MATHEMATICAL MODEL OF NOISE IN X-RAY AND PHOTOGRAPHIC FILMS

L. M. Ananyev, A. S. Moskaleva, V. A. Schlotgauer

When a uniformly exposed and developed sample of X-ray film is analyzed with a photoelectric microphotometer, the nonuniformity of blackening is recorded as fluctuations of the electrical signal obtained as a result of conversion of the luminous flux that has passed through a layer of photographic or X-ray film. The fluctuations of blackening, transformed into oscillations of the electrical signal, have a definite frequency spectrum, expressing the distribution of the power of the random oscillation over the frequency range. The power spectrum is one of the most important characteristics of the nonuniformity of blackening.

For a theoretical analysis of the spectral characteristics of granularity and for clarifying the qualitative dependence of the character and magnitude of the spectrum on the physical parameters of the photographic layer, it is proposed to consider the simplest one-dimensional model of the resulting blackening (Fig. 1). To describe a random function in the general case it is necessary to specify its multidimensional distribution, but in view of the isotropy of the nonuniformity of blackening one may restrict oneself to considering a random function of one argument—the coordinate \(x\) of the layer.

Models of granular structure are known in the literature [1, 2], used for calculating the correlation and spectral characteristics of the nonuniformity of blackening. However, one of these models is based on highly simplified notions of graininess [1], while the other, although more consistent with reality, is extremely complicated and in calculations entails cumbersome mathematical derivations. It should be noted that photographic blackening, when subjected to modeling, is in a very complex dependence on a multitude of different factors that are practically impossible to take into account; therefore the desire to increase the accuracy of calculation of the model is not always justified.

The proposed model of granular structure is simple, but at the same time reflects the main features of the original sufficiently well. The model under consideration is obtained under the assumptions that: a) the developed grains located in separate elementary layers, into which the blackening is conventionally divided solely for the purpose of showing the mechanism by which the resulting blackening is obtained (Fig. 1), are absolutely opaque; b) differences in the sizes of the developed grains \(l\) within a layer are insignificant, and for each given emulsion they may, for simplicity, be taken as identical.

Figure 1 presents a one-dimensional model of blackening in the form of a sum of elementary layers which, adjoining one another closely, create the resultant blackening. The developed grains—\(l\) in the layer—are arranged along the coordinate \(x\) at random, but in the aggregate all layers, superposed on one another, form, as it were, projections of groups

Fig. 1. One-dimensional model of blackening: a) elementary layers; b) resultant blackening; c) random sequence of pulses when the blackening is scanned with a point aperture

of grains \(l_i\) and gaps between grains \(l_i\) of the total blackening. Such a picture is observed in microphotometry of the photosensitive layer of an X-ray or photographic film. The total blackening will be inhomogeneous, and when it is considered over elementary sections along the coordinate \(x\), the picture will change, since the magnitude of the blackening will fluctuate around some mean value—\(\overline{T}\). Suppose that such blackening is analyzed by the point aperture of a photoelectric microphotometer. In this case a random sequence of rectangular pulses will be obtained (Fig. 1c), characterized by a constant amplitude \(A_0\), durations \(\tau_i\) corresponding to the sizes of individual groups of grains—\(l_i\), and intervals between pulses—\(\Delta_i\), corresponding to the sizes of the gaps between groups of grains \(l_i\).

Such a pulse sequence has a spectral density described by the expression [3]:

\[ S(\omega)= \frac{2A_0^2}{\omega^2(\langle \tau\rangle+\langle \Delta\rangle)} \, Re \, \frac{[1-\Theta_1(\omega)]\cdot[1-\Theta_2(\omega)]} {1-\Theta_1(\omega)\cdot\Theta_2(\omega)} . \tag{1} \]

Here \(A_0=\mathrm{const}\), the amplitude of the independent rectangular pulses of the sequence. \(\langle \tau\rangle\) and \(\langle \Delta\rangle\) are the mathematical expectations of the random variables \(\tau_i\) and \(\Delta_i\). \(\omega\) is the angular frequency. \(\Theta_1(\omega)\) and \(\Theta_2(\omega)\) are the characteristic functions, respectively, of the pulse durations and of the intervals between them, which are defined as the mathematical expectations of the corresponding random variables \(e^{j\omega\tau}\) and \(e^{j\omega\Delta}\), i.e.

\[ \Theta_1(\omega)=M\left(l^{j\omega\tau}\right)=\int_{-\infty}^{\infty} e^{j\omega\tau}\cdot W(\tau)\cdot d\tau; \tag{2} \]

\[ \Theta_2(\omega)=M\left(l^{j\omega\Delta}\right)=\int_{-\infty}^{\infty} e^{j\omega\Delta}\cdot W(\Delta)\cdot d\Delta, \]

where \(W(\tau)\) and \(W(\Delta)\) are the probability densities of the random variables \(\tau\) and \(\Delta\).

In the resulting blackening, just as in the grain model under consideration, the grain sizes and the intervals between them will be distributed according to random laws; accordingly, the pulse durations \(\tau_i\) and the intervals between them \(\Delta_i\) will be distributed according to the same laws. Suppose that the grain sizes and the intervals between them are distributed according to an exponential law. This assumption is not of fundamental importance in the present case, since our task includes only obtaining qualitative characteristics of graininess (granularity). For an exponential distribution law, the probability densities and the corresponding characteristic functions are expressed as:

\[ W_1(\tau)=\alpha\cdot e^{-\alpha\tau},\quad W_2(\Delta)=\beta\cdot e^{-\beta\Delta}. \tag{3} \]

\[ \Theta_1(\omega)=\frac{\alpha}{\alpha-j\omega},\quad \Theta_2(\omega)=\frac{\beta}{\beta-j\omega}, \tag{4} \]

where \(\alpha\) and \(\beta\) are the parameters of the exponential distribution law for the corresponding random variables \(\tau_i\) and \(\Delta_i\). Substituting into equation (1) the values of \(\Theta_1(\omega)\) and \(\Theta_2(\omega)\), as well as the values of \(m_\tau\) and \(m_\Delta\), which for the exponential law are respectively equal to \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\), we obtain an expression for the energy spectrum of the process under consideration:

\[ S(\omega)=\frac{2\alpha\cdot 3A_0^2}{\alpha+\beta}\cdot \frac{1}{\omega^2+(\alpha+\beta)^2}. \tag{5} \]

To find the graininess spectrum, according to the proposed model, it is necessary to introduce into the expression for the energy spectrum (5) the principal blackening parameters: \(\overline{T}\), \(l\), \(l_z\), \(l_0\). The mean blackening \(\overline{T}\) on the segment \(x\) may be defined as the ratio of the length of all darkened segments (groups of grains) to the entire length of the segment over which scanning has been performed with a point aperture,

\[ \overline{T}=\frac{\sum l_{i_z}}{\sum l_{i_z}+\sum l_{i_0}}, \tag{6} \]

where \(l_{i_z}\) is the extent of the \(i\)-th group of grains,
\(l_{i_0}\) is the extent of the \(i\)-th window.

On the scanning segment \(x\), let us define the mean grain extent \(\overline{l}_z\) and the mean window extent \(\overline{l}_0\).

\[ \overline{l}_z=\frac{\sum_{i=1}^{m} l_{i_z}}{m},\quad \overline{l}_0=\frac{\sum_{i=1}^{n} l_{i_0}}{n}, \tag{7} \]

where \(m\) and \(n\) are the number of blackened grain groups and the number of intervals (windows), respectively. For a large number of grains, it may be taken that \(m=n\). Then the mean blackening may be expressed through the mean quantities \(\overline{l}_z\) and \(\overline{l}_0\).

\[ \overline{T}=\frac{\overline{l}_{3}}{\overline{l}_{3}+\overline{l}_{0}}. \tag{8} \]

From expression (8) one can find the value \(\overline{l}_{0}\)

\[ \overline{l}_{0}=\overline{l}_{3}\cdot \frac{1-\overline{T}}{\overline{T}}. \tag{9} \]

The parameters of the pulse sequence are related to the parameters of the resulting blackening as follows:

\[ \tau_{\mathrm{avg}}=\frac{\overline{l}_{3}}{v}, \qquad \Delta_{\mathrm{avg}}=\frac{\overline{l}_{0}}{v}, \tag{10} \]

where \(v\) is the scanning speed in mm/sec, and \(\tau_{\mathrm{avg}}\) and \(\Delta_{\mathrm{avg}}\) are the mean values of the pulse duration and of the interval between pulses, respectively.

From probability theory it is known that the mathematical expectation of a random variable under an exponential law of its distribution is expressed as

\[ m_{\tau}=\int_{0}^{\infty}\tau\cdot \alpha\cdot e^{-\alpha \tau}\cdot d\tau=\frac{1}{\alpha} \]

\[ m_{\Delta}=\int_{0}^{\infty}\Delta\cdot \beta e^{-\beta \Delta}\cdot d\Delta=\frac{1}{\beta}. \tag{11} \]

Taking into account expressions (8) and (9), the values of \(\alpha\) and \(\beta\) can be written as:

\[ \alpha=\frac{1}{m_{\tau}}=\frac{1}{\tau_{\mathrm{avg}}}=\frac{v}{\overline{l}_{3}}, \tag{12} \]

\[ \beta=\frac{1}{m_{\Delta}}=\frac{1}{\Delta_{\mathrm{avg}}} =\frac{v\cdot \overline{T}}{(1-\overline{T})\overline{l}_{3}}. \tag{13} \]

After substituting into expression (5) the values of \(\alpha\) and \(\beta\), expressed through the parameters of the resulting blackening of the model, we obtain the expression for the ideal granularity spectrum:

\[ G(n)=\frac{A_{0}^{2}}{\pi\cdot \omega_{0}}\cdot \frac{\overline{T}}{n^{2}+\dfrac{1}{(1-\overline{T})^{2}4\pi^{2}}}, \tag{14} \]

where \(\omega_{0}=2\pi \dfrac{v}{\overline{l}_{3}}\) is the natural frequency of the sample, associated with the parameters of the blackening; \(n^{2}=\dfrac{\omega^{2}}{\omega_{0}^{2}}\) is the number of the harmonic of the spectrum; \(\omega\) is the current frequency constituting the granularity spectrum.

The mean value of the square of the noise amplitude, or the mean power, is equal to the integral of the spectral density:

\[ \sigma^{2}=\frac{1}{\pi}\int_{0}^{\infty}G(n)\cdot dn =A_{0}^{2}\overline{T}(1-\overline{T}) \tag{15} \]

Figure 2 shows the dependence of the root-mean-square deviation of the noise on the mean level of blackening \(\overline{T}\). This characteristic agrees well with the data given in the literature [4], which indicates the validity of the adopted model of photographic blackening. On the basis of formula 14, families of characteristics of the granularity spectrum (Figs. 3, 4) were constructed as functions of \(\overline{T}\) and \(n\). Analysis of these characteristics makes it possible to draw the following conclusions:

Fig. 2. Dependence of blackening noise on its mean level

1. The proposed model made it possible to theoretically determine the nature of the granularity spectra as noise in photographic and X-ray images, and may serve as a basis for further analysis of this category of noise in order to clarify the possibility of reducing its influence when viewing an X-ray or photographic image.

Fig. 3. Dependence of the granularity spectrum at different blackening levels

Fig. 4. Dependence of the components of the spectrum on blackening

1. The composition of the granularity spectrum changes in the transition from one value of blackening \(\overline{T}\) to another; in this case, the low-frequency components of the spectrum everywhere have the greatest power (Fig. 3).

2. The magnitude of the spectral components changes its maximum value depending on the magnitude of the mean blackening level; moreover, as the blackening increases, the maxima shift to the right (Fig. 4).

3. The granularity characteristics are in qualitatively good agreement with the experimental data obtained by us for X-ray film of type RT-1 [5], as well as with data given in the literature [6].

REFERENCES

1. P. Fellgett, J. Opt. Soc. Amer. 1953, 43, 271.
2. E. L. O’Neill, J. Opt. Soc. Amer. 1958, 48, 945.
3. V. I. Tikhonov. Statistical Radio Engineering. “Sovetskoe radio,” 1966.
4. R. B. Gurevich, I. I. Breido, G. A. Gavrilov. Advances in Scientific Photography, 10, 171, 1964.
5. A. S. Moskaleva. Izvestiya VUZov, 1970, vol. 225.
6. E. J. B. Wall and B. G. Stell. J. Photographic Science, 1964, 12, 34—46.