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Laboratory of Automated Image Processing Systems
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 Inverse problems of Diagnostics

  X-ray tomography     Ultrasonic tomography     Synthetic aperture  
 

Synthetic aperture

The idea of synthetic aperture is closely linked to computer tomography. The key problem in the use of sensing radars is how to improve their resolution. If we have an ideal parabolic mirror with aperture R that focuses radiation to a point at distance h from the mirror plane the result in the focal plane would be not a point but a diffraction structure with a characteristic size d given by formula: d=2λh/R.

With a 1-m large mirror mounted on a spacecraft at a distance of about 200 km from the Earth surface and operating at optical wavelengths (λ=0.5 microns) the resolution is about 20 cm. The only downside of optical spaceborne imaging systems is that they cannot observe through clouds or during nighttime. Radars operating at a wavelength of about 10 cm can allow observations to be performed at any time of day or night and irrespective of the state of the atmosphere. A drawback of radar systems is their poor resolution — a factor of 200 thousand lower than in the example above and equal to several kilometers, which is absolutely unacceptable for practical applications.

The resolution of radar systems can be improved by using a special technique of observations, which is based on the principles of synthetic aperture. Figure 1 illustrates the idea of synthetic aperture for the case of side-looking synthetic aperture radars (SAR). Sounding electromagnetic signals propagate from the spacecraft, then reflect from the surface, and arrive at the detector of the radar. Let us describe the reflective properties of the Earth surface by function g(x,y) of Cartesian coordinates x,y in the Earth surface region studied. After detection reflected signals u(t) are recorded as input data. If recorded over a certain (several tens of kilometers long) portion of the spacecraft trajectory these signals can be used to solve the inverse problem of the reconstruction of complex-valued function g(x,y) in order to substantially improve the resolution of the radar system. Mathematically the inverse problem of SAR image reconstruction reduces to solving a linear operator equation for an unknown function of two variables g(x,y).

 

(a) (b) (c)
Figure 1. SAR:(a) sounding scheme; (b) input data; (c) reconstructed image

 

Figure 1 shows: (a) the scheme of operation of a SAR; (b) input data u(t) for solving the inverse problem, which have been transmitted from the "Almaz" spacecraft, and (c) the results of the solution of the inverse problem - the reconstructed image of an Earth surface area (outskirts of Moscow and Moscow vicinity). Black color in Figure 1(c) means the absence of reflected signal. In the case of side-looking SARs pulses reflect from smooth water surface like from a mirror and therefore do not arrive at radar detectors. As a result, water surface appears black. The resolution in this example is equal to 11 - 15 m. [1] In the mathematical models used with SARs it is important that the space between the radar and the object examined be uniform and isotropic. Such a model is well suited for electromagnetic Earth sounding from space and acoustic sounding of the ocean floor.

References

1. A. Tikhonov, A. Goncharskii, A. Matviyenko et al. Image-reconstruction problems for synthetic aperture radars. // Doklady Akademii nauk SSSR. Ч 1992. Ч Vol. 322, no. 5. Ч P. 884Ц886