# Speckle Correlation Analysis

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```					Adaptive Imaging Preliminary:
Speckle Correlation Analysis
Speckle Formation
• Speckle results from coherent interference of un-
resolvable objects. It depends on both the
frequency and the distance.

sample
transducer                      volume
Speckle Second-Order Statistics
• The auto-covariance function of the
envelope detection) is simply the
convolution of the system’s point spread
function if the insonified region is
– macroscopically slow-varying.
– microscopically un-correlated.
Speckle Second-Order Statistics

• The shape of a speckle spot (assuming fully
developed) is simply determined by the
shape of the point spread function.
• The higher the spatial resolution, the finer
the speckle pattern, and vice versa.
Speckle Statistics
• The above statements do not hold if the
object has structures compared to or larger
than the ultrasonic wavelength.
• Rician distribution is often used for more
general scatterer distribution.
• Rayleigh distribution is a special case of
Rician distribution.
van Cittert-Zernike Theorem
• A theorem originally developed in statistical
optics.
• It describes the second-order statistics of the field
produced by an in-coherent source.
• The insonification of diffuse scatterers is
assumed in-coherent.
• It is different from the aforementioned lateral
displacement.
van Cittert-Zernike Theorem
• The theorem describes the spatial
covariance of signals received at two
different points in space.
• For a point target, the correlation of the two
signals should simply be 1.
• For speckle, correlation decreases since the
van Cittert-Zernike Theorem

• The theorem assumes that the target is
microscopically un-correlated.
• The spatial covariance function is the
Fourier transform of the radiation pattern at
the point of interest.
van Cittert-Zernike Theorem

van Cittert-Zernike Theorem
• The theorem states that the correlation
coefficient decreases from 1 to 0 as the
distance increases from 0 to full aperture
size.
• The correlation is independent of the
frequency, aperture size, …etc.
van Cittert-Zernike Theorem
• In the presence of tissue inhomogeneities,
the covariance function is narrower since
• The decrease in correlation results in lower
accuracy in estimation if signals from
different channels are used.
van Cittert-Zernike Theorem

correlation

distance
van Cittert-Zernike Theorem

RF Signals
Channel

Time (Range)
van Cittert-Zernike Theorem
(Focal length 60mm vs. 90mm)
van Cittert-Zernike Theorem
(16 Elements vs. 31 Elements)
van Cittert-Zernike Theorem
(2.5MHz vs. 3.5MHz)
van Cittert-Zernike Theorem
(with Aberrations)
Lateral Speckle Correlation

correlation coefficient

displacement
L/2
Lateral Speckle Correlation
• Assuming the target is at focus, the
correlation roughly decreases linearly as the
lateral displacement increases.
• The correlation becomes zero when the
displacement is about half the aperture size.
• Correlation may decrease in the presence of
non-ideal beam formation.
Lateral Speckle Correlation
14.4 mm Array
Lateral Speckle Correlation
Lateral Speckle Correlation
Lateral Speckle Correlation
Lateral Speckle Correlation:
Implications on Spatial Compounding
Speckle Tracking
• Estimation of displacement is essential in
many imaging areas such as Doppler
imaging and elasticity imaging.
• Speckle targets, which generally are not as
ideal as points targets, must be used in
many clinical situations.
Speckle Tracking
• From previous analysis on speckle analysis,
we found the local speckle patterns simply
translate assuming the displacement is small.
• Therefore, speckle patterns obtained at two
instances are highly correlated and can be
used to estimate 2D displacements.
Speckle Tracking
• Displacements can also be found using
phase changes (similar to the conventional
Doppler technique).
• Alternatively, displacements in space can be
estimated by using the linear phase shifts in
the spatial frequency domain.
Speckle Tracking
• Tracking of the speckle pattern can be used
for 2D blood flow imaging. Conventional
Doppler imaging can only track axial
motion.
• Techniques using phase information are still
inherently limited by the nature of Doppler
shifts.
Correlation-Based Approach
Sound Velocity Inhomogeneities

body wall   viscera

point of interest

v1 v2 v3
transducer array
Sound Velocity Inhomogeneities

Velocity (m/sec)
water            1484
blood            1550
myocardium          1550
fat            1450
liver           1570
kidney           1560
Sound Velocity Inhomogeneities
• Sound velocity variations result in arrival
time errors.
• Most imaging systems assume a constant
sound velocity. Therefore, sound velocity
variations produce beam formation errors.
• The beam formation errors are body type
dependent.
Sound Velocity Inhomogeneities

no errors         with errors
• Due to beam formation errors, mainlobe
may be wider and sidelobes may be higher.
• Both spatial and contrast resolution are
affected.
Near Field Assumption
beam formation
geometric delay
aligned

velocity
correction
variations
• Assuming the effects of sound velocity
inhomogeneities can be modeled as a phase
screen at the face of the transducer, beam
formation errors can be reduced by
correcting the delays between channels.
Correlation-Based Aberration Correction
No Focusing
Correlation-Based Aberration Correction
Transmit Focusing Only
Correlation-Based Aberration Correction
Correlation-Based Aberration Correction

Wire: Before Correction   Wire: After Correction
Correlation-Based Aberration Correction

Diffuse Scatterers: Before   Diffuse Scatterers: After
Correlation Based Method
1 T
Cn (t )   S n ( ) S n 1 (t   )d
T 0

tn  max Cn (tn )
t n

• Time delay (phase) errors are found by
finding the peak of the cross correlation
function.
• It is applicable to both point and diffuse
targets.
Correlation Based Method
n
Tn     t
i 1
i

• The relative time delays between adjacent
channels need to be un-wrapped.
• Estimation errors may propagate.
Correlation Based Method

filter   correlator

x

• Two assumptions for diffuse scatterers:
– spatial white noise.
– high correlation (van Cittert-Zernike theorem).
Correlation Based Method

• Correlation using signals from diffuse
scatterers under-estimates the phase errors.
• The larger the phase errors, the more severe
the underestimation.
• Iteration is necessary (a stable process).
Alternative Methods

• Correlation based method is equivalent to
minimizing the l2 norm. Some alternative
methods minimize the l1 norm.
• Correlation based method is equivalent to a
maximum brightness technique.
Baseband Method

1 T                            1 j0 tn T
Cn (t )   BBn ( ) BBn 1 (t   )d  e
*
0 A( ) A(t    tn )d
T 0                            T
tan 1 (Im( Cn (0)) / Re( Cn (0)))
t n 
0

• The formulation is very similar to the
correlation technique used in Color Doppler.
Baseband Method
Cn (0)                      *
BBn (m) BBn 1 (m)
mregion of interest

I                                        acc.
CORDIC
Q

sign control      Q sign bit

I                                        acc.
CORDIC
Q                                        acc.
One-Dimensional Correction:
Problems
• Sound velocity inhomogeneities are not
restricted to the array direction. Therefore, two-
dimensional correction is necessary in most
cases.
• The near field model may not be correct in
some cases.
One-Dimensional Correction:
Problems
One-Dimensional Correction:
Problems
Two-Dimensional Correction
• Using 1D arrays, time delay errors can only
be corrected along the array direction.
• The signal received by each channel of a 1D
array is an average signal. Hence,
estimation accuracy may be reduced if the
elevational height is large.
• 2D correction is necessary.
Two-Dimensional Correction
• Each array element has four adjacent
elements.
• The correlation path between two array
elements can be arbitrary.
• The phase error between any two elements
should be independent of the correlation
path.
Full 2D Correction
(1,1)   corr   (1,2)   corr   (1,3)
corr

corr

corr
(2,1)   corr   (2,2)   corr   (2,3)
corr

corr

corr
(3,1)   corr   (3,2)   corr   (3,3)
Row-Sum 2D Correction
(1,1)   corr   (1,2)   corr   (1,3)
corr

(2,1)   corr   (2,2)   corr   (2,3)
corr

(3,1)   corr   (3,2)   corr   (3,3)
Correlation Based Method: Misc.
• Signals from each channel can be correlated
to the beam sum.
• Limited human studies have shown its
efficacy, but the performance is not
consistent clinically.
• 2D arrays are required to improve the 3D
resolution.
Displaced Phase Screen Model
• Sound velocity inhomogeneities may be
modeled as a phase screen at some distance
from the transducer to account for the
distributed velocity variations.
• The displaced phase screen not only
produces time delay errors, it also distorts
ultrasonic wavefronts.
Displaced Phase Screen Model

phase screen

• Received signals need to be “back-propagated”
to an “optimal” distance by using the angular
spectrum method.
• The “optimal” distance is determined by using
a similarity factor.
Displaced Phase Screen Model
TSC + BP
Time-shift compensation with back-propagation
TSC + BP
Time-shift compensation with back-propagation
TSC + BP
Time-shift compensation with back-propagation
TSC + BP
Time-shift compensation with back-propagation
Abdominal Wall Measurements
Abdominal Wall Measurements
Abdominal Wall Measurements
Displaced Phase Screen Model

• After the signals are back-propagated,
correlation technique is then used to find
errors in arrival time.
• It is extremely computationally extensive,
almost impossible to implement in real-time
using current technologies.
Wavefront Distortion

• Measurements on abdominal walls, breasts
and chest walls have shown two-
dimensional distortion.
• The distortion includes time delay errors
and amplitude errors (resulting from
wavefront distortion).
Phase Conjugation

phase screen at face   displaced phase
of transducer       screen

phase                phase

f                    f
Phase Conjugation
Phase Conjugation
No aberration

At 0 mm

At 20 mm

At 40 mm

At 60 mm
Phase Conjugation

• Simple time delays result in linear phase
shift in the frequency domain.
• Displaced phase screens result in wavefront
distortion, which can be characterized by
non-linear phase shift in the frequency
domain.
Phase Conjugation

• Non-linear phase shift can be corrected by
dividing the spectrum into sub-bands and
correct for “time delays” individually.
• In the limit when each sub-band is
infinitesimally small, it is essentially a
phase conjugation technique.
End 4/13/2005
Some of the Recent Developments
Real-Time In Vivo Imaging[15]
Real-Time In Vivo Imaging
Real-Time In Vivo Imaging
Real-Time In Vivo Imaging
Real-Time In Vivo Imaging
Real-Time In Vivo Imaging

Distribution of time delay corrections
Clinical Imaging Using 1-D Array [16]
Clinical Imaging Using 1-D Array

Before Correction   After Correction
Clinical Imaging Using 1-D Array

Before Correction   After Correction
Clinical Imaging Using 1-D Array

Channel Data   Complex Scattering Structures
1.75D, High Frequency Arrays [17]

1D and 2D Least Squares Estimation
1.75D, High Frequency Arrays

Before Correction   After Correction
1.75D, High Frequency Arrays

Before Correction   After Correction
1.75D, High Frequency Arrays

Original    1 iteration   4 iterations
1.75D, High Frequency Arrays

2D Correction Using 1.75d Array On
Breast Microcalcifications [18]
2D Correction Using 1.75d Array On
Breast Microcalcifications
2D Correction Using 1.75d Array On
Breast Microcalcifications

(also with a 60% brightness improvement)
2D Correction Using 1.75d Array On
Breast Microcalcifications
2D Correction Using 1.75d Array On
Breast Microcalcifications

(a)   1D
(b)   1D with correction
(c)   1.75D
(d)   1.75D with correction
Aperture Domain Processing
Single Transmit Imaging
• Fixed direction transmit, all direction receive
Measuring Source Profile
Removing Focusing Errors
Focusing Errors

No Aberrations       With Aberrations
Single Transmit Imaging

No Aberrations   With Aberrations
PARCA

No Correction           With Correction
Simplifications:
1. DFT vs. Single Transmit Imaging
2. Weighting vs. Complex Computations
DFT vs. Single Transmit Imaging

Single Transmit Imaging   DFT
Frequency Domain Interpretation
of the Aperture Data Coherent

Speckle

Incoherent

Aberrated

*P.-C. Li and M.-L. Li, “Adaptive Imaging Using the Generalized Coherence Factor”,
IEEE UFFC, Feb., 2003.
Coherence Factor (CF)
•    A quantitative measure of coherence of the
energy in the low - frequency region
General Definition :
total energy
N 1          2

 C (i, t )
i 0
Coherent sum (DC)
CF(t )      N 1

The larger, the better? N  C (i, t )
2
Total energy (times N)
i 0

N: the number of array channels used in beam sum
C(i,t) : the received signal of channel i
Determination of the Optimal

Object of
Interest       Enhance

Unwanted
Suppress
Sidelobes

Classify “object types”     Optimize the receive aperture size
Experimental Results:
Tissue Mimicking Phantoms
Azimuth
0         1X   2X
Range

28.6 mm
Original

–40           40        96.2 mm
Aperture

Dynamic range: 60 dB

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