Reducing Redundancy in the Description of Defectograms
Abstract Generated abstract
This paper addresses redundancy reduction in automatic interpretation of defectogram images, focusing on economical description through detection, filtering, and contour extraction. It formulates a Walsh-function-based filtering approach, defining amplitude attenuation coefficients through Walsh expansion terms and outlining a technically simple implementation using switches, integrators, memory elements, and adjustable attenuation. The authors then apply the same apparatus to contour extraction by expressing finite-difference gradient and Laplacian operations in terms of Walsh functions. The resulting scheme combines filtering and contour detection in one programmable filter, suggesting that image contours can be obtained by substituting Walsh gradient or Walsh Laplacian functions for the ordinary Walsh functions at the filter output stage.
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REDUCING REDUNDANCY IN THE DESCRIPTION OF DEFECTOGRAMS
B. N. EPIFANTSEV, Yu. P. ZABASHTA
(Presented by the scientific seminar of the Research Institute of Electronic Introscopy)
In constructing systems for the automatic deciphering of images, the problem of reducing redundancy in their description becomes of considerable importance. The need is obvious for transforming the initial information in such a way that it makes it possible not only to extract from the totality of objects those of interest (the detection problem), but also to describe them economically.
Recently, among the methods that make it possible to solve this problem [1], preference has been given to methods most acceptable from the standpoint of their technical implementation by modern means, including integrated circuits. In this respect the decisive role belongs to methods based on the orthogonal discrete Walsh functions.
A distinctive feature of these functions is that they take only two values, \(+1\) and \(-1\); the system of Walsh functions \(\{W_k(t)\}\) is orthogonal and complete in \(L^2\), and the functions are extremely simple to compute and implement.
The graph of the first 16 Walsh functions is shown in Fig. 1.
Let us consider the solution of the problems of detecting (filtering) and extracting the contours of images of objects to be deciphered when using a Walsh apparatus. Since real signals admit representation through various systems of orthogonal functions, there exists the possibility of constructing filters that differ from the generally accepted ones.
Fig. 1. Graphs of Walsh functions
The electrical properties of a filter (for simplicity of exposition the one-dimensional case is considered) are characterized by its amplitude and phase characteristics, the first of which determines the attenuation of the signal components, and the second—their phase shifts.
Let us determine the amplitude and phase characteristics for the Walsh filter. Let a signal \(f(t)\) act at the input of the filter; then at the output of the filter it will have the form \(\varphi(t-\tau)\). Represent \(f(t)\) and \(\varphi(t-\tau)\) in the form of a series
\[ f(t)=\sum_{k=0}^{\infty} a_k W_k(t), \qquad \varphi(t-\tau)=\sum_{k=0}^{\infty} b_k W_k(t-\tau), \]
where
\[ a_k=\int_{0}^{\tau} f(t) W_k(t)\,dt, \]
\(T\) is the period of the Walsh function.
\[ b_k=\int_{0}^{\tau} \varphi(t) W_k(t)\,dt. \]
The attenuation coefficient for the components of the signal \(f(t)\) (the amplitude characteristic of the filter) is determined as
\[ \mu_k=\frac{b_k}{a_k}. \]
The general expression for the output signal \(\varphi(t-\tau)\) will be
\[ \varphi(t-\tau)=\sum_{k=0}^{\infty} \mu_k \left[\int_{0}^{0<\tau<T} f(t) W_k(t)\,dt\right] W_k(t-\tau). \]
Figure 2 presents a block diagram of the Walsh filter implementing the last expression. The Walsh-function generator supplies \(W_k(t)\) in the form of periodic sequences with period \(T\). The signal \(f(t)\) entering the
Fig. 2. Block diagram of the Walsh filter
input is multiplied in the switches \(k_0—k_n\) by the function \(W_k(t)\). Since the latter take only two values, \(\pm 1\), in practice the switches \(k_0—k_n\), in accordance with the sign of the Walsh function, connect the signal to the direct or inverse input of the integrator. During the integration time \(\tau\), the voltages from the outputs of the integrators, corresponding to the expansion coefficients \(a_k\), are fed to
memory elements of the electronic computer. The periodically repeating functions \(W_{\kappa}(t-\tau)\) in the keys \(\kappa'_0—\kappa'_n\) are multiplied by the values \(a_{\kappa}\) and, with the attenuation coefficient \(\mu_{\kappa}\), are fed to an adder, from whose output \(\varphi(t-\tau)\) is taken. By changing the values of the corresponding resistances \(R_0—R\), and thereby changing the attenuation coefficient \(0 \leqslant \mu_{\kappa} \leqslant \infty\) for \(\infty \leqslant R_{\kappa} < 0\), it is not difficult to obtain the required transfer characteristic of the filter. The total delay of the output signal \(\tau\) is determined by the integration time and can be regulated within the limits \(0 < \tau \leqslant T\).
The technical implementation of such a filter presents no difficulties. It is also not difficult to obtain a programmable filter; in this case it is possible to program both the frequency range of the filter (by changing the period \(T\) of the Walsh functions and the corresponding numbers of the functions “\(K\)”) and the transfer function of the filter within the range, by changing the attenuation coefficients \(\mu_{\kappa}\).
Let us consider the use of the Walsh filter for extracting image contours. Analysis of defectogram images shows that the most informative feature, i.e., the one carrying practically all information about the object, is the image contour.
Fig. 3. Graphs of the Walsh gradient and Laplacian operators
Existing methods for extracting contours [1] can be reduced to two: the gradient method and the method using the Laplace operator.
The gradient of the image \(B(x,y)\) is a vector signal and has two components
\[ \nabla B(x,y)=\frac{\partial B(x,y)}{\partial x}\,\mathbf{i} +\frac{\partial B(x,y)}{\partial y}\,\mathbf{j}. \]
and in finite differences has the following form:
\[ \nabla B= \frac{B(x_{i+1},y)-B(x_i,y)}{x_{i+1}-x_i}\,\mathbf{i} + \frac{B(x_i,y_{j+1})-B(x_i,y_j)}{y_{j+1}-y_j}\,\mathbf{j}. \]
Again consider the one-dimensional case
\[ \nabla f(t)=f(t+\eta)-f(t) \]
or
\[ \nabla f(t)= \sum_{k=0}^{\infty} a_k W_k(t+\eta) - \sum_{k=0}^{\infty} a_k W_k(t) = \sum_{k=0}^{\infty} a_k \nabla W_k(t), \]
where
\[ \nabla W_k(t)=W_k(t+\eta)-W_k(t). \]
The Laplacian is defined analogously as
\[ \Delta B=\frac{\partial^2 B}{\partial x^2} +\frac{\partial^2 B}{\partial y^2}, \]
and in finite differences for the one-dimensional case—
\[ \Delta f(t)=\nabla f(t+\eta)-\nabla f(t) = \sum_{k=0}^{\infty} a_k \Delta W_k(t). \]
In Fig. 3 the operators \(\nabla W_{\kappa}(t)\)—the Walsh gradient—and \(\Delta W_{\kappa}(t)\)—the Walsh Laplacian—are represented graphically.
Combining the processes of filtering and contour extraction in a single filter, we obtain the following expression for the contour signal at the filter output:
\[ \Delta[\nabla]\varphi(t-\tau) = \sum_{\kappa=0}^{\infty} \mu_{\kappa} \left\{ \int\limits_{0}^{0<\tau<T} f(t)W_{\kappa}(t)\,dt \right\} \Delta[\nabla]W_{\kappa}(t-\tau). \]
Analyzing the last expression, it is not difficult to see that, in order to obtain the contour of an image, it is sufficient in the filter in Fig. 2 to feed to the switches \(\kappa'_0—\kappa'_n\), instead of the Walsh functions, their gradient \(\nabla W_{\kappa}(t-\tau)\) or Laplacian \(\Delta W_{\kappa}(t-\tau)\), whose generation presents no difficulty.
REFERENCES
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