Speckle Correlation Analysis by dfhdhdhdhjr


									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.

          transducer                                  volume

                       Speckle Correlation Analysis            2
 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 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


          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
• 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
  received signal changes.

                 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

 radiation pattern                             correlation

                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
• 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 radiation pattern is wider.
• 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



            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
• Techniques using phase information are still
  inherently limited by the nature of Doppler

                Speckle Correlation Analysis   18
Adaptive Imaging Methods

       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

                 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
                    Speckle Correlation Analysis            23
       Near Field Assumption
             geometric delay    beam formation

                        velocity            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.

                    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
• It is applicable to both point and diffuse
                    Speckle Correlation Analysis    25
   Correlation Based Method
              Tn           t
                            i 1

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

              Speckle Correlation Analysis   26
     Correlation Based Method

                  filter           correlator


• 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 

• 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.

    sign control       Q sign bit

I                                                 acc.
Q           CORDIC

                   Speckle Correlation Analysis          31
  One-Dimensional Correction:
• 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
• The correlation path between two array
  elements can be arbitrary.
• The phase error between any two elements
  should be independent of the correlation

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


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


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

                  Speckle Correlation Analysis            35
       Row-Sum 2D Correction
         (1,1)    corr            (1,2)         corr   (1,3)

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

         (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

                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

                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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