A Power Law Reconstruction of Ultrasound Backscatter Images
Abstract
:1. Introduction
2. Theory
- ;
- a power law function of discrete frequency in the expected value of the ensemble average;
- a power law distribution of amplitudes, i.e., Burr distribution;
- a Lomax distribution of the intensity;
- the spatial autocorrelation function of approaches a power law function of lag.
3. Methods
- (i)
- Initial conditions: From the received echo , calculate the analytical envelope and intensity ; optionally check the histogram of within a region of interest in tissue.
- (ii)
- Generate a random uncorrelated noise distribution from Gaussian or uniform zero-mean white noise adjusted locally such that the mean is equal to and where is a scale factor adjusted such that has approximately the same energy as . In two dimensions, is formed as a discrete and whitened version of .
- (iii)
- Filter with a two dimensional, isotropic filter in the frequency domain (or by convolution with the equivalent filter in the sampled time domain) to produce , which then would have an ensemble-averaged power law spectrum approaching .
- (iv)
- The magnitude of serves as the first iteration estimate of the local scattering density and can be used as an initial image. The above steps can be repeated and averaged, to obtain an ensemble average estimate of the inherent scattering distributions, and, optionally, further iterations can be taken as follows.
- (v)
- Calculate the new image lines as , and the new intensity .
- (vi)
- Adjust the local amplitudes of so that the convolution with the pulse yields a better match to the original envelope: .
- (vii)
- Optionally enforce any of the constraints on the intensity, autocorrelation, and power spectra listed in Section 2.
- (viii)
- Repeat steps (v)–(vii) until a convergence criteria is met.
4. Results
5. Discussion and Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
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Parameter | Definition | Additional Information |
---|---|---|
Analytic envelope of the received signal | ||
Intensity | Specified here as the square of the analytic envelope. | |
Power spectrum of | Specified here as , where is the Fourier transform operator. | |
Reflection coefficient | Unknown at high resolution and is the object of our constrained reconstruction. In the theory of multiscale weak scattering, is distributed as a zero-mean random variable with a magnitude described by the power law (Pareto) distribution, so the ultrasound pulse will encounter few strong scatterers and many smaller weaker scatterers according to a power law. | |
Histogram of intensity of the echoes | In theory approaching a Lomax distribution, or concurrently, a Burr distribution for amplitude. | |
Autocorrelation function of the reflections | In theory approaching a power law function. | |
Power spectrum of the scatterers | In theory and from scattering experiments the ensemble average approaches a power law function of frequency. |
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Parker, K.J. A Power Law Reconstruction of Ultrasound Backscatter Images. Acoustics 2024, 6, 782-791. https://doi.org/10.3390/acoustics6030043
Parker KJ. A Power Law Reconstruction of Ultrasound Backscatter Images. Acoustics. 2024; 6(3):782-791. https://doi.org/10.3390/acoustics6030043
Chicago/Turabian StyleParker, Kevin J. 2024. "A Power Law Reconstruction of Ultrasound Backscatter Images" Acoustics 6, no. 3: 782-791. https://doi.org/10.3390/acoustics6030043