ECE184
4F OPTICAL PROCESSING SYSTEM (part B)
B. Coherent Complex Spatial Filtering
Using special filters with selected impulse responses in the filter plane P2 of the 4F
optical processing system, operations such as addition, subtraction, differentiation, and character
recognition can be performed.
1. Addition and Subtraction with a Single-Grating Filter
Fig. 2. Complex amplitude addition and subtraction with a grating.
Consider the object shown in Fig.2. The transmittance function in the object plane P1 can be
expressed as
In front of the filter plane the field distribution becomes
where
and
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The filter is a grating with transmittance
which can be made by interfering two plane wave incident on a piece of film, with a small angle
between the two propagating directions. These two wave planes can be generated with two
off-axis point sources as shown on Fig.3.
Fig. 3. Optical system for producing a cos2(πυyb) grating
The amplitude distribution immediately behind the grating is
The lens L2 forms the Fourier transform of u2+(υy) at its back focal plane P3, so
Hence, at the center portion of the image plane, we have the addition of the two complex
functions f1 and f2.
If the transmittance of the grating is
which corresponds to having the maximum transmittance of the grating shifted by a quarter of a
fringe spacing from the optical axis, and
2
and thus
Therefore, at the center portion of the image plane, we have the subtraction of the two complex
functions f1 and f2.
2. Differentiation with a Double-Grating Filter
A filter which is composed of two gratings of slightly different frequencies, and shifted
by one half of a fringe spacing, will have the transmittance function
where ε is small positive number. By inserting this filter in the Fourier plane P2 , and using an
object with amplitude distribution f(x,y) at the input plane P1, the light field at the output is
The last terms are proportional to the differentiation operation ∂f/∂y since
3. Vander Lugt Filter
First consider how this particular filter is synthesized, see Fig.4(a). Lens L1 collimates
light from the point source S. The lower portion of this collimated beam strikes the mask which
has the desired impulse response h. Lens L3 Fourier transforms the amplitude distribution h,
yielding an amplitude distribution
3
incident on the film. A smaller lens L2 focuses the upper portion of the collimated beam to a
bright spot at the focal plane of lens L3. When the spherical wave generated by this reference
point passes through L3, it is collimated to produce a tilted plane wave at the film with the field
distribution:
Fig.4. (a) a modified Rayleigh interferometer system to produce the Vander Lugt filter,
(b) location of the four terms of the processor output where Wh and Wg are the maximum width
of h and g, respectively.
where the spatial frequency α is given by
and ϴ is the tilt angle. The total intensity distribution on the film is
The film is then exposed and developed to produce a transparency with an amplitude
transmittance proportional to the intensity distribution incident on the film during exposure.
Thus
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Once the film of transmittance t(x2, y2) has been synthesized, it may be inserted in the
transform plane P2 of the 4F optical processing system. If the input function is g(x1, y1), then the
amplitude distribution incident on the transform plane is
The field amplitude after the filter is
The final lens L3 Fourier transforms u2. Taking note of the reflected coordinated system in the
plane P3, the field in that plane is
The third term in u3 has the form
This portion of the output is seen to yield the convolution of h and g at coordinates (0, -αλF) in
the x3y3 -plane, as shown in Fig.4(b).
The forth term has the form
which is the cross-correlation of g and h centered at coordinates (0, αλF) in the x3y3 –plane.
From the Fig.4.(b) it is clear that complete separation will be achieved if
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EXPERIMENTS
B. Coherent Complex Spatial Filtering
1. Coherent Addition and Subtraction Filter.
- A spatial filter consisting of a cos2 - grating has an impulse response of three delta functions.
This filter can be used in the system of Fig. 2 to perform addition and subtraction.
- Place filter marked "addition/subtraction" in filter plane. Using the twin-triangle objects (one
with triangles spaced 2b apart and another with triangles spaced 4b apart) in the object plane,
perform addition and subtraction of the images.
2. Differentiation Filter.
By superimposing two gratings of slightly different frequencies and shifted by a half of a period,
a filter capable to perform the differentiation can be generated. When this filter is placed in the
filter plane, spatial differentiation is performed on the image in the first diffraction order.
- Place filter marked "differentiation" in the filter plane and perform differentiation in xdirection on letter "T".
- Perform differentiation on letter T in y-direction.
3. Vander Lugt Filter
Note: The character "AE" was used as the desired impulse response in Fig. 4(a) to
generate a Vander Lugt filter.
- Place slide with symbol AE in the object plane.
- Look on VDL filter and find the spot where the object recorded (“sparkling dot” on the slide)
- Place the VDL filter in the filter plane to perform character recognition by correlation.
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- Adjust filter position to illuminate the “sparkling dot”. As result you should see three spots on
the screen before the image plane.
- Place CCD camera into upper spot (in our case it is the cross-correlation area) and observe the
response. Fine tune filter position to get the brightest intense small spot. (Reduce light intensity
to prevent saturation of CCD if needed.)
- Replace symbol AE by letter A (E, and S) and observe the response.
- Analyze your observed results. Does symbol AE give the brightest spot?
- Verify the content of VDL filter by placing the filter in the object plane and observing its
Fourier transform.
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