ultrasound lecture 2 post (3)

8/22/2019 Ultrasound Lecture 2 Post (3)

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Ultrasound Imaging: Lecture 2

Absorption

Reflection

Scatter

Speed of sound

Signal modeling

Signal Processing

Statistics

Interactions of ultrasoundwith tissue

Image formation

Jan 14, 2009

Steering

Focusing

Apodization

Design rules

Beams and Arrays


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Anatomy of an ultrasound beam Near field or Fresnel zone

Far field or Fraunhofer zone Near-to-far field transition,L

2aL

L


3/62

Anatomy of an ultrasound beam

Lateral Resolution (FWHM)

FWHM

numberFR

aFWHM

2


4/62


Depth of Field (DOF)

DOF

2)(7 numberFDOF


5/62

Array Geometries

Schematic of a linear phased array

Definition of azimuth, elevation

Scanning angle shown, q, in negative scan

direction.

ya (elevation)

xa (azimuth)

za (depth)

array pitch

Acoustic beamq

t

trtrp

,

,

N

ii ttrhWtrh 1 ),(,


6/62

Some 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

x

z

x

rqr

0

rx

crr x

rrrxxc

q 22 cos21

fs


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Far field beam steering

For beam steering: far field calculation

particularly easy

often implemented as a fixed

delay

c

xs

qsin

x

z

x

r

0

q


8/62

Beamformation: Focusing Basic focusing type beamformation Symmetrical delays about phase center.

-4 0 -3 0 -2 0 -1 0 0 10 20 30 40

-2 0

-1 0

0

10

20

point

source

wavefront s

befo re correct ion

t ransducer

elements

delay

lines

wavefront s

after correction

summing

stage


9/62

Beamformation: Beam steering

Beam steering with linear phased arrays.

Asymmetrical delays, long delay lines

-4 0 -3 0 -2 0 -1 0 0 10 20 30 40

-2 0

-1 0

0

10

20

po int

source

wavefront s

befo re correction

array

element s

delay

lines

wavefront s

after beam

steering and focusing

summing

stage


10/62


Electronic Focusing


11/62

Grating Lobes

Linear array: 32 element array

3 MHz

pitch l= 0.4 mm

= 0.51 mm

L= N l= 13 mm

How to avoid:

design for horizon-to-

horizon safety275.1

4.

51.)(

)(

g

g

Sin

lSin

q

q

How many elements?

What Spacing?

gql

gq

l

Main Lobe


12/62

Array design

Linear array: 32 element array

3 MHz

pitch l= 0.4 mm

= 0.51 mm

Larray= N l= 13 mm

How to avoid:

design for horizon-to-

horizon safety2

l

How many elements?

What Spacing?


13/62

Apodization

Same array:

32 element array

3 MHz

pitch l= 0.4 mm

= 0.51 mm

Larray = N l= 13 mm

With & w/o Hanning wting.

Sidelobes way down. Mainlobe wider

No effect on grating lobes.


14/62

Summary of Beam Processing

Beam shape is improved by several

processing steps:

Transmit apodization

Multiple transmit focal locations Dynamic focusing

Dynamic receive apodization

Post-beamsum processing

Upper frame: fixed transmit focus

Lower frame: the above steps.


15/62

I INTERACTIONS OF ULTRASOUND WITH TISSUE

Some essentials of linear propagation

Recall the equation of motion

t

v

x

p

0 (1)

Assume a plane progressive wave in the +x direction that

satisfies the wave equation

ie)(

0

kxtepp (2)


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Substituting 2 into 1 we have

t

vjkep kxtj

0

)(

0

Z

p

c

pv

ep

f

ejk

j

pv

dtejkp

v

kxtj

kxtj

kxtj

0

0

0

)(

0

0

)(

0

0

2

2

Acoustic impedance

(3)


17/62

cZ 0Where

= Characteristic Acoustic Impedance

Define a type of Ohms Law for acoustics

Electrical:

Acoustical:

Extending this analogy to Intensity we have

vZp

IRV

2

0

2

0

2

1

2

1ZvZ

pI


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Propagation at an interface between 2 media

111 cZ 222 cZ

iP

rP

tP

xktj

tt

xktj

rr

xktj

ii

eP

p

ePp

ePp

2

1

1


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Define Reflection/Transmission Coef

i

t

i

r

p

pT

ppR ,

You will show:

212

1212 2

ZZZT

ZZZZR

Example: Fat Bone interface

38.16.7)6.7(2

38.16.738.16.7

TR

70.0 69.1

(4)

(5)


20/62

THE DECIBEL (dB) SCALE

refsBd A

A

LogA 10)( 20

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)

(6)

R fl ti C ffi i t


21/62

0

-10

-20

-30

-40

-50

Reflection Coefficients

Air/solid or liquid

Brass/soft tissue or water

Bone/soft tissue or water

Perspex/soft tissue or water

Tendon/fat

Lens/vitreous or aqueous humour

Fat/non-fatty soft tissuesWater/muscle

Fat/water

Muscle/blood

Muscle/liver

Kidney/liver, spleen/blood

Liver/spleen, blood/brain

Water/soft tissues

R = 1.0

R = .1

R = .01

Reflection

Coef.dB


22/62

3) 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

a

- specular reflection

-

- reflection coefficient 12

12

ZZ

ZZ

R

where cZ

- strong angle dependance

- refraction effects

density speed of sound


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b) Small interfaces

a-

- Rayleigh scattering

Cos

akD

0

0

0

0

32

2

33

3

Compressibility Density

and Arp , Dr

eikr (7)

Morse and Ingard Theoretical Acoustics

p. 427

*


24/62

SCATTER FROM A RIGID SPHERE

Cosr

a

cDs 31

3

4 32

*

*


25/62

SCATTER FROM A RIGID SPHERE (Mie Scatter)

*


26/62

ATTENUATION

= absorption component + reflectivity component

xepxp 0

The units of are cm-1 for this equation. However attenuation

is usually expressed in dB/cm. A simple conversion is given

by

1686.8 cmcm

dB


27/62

Attenuation in

Various Tissues

15%


28/62

Speed of

Sound in

VariousTissues

0%

5%

10%

15%

-5%

-10%

Assumed speedof sound = 1540

m/s


29/62

SUMMARY ULTRASONIC PROPERTIES

Table 1

Material Speed of Sound Impedance Attenuation Frequency

ms-1 Kg m-2 s-1

X 106

At 1 MHz (dB

cm-1) Dependency

water 1490 @ 23C 1.49 0.002 2

muscle 1585 @ 37C 1.70 1.3-3.3 1.2

fat 1420 @ 37C 1.38 0.63 1.5-2

liver 1560 @ 37C 1.65 0.70 1.2

breast 1500 + 80 @ 37C ------ 0.75 1.5

blood 1570 @ 37C 1.70 0.18 1.2

skull bone 4080 @ 37C 7.60 20.00 1.6air 331 @ STP 0.0004 12.00 2

PZT 4300 @ STP 33.00 ------ --

smc /1540


30/62

2.2 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:

r


31/62

trhtrhtstgtgt

VktrV rtout ,*,***,*, 2

0

transmit + receive

electromechanical

IRs

scatterer

IR

transmit

IR

receive

IR

pulse (t)

trHtpulse

trhtrhtpulsetrV rtout,*)(

,*,*,

easily

measured


32/62

Now consider a complex distribution of scatterers

Isochronous

volume

rx ri

(1)

(2)

(3)

(4)

At any point in the isochronous volume there exists a transmit

receive path length divided by c for a time, t, such that

c

zt

c

ll

21

zl1

l2


33/62

If we look at the four field points shown on the previous page

we would see the following impulse responses

(1)

(2)

(3)

(4)


34/62

The total signal for a given ray position rx is given by

trHrWtpulsetrVout xiN

iiix ,

1

*)(,

(9)

scatterer

strength


35/62

The 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:

ttCostatytVout 2

Envelope Modulated

carrier

Phase

Grayscale informationfor B-scan Image

How do we calculate a(t) and (t)?

Velocity informationfor Doppler

(10)


36/62

3.3 Hilbert Transform

The Hilbert transform is an unusual form of filtration in which thespectral magnitude of a signal is left unchanged but its phase

is altered by for negative frequencies and for

positive frequencies2

2

Definition

)(*1

1

xfx

xdxx

xfxFH

(11)


37/62

In the frequency domain

sH FsjxF )sgn(

Consider the Hilbert transform of Cos x

RE RE

(12)

IMIM

xCos sjSgn


38/62

The application of two successive Hilbert transforms results

in the inversion of the signal we have 2 successiverotations in the negative frequency range and 2

rotations in the positive frequency range. Thus the total

shift in each direction is .

2

2

1

II


39/62

xfxx

xF

xH

111

sFsjsj sgnsgn

xfF

sF

s

1

1


40/62

The 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.

tjztytf tz(13)

ty


41/62

Just 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.

0,2

0,0

][

sY

s

YsSgnY

YsSgnjjYtjzty

s

s

ss

s sy

0

2

s

Ys

(14)


42/62

Equation 14 gives us an easy way to calculate the analytic

signal of a function:

1) Fourier transform function

2) Truncate negative frequencies to zero

3) Multiply positive frequencies by 2

4) Inverse Fourier Transform

Recall that our resultant ultrasound signal can be expressed

as:

ttCostaty 2

Its analytic signal is then

ttetatf 2(15)


43/62

which on the complex plane looks like:

IM

RE

ty

t

ta tz

Where

and the phase is given by

tztyta22

ty

tzTant )(1

(16)

(17)

a(t) envelope


44/62

Demodulation: estimate using

1) Analytic signal method using FFT (slow)

2) Analytic signal using baseband quadrature approach

3) Sampled quadrature

)(),( tta QI,

Baseband Quadrature Demodulation

X

X

Low

Pass

Low

Pass

tCos 2

tQtSin

2 ty

tIt )Re(

Baseband

Inphase Signal

)()Im( tQt

Baseband

Quadrature Signal


45/62

tCostCosa

tCostCosaI

tt

ttt

22

2

22

Use shift and convolution theorems to calculate spectra

ttnote :

2

2 tjt eAI

(slowly varying)

tjeA 2 221

tjt eAI 21


46/62

tjtj

t eAeAI

2

1

2

1

tt CostaI 2

1 tj

t etas)(

2

1

Similarly

tt SintaQ )(2

1 Baseband

Analytic

SignalNo carrier

Phase preserved


47/62

ttt

t

ttt

QIa

a

CosSinaQI

22

2

22222

2

41

)(4

1

Thus

)(

)(

)()(

)(

tI

tQArcTan

I

Q

Tan

t

t

tt

and

Sampled Quadrature

Begin with the signal of the ultrasound waveform

ttt Cosay 2


48/62

Sample with period 1T

* **

)(nTI

T

tIIIy t

)(nTQ

T

tIIIy t

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 . . .

nTIT

tIIIy t


49/62

If the inphase and quadrature signals are slowly varying

we can get the quadrature signal simply by sampling the

inphase signal 90 or period later

Sampling t = nT for I samples

t = nT+T/4 for Q sample

1

42)()(

)2()()(

T

nTTnTCosnTanTQnTrTCosnTanTI

let

nTSinntanTnCosnTanTQ

TnCosnTanTnCosnTanTI

)()2

2()()(

)()()2()()(

(18)


50/62

1

Overall Imager Block Diagram

Doppler

Beamformer

Digital

Receive

Beamformer

Beamformer

Central

Control

Digital

Transmit

Beamformer

Transmit

Demux

Receive

Mux

Transducer

Connectors

System

Control

Image

Proces-

sing

2 3 4 5 6

Imaging System Signals


51/62

Imaging System Signals

Doppler

Beamformer

Digital

Receive

Beamformer

Beamformer

Central

Control

Digital

Transmit

Beamformer

Transmit

Demux

Receive

Mux

Transducer

Connectors

System

Control

Image

Proces-

sing

23 4 5

6

1


52/62

Coarse and Fine Beamforming Delays

Ho()e-j/4

Ho()e-j/2

Ho()e-j3/4

Ho()

MUXFIFO

Input from

ADC at 20to 40 MHz,

8 to 12 bits

Output withdelay accuracy

up to 160 MHz

To apodization

and further

processing

Coarse

Delay

Control

Fine

Delay

Control


53/62

SIGNAL STATISTICS

Recall that the ultrasound signal is the sum of harmoniccomponents 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.

2

2

221)(

y

eyp

(19)

The quadrature signal will also be Gaussian with the

same standard deviation

2

2

22

1)(

z

ezp

(20)


54/62

Since p(y) and p(z) are independent random variables the joint

probability density function is given by

2

22

2

2

2

2

22

22

2

1

2

1

2

1),(

zy

zy

e

eezyp

(21)

The probability of a joint event (corresponding to a particularamplitude of the envelope) is the probability that:

)(zp


55/62

)(zp

)(yp

total area = ada2

The probability thata lies between

a and a + da is

222 zya

dae

a

daap

a2

2

222

2

)(

adad

adad


56/62

So that the probability density function for the radio

frequency signal is given by

22

22

a

ea

ap

Rayleigh Prob.

Density function

aa

)(ap

few white pixels

many gray pixels

few

blackpixels


57/62

The 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

arms

2122

12aaaN Thus

daea

daapaa

a

o

o

2

2

22

2

Recall

2a


58/62

Thus:

21

21

22

22

2

22

2

N

aSNR

SNR = 1.91 and is invariant (25)

Note that the SNR in ultrasound imaging is independent of

signal level. This is in contrast to x-ray imaging where the

noise is proportional to the square root of the number of

photons.

S kl N i i Ult d I


59/62

Speckle Noise in an Ultrasound Image


60/62

a

ias

as

i

00

x

Lets make several independent measurements of

so and si

These measurements will form distributions

i 0

is 0s


61/62

The 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)

0

0

s

ss i

0

2122

0

si

22

0

0

i

issCSR

(26)


62/62

Suggested 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.

ultrasound lecture 2 post (3)

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