Adaptive Imaging Preliminary: Speckle Correlation Analysis

1. Adaptive Imaging Preliminary:Speckle Correlation Analysis

2. sample volume transducer Speckle Formation • Speckle results from coherent interference of un-resolvable objects. It depends on both the frequency and the distance.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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 received signal changes.

8. 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.

9. correlation radiation pattern van Cittert-Zernike Theorem

10. 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.

11. van Cittert-Zernike Theorem • 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.

12. van Cittert-Zernike Theorem correlation distance

13. van Cittert-Zernike Theorem RF Signals Channel Time (Range)

14. van Cittert-Zernike Theorem(Focal length 60mm vs. 90mm)

15. van Cittert-Zernike Theorem(16 Elements vs. 31 Elements)

16. van Cittert-Zernike Theorem(2.5MHz vs. 3.5MHz)

17. van Cittert-Zernike Theorem(with Aberrations)

18. correlation coefficient displacement L/2 Lateral Speckle Correlation

19. 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.

20. Lateral Speckle Correlation 14.4 mm Array

21. Lateral Speckle Correlation

22. Lateral Speckle Correlation

23. Lateral Speckle Correlation

24. Lateral Speckle Correlation: Implications on Spatial Compounding

25. 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.

26. 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.

27. 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.

28. 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.

29. Adaptive Imaging Methods:Correlation-Based Approach

30. body wall viscera point of interest v1 v2 v3 transducer array Sound Velocity Inhomogeneities

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

32. 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.

33. Sound Velocity Inhomogeneities • 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

34. beam formation geometric delay aligned velocity variations correction 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.

35. Correlation-Based Aberration Correction No Focusing

36. Correlation-Based Aberration Correction Transmit Focusing Only

37. Correlation-Based Aberration Correction Transmit and Receive Focusing

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

39. Correlation-Based Aberration Correction Diffuse Scatterers: Before Diffuse Scatterers: After

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

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

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

43. 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).

44. 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.

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

46. CORDIC acc. I I Q Q sign control Q sign bit acc. CORDIC acc. Baseband Method

47. 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.

48. One-Dimensional Correction:Problems

49. One-Dimensional Correction:Problems

50. 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.