N l 13 mm
 How to avoid
 design for horizontohorizon safety
Main Lobe
12Array design
How many elements? What Spacing?
 Linear array
 32 element array
 3 MHz
 pitch l 0.4 mm
 l 0.51 mm
 Larray N l 13 mm
 How to avoid
 design for horizontohorizon safety
13Apodization
 Same array
 32 element array
 3 MHz
 pitch l 0.4 mm
 l 0.51 mm
 Larray N l 13 mm
 With w/o Hanning wting.
 Sidelobes way down.
 Mainlobe wider
 No effect on grating lobes.
14Summary of Beam Processing
 Beam shape is improved by several processing
steps  Transmit apodization
 Multiple transmit focal locations
 Dynamic focusing
 Dynamic receive apodization
 Postbeamsum processing
 Upper frame fixed transmit focus
 Lower frame the above steps.
15I INTERACTIONS OF ULTRASOUND WITH TISSUE Some
essentials of linear propagation Recall the
equation of motion
(1)
Assume a plane progressive wave in the x
direction that satisfies the wave equation ie
(2)
16Substituting 2 into 1 we have
(3)
Acoustic impedance
17Where
Characteristic Acoustic Impedance Define a
type of Ohms Law for acoustics Electrical Acous
tical Extending this analogy to Intensity we
have
18Propagation at an interface between 2 media
19Define Reflection/Transmission Coef
(4)
You will show
(5)
Example Fat Bone interface
20THE DECIBEL (dB) SCALE
(6)
Where A measured amplitude Aref reference
amplitude
In the amplitude domain 6 dB is a factor of
2 6 dB is a factor of .5 (i.e. 6dB down) 20 dB
is a factor of 10 20 dB is a factor of .1 (i.e.
20dB down)
21Reflection Coefficients
0
Air/solid or liquid
R 1.0
Brass/soft tissue or water
Reflection Coef. dB
Bone/soft tissue or water
10
Perspex/soft tissue or water
Tendon/fat
Lens/vitreous or aqueous humour
R .1
20
Fat/nonfatty soft tissues
Water/muscle
Water/soft tissues
Fat/water
30
Muscle/blood
Muscle/liver
R .01
40
Kidney/liver, spleen/blood
50
Liver/spleen, blood/brain
223) ULTRASOUND IMAGING AND SIGNAL PROCESSING
Thus far we have been concerned with the
ultrasound transducer and beamformer. Lets now
start considering the signal processing aspects
of ultrasound imaging. Begin by considering the
sources of information in an ultrasound
image a) Large interfaces, let a structure
dimension

 specular reflection
 reflection coefficient
where
density
speed of sound
 strong angle dependance
 refraction effects
23 b) Small interfaces

 Rayleigh scattering
Compressibility Density
and
(7)
Morse and Ingard Theoretical Acoustics p. 427
24SCATTER FROM A RIGID SPHERE
25SCATTER FROM A RIGID SPHERE (Mie Scatter)
26ATTENUATION
absorption component reflectivity component
The units of are cm1 for this equation.
However attenuation is usually expressed in
dB/cm. A simple conversion is given by
27Attenuation in Various Tissues
2815
Speed of Sound in Various Tissues
10
5
Assumed speed of sound 1540 m/s
0
5
10
29SUMMARY ULTRASONIC PROPERTIES Table 1
302.2 Modeling the signal from a point
scatterer Imagine that we have a transducer
radiating into a medium and we wish to know the
received signal due to a single point scatterer
located at position By modifying the impulse
response equation (Lecture 1 Equ. 25 ) we can
write
31transmit receive electromechanical IRs
scatterer IR
transmit IR
receive IR
pulse (t)
easily measured
32Now consider a complex distribution of scatterers
Isochronous volume
(4)
(1)
z
l1
(2)
l2
rx
ri
(3)
At any point in the isochronous volume there
exists a transmit receive path length divided
by c for a time, t, such that
33If we look at the four field points shown on the
previous page we would see the following impulse
responses
(1)
(2)
(3)
(4)
34The total signal for a given ray position rx is
given by
(9)
scatterer strength
35The resultant signal is the coherent sum of
signals resulting from the group of randomly
positioned scatterers that make up the
isochronous volume as a function of time. A
useful model of the signal is
(10)
Envelope
Modulated carrier
Phase
Grayscale information for Bscan Image
Velocity information for Doppler
How do we calculate a(t) and (t)?
363.3 Hilbert Transform The Hilbert transform is
an unusual form of filtration in which
the spectral magnitude of a signal is left
unchanged but its phase is altered by
for negative frequencies and for positive
frequencies
Definition
(11)
37In the frequency domain
(12)
Consider the Hilbert transform of Cos
38The application of two successive Hilbert
transforms results in the inversion of the signal
we have 2 successive rotations in the
negative frequency range and 2 rotations
in the positive frequency range. Thus the total
shift in each direction is .
39(No Transcript)
40The Hilbert transform is interesting but what
good is it? ANALYTIC SIGNAL THEORY Consider a
real function . Associate with this
function another function called the analytic
signal defined by where Hilbert
Transform The real part of the analytic signal
is the function itself whereas the imaginary
part is the Hilbert transform of the
function. Note that the real and imaginary
components of the analytic signal are often
called the in phase, I, and quadrature,
Q, components.
(13)
41Just as complex phasors simplify many problems in
AC circuit analysis the analytic signal
simplifies many signal processing problems. The
Fourier transform of the analytic signal has an
interesting property.
(14)
42 Equation 14 gives us an easy way to calculate the
analytic  signal of a function
 Fourier transform function
 Truncate negative frequencies to zero
 Multiply positive frequencies by 2
 Inverse Fourier Transform
 Recall that our resultant ultrasound signal can
be expressed  as
Its analytic signal is then
(15)
43which on the complex plane looks like
(16)
Where and the phase is given by
(17)
a(t) envelope
44 Demodulation estimate using
 Analytic signal method using FFT (slow)
 Analytic signal using baseband quadrature
approach  Sampled quadrature
Baseband Quadrature Demodulation
Low Pass
Baseband Inphase Signal
Low Pass
Baseband Quadrature Signal
45(slowly varying)
Use shift and convolution theorems to calculate
spectra
46Similarly
Baseband Analytic Signal No carrier Phase
preserved
47Thus
and
Sampled Quadrature Begin with the signal of the
ultrasound waveform
48Sample with period
Recall that the quadrature signal is the Hilbert
Transform of the inphase component of the
analytic signal i.e. for a cos wave it is a
negative sine wave. Thus we see that . . .
49If the inphase and quadrature signals are slowly
varying we can get the quadrature signal simply
by sampling the inphase signal 90º or ¼ period
later Sampling t nT for I samples t
nTT/4 for Q sample
(18)
let
50Overall Imager Block Diagram
6
2
3
4
5
1
51Imaging System Signals
52Coarse and Fine Beamforming Delays
53SIGNAL STATISTICS Recall that the ultrasound
signal is the sum of harmonic components with
random phase and amplitude. It can be shown that
the probability density function for such a
situation is Gaussian with zero mean i.e.
(19)
The quadrature signal will also be Gaussian with
the same standard deviation
(20)
54Since p(y) and p(z) are independent random
variables the joint probability density function
is given by
(21)
The probability of a joint event (corresponding
to a particular amplitude of the envelope) is the
probability that
55total area
The probability that a lies between a and a da
is
56So that the probability density function for the
radio frequency signal is given by
Rayleigh Prob. Density function
many gray pixels
few black pixels
few white pixels
57The speckle in an ultrasound image is described
by this probability density function. Lets
define the signal as and the noise as the
deviation from this value
Thus
Recall
58Thus
SNR 1.91 and is invariant
(25)
Note that the SNR in ultrasound imaging is
independent of signal level. This is in
contrast to xray imaging where the noise is
proportional to the square root of the number
of photons.
59Speckle Noise in an Ultrasound Image
60Lets make several independent measurements of
so and si These measurements will form
distributions
61The parameter used to define image quality
includes both the observed contrast and the noise
due to speckle in the following
fashion Define Contrast Define
Normalized speckle noise as and finally,
define our quality factor as the contrast to
speckle noise ratio (CSR)
(26)
62Suggested Ultrasound Book References General
Biomedical Ultrasound (and physical/mathematical
foundations) Foundations of Biomedical
Ultrasound, RSC Cobbold, Oxford Press
2007. General Biomedical Ultrasound (bit more
applied) Diagnostic Ultrasound Imaging inside
out TL Szabo Academic Press
2004. Ultrasound Blood flow detection/imaging
Estimation of blood velocities with ultrasound
JA Jensen Cambridge university press
1996 Basic acoustics Theoretical Acoustics PM
Morse and KU Ingard, Princeton University Press
(many editions). Bubble behaviour The
Acoustic bubble TG Leighton Academic Press
1997. Nonlinear Acoustics Nonlinear Acoustics
Hamilton and Blackstock, Academic Press 1998.