PROCEEDINGS

OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC INSTITUTE named after S. M. KIROV

Vol. 162　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1967

ON THE QUESTION OF DECORRELATION PROCESSING OF AN IMAGE THAT INCREASES VIEWING EFFICIENCY

L. M. ANAN'EV, V. Ya. KRIVCHIK

(Presented by the scientific seminar of the Department of Industrial Electronics)

To improve the quality of viewing an image and to increase the probability of detecting its individual elements, noisy low-contrast images may be subjected to various kinds of processing. The effect of image contrast enhancement on improving visual perception is sufficiently well known. We introduce for consideration the concept of the “contrast factor” $\xi$ [8] as an indicator of image quality during viewing:

\[ \xi=\frac{\Delta B}{\Delta x}, \tag{1} \]

where $\Delta B$ is the brightness jump in the image over the segment $\Delta x$ (Fig. 1, a).

It can be shown that the contrast of an image can be increased, respectively, by decreasing $\Delta x$ (Fig. 1, b) or by increasing $\Delta B$, which is accomplished by emphasizing the brightness transition by a method of obtaining an outline drawing with its subsequent superposition on the original image (Fig. 1, c, d, e) [4, 5, 6].

The operation of obtaining an outline signal corresponding to an element of the outline image from the signal corresponding to the original brightness transition in the image can be represented by the expression

\[ F(x,y)=A(x,y)f(x,y), \tag{2} \]

where $F(x,y)$ is an element of the outline image;
$f(x,y)$ is the original image;
$A(x,y)$ is an operator corresponding to the operation of obtaining an outline element from the original image;
$x,y$ are coordinates.

We apply a method of analyzing a stationary image by means of its successive element-by-element decomposition. Expression (2) is rewritten in the form:

\[ F(t)=A(t)f(t), \tag{3} \]

where

\[ f(t)=\iint f(x,y)D_r(xyt)\,dxdy, \tag{3a} \]

in this case $D_r(x,y,t)$ describes the law of displacement of the ideal zon-

of the differentiating element of the decomposition in the space \(x,y\) as a function of the current time \(t\), \(F(t)\) in the general case is a signal of duration \(\tau_k=(3—5)\tau_0\), where \(\tau_0\) is the duration of the decomposition element,

where

\[ \tau_k \ll t_c, \tag{3б} \]

\(t_c\) is the duration of the video signal corresponding to the image detail being analyzed.

Fig. 1. Boundary of a brightness transition in an image:
\(a\)—original image, \(б\)—image emphasized by reducing the duration of the boundary, \(в\)—image emphasized by superimposing a unipolar contour image, \(д\)—image emphasized by superimposing a bipolar contour image, \(г\)—bipolar contour image, \(е\)—unipolar contour image.

Let us consider, as an example, a contour element representing a rectangular pulse of duration \(\tau\) with amplitude \(q\). The spectrum \(s(\omega)\) of such a pulse is determined by the expression

\[ s(\omega)=q/j\omega[1-e^{-j\omega\tau}]. \tag{4} \]

Expression (4) in the spectral representation is a special case of the general expression (3), where, respectively,

\[ q/j\omega \]

is the original image (a Heaviside step with amplitude \(q\)),

\[ [1-e^{-j\omega\tau}] \]

is the operator corresponding to the operation of obtaining a contour element from the original image.

It is not difficult to see that the obtained operator

\[ A(\omega)=[1-e^{-j\omega\tau}] \tag{5} \]

makes it possible to form a contour element from any other video pulse \(f(\omega)\) satisfying condition (3б) and having a front different from a rectangular one.

Thus, the spectral function of the signal \(S_{\text{ke}}(\omega)\) corresponding to the contour element will be

\[ S_{\text{ke}}(\omega)=f(\omega)\,[1-e^{-j\omega\tau}], \tag{6} \]

and (5) may be regarded [1, 3] as an optimal operation for obtaining a contour element with respect to such operations as differentiation of the video signal, crispening, etc. [1, 5, 6].

A linear device for obtaining a contour element with a transfer function corresponding to (5) may be represented by the block diagram shown in Fig. 2, \(a\).

We shall assume that at the input of this device, in addition to the signal, there acts only “white” Gaussian noise with spectral intensity

\[ F_{\text{n}}(\omega)=A_0. \]

The expression relating the spectral density of a signal to its total energy \(W\) [10] is

\[ W=\frac{1}{2\pi}\int_{-\omega}^{\omega} S_1(\omega)S_1^{*}(\omega)\,d\omega, \tag{7} \]

where

\[ S_1(\omega)=g(\omega)S_{\text{ke}}(\omega), \tag{8} \]

\(S_1^{*}(\omega)\)—the function complex-conjugate to the spectral density of the contour signal \(S_1(\omega)\),

\(\omega\)—the cutoff frequency of the frequency band in which the principal part of the signal energy is contained,

\(g(\omega)\)—the transfer function of a filter providing, for the given class of image under study, the maximum \(W\) in the minimum frequency band.

Then one can find the optimal \(g(\omega)\) for achieving maximum noise immunity when obtaining the contour element by the method indicated above. In this case the operation of obtaining the contour element may be represented by the expression

\[ \Lambda_1(\omega)=g(\omega)A(\omega)=g(\omega)[1-e^{-j\omega\tau}], \tag{9} \]

and the linear device implementing this operation by the block diagram in Fig. 2, \(b\).

Defining the operation of obtaining a contour element, described by expressions (5) and (9), as the operation of obtaining the first finite difference, or “single-polarity retouching” (it should be noted that, in order to achieve isotropy in image processing, this operation is repeated twice—obtaining the second finite difference, or “double-polarity retouching” [6]), it is useful to give a certain estimate of the effectiveness of applying the above and subsequent processing methods. In transforming an image into the corresponding video signal, the effectiveness of the transformation and of subsequent video-signal processing is determined as a function of the resulting value of the signal/noise ratio of the image and of the video-signal sensor.

The distribution of noise in sensors of the transmitting television-tube type, to a certain approximation, obeys the normal law [11]. The distribution of noise in the difference signal also obeys this same law [9]

\[ P(y,\tau)=\frac{1}{2\sqrt{\pi}\sigma_0}\sqrt{1-R(\tau)}\cdot e^{-\frac{y^2}{4\sigma_0^2[1-R(\tau)]}}, \tag{10} \]

where \(\tau\) is the duration of the contour element (delay time),
\(\sigma_0\) is the root-mean-square value of the noise in the video signal,
\(R(\tau)\) is the correlation coefficient, whose value in the case of “white noise” for \(\tau \ge \tau_0\) is considerably less than unity (for \(\tau = n\tau_0\), \(R(\tau)=0\)).

Fig. 2. Block diagram of devices for forming a contour element:
\(I\) — inverting stage, \(DL\) — delay line, \(\Sigma\) — summing device, \(\Phi\) — filtering device

Consequently, the root-mean-square value of the noise in the first-difference signal is

\[ \sigma_{1\,\mathrm{diff.\ noise}}=\sqrt{2}\,\sigma_0 . \tag{11} \]

Similarly, the root-mean-square value of the noise in the second-difference signal is

\[ \sigma_{2\,\mathrm{diff.\ noise}}=2\sigma_0, \tag{12} \]

or, in the general case, the noise dispersion is determined as

\[ P_{\mathrm{noise}}(y)= \frac{1}{\sqrt{2\pi}\,\sigma_{\mathrm{diff.\ noise}}} e^{-\frac{y^2}{2\sigma_{\mathrm{diff.\ noise}}^2}} . \tag{13} \]

An increase in \(\sigma\) in the region of small values of the signal-to-noise ratio in a single reading significantly increases the probability of an error \(P_{\mathrm{err}}\) in detecting an image element and its contour signal. Defining the signal-to-noise ratio by the expression

\[ \rho=\frac{a^2}{\sigma^2}, \tag{14} \]

where \(a\) is the value of the signal magnitude, and using for the calculations the expression

\[ P_{\mathrm{err}}=\frac{1}{2}\,[1-\Phi(z)], \tag{15} \]

where \(\Phi(z)\) is the Kramp function and \(z=\sqrt{\frac{1}{8}\rho}\) [1], one can calculate \(P_{\mathrm{err}}\).

If \(\rho_{\text{video signal}} = 16\) and, correspondingly, \(\rho_{\text{diff}_1}=8\), \(\rho_{\text{diff}_2}=4\), then in this case \(P_{\text{err}_0}=2.23\%\), \(P_{\text{err. diff}_1}=7.87\%\), \(P_{\text{err. diff}_2}=15.77\%\). In view of the anisotropy in the direction of line scanning, and consequently in view of the appearance of the false information connected with it during processing, the “single-polarity retouching” method is used very rarely. Usually the “two-polarity retouching” method is used; it is free from this drawback, but it correspondingly assumes an increase of \(\sigma_{\text{sh}}\) to the level \(2\sigma_0\), while remaining the optimal operation for obtaining an isotropic contour element in the direction of line scanning. In accordance with what has been said, the basic conclusions and expressions (4–9) given earlier remain valid also for the “two-polarity retouching” method.

Let us consider the device in Fig. 2a as an image decorrelation device intended to eliminate statistical redundancy between image elements along the scanning line [4].

To achieve isotropy and processing efficiency, the principle of decorrelation processing of the image along the scanning line can be extended to the case of its spatial processing in the \(x,y\) plane—“spatial decorrelation processing of an image.” Such processing, in turn, presupposes analysis of the image by a complex aperture*, consisting of an outer aperture, for obtaining the response from neighboring image elements in the image plane, and an inner aperture of the ordinary scanning element for obtaining the video signal. Construction of the complex analyzing aperture is carried out in accordance with the expression

\[ R_{\text{pr}}=\frac{1}{\pi}\int_{0}^{2\pi} R_{\Theta}\,d\Theta \Rightarrow 0, \tag{16} \]

where \(R_{\text{pr}}\) is the spatial mutual-correlation function between the values of the image elements that fall, during analysis, respectively into the region of the outer and inner aperture of the complex analyzing aperture;

\(R_{\Theta}\) is the mutual-correlation function between the values of the elements in the region of the outer and inner apertures in the direction \(\Theta\),

\(\Theta\) is the angular coordinate in the plane of the polar coordinate system \(\Theta R\).

The great variety in the character and parameters of image details and of the images themselves, in order to achieve optimal decorrelation during processing \((R_{\text{pr}}=1)\), requires that, when constructing the analyzing aperture, the principal characteristic features of the class of images being processed be taken into account in each specific case. Let us consider the case of processing a class of images with sharply expressed background uniformity, low detail, and image details statistically homogeneous in shape and size. Such a class may include images obtained in flaw detection of products by transillumination, X-ray diagnostics, aerial photography, astronomical observations, etc. The mechanism of spatial decorrelation processing of these images is explained using the example

* By analogy with the method described above, where the implementation of operation 5 can be interpreted as analysis of the image by two probing elements separated by a distance equivalent to \(\tau\).

construction of the complex analyzing aperture of Fig. 3, consisting of the decorrelation contour 1 (outer aperture) and the decomposition element 2, for a detail of a defectoscopic image of the class “round—nearly round” 3.

A device implementing the principle of spatial decorrelation, to whose inputs 1 and 2 signals from apertures 1 and 2 (Fig. 3) are applied, is represented by the block diagram shown in Fig. 4, and the

Fig. 3. Superposition of the analyzing aperture with an image detail:
a — graph of the magnitude of the mutual-correlation function between image elements in the direction \(\Theta\).

method of spatial decorrelation, in all its indicators, plus the achievement of isotropy in image processing, is similar to the method of “unipolar retouching,” i.e.

\[ \sigma_{\text{sh pr. decorr}}=\sqrt{2}\,\sigma_0 . \]

Fig. 4. Block diagram of a device implementing the principle of spatial decorrelation:
\(И\) — inverse cascade, \(\Sigma\) — summing device

In this aspect it is advisable to compare, in terms of sensitivity, the method of spatial decorrelation processing described in the present work, which consists in analyzing an image with a complex analyzing aperture “decorrelation contour—decomposition element,” with the known [6, 12] methods of spatial decorrelation processing, such as the “logetron” and a two-dimensional filter constructed with the aid of delay lines. If \(\rho_{\text{pr. dec.}}\) is the signal-to-noise ratio in the contour signal obtained when analyzing an image with the aperture “decorrelation contour—scanning element,” and \(\rho_{\text{logetron}}\) is the corresponding ratio when analyzing an image by the “logetron” method, then the relative decrease in sensitivity for the case of applying the logetron can be determined from the expression

\[ \frac{\rho_{\text{logetron}}}{\rho_{\text{pr. dec}}}=1-\frac{S_1}{S_2}, \tag{17} \]

where \(S_1\) is the area of the image detail inscribed within the area of the external aperture,
\(S_2\) is the area of the external aperture.

Here it is assumed that \(\sigma_{\text{pr.dec}}=\sigma_{\text{logetron}}\).

Thus, as a result of analyzing the possibilities for increasing the effectiveness of viewing low-contrast images by emphasizing brightness transitions in the image, it has been shown that the most effective method is that of obtaining finite differences with their subsequent superposition on the original image.

An expression has been obtained which is used as a rule for constructing a complex analyzing aperture.

An example is given of constructing a complex aperture for the analysis of defectoscopic and other images for the class of “round—almost round” parts, and it is shown that this method is more effective in comparison with the logetron.

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