# Speckle Correlation Analysis by dfhdhdhdhjr

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```									Adaptive Imaging Preliminary:
Speckle Correlation Analysis

Speckle Correlation Analysis   1
Speckle Formation
• Speckle results from coherent interference of un-
resolvable objects. It depends on both the
frequency and the distance.

sample
transducer                                  volume

Speckle Correlation Analysis            2
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 Correlation Analysis   3
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 Correlation Analysis     4
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.

Speckle Correlation Analysis   5
Lateral Speckle Correlation

correlation coefficient

displacement
L/2

Speckle Correlation Analysis      6
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.

Speckle Correlation Analysis   7
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.
Speckle Correlation Analysis   8
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

Speckle Correlation Analysis      9
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.

Speckle Correlation Analysis   10
van Cittert-Zernike Theorem

Speckle Correlation Analysis                 11
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.

Speckle Correlation Analysis   12
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.

Speckle Correlation Analysis    13
van Cittert-Zernike Theorem

correlation

distance

Speckle Correlation Analysis              14
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 Correlation Analysis   15
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 Correlation Analysis   16
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 Correlation Analysis   17
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.

Speckle Correlation Analysis   18

Speckle Correlation Analysis   19
Sound Velocity Inhomogeneities
body wall           viscera

point of interest

v1 v2 v3
transducer array

Speckle Correlation Analysis                   20
Sound Velocity Inhomogeneities

Velocity (m/sec)
water                      1484
blood                      1550
myocardium                   1550
fat                      1450
liver                     1570
kidney                     1560

Speckle Correlation Analysis    21
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.

Speckle Correlation Analysis   22
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.
Speckle Correlation Analysis            23
Near Field Assumption
geometric delay    beam formation
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.

Speckle Correlation Analysis               24
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.
Speckle Correlation Analysis    25
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.

Speckle Correlation Analysis   26
Correlation Based Method

filter           correlator

x

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

Speckle Correlation Analysis        27
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).
Speckle Correlation Analysis   28
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.

Speckle Correlation Analysis   29
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.

Speckle Correlation Analysis                30
Baseband Method
Cn (0)                        *
BBn (m) BBn 1 (m)
mregion of interest

I                                                 acc.
CORDIC
Q

sign control       Q sign bit

I                                                 acc.
Q           CORDIC
acc.

Speckle Correlation Analysis          31
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.

Speckle Correlation Analysis     32
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.

Speckle Correlation Analysis   33
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.

Speckle Correlation Analysis   34
Full 2D Correction
(1,1)
corr       corr             (1,2)          corr   (1,3)

corr

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

corr

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

Speckle Correlation Analysis            35
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)

Speckle Correlation Analysis                  36
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.

Speckle Correlation Analysis   37
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.

Speckle Correlation Analysis   38
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.
Speckle Correlation Analysis   39
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.

Speckle Correlation Analysis   40
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).

Speckle Correlation Analysis   41
Phase Conjugation

phase screen at                               displaced
face of transducer                             phase screen

phase                                      phase

f                                            f

Speckle Correlation Analysis                      42
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.

Speckle Correlation Analysis   43
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.

Speckle Correlation Analysis   44

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