BME/EECS 516 (2014)
FT Notes
1
Notes on the Fourier Transform
Definition. The Fourier Transform (FT) relates a function to its frequency domain equivalent.
The FT of a function g(x) is defined by the Fourier integral:
G ( s)
g ( x)e i 2 xs dx
Fcfw_g ( x)
Ultrasound Imaging
Lecture #7
BME/EECS 516
Zhen Xu
Last Lecture
Array
l
Systems
Transmit mode: focusing and steering
2
xn x z 0
xn
For Focusing and Beam Steering: ' ( xn ) =
2 z0 c
z0 c
2
xn sin 0 xn cos 2 0
'n = ' ( xn , r0 , 0 ) =
+
c
l
2cr0
Receive
Ultrasound Imaging
Lecture #5
BME/EECS 516
Zhen Xu
Last Lecture
Lateral
resolution Beam Pattern,
diffraction analysis
a
ikr0 z
ze
p( z, x z ) =
dx 0
Transmit only:
2
a (r0 z )
l Transmit and receive:
l
2
B ( x x z , z ) = [ p ( z , x x z )]
2
Fresnel
Ultrasound Imaging #6
BME/EECS 516
Zhen Xu
Last Lecture
Beam
Steering in Cartesian
Coordinates
2
s ' ( x0 ) = s ( x0 ) e
ik
x0
2 z0
e
ik
x z 0 x0
z0
Diffraction
in Polar coordinates
Fresnel Approximation:
cos e
p(r , ) =
r
ikr
s( x0 )e ikx0 sin e
( x0
ULTRASOUND
IMAGING
BME/EECS 516
Instructor: Zhen Xu
Lecture #1
This Lecture
Ultrasound Imaging Introduction
Ultrasound signal source
Wave Equation
Basics
Sound:
Mechanical wave propagate
through vibration of particles in
the media
Sound propagation
Does n
BME/EECS 516
US Notes 4: page 1
Ultrasound Notes 4: Doppler
Imaging of Blood Flow with US
One of the more interesting and useful applications of US is for the imaging of flow in
the body. One way to do this is track speckle patterns over time using a cros
ULTRASOUND IMAGING
Lecture #3
BME/EECS 516
Zhen Xu
Last Lecture
Add dissipation and dispersion to wave equation
Add exp(-(f)l)
Bandpass signal
Gaussian pulse for transmitted signal received signal
maintain the same shape, dispersion effect can be negl
Ultrasound Imaging #4
Lecture
BME/EECS 516
Zhen Xu
Last Lecture
2D
l
l
Imaging
What determines the resolution
2( z z 0 )
I ( x, z ) = B( x, z 0 )a
c
Time compensated Gain
Issues
l
Determined by a(t)
Issues
l
related to depth resolution
related to la
Ultrasound Imaging
Lecture #8
BME/EECS 516
Zhen Xu
Last Lecture
Array
l
Systems
Point spread function (system impulse
response) for an array transducer
'1
!x$
! x $*
!x$
s(x) = ) comb # &*rect # &, rect # &
"d%
" w %+
"D%
(d
sin
sin n
p(r , ) = K cos