Lecture from K-Space Speckle Class

k_space lecture.md

K-Space Lecture

2015/02/05

Ultrasound and laser light

• Both are broadband coherent radiation sources
• The more narrowband a laser, the more its image is like a speckle ultrasound image

Speckle Statistics

• First-order Statistics
  • Mean speckle brightness and distribution
  • Ratio of the mean to standard deviation, mu / sigma, SNR
  • Function of # of scatterers per volume resolution cell
  • Fully-developed speckle?
• Second-order statistics
  • Need two 'darts' connected together
  • Speckle 'size' found by autocorrelation
  • Axial and lateral autocorrelation of speckle pattern
  • Function of mean number of scatterers per resolution cell
• Both types of statistics will asymptote toward an answer with increasing #s scatterers per res. cell
• Do this for both RF and env-detected data

K-space

• Related elements in physical space to regions of support in k-space
• Consider an element doing both TX & RX
• If you space elements by lambda / 2, you can weigh the individual responses to recover a larger array

Power spectral density

• Power spectrum discards the phase of the signal
• Square of the magnitude spectrum
• Can take the autocorrelation of the image speckle pattern

Cross power spectral density

• Product of the two magnitude spectra
• PSFs or speckle patterns
• Cross correlation function and product of spectral density function ore FT pair
• Also disregards phase information
• Normalized cross-correlation between two PSFs

Multiplication in k-space

• Sometimes easier and more intuitive than convolution in aperture space
• Used to find PSDs of different imaging systems
• Analytic expression of k-space overlap

References

• Weiner-Kinschen Theorem
• Wagner '88
• Gahlbach '89?

Equivalency

• Measure 2-D PSFs of a point target and correlate
• Measure a diffuse speckle field's 1-D/2-D RF lines (axial or lateral) or 2-D speckle patterns, and correlate over many trials
• Take normalized product in k-space

Envelope detected data

• Quantity squared of the correlation of the RF data
• Removes the effect of the carrier frequency

Translate Transmitter and Receiver apart laterally

• No change in K-Space representation
• No change in correlation

Change size of aperture(s)

• Start with width D
• Double to 2D
• K-Space limits go to +/-4D/(lambda*z)
• Correlation can be found by finding the product of the K-space regions
• Square that to find the correlation for the envelope detected signal

Move 1 element receiver laterally

• Correlation between elements goes down in a triangle from 1 to 0
• Width of triangle is width of transmit aperture
• VCZ Theorem (developed for stochastic process)
• Works for point target or speckle pattern
• Acoustic reciprocity means TX and RX can be exchanged
• Important for aberration correction, SLSC, etc.

Move both TX & RX over

• Move triangle in k-space
• Normalized product of two shifted triangles
• Autocorrelation of triangle function

Speckle Reduction via Compounding

• Add together detected, incoherent image patterns
  • Spatial compounding with sub-arrays
  • Frequency compounding with sub-band filtering of broadband received signal
• N uncorrelated transmits reduces noise by root N
• Many partially correlated transmits can get you to root 3.2 reduction in speckle
• But decreases reolution, introduces motion artifact

Lateral target translation

• Correlation curve indicates lateral resolution or lateral speckle size autocorrelation
• Reciprocity between moving target or translating active aperture on linear array