Spatial filtering was done almost one hundred years ago by Ernst Abbe[1] when he performed his famous experiments in support of his Fourier theory of coherent image formation. Recently, a special type of a spatial filter, the so-called matched filter, has been used for character recognition and signal detection.[2] From the wave-optical point of view, these matched filters are not really different from formerly investigated spatial filters[3]; but they are, in general, much more difficult to produce since the complex filter function might vary widely both in amplitude and in phase. We intend to present some easy ways to produce such complex spatial filters. The explanation is presented in intuitive optical terms, leaving a more rigorous treatment for a later date. Some experiments, including Fraunhofer holography and character recognition, are presented in order to confirm the theoretical model. The objects, which are indirectly contained in our holograms or matched filters, do not have to exist physically.

An unusual, but fruitful, way of entering the field of matched spatial filtering is this: it is our task to find a diffraction object that yields a predefined Fraunhofer diffraction pattern. This task is the same as finding a Fraunhofer hologram for a predefined image. But our task is the opposite from the usual diffraction problem, where the diffraction object is given and one wants to find out what the diffraction pattern will look like. Our task is illustrated in Fig. 1. The unknown diffraction object that we will call the filter F(ν_x,ν_y), is illuminated by a monochromatic plane wave, coming from a point source in plane O. The resulting Fraunhofer diffraction pattern u(x,y) will appear in plane I. Knowing that the effect of Fraunhofer diffraction on the complex light amplitudes is described by a Fourier transformation, we can calculate the filter F from the prescribed diffraction pattern u:

The reduced coordinates (ν_x, ν_y) and the genuine coordinates (x_f,y_f) in plane F are connected according to x_f = λfν_x, y_f = λfν_y.

From the theoretical point of view, our task is completed. Next, we have to find a way of implementing the complex amplitude F(ν_x, ν_y) = A(ν_x, ν_y)e^iα(νx,νy). The amplitude factor A(ν_x, ν_y) can be realized by a photographic plate of proper transmittance distribution. However, we want to avoid the somewhat difficult problem of grey-tone control. Therefore, we replace the continuous amplitude factor by an array of tiny dots, much as halftone simulation is done in the printing industry. A quantitative description of proper shapes and positions of these dots is given after we devise a way for implementing the phase α (ν_x, ν_y).

There are three possibilities to influence the phase of a light wave: retardation while traveling through a dielectric, phase jump at reflection, and detour phase. The detour phase is best described by referring to the elementary explanation of grating diffraction. Two rays leaving two adjacent grating slits and going into the first diffraction order have a path difference of one wavelength. Their detour phase is 2π. If some of the grating slits are not at their perfect positions, as in Fig. 2, the diffracted wave front will be deformed. This is undesirable for an ordinary diffraction grating, but for us it will be a convenient and achromatic way to implement the phase α(ν_x, ν_y) of our diffraction object or filter in the form of a detour phase. Although the term is new, the detour phase has been used previously for grating spectroscopy by Hauk et al.,[4] for holography by Leith and Upatnieks[5] and in the earlier work on matched spatial filters.[2]

Now we synthesize the complex filter function F(ν_x, ν_y) which we assume to be continuous. In sufficiently small cells of a size (Δν)², the function F will be almost constant: F(ν_x, ν_y) ≈ F(nΔν,mΔν) = F_nm = A_nme^iαnm, where |ν_x − nΔν| ≤Δν/2, and |ν_y − mΔν| ≤ Δν/2, and n,m = 0, ±1, ±2, …. These cells may be so small that their microstructure is unimportant. All that matters is that from this cell a complex light amplitude F_nm will emerge in total. If, for example, the cell (n,m) consists of a vertical slit, as in Fig. 3(a), and if this cell is illuminated by a tilted plane wave e^{−2πix₀ν_x} coming from the point (x₀,O) in the plane O (Fig. 1), the emerging complex amplitude will be

This we want to be equal to F = Ae^iα. Hence, A = (Δν/πx₀) sin(πx₀WΔν); α = −2πx₀(n + p)Δν. A, α, W, and p all have indices (n,m) referring to that cell. It is convenient to choose x₀ so that a phase α = 0 (mod 2π) can be achieved by p = 0 which means the slit is at the center of the cell for α = 0. This is obtained by x₀Δν = N (integer); A = ((Δν)²/πN) sin(πNW); α = −2πNp. This result tells us how we can indirectly realize a complex filter function F(ν_x,ν_y) by means of an array of many little parallel slits of proper width and proper position. If the shape of the cells had been as in Fig. 3(b) and 3(c), the result for the phase would be again α = −2πNp, but differently for the amplitudes:

Now we verify the validity of our intuitive arguments by showing some experiments. In Fig. 4(a), an actual filter of the type in Fig. 3(a) is shown. It was calculated in order to get the capital letter E as a diffraction pattern (see Fig. 5). In other words, F(ν_x, ν_y) is the Fourier transform of the pattern E. The cell width Δν can be interpreted as an averaged grating constant λfΔ measured in the actual coordinate x_f = λfν_x of plane F. In order to avoid overlap of adjacent grating diffraction orders in plane I, their shift 1/Δν should exceed the size Δx of the diffraction pattern E, which means that ΔνΔx ≤ 1. This requirement is intimately related to the requirement that F should be almost constant within each (Δν) cell. The proof, which employs Cauchy’s 125-year-old sampling theorem,[6] is beyond the scope of this paper. The actual production of the filter or hologram was done first at large scale, and then photographically reduced. In one case [Fig. 4(a)] the drawing was made by hand; in another case [Fig. 4(b)] by a computer-guided plotter. Because of the binary nature of our filters, no precautions were necessary against the dielectrical phase owing to the gelatine profile of the photographic film.

Finally, we show a character recognition experiment with the same filter [Fig. 3(a) ] that had been used for producing Fig. 5. Now we replace the single light point in O by an array of nine different capital letters, one among them being the E. As shown in Fig. 6, the zeroth diffraction order is simply an image of these letters. The first diffraction order consists of nine more- or less-pronounced light points at positions corresponding to the nine letters. The most intense light point tells us about the position of the letter E at which our matched filter is aimed. The next highest intensity owing to the B goes up to 81%.

We would like to acknowledge stimulating discussions with A. Kozma, Ann Arbor, about his earlier work on binary matched filtering.[2]

Figures

 

Fig. 1 Fraunhofer diffraction with the complex filter F as diffraction object and μ as diffraction pattern. This setup can serve also for the readout of the Fraunhofer hologram F, yielding a reconstructed image μ.

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Fig. 2 The detour phase. The two lower grating slits are slightly out of position. Therefore, the complex amplitudes in these slits lag behind in phase, causing a deformation of the wave front.

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Fig. 3 Three ways for constructing the cell (n,m). The parameters W and p that control amplitude and phase of the light emerging from this cell carry indices (n,m). The slit width c is a constant. Cases b and c are particularly suitable for automatic plotting.

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Fig. 4 Two matched filters or Fraunhofer holograms for E: (a) according to Fig. 3(a); (b) according to Fig. 3(b).

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Fig. 5 Filter output or image reconstruction from filter in Fig. 4(a).

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Fig. 6 Character recognition by matched filtering with nine input letters and E filter from Fig. 4(a). The B–E correlation was measured to be 81% of the E–E autocorrelation.

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References

1. O. Lummer and F. Reiche, Die Lehre von der Bildentstehung im Mikroskop (von Ernst Abbe, Braunschweig1910); A. B. Porter, Phil. Mag. 11, 154 (1906); M. Wolfke, Ann. Physik 34, 277 (1911); [CrossRef]  Ann. Physik37, 96; Ann. Physik38, 385; Ann. Physik 39, 569 (1912); Ann. Physik 40, 194 (1913).

2. A. Vander Lugt, Proc. IEEE IT-10, 139 (1964): A. Kozma and D. L. Kelly, Appl. Opt. 4, 387 (1965): [CrossRef]  A. Vander Lugt, F. B. Rotz, and A. Klooster, in Optical and Electro-Optical Information Processing, J. T. Tippett, D. A. Berkowitz, L. C. Clapp, C. J. Koester, and A. Vanderburgh, Eds. (MIT Press, Cambridge, Mass., 1965), p. 125: E. N. Leith, A. Kozma, and J. Upatneiks, in Optical and Electro-Optical Information Processing, J. T. Tippett, D. A. Berkowitz, L. C. Clapp, C. J. Koester, and A. Vanderburgh, Eds. (MIT Press, Cambridge, Mass., 1965), p. 143.

3. A. Marechal and P. Croce, Compt. Rend. Acad. Sci. (Paris) 237, 607 (1953); E. L. O’Neill, Proc. Inst. Radio Engrs. IT-2, 56 (1956); and J. Tsujiuchi, in Progress in Optics, E. Wolf, Ed., Vol. 2, p. 133 (1963) [CrossRef]  .

4. D. Hauk and A. Lohmann, Optik 15, 275 (1958).

5. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962); [CrossRef]  J. Opt. Soc. Am. 53, 1377 (1963); J. Opt. Soc. Am. 54, 1295 (1964).

6. H. S. Black, Modulation Theory (D. Van Nostrand Co., Inc., New York, 1953).