STUDY OF NOISES IN X-RAY FILM

B. N. Epifantsev, L. V. Startseva, V. V. Tarasov

(Presented by the scientific seminar of the Research Institute of Electronic Introscopy)

A necessary element of a system for the automatic interpretation of radiographic images is an information-filtering device whose task is to extract the useful signal from a mixture of the defect signal, accompanying signals, and noise both of the information carrier itself and of the recording apparatus [1].

To construct optimal filters for signal extraction, it is necessary to know the granularity characteristics of X-ray film, which constitute the noise of the information carrier.

The average powers of granularity of domestic X-ray films reported in [2] amount to 25–45% of the mean blackening density of the negative at an aperture of \(0.1 \times 0.1\ \mathrm{mm}^2\). This result is an order of magnitude greater than analogous estimates for X-ray films of foreign firms [3, 4] and requires verification.

In studying the frequency characteristics of X-ray film, some researchers [3, 4] observed a substantial deviation from uniformity of its granularity spectrum, in contrast to black-and-white cine films, whose granularity spectrum is practically uniform. However, with respect to the frequency at which the spectral falloff begins, there are considerable discrepancies among the authors. Thus, Kholmshou [4] obtained a sharp falloff of the granularity spectrum in the range of spatial frequencies from 20 to 40 \(\mathrm{mm}^{-1}\), whereas the results of Zweig [3] permit the spectrum to be regarded as uniform up to 100–200 \(\mathrm{mm}^{-1}\) (depending on the blackening density of the negative). Soviet researchers who studied the granularity of films irradiated by electrons of medium energies obtained a uniform granularity spectrum up to 60 \(\mathrm{mm}^{-1}\) [5].

Thus, the literature data on the noises of X-ray films are contradictory and insufficient, which gave rise to the need for the present study.

The objective of the present study is to obtain, on the basis of measurements of the noises of RT-1 film, the granularity characteristics of radiographic film for the development of filters for extracting the contours of defects.

Uniformly exposed samples of coarse-grained X-ray film RT-1 were scanned on an MF-4 microphotometer with apertures most widely used in the automatic interpretation of images: \(0.05 \times 0.05\ \mathrm{mm}^2\); \(0.1 \times 0.1\ \mathrm{mm}^2\); \(0.2 \times 0.2\ \mathrm{mm}^2\). The microphotometry data, in units of optical density, were output to a PL-80 tape punch and processed on an “M-220” electronic digital computer.

As granularity characteristics, the autocorrelation function \(R(k)\), the power spectrum \(S(\omega_r)\), the root-mean-square deviation \(\sigma_D\), and the grain-size distribution were chosen. In the calculations the following algorithms were used:

\[ R(k,n)=\frac{1}{n-m}\sum_{i=1}^{n-m}F(i\Delta x)\cdot F[(i+k)\Delta x]; \tag{1} \]

\[ S(\omega_r,n)=\Delta x\left[R(0)+2\sum_{k=1}^{m-1}R_k\cos\frac{kr\pi}{m}+R_m\cos r\pi\right], \tag{2} \]

\[ \sigma_D^{2}=\sum_{r=0}^{n}S(\omega_r,n), \]

where \(F(x)\) is a microphotometric section of the image;

\[ n=\frac{X}{\Delta X} \]

is the total number of readings on the interval \([0,X]\);

\[ \omega_r=\frac{r\pi}{(m-1)\Delta X} \]

is the spatial frequency.

Measurement of the simplest characteristic of the film—the root-mean-square deviation \(\sigma_D\)—revealed the dependence of \(\sigma_D\) on the mean optical density level of the negative \(D\); the obtained values \(\sigma_D=f(D)\) with 99.7% confidence limits are plotted in Fig. 1. As \(D\) increases, the mean power of the blackening fluctuations increases noticeably; the dependence is mainly linear in character, which agrees with the results obtained by Zweig [3]. The values of the root-mean-square deviation of PT-1 film coincide with the values of the mean power of fluctuations, described in the literature [3, 4], for coarse-grained films produced by foreign firms.

Fig. 1. Smoothed dependence \(\sigma_D=f(D)\); — — — confidence limits for confidence probability 0.997; —○— \(a=0.05\) mm; —△— \(a=0.1\) mm; —□— \(a=0.2\) mm.

Analysis of the experimental dependence of \(\sigma_D\) on the scanning aperture showed that, for a fixed blackening density \(D\), Selwyn’s law is satisfied: \(\sigma_D\cdot a=G=\mathrm{const}\).

To investigate the frequency properties of granularity, using formulas (1, 2), sample autocorrelation functions and sample spectral estimates of the power of fluctuations in the density of blackening of specimens of PT-1 film were calculated (Fig. 2). The autocorrelation of the process under study

Fig. 2.

drops sharply already at the first step (equal to 0.05 mm) and subsequently remains low, which is characteristic of a purely random series or white noise.

When the number of terms of a purely random series \(N\) is sufficiently large, it is permissible to assume that the sample correlation function \(r_{xx}(k)\) is normally distributed with zero mean and variance \(1/N\). According to the criterion proposed in [6], if no more than 5% of the individual correlations of the sample correlation function of the series being tested fall outside the 95% confidence limits \(r_{xx}(k) \pm \frac{1}{\sqrt{N}}\cdot 1.96\), then the series being tested may be considered white noise. The result of testing by this criterion is consistent with the hypothesis that the granularity in the frequency range under consideration is white noise.

As is known, the spectrum of white noise is uniform. However, the experimentally obtained spectra have a clearly pronounced decrease at high frequencies, and the region of the decrease depends on the scanning aperture (Fig. 2). The observed nonuniformity of the spectrum can be explained by smoothing of the higher harmonics when scanning with apertures of finite smallness. Such processing of noise in the one-dimensional case is analogous to the operation of a linear system that computes the mean value of the input process over a finite interval \(\alpha\).

If a purely random process with variance \(\sigma_x^2\) is fed to the input of the system, then at the output a linear process is obtained with spectral density [9], where \(H(\omega)\) is the frequency response of the system.

If one uses the frequency response of the transform for a square aperture, given in [7],

\[ |H(\omega)|^2 = \left( \frac{\sin \dfrac{\alpha\omega}{2}} {\dfrac{\alpha\omega}{2}} \right)^2, \]

then the spectral density of the system’s output process can be written in the form

\[ G(\omega)=\sigma_x^2\left(\frac{\sin \dfrac{a\omega}{2}}{\dfrac{a\omega}{2}}\right)^2, \]

the variance of the output process is \(\sigma_Z^2=\dfrac{\sigma_x^2}{a}\).

The formula for the normalized spectrum will therefore have the form:

\[ g(\omega)=a\left(\frac{\sin \dfrac{a\omega}{2}}{\dfrac{a\omega}{2}}\right)^2 . \]

The theoretically obtained curve \(g(\omega)\) of the spectral density, characterizing the result of processing white noise by a linear system, gives good agreement with the experimental smoothed spectral estimate of granularity \(s(\omega)=\dfrac{S(\omega)}{\sigma_D^2}\) (Fig. 2).

Fig. 3. a — distribution of intervals between grains at \(D=0.4\); б — distribution of grain sizes at \(D=0.4\); в — distribution of intervals between grains at \(D=1.15\); г — distribution of grain sizes at \(D=1.15\); д — distribution of intervals between grains at \(D=1.85\); е — distribution of grain sizes at \(D=1.85\).
Designations: — \(a=0.05\) mm; - - - \(a=0.1\) mm; —— \(a=0.15\) mm.

In addition, probability distributions were constructed for the lengths of positive $\tau_1$ and negative $\tau_2$ segments of a centered microphotometric trace (Fig. 3), reflecting the distribution by grain size of the X-ray film and of the gaps between them.

The histograms of the distributions obtained when scanning negatives of different optical density with the same aperture contain no noticeable differences. With an aperture of $0.05 \times 0.05\ \mathrm{mm}^2$ (quantization step $0.05\ \mathrm{mm}$), the greatest relative weight is possessed by unipolar segments $1 \div 2$ counts long; increasing the scanning aperture while maintaining the same discretization step leads to an increase in the number of unipolar segments of greater length. The form of the histogram when the aperture is equal to the step corresponds fairly accurately to the distribution of the lengths of unipolar segments of a purely random series (shown by the dotted line in Fig. 3). If the aperture is larger than the step, correlation appears between counts, as a result of which the distribution is deformed.

Conclusions

1. When uniformly exposed X-ray-film samples are scanned with apertures of $0.05 \div 0.2\ \mathrm{mm}$, the power spectrum of the granularity may be regarded as uniform. Consequently, for designing optimal filters it is sufficient to establish the form of the dependence $\sigma_D = f(D)$.

2. The dependence $\sigma_D = f(D)$ at densities $0.4—1.85$ is linear in character, with a tendency to increase as the optical density of the negative increases.

3. Since increasing the aperture of the reading device leads to a proportional decrease in noise power, the most effective separation of signal and noise by amplitude is achieved with an aperture equal to the minimum size of the defect being detected.

REFERENCES

1. L. I. Kosarev et al. The state and prospects of automation of certain radiographic processes. “Defektoskopiya,” 1972, No. 1, p. 117.

2. V. P. Abramov, V. I. Gorbunov. Noises caused by the graininess of X-ray film and their influence on the choice of the optimal size of the reading aperture of a scanning device. “Defektoskopiya,” 1972, No. 1, p. 107.

3. H. J. Zweig, “J. Opt. Soc. Amer,” 1959, 49, 238.

4. R. Halmshaw. “J. Photogr. Sci,” 1971, 19, 167.

5. A. A. Aleksandrov et al. Photographic recording of electron-microscopic images. “Journal of Scientific and Applied Photography and Cinematography,” 1972, No. 4, p. 289.

6. G. Jenkins, D. Watts. Spectral Analysis and Its Applications. Vol. 1. Moscow, “Mir,” 1972, pp. 229—230.

7. I. I. Tsukkerman. Transformations of electron images. Leningrad, “Energiya,” 1972, p. 131.