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Adaptive Imaging Preliminary: Speckle Correlation Analysis

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Adaptive Imaging Preliminary:Speckle Correlation Analysis

sample volume

transducer

- Speckle results from coherent interference of un-resolvable objects. It depends on both the frequency and the distance.

- 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.

- 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.

- 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.

- 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.

correlation

radiation pattern

- 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.

correlation

distance

RF Signals

Channel

Time (Range)

correlation coefficient

displacement

L/2

- 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.

14.4 mm Array

- 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.

- 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.

- 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.

- 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:Correlation-Based Approach

body wall

viscera

point of interest

v1 v2 v3

transducer array

Velocity (m/sec)

water

1484

blood

1550

myocardium

1550

fat

1450

liver

1570

kidney

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

beam formation

geometric delay

aligned

velocity variations

correction

- 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.

No Focusing

Transmit Focusing Only

Transmit and Receive Focusing

Wire: Before Correction

Wire: After Correction

Diffuse Scatterers: Before

Diffuse Scatterers: After

- Time delay (phase) errors are found by finding the peak of the cross correlation function.
- It is applicable to both point and diffuse targets.

- 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

Dx

- 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).

- 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.

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

CORDIC

acc.

I

I

Q

Q

sign control

Q sign bit

acc.

CORDIC

acc.

- 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.

- 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.

- 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.

(1,1)

(3,1)

(2,1)

corr

corr

corr

(1,2)

(2,2)

(3,2)

corr

corr

corr

(2,3)

(3,3)

(1,3)

corr

corr

corr

corr

corr

corr

(1,1)

(3,1)

(2,1)

corr

corr

corr

(1,2)

(2,2)

(3,2)

corr

corr

corr

(1,3)

(3,3)

(2,3)

corr

corr

- 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.

- 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.

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.

- 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.

- 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

phase

f

f

phase screen at face of transducer

displaced phase screen

No aberration

At 0 mm

At 20 mm

At 40 mm

At 60 mm

- 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.

- 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

Distribution of time delay corrections

Before Correction

After Correction

Before Correction

After Correction

Channel Data

Complex Scattering Structures

1D and 2D Least Squares Estimation

Before Correction

After Correction

Before Correction

After Correction

Original

1 iteration

4 iterations

Original

Receive Only

(also with a 60% brightness improvement)

- 1D
- 1D with correction
- 1.75D
- 1.75D with correction

Adaptive Imaging Methods:Aperture Domain ProcessingParallel Adaptive Receive Compensation Algorithm

- Fixed direction transmit, all direction receive

With Aberrations

No Aberrations

With Aberrations

No Aberrations

With Correction

No Correction

Simplifications:1. DFT vs. Single Transmit Imaging2. Weighting vs. Complex Computations

Single Transmit Imaging

DFT

Coherent

Speckle

Incoherent

Aberrated

*P.-C. Li and M.-L. Li, “Adaptive Imaging Using the Generalized Coherence Factor”,

IEEE UFFC, Feb., 2003.

Coherent sum (DC)

The larger, the better?

Total energy (times N)

N: the number of array channels used in beam sum

C(i,t) : the received signal of channel i

- A quantitative measure of coherence of the received array signals.

Object of Interest

Enhance

Unwanted Sidelobes

Suppress

Optimize the receive aperture size

Classify “object types”

1X

0

2X

Azimuth

28.6 mm

Range

Original

96.2 mm

–40

40

Adaptive Receive Aperture

Dynamic range: 60 dB