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 .
(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.
Five first spheroidal prolate functions amplitude for (left) and (right).
Analytical (solid) and simulated (dashed) singular values for an object with . The difference is likely due to the element directivity.
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.
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 .
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.
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.
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.
Backpropagation of the first invariants for the object with . Because of the narrow amplitude of the first prolate function, the resolution is poor.
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).
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.
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.
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