THEORETICAL STUDIES OF FLUCTUATION PHOTOGRAPHIC NOISE TRANSFORMED BY A MEASUREMENT APERTURE

L. M. Anan’ev, A. S. Moskaleva

Photographic noise, like any other fluctuation noise, has a negative effect in the recognition of fine image details. At present, the study of this type of noise and the reduction of its influence is an important problem. The characteristics of photographic noise in the form of the root-mean-square value $\sigma_0$ and the spectral density $G_0(\nu)$, obtained by us earlier as a result of a theoretical study of the model of blackening nonuniformity, are idealized, since they were derived under the assumption that the image is analyzed by a point aperture [1]. In reality, the amount of blackening measured at any given moment will be averaged over the area of the analyzing aperture, and the quantities $\sigma_0$ and $G_0(\nu)$ will be transformed in accordance with its characteristic.

In general form, the two-dimensional energy spectrum transformed by the measurement aperture is determined by the expression:

\[ G(\nu_x,\nu_y)=G_0(\nu)\cdot |U(\nu_x,\nu_y)|^2, \tag{1} \]

where

$G_0(\nu)$ is the noise spectrum for a point aperture [1];

$|U(\nu_x,\nu_y)|^2$ is the squared modulus of the transfer characteristic of the measurement aperture;

$\nu_x,\nu_y$ are the spatial frequencies along the coordinates $x$ and $y$, the range of which is determined by the aperture dimensions.

For a measuring device with an analyzing aperture in the form of a rectangular slit, within which the receiver sensitivity may be considered constant, the frequency characteristic is expressed by the product:

\[ U(\nu_x,\nu_y)=\operatorname{sinc}\pi\nu_x h\cdot \operatorname{sinc}\pi\nu_y l, \tag{2} \]

where

$h$ and $l$ are the thickness and length of the slit, respectively, in the directions $x$ and $y$.

Since photographic blackening is an isotropic medium and the noise spectrum will be determined only by the aperture size in the scanning direction $x$ of the specimen, a long narrow slit elongated along the $y$ axis will give us an exhaustive characteristic of the spectrum.

of an isotropic medium. Let us consider the process under the action, on the idealized spectrum of aperture noise, of an aperture in the form of a long and narrow slit. Then, according to the expressions for the idealized spectrum (1) and for the frequency characteristic of the aperture, one may write:

\[ G(\nu_x,\nu_y)= \frac{2A_0^{\,2}\nu_0(\sin c\,\pi \nu_x h)^2} {\dfrac{\nu_x^2}{\nu_0^2}+\dfrac{1}{T^2(1-\overline{T})^2}} \cdot \frac{2A_0^{*}\nu_0(\sin c\,\pi \nu_y l)^2} {\dfrac{\nu_y^2}{\nu_0^2}+\dfrac{1}{T^2(1-\overline{T})^2}}, \tag{3} \]

where

\[ A_0,\ A_0^{*} \text{ are constant quantities,} \]

\[ \nu_0 \text{ is the intrinsic parameter of the photographic film,} \]

\[ \nu_x,\nu_y \text{ are the current spatial frequency,} \]

\[ \overline{T} \text{ is the mean blackening over the aperture area.} \]

Fig. 1. Dependence of the noise spectrum on the dimensions of the analyzing aperture.

In Fig. 1 are presented calculated curves of the spectra of photographic noises transformed by the measuring aperture.

In addition to spectral characteristics, there are other kinds of estimates of photographic noise. The most widespread estimate, and the one convenient from a practical point of view, is the rms value \(\sigma\). The total power \(\sigma\) is expressed as the integral of expression (3) over spatial frequency.

\[ \sigma_T^2=\frac{1}{\pi^2}\iint_S \sigma(\nu_x,\nu_y)\,d\nu_x\,d\nu_y . \tag{4} \]

Carrying out the integration under the assumption that the linear dimensions of the aperture vary along the coordinates \(x\) and \(y\) from 0 to \(\infty\), we obtain the expression for \(\sigma_{Txy}^{2}\) in the form

\[ \sigma_{Txy}^{2}= \frac{2A_0^{\,2}\overline{T}^{\,2}(1-\overline{T})^2}{2\pi\nu_0 h} \cdot \left[ 1-\frac{\overline{T}(1-\overline{T})}{2\pi\nu_0 h} \left(1-e^{-\frac{2\pi\nu_0 h}{\overline{T}(1-\overline{T})}}\right) \right]\cdot \tag{5} \]

\[ \frac{2A_0^{*\,2}\overline{T}(1-\overline{T})^2}{2\pi\nu_0 l} \cdot \left[ 1-\frac{\overline{T}(1-\overline{T})}{2\pi\nu_0 l} \left(1-e^{-\frac{2\pi\nu_0 l}{\overline{T}(1-\overline{T})}}\right) \right]. \]

Next, after carrying out some simplifications and transformations in expression (5), we obtain a formula for the rms value of the noises as a function of the aperture dimensions:

\[ \sigma_{Txy}=\sigma_{0x} \sqrt{ \frac{2\overline{T}(1-\overline{T})}{a h} \left[ 1-\frac{\overline{T}(1-\overline{T})}{a h} \left(1-e^{-\frac{a h}{\overline{T}(1-\overline{T})}}\right) \right] } \cdot \tag{6} \]

\[ \sigma_{0y} \sqrt{ \frac{2\overline{T}(1-\overline{T})}{a l} \left[ 1-\frac{\overline{T}(1-\overline{T})}{a l} \left(1-e^{-\frac{a l}{\overline{T}(1-\overline{T})}}\right) \right] }, \]

where the following notation has been adopted:

\[ \sigma_{0x}=A_0\sqrt{\overline{T}(1-\overline{T})}, \]

\[ \sigma_{0y}=A_0^{*}\sqrt{\overline{T}(1-\overline{T})}, \]

\[ \alpha=2\tau\nu_0 . \]

From expression (6) it is seen that, as \(h\) and \(l\) tend to infinity, \(\sigma_{0x}\) and \(\sigma_{0y}\) tend to zero, and consequently \(\sigma_{Txy}\) will also tend to zero; i.e., as the aperture area is increased to infinitely large dimensions, the sensitivity threshold of the receiving device is, as it were, lowered, and no fluctuations of blackening will be detected. As \(h\) and \(l\) tend to zero, expression (6) becomes an indeterminacy of the type \(\frac{0}{0}\). Resolving the indeterminacy by expanding \(l^{-z}\) in a series and

Fig. 2. Dependence of photographic noise on the shape of the analyzing aperture

Fig. 3. Dependence of photographic noise on mean blackening for different aperture areas

limiting ourselves to the third term of the series, the expression for \(\sigma_{Txy}\) at \(h\) and \(l \to 0\) can be written in the form:

\[ \sigma_{Txy}=\sigma_{0x}\sqrt{1-\frac{\alpha\sqrt{S}}{3\overline{T}(1-\overline{T})\sqrt{K}}}\cdot \sigma_{0y}\sqrt{1-\frac{\alpha\sqrt{S\cdot K}}{3\overline{T}(1-\overline{T})}}, \tag{7} \]

where

\[ S=h\cdot l \]
is the area of the analyzing aperture,

\[ K=\frac{l}{h} \]
is the proportionality coefficient.

It is now easy to show that, for \(h\to 0\) and \(l\to 0\), the value \(\sigma_{T}x\to\sigma_0 x\) and \(\sigma_{T}y\to\sigma_0 y\); i.e., we arrive at the idealized value of the noise obtained when analyzing the specimen with a point aperture.

Figure 2 gives the dependence of the noise level on the shape of the analyzing aperture, calculated on the basis of expression (6). It is seen from the figure that, when the ratio of the aperture sides, \(K\), is changed, the noise level does not change. This agrees well with the experimental data cited in the literature [2].

Figure 3 shows the dependence of the rms noise level on the mean blackening \(T\) for various apertures. Curve 1 represents the dependence of the noise on the blackening level without the action of the aperture. Curves 2, 3, and 4 show that the analyzing aperture reduces the noise over the entire range of blackening according to one and the same law.

Thus, the analyzing aperture acts on the noise, changing its spectral composition and reducing its power; i.e., the larger the aperture, the lower the noise level. The latter leads, as it were, to an increase in the response threshold of the photographic system and does not always make it possible to detect information about the presence of a useful signal in the noise. For this purpose, additional image processing is necessary, implementing, for example, such a feature of photographic noise as the dependence of its level on the magnitude of the useful signal.

References

1. L. M. Ananyev, A. S. Moskaleva. Izvestiya TPI, vol. 213, Tomsk, 1972.
2. G. A. Gavrilov, S. B. Gurevich, I. I. Breido. Advances in Scientific Photography, vol. 10, 1964.