Phase-Derivative Imaging III: Theoretical Derivation of First and Second Order Statistics

Abstract

The echo signal obtained from a homogenous and isotropically scattering medium can be described as a Poisson time series which is convolved with the transmission pulse of the transducer. The probability density function (pdf) of this signal approximates to a Gaussian pdf for a narrowband pulse waveform. Methods to derive the phase-derivative (PD) signal from the complex envelope and the preenvelope of the echo signal are described. The first order pdf of the PD asymptotically becomes a Gaussian pdf by smoothing. Since the rectified PD is employed to obtain 2-dimensional grey scale images, the first order pdf as well as the signal-to-noise ratio (SNR) of this signal are also derived. The rectified PD is further smoothed by a cosine time window prior to the imaging. The SNR and the autocorrelation function (in the axial direction) of this latter signal can be derived under the assumption of a Gaussian spectrum of the transmission pulse. These first and second order characteristics of the PD images are calculated for the conditions employed in simulations and experiments reported previously and are quantitatively compared to the values obtained from these.

Keywords

Acoustic scattering medium demodulation FM instantaneous frequency phase-derivative imaging scattering medium

Get full access to this article

View all access options for this article.

References

Ferrari

L.A.

Jones

J.P.

Gonzalez

Behrens

, Acoustical imaging using the phase of echo wave forms, in Acoustical Imaging, Vol. 12, Ash

E.A.

Hill

C.R.

, eds., pp. 635–641 (Plenum Press, New York, 1982).

Ferrari

L.A.

Jones

J.P.

Gonzalez

Behrens

, Ultrasound B-mode images derived from the phase of A-line echosignals, Ultrasonic Imaging 4, 192 (1982) (abstract only).

Ferrari

L.A.

Sankar

P.V.

Fink

Shin

S.B.

Chandler

, Use of signal phase in medical ultrasound, Acta Electronica 26, 111–120 (1984).

Seggie

D.A.

Leeman

Shin

S.B.

Sankar

P.V.

Ferrari

L.A.

, The use of phase information in ultrasound imaging; in Ultrasonic Tissue Characterization, Vol. 4, Thijssen

J.M.

Smith

, eds., pp. 59–66 (Fac. of Medicine Printing Office, Nijmegen, 1985).

Seggie

D.A.

Doherty

G.M.

Leeman

Ferrari

L.A.

, Ultrasonic imaging using the instantaneous frequency of pulse-echo signals, in Acoustical Imaging, Vol. 14, Berkhout

A.F.

, Ridder

van der Wal

L.F.

, eds., pp. 487–496 (Plenum Press, New York, 1985).

Komaili

Sankar

P.V.

Ferrari

L.A.

Leeman

, The instantaneous frequency of a sinewave squelched bandlimited signal, Ultrasonic Imaging 8, 285–295 (1986).

Tsao

Morimura

Itoh

Konishi

, Comparisons of images derived from amplitude, mean frequency and attenuation coefficient of pulse echo signals, Jpn. J. Med. Ultrasonics 14, 393–406 (1987).

Romijn

R.L.

Oosterveld

B.J.

Thijssen

J.M.

, Calculation and stabilization methods of instantaneous frequency echograms, Ultrasonic Imaging 8, 42–43 (1986) (abstract only).

Oosterveld

B.J.

Romijn

R.L.

Thijssen

J.M.

, Diffraction analysis of instantaneous frequency (FM) images, Ultrasonic Imaging 8, 43 (1986) (abstract only).

10.

Romijn

R.L.

Oosterveld

B.J.

Thijssen

J.M.

, Phase-derivative imaging I: Methods and stabilization analysis, Ultrasonic Imaging 9, 147–161 (1987).

11.

Oosterveld

B.J.

Romijn

R.L.

Thijssen

J.M.

, Phase-derivative imaging II: Effects of beam diffraction and of scatterer density, Ultrasonic Imaging 11, 153–174 (1989).

12.

Thijssen

J.M.

Oosterveld

B.J.

, Texture in B-mode echograms: a simulation study of the effects of diffraction and of scatterer density on gray scale statistics, in Acoustical Imaging, Vol. 14, Berkhout

A.F.

Ridder

van der Wal

L. F.

, eds., pp. 481–486 (Plenum Press, New York, 1985).

13.

Oosterveld

B.J.

Thijssen

J.M.

Verhoef

W.A.

, Texture of B-mode echograms: 3-D simulations and experiments of the effects of diffraction and scatterer density, Ultrasonic Imaging 7, 142–160 (1985).

14.

Thijssen

J.M.

Oosterveld

B.J.

, Performance of echographic equipment and potential for tissue characterization, Proc. NATO-ASI Mathematics Computer Sci. Med. Imag., Viergever

M.A.

Todd-Prokopek

, eds., pp. 455–468, (Springer, Berlin, 1987).

15.

Thijssen

J.M.

Oosterveld

B.J.

Romijn

R.L.

, Texture in amplitude modulated (AM) and phase-derivative (PD) echograms, Proc. Int. Symp. Pattern Recognition Acoust. Imag., Vol. 768, Ferrari

L.A.

, ed., pp. 162–167 (SPIE, Bellingham, 1987).

16.

Requicha

A.A.G.

, The zeroes of entire functions: theory and engineering applications, Proc. of the IEEE 68, 308–328 (1974).

17.

Middleton

, On the theory of random noise. Phenomenological models, I, II, J. Appl. Phys. 22, 1143–1152, 1153–1161 (1951).

18.

Middleton

, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

19.

Papoulis

, Probability, Random variables and Stochastic Processes (McGraw-Hill, New York, 1965).

20.

Whalen

A.D.

, Detection of Signals in Noise (Academic Press, New York, 1971).

21.

Rice

S.O.

, Statistical properties of a sine wave plus random noise, Bell Syst. Techn. J. 27, 109–157 (1948).

22.

Angelsen

B.A.J.

, Instantaneous frequency, mean frequency, and variance of mean frequency estimators for ultrasonic blood velocity Doppler signals, IEEE Trans. Biomed. Eng. BME-28, 733–741 (1981).

23.

Broman

, The instantaneous frequency of a Gaussian signal: the one-dimensional density function, IEEE Trans. Acoust. Speech Signal Processing ASSP-29, 108–111 (1981).

24.

Spiegel

M.A.

, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968).

25.

Papoulis

, Signal Analysis, Chap. 9 (McGraw-Hill, New York, 1977).

26.

Wagner

R.F.

Brown

D.G.

, Unified SNR analysis of medical imaging systems, Phys. Med. Biol. 30, 489–518 (1985).

27.

Thijssen

J.M.

Oosterveld

B.J.

Wagner

R.F.

, Gray level transforms and lesion detectability in echographic images, Ultrasonic Imaging 10, 171–175 (1989).

28.

Romijn

R.L.

, On the quantitative analysis of Ultrasound Signals and Application to Intraocular Melanomas (Fac. Medicine Printing Office, Nijmegen, 1990).