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.
The prolate spheroidal wave functions as invariants of the time reversal operator for an extended scatterer in the Fraunhofer approximation
Rent this article for
View: Figures


Image of FIG. 1.
FIG. 1.

Setup showing the array and the extended scatterer whose length is . The coefficient of the transfer matrix is acquired by transmitting a pulse from element and receiving with element .

Image of FIG. 2.
FIG. 2.

(a) Amplitude of the coefficients of the matrix for an object of size , such as (Fraunhofer approximation). In this case, the amplitude is given by the FT of the object. For a rectangular object, this yields a sinc function. (b) Same plot for a larger object (, while ) The phase term in the Fresnel transform can no longer be neglected. Bottom: Amplitude along the diagonal in the two cases. The sinc pattern (c) and Fresnel transform of a square aperture pattern (d) are noticed.

Image of FIG. 3.
FIG. 3.

Five first spheroidal prolate functions amplitude for (left) and (right).

Image of FIG. 4.
FIG. 4.

Analytical (solid) and simulated (dashed) singular values for an object with . The difference is likely due to the element directivity.

Image of FIG. 5.
FIG. 5.

Left panel: Singular values as a function of the frequency. The bell shape is due to the bandwidth of the transducer. On the right panel, the singular values have been normalized to get rid of the bandwidth effect. As the frequency increases, more points are resolved in the object, and therefore the number of nonzero singular values increases.

Image of FIG. 6.
FIG. 6.

Left: Absolute value of the amplitude of the prolate functions for (first, third, sixth, and tenth functions are shown). Right: Amplitude of the time reversal invariants obtained in simulation for an object such as .

Image of FIG. 7.
FIG. 7.

Analytical (left) and simulated (right) singular values for an object in the so-called resonance region (the object is about the size of the resolution cell) with a parameter in this case. One singular value is dominant but a second singular value has also a relatively large value. This second singular value is important in order to differentiate the object from a point scatterer.

Image of FIG. 8.
FIG. 8.

Absolute value of the amplitude of the first two invariants for . The first invariant is on top. Left: Analytical solutions (the two first prolate functions) Right: Simulation results. The agreement is very good.

Image of FIG. 9.
FIG. 9.

Field resulting from the backpropagation of the first four invariants for the small object (resonance region) The first invariant (top left) yields a focusing comparable to the Green’s function of a point. The other invariants cannot be used for focusing because of the lobes.

Image of FIG. 10.
FIG. 10.

Backpropagation of the first invariants for the object with . Because of the narrow amplitude of the first prolate function, the resolution is poor.

Image of FIG. 11.
FIG. 11.

Selected invariants for a simulated rectangular plate imaged with a 2D array. The object size was roughly five resolution cells in azimuth resolution cells in elevation. The invariants can be described by a Cartesian product of prolate functions. Here the modes are shown: (a) (1,1), (b) (1,2), (c) (2,3), and (d) (3,5).

Image of FIG. 12.
FIG. 12.

Profile of the object at the center frequency. It is the object convolved by the beam pattern (sinc). It is then a low-pass filtered version of the object (propagation act as a low-pass filter). Right: Profile obtained with the minimum variance algorithm. The real object is in dashed line.

Image of FIG. 13.
FIG. 13.

Same as Fig. 12, but for a point scatterer. With the conventional profile, it is very difficult to distinguish between the point and the extended object. With the minimum variance method, the difference is obvious, and it is easier to assess the size of the scatterer.


Article metrics loading...


Full text loading...

This is a required field
Please enter a valid email address
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: The prolate spheroidal wave functions as invariants of the time reversal operator for an extended scatterer in the Fraunhofer approximation