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Chapter 2:
Statistical Description of Coherent Images
2.1 Introduction In this chapter, we use the models of coherent fields scattered by extended objects, which were studied in Chapter 1, for the statistical description of coherent images. According to our definition, an image will be called coherent if each point of it is formed by interfering contributions of all waves coming from the smallest part of the object's surface that is resolvable by the imaging system. In this case, a speckle pattern is formed in the image plane. The conditions for the formation of coherent images will be considered in detail in Sec. 2.5. We consider the general theory of remote object coherent imaging, which takes into account the correlation properties of roughness height distribution on the object's surface as well as the spectrum S r (ω) of the illuminating radiation. The interest in coherent images lies in their wide use in remote sensing, and because the formation of coherent images provides the most complete information about an object in the presence of additional noise. This was first shown in Ref. 11 from the viewpoint of the theory of statistical decisions, by means of processing the fields scattered by rough objects and calculating the probability density function for these fields (see Sec. 1.2). In most works on coherent images, one considers flat rough objects placed on the axis of the optical system (the lens), which forms coherent images in monochromatic light. At each point of the image, the field is assumed to have Gaussian distribution. However, in practice, the objects are usually not flat and their positions with respect to the optical system are arbitrary. In addition, sometimes it is necessary to use images formed in quasi-monochromatic light. For these reasons, one should take into account that the correlation properties of the object surface and the spectral properties of the illuminating source influence the properties of coherent images. In this chapter, this influence is studied in detail, which provides a more accurate and complete description of coherent images of objects with arbitrary shapes. In particular, the contrast and the correlation radius of the speckle patterns are defined more accurately. If the imaging system resolves the object surface roughness, then, as we will show, the field distribution in the coherent image is essentially non-Gaussian. We dealt with such deviations in the previous chapter when we analyzed fluctuation characteristics of the scattered radiation in the case of illumination with a narrow monochromatic beam.
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