Filtering in optics

The filter of Vander Lugt (1963)

Introduction

The filters produced have the remarkable property to permit the efficient control of the amplitude and of the phase of a transfer function at the same time even if they are only formed in purely absorbent figures.

Achievement of the function of transfer

The frequent mask (of the filter of Vander Lugt) is produced with the help of a interferometric system that can be seen in the figure II-15.



   

    Figure II-15 : Achievement of the Vander Lugt filtration by an interferometric method
Figure II-15 : Achievement of the Vander Lugt filtration by an interferometric method [zoom...]

The information coding the phase finds itself in the term of interference. L1 serves as a collimator. A part of this light falls on the plane P1 in which a transmittance in amplitude equal to the impulsion desired response h is found. The lens L2 produces the TF of h on the film placed in P2. The other part of the light crosses the prism P and arrives at P2 under the incidence .

The total incident intensity at the level of the film is determined by the interference of two distributions of amplitude (coming from L2 and from P). The inclined wave plane is represented by its complex amplitude in the plane P2 .

The distribution of the intensity is therefore written as :

being complex, in the general case, one can pose that :

Which gives the intensity :

This form shows how the interferometric process permits the registration of a complex function on a detector sensitive to the intensity. The information relative to the amplitude and to the phase is registered under the form of a amplitude modulation and of a wave phase carrying a high frequency that is introduced by the wave of inclined “reference” coming from the prism. To produce this filter, the film of such a type is developed so that its transmittance in the amplitude is proportional to the intensity of exposition :

In the term of interference (c. to d. in t) we have the information required to make the impulsion response filter equal to h

The remaining problem is to show how and under what conditions this information can be extracted from the other terms that are present in I.

Treatment of the given

Let us introduce the mask  that we just produced in the montage 4-f (in the plane P2) (see the figure II-16).



   

    Figure II-16 : Treatment of the givens in the system 4-f
Figure II-16 : Treatment of the givens in the system 4-f [zoom...]

Where g is the function of entry that we wish to filter.

The amplitude U2 of the transmitted field by the mask is written :

L3 produces the TF of U2 in P3 and knowing that , the amplitude of the field is written :

L2 divides the amplitude of the spectrum by and reduces the coordinates by following each axis. L3 always divides the amplitude of the spectrum by ut puts back this time the coordinates at the same level. The theorem of similarity tells us that it is necessary to multiple by . At the total, the result comes back to multiply by .

Knowing that is the natural neutral element of the product of convolution with which the following translation is done following the third and fourth term being :

The first two terms centered at the origin do not have any particular interest for filtration. In P3 three regions where the amplitude is different from 0 are observed. These regions do not cover themselves if is sufficiently large before the spatial extension of h and of g (voir figure II-17).



   

    Figure II-17 : Position of different terms at the exit of an optical processor
Figure II-17 : Position of different terms at the exit of an optical processor [zoom...]

Here the principal advantage of this filter is to produce a function of transfer at complex values with an absorbent filter. The transmittance of phase in the Fourier plane is more simple to produce technically (no control of dimension, of thickness, or of indication).

Application to the recognition of forms

It is about knowing the placement of a letter (the P) presented at the entry of a system 4-f among different letters (three in this example).



   

    Figure II-18 : Optical recognition of the character P
Figure II-18 : Optical recognition of the character P [zoom...]

In the figure II-18, the image of the impulsion response of a filter of Vander Lugt (at the left) can be seen that will be introduced in the filtration plane. The response of this filter to the letters Q, W and P (image at the right) at the exit of the system. Note the presence of a intense spot under P (at the level of the term of inter correlation) indicating a degree of resemblance elevated between this letter and the chosen filter.

It is necessary to note that the inconvenient principal of such a method of recognition using the transformation of Fourier is its big sensitivity to the rotation and to the changing of levels. Others that are transformed (such as that of Mellin) offer less sensitivity to the enlargement [].

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