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4.1.2. DISCRETIZATION OF RANDOM IMAGES

Lecture



Above, when analyzing questions about discretization and reconstruction, images it was assumed that the function   4.1.2.  DISCRETIZATION OF RANDOM IMAGES describing the original image is deterministic. It was shown that if this image has a spectrum of limited width, then the set of image samples taken at the Nyquist frequency at discrete points is sufficient to reconstruct an exact copy of the original image from samples. A similar conclusion holds in the case of discretization of random two-dimensional fields. Let be   4.1.2.  DISCRETIZATION OF RANDOM IMAGES denotes a continuous two-dimensional stationary random process with known mean values   4.1.2.  DISCRETIZATION OF RANDOM IMAGES and autocorrelation function

  4.1.2.  DISCRETIZATION OF RANDOM IMAGES (4.1.17)

Where   4.1.2.  DISCRETIZATION OF RANDOM IMAGES and   4.1.2.  DISCRETIZATION OF RANDOM IMAGES . Using a set of delta functions, we will discretize this process. As a result, we get

  4.1.2.  DISCRETIZATION OF RANDOM IMAGES (4.1.18)

The autocorrelation function of the discretized process is

  4.1.2.  DISCRETIZATION OF RANDOM IMAGES (4.1.19)

The first factor of the right side of this equality is the autocorrelation function of the stationary source image. It should be noted that the product on the right-hand side of equality (4.1.19) of two discretizing delta functions is a delta function of the form

  4.1.2.  DISCRETIZATION OF RANDOM IMAGES (4.1.20)

Consequently, the discretized random field representing the image is also stationary and has an autocorrelation function.

  4.1.2.  DISCRETIZATION OF RANDOM IMAGES (4.1.21)

Using the two-dimensional Fourier transform of the autocorrelation function (4.1.21), we find the energy spectrum of a discretized random field. According to the spectrum theorem,

  4.1.2.  DISCRETIZATION OF RANDOM IMAGES (4.1.22)

Where   4.1.2.  DISCRETIZATION OF RANDOM IMAGES and   4.1.2.  DISCRETIZATION OF RANDOM IMAGES designate, respectively, the spectral density of the original and discretized images, and   4.1.2.  DISCRETIZATION OF RANDOM IMAGES - the result of the Fourier transform of the sampling delta functions. Further, repeating the steps taken in deriving the relation (4.1.7), we obtain that the energy spectrum of the discretized image can be written in the following form:

  4.1.2.  DISCRETIZATION OF RANDOM IMAGES (4.1.23)

Thus, the energy spectrum of the sampled image is formed by repetition of the energy spectrum of the continuous source image in the spatial frequency domain at intervals that are multiples of the spatial sampling frequency   4.1.2.  DISCRETIZATION OF RANDOM IMAGES . If the width of the energy spectrum of the continuous image is limited so that   4.1.2.  DISCRETIZATION OF RANDOM IMAGES at   4.1.2.  DISCRETIZATION OF RANDOM IMAGES and   4.1.2.  DISCRETIZATION OF RANDOM IMAGES Where   4.1.2.  DISCRETIZATION OF RANDOM IMAGES and   4.1.2.  DISCRETIZATION OF RANDOM IMAGES - boundary frequencies, then in the sum (4.1.23) the individual spectra will not overlap in the case when the sampling intervals satisfy the conditions   4.1.2.  DISCRETIZATION OF RANDOM IMAGES and   4.1.2.  DISCRETIZATION OF RANDOM IMAGES . Continuous random image   4.1.2.  DISCRETIZATION OF RANDOM IMAGES can be recovered from samples of the original random image using interpolation using the formula

  4.1.2.  DISCRETIZATION OF RANDOM IMAGES (4.1.24)

Where   4.1.2.  DISCRETIZATION OF RANDOM IMAGES - deterministic interpolation function. It can be achieved that the reconstructed field and the original image are equivalent in the mean square sense [5, p. 284], i.e.

  4.1.2.  DISCRETIZATION OF RANDOM IMAGES (4.1.25)

For this, it suffices that the Nyquist criterion is satisfied and a sufficiently “good” function was chosen as an interpolation function, such as, for example, the Bessel or   4.1.2.  DISCRETIZATION OF RANDOM IMAGES , defined by the formulas (4.1.16) and (4.1.14).

  4.1.2.  DISCRETIZATION OF RANDOM IMAGES

Fig. 4.1.4. Spectra of a discretized image containing signal and interference.

a - signal: b - interference; в - sampled signal; g - discrete interference.

The results can be directly applied to the practical problem of discretization of the sum of the deterministic image and additive noise, the model of which is a random field. In fig. 4.1.4 shows an example of the energy spectrum of a discretized noisy image. This example indicates the possibility of serious difficulties. The noise spectrum may be wider than the spectrum of the useful image, and if the noise is sampled with insufficient frequency, then the "tails" of the noise spectrum will fall into the passband of the recovery filter. This will increase the distortion caused by noise. To eliminate this difficulty, before sampling, you should, of course, filter the noisy image in order to narrow the noise spectrum.


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Digital image processing

Terms: Digital image processing