banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
Time reversal operator decomposition with focused transmission and robustness to speckle noise: Application to microcalcification detection
Rent this article for


Image of FIG. 1.
FIG. 1.

(a) DORT: Each element of the transmit array is fired individually: Insonification of the medium by element m of the transmit array and reception of the corresponding echo by element n of the receive array gives . (b) FDORT: Groups of elements of the transmit array are used to transmit a focused pulse; reception of the echo by element of the receive array gives .

Image of FIG. 2.
FIG. 2.

The time reversal process can be seen as a full cycle between 3 actors: The array, the array and the scatterers. Propagation between the actors is described by and . Reflection from the scatterers is equivalent to a multiplication by , and the backpropagation by the arrays is equivalent to a phase conjugation, included in the hermitian transpose . describes the one-way propagation between and , represented by the solid line. The time reversal operator can be expressed mathematically from the point of view of any of these 3 actors. Then one has to start from the desired actor and make a full cycle.

Image of FIG. 3.
FIG. 3.

(a) Magnitude of the field resulting from focusing on a scatterer (“the target”) located at depth . The shadow of the transducer, in gray, is the coupling area. Any scatterer located in that area is coupled with the first one: The eigenvectors of the time reversal operator are then linear combination of the scatterers’ Green functions. If the second scatterer is located outside of the shadow in the black area, the scatterers are well resolved and the Green functions are separated, each corresponding to one eigenvector. (b) Map of scatterers coupled to a given target (, ) in a medical image. The backscatter signal has been projected onto the target Green function before the beamforming and then only the scatterers whose Green functions are not orthogonal to the target’s are imaged with an intensity proportional to the coupling.

Image of FIG. 4.
FIG. 4.

Coupling condition with focused transmit. (a) The distance between the scatterers is greater than the width of the PSF; each beam insonifies only one scatterer, and they are not coupled. (b) the scatterers have a separation smaller than the point spread function; consequently, a beam may insonify both scatterers. In this case they are coupled.

Image of FIG. 5.
FIG. 5.

Results of FDORT performed on simulated data with 2 scatterers and an aberrator. (a) Eigenvalues as a function of frequency: Two eigenvalues have a significant magnitude, corresponding to the two scatterers. (b) Phase of the corresponding eigenvectors at the central frequency. It is proportional to the focal delay law to focus on each of the targets. In particular, it contains the delay law introduced by the aberration. (c) Intensity of the field transmitted in the medium when a simple geometric delay law is used to focus on one of the targets. The aberration results in poor focusing. (d) Intensity of the transmitted field when one of the eigenvectors is used to focus on the target.

Image of FIG. 6.
FIG. 6.

Eigenvalues of DORT (thin line) and FDORT (thick line) vs distance for two scatterers located at the same depth (a) in the focal plane of the array , (b) in the focal plane but in presence of aberration, and (c) out of the focal plane, at depth . When the scatterers are too close, coupling occur and results in a separation of the eigenvalues. The importance of coupling depends on how well the scatterers are resolved by both transmit and receive array. It occurs over a larger range with FDORT because only half the aperture is used in transmission. FDORT’s ability to separate two targets response is not degraded by aberration or for scatterers out of the focal plane. DORT and FDORT eigenvalues have been normalized to appear on the same scale.

Image of FIG. 7.
FIG. 7.

Tissue-mimicking phantom used for the experiments. The zone of coupling for one scatterer is indicated by the white conical shape. Scatterers inside this area are coupled with the scatterer. The box drawn with dashed lines indicates the area used in Sec. III, obtained by time gating the signals. This reduces the influence of the coupled scatterers. The box in solid lines indicates the area used for Local FDORT in Sec. IV.

Image of FIG. 8.
FIG. 8.

The FDORT method has been performed on a tissue-mimicking phantom with wires. Top: Eigenvalues vs frequency (a) and intensity (b) of the field resulting from the backpropagation of the second eigenvector, when FDORT is restricted to a slice around depth where 9 wires are located, using time gating. Nine significant eigenvalues are observed and the eigenvectors focus accurately on the wires. Bottom: The same experiment but without time gating. The focusing property of the eigenvectors is dramatically reduced, because of the coupling with the speckle.

Image of FIG. 9.
FIG. 9.

Shows how FDORT can be restricted to a small area (the black window depicted): Windowing in depth is achieved by time gating of the signals and windowing in azimuth is achieved by the number of consecutive lines used: On the picture, three lines are used. is the distance between two consecutive lines (the beam spacing).

Image of FIG. 10.
FIG. 10.

(a) Eigenvalues in pure speckle in decreasing order: The spectrum is continuous, thus the first two eigenvalues have similar magnitude. FDORT was performed locally in a deep and wide area at several azimuth along the white line depicted on the breast ultrasound image. (b) A scatterer identified as a microcalcification is shown by the white arrow. (c) The first two eigenvalues are plotted vs the azimuth. The first eigenvalue is proportional to the echogenicity, but considering the second eigenvalue adds additional information. (d) Ratio of the first two eigenvalues vs the azimuth: The position of the microcalcification is indicated by a high ratio.

Image of FIG. 11.
FIG. 11.

Focusing achieved in breast clinical data using a microcalcification as a point scatterer. The first eigenvector has been numerically backpropagated. Left: Intensity of the resulting field. Right: Intensity vs azimuth at the depth of the microcalcification.


Generic image for table

Comparison of DORT and FDORT.


Article metrics loading...


Full text loading...

This is a required field
Please enter a valid email address
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Time reversal operator decomposition with focused transmission and robustness to speckle noise: Application to microcalcification detection