Speckle Correlation Analysis1 Adaptive Imaging Preliminary: Speckle Correlation Analysis

Speckle Correlation Analysis 1

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 volumetransducer

Speckle Second-Order Statistics

• The auto-covariance function of the received phase-sensitive signals (i.e., before 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.

Lateral Speckle Correlation

correlation coefficient

displacementL/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.

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.

• 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 received signal changes.

• 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.

radiation pattern correlation

• 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.

• In the presence of tissue inhomogeneities, the covariance function is narrower since the radiation pattern is wider.

• The decrease in correlation results in lower accuracy in estimation if signals from different channels are used.

distance

correlation

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.

Adaptive Imaging Methods

Sound Velocity Inhomogeneities

transducer arrayv1 v2 v3

point of interest

body wall viscera

Velocity (m/sec)

water 1484blood 1550

myocardium 1550fat 1450

liver 1570kidney 1560

• 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.

• Due to beam formation errors, mainlobe may be wider and sidelobes may be higher.

• Both spatial and contrast resolution are affected.

no errors with errors

Near Field Assumption

• 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.

beam formation

correction

geometric delay

velocity variations

aligned

Correlation Based Method

nn )()(1

)(max nnt

n tCtn

•Time delay (phase) errors are found by finding the peak of the cross correlation function.

• It is applicable to both point and diffuse targets.

iin tT

• The relative time delays between adjacent channels need to be un-wrapped.

• Estimation errors may propagate.

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

filter correlator

• 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

• The formulation is very similar to the correlation technique used in Color Doppler.

nnn dttAAeT

dtBBBBT

*1 )()(

1 )))0(Re(/))0((Im(tan

Baseband Method

)()()0( *1

interest ofregion

mBBmBBC nm

CORDIC

Q sign bitsign control

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.

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

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

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

Row-Sum 2D Correction

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

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

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

corrcorr

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.

• 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.

phase screen

• 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 of transducer

displaced phase screen

phase phase

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.

Speckle Correlation Analysis1 Adaptive Imaging Preliminary: Speckle Correlation Analysis

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