Image analysis – Image transformation or preprocessing – Fourier transform
Reexamination Certificate
1998-05-11
2001-11-06
Boudreau, Leo (Department: 2621)
Image analysis
Image transformation or preprocessing
Fourier transform
C359S559000
Reexamination Certificate
active
06314210
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a multiplexing optical system. More particularly, the present invention relates to an apparatus capable of performing a plurality of filtering operations simultaneously on an input image and obtaining a result of the filtering process.
2. Discussion of Related Art
In the field of image processing, frequency filtering is frequently executed to change the spatial frequency distribution of an input image for the purpose of emphasizing a part of the input image or extracting only a specific component which is obscured by noise. To perform such frequency filtering, a Fourier transformed image of the original image must be obtained. Let us express the amplitude distribution of an input image by f(x,y) and the Fourier transform F(&mgr;,&ngr;) thereof by
F{f
(
x,y
)}=
F
(&mgr;,&ngr;) (1)
When the Fourier transform F(&mgr;,&ngr;) is subjected to filtering expressed by the function H(&mgr;,&ngr;), the following relationship holds:
F{F
(&mgr;,&ngr;)
H
(&mgr;,&ngr;)}=
h
(−
x, −y
)
f
(−
x,−y
) (2)
where “*” represents convolution calculation.
In the above expression, H(&mgr;,&ngr;) is the Fourier transform of h(x,y). In other words, the convolution of the input image f(−x,−y) with h(−x,−y) is the Fourier transform of the product of their respective Fourier transforms F(&mgr;,&ngr;) and H(&mgr;,&ngr;). However, this processing requires an exceedingly large quantity of computation. Therefore, in the case of sequential processing as in an electronic computer, it takes a long time to process data having a massive amount of information such as two-dimensional images.
Meanwhile, by virtue of its high-speed nature and parallelism, light makes it possible to obtain a Fourier transformed image of a two-dimensional image, which has a large amount of information, at a high speed which is absolutely impossible to attain with an electronic computer. By disposing a filter corresponding to H(&mgr;,&ngr;) in a plane where the Fourier transformed image is formed, filtering in the frequency space of the image can be readily performed. Optical systems that perform such filtering are shown in
FIGS. 12 and 13
. It should be noted that in the following description a side where principal rays enter a lens is referred to as the “front side”, and a side where the principal rays exit from the lens is referred to as the “back side”.
First, the optical system shown in
FIG. 12
will be explained. Light from a light source
11
passes successively through a condenser lens
12
and a collimator lens
13
to form collimated light having an enlarged beam width. The collimated light enters a lens array
14
having a focal length f
a
. A lens
15
has a focal length f
2
. The distance between the lens array
14
and the lens
15
is equal to the sum of the focal lengths f
a
and f
2
. Consequently, light beams emanating from the lens
15
form collimated light, and the light beams are incident from various directions on an input plane F
1
, which is the back focal plane of the lens
15
. A spatial light modulator
21
is disposed such that the read surface thereof is coincident with the input plane F
1
. An input image f(x,y)
211
is displayed on the read surface of the spatial light modulator
21
. A Fourier transform lens
31
is disposed such that the front focal plane thereof is coincident with the read surface of the spatial light modulator
21
. Therefore, each light beam from the spatial light modulator
21
forms a Fourier transformed image F(&mgr;,&ngr;)
311
of the input image f(x,y)
211
on a Fourier transform plane F
2
, which is the back focal plane of the Fourier transform lens
31
.
The above-described processing is carried out for each parallel light beam formed by the combination of the lens array
14
and the lens
15
. Accordingly, a plurality of Fourier transformed images F(&mgr;,&ngr;)
311
of the input image f(x,y)
211
are reproduced on the Fourier transform plane F
2
. A lens array
331
for performing an inverse Fourier transform on each of the reproduced Fourier transformed images
311
is placed such that the front focal plane thereof is coincident with the Fourier transform plane F
2
. Consequently, the input image
211
is reproduced on a reproducing plane F
3
, which is the back focal plane of each lens element
91
of the lens array
331
. In this optical system, a variety of different filters
321
H
i
(&mgr;,&ngr;) (i=1, 2, 3 . . . ) are disposed for a plurality of Fourier transformed images
311
formed on the Fourier transform plane F
2
, and thus filtered reproduced images F{F(&mgr;,&ngr;)H
i
(&mgr;,&ngr;)}=h
i
*f are formed on the reproducing plane F
3
.
There has also been proposed an optical system such as that shown in
FIG. 13
(Dec. 10, 1995/Vol.34, No.35/Applied Optics). A parallel light beam emitted from a light source
11
reads an input image f(x,y)
221
on an input plane F
1
and then enters a Damman grating
23
. The Damman grating
23
is a grating with a binary transmittance which is designed so that some different orders of diffracted light have a uniform intensity. The Damman grating
23
causes diffracted light to enter a Fourier transform lens
31
at an angle unique to each order of diffraction, and the number of Fourier transformed images F(&mgr;,&ngr;)
311
of the input image
221
which is equal to the number of orders of diffraction are reproduced for each order of diffraction on a Fourier transform plane F
2
, which is the back focal plane of the Fourier transform lens
31
. On the Fourier transform plane F
2
, each of the reproduced Fourier transformed images F(&mgr;,&ngr;)
311
is filtered by each matched filter H
i
(&mgr;,&ngr;) of a matched filter array
322
having different Fourier transforms recorded for respective filters. Further, an inverse Fourier transform lens
332
forms filtered images of the input image on a reproducing plane F
3
.
If each matched filter of the matched filter array
322
is a spatial frequency cutoff filter, light beams from each matched filter form convolution images at the same positions in the reproducing plane F
3
by the action of the inverse Fourier transform lens
332
. However, if each matched filter of the matched filter array
322
is a spatial frequency cutoff filter, a plurality of convolution images overlap each other on the reproducing plane F
3
, making it impossible to obtain desired images. Therefore, to prevent this problem, a holographic filter is used as each matched filter of the matched filter array
322
, and a result of the convolution of the input image
221
with the filter is obtained by performing an inverse Fourier transform on −1st-order diffracted light through the inverse Fourier transform lens
332
. The direction of −1st-order diffraction of the holographic filter used as each matched filter is effectively adjusted so that a plurality of reproduced convolution images do not overlap each other on the reproducing plane F
3
.
However, the arrangement of the first conventional optical system, which is shown in
FIG. 12
, needs to prepare two lens arrays
14
and
331
for producing parallel light beams different in the read direction from each other to read the input image
211
displayed on the spatial light modulator
21
and for performing an inverse Fourier transform on each Fourier transformed image
311
formed on the Fourier transform plane F
2
. It costs a great deal to produce a lens array with high accuracy in terms of the pitch between the lens elements and the optical performance of the lens, and it takes a great deal of effort to effect alignment for the entire optical system. Further, the whole lens array
14
must be illuminated in order to produce each parallel light beam for reading the input image
211
. Therefore, it is necessary to diverge the light beam to a considerable extent by the combination of the condenser lens
12
and the collimator lens
1
Fukushima Ikutoshi
Namiki Mitsuru
Boudreau Leo
Olympus Optical Co,. Ltd.
Patel Kanji
Pillsbury & Winthrop LLP
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