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Optical processing of one-dimensional signals
Volume 42, number 2
OPTICS COMMUNICATIONS
15 June 1982
OPTICAL PROCESSING OF ONE-DIMENSIONAL SIGNALS H. BARTELTand A.W. LOHMANN Physikalisches Institut der Universitiit, 8520 Erlangen, Fed. Rep. Germany Received 19 February 1982
Long one-dimensional signals can be processed efficiently by optical methods if they are represented in a two-dimensional version. We show that by choosing an appropriate scanning pattern either new types of operations can be synthesized or classical operations like Fourier transformation or correlation can be implemented for this type of signals. Besides coherent illumination, incoherent light can also be used in order to reduce stability requirements, to increase the signal to noise ratio and to accommodate self-luminous inputs.
1. Efficient signal processing by optics One-dimensional time signals such as speach signals are mostly processed by electronic systems. On the other hand, processing by optical systems may become interesting if very long signals or signals with a very high number of samples have to be processed. The number of degrees of freedom in an optical system can easily be 106 or even 108 due to the twodimensional, parallel processing capacity. The input of an optical system has a two-dimensional character of course. For optimal use of the space-bandwidth product the input functions therefore have to be arranged in a two-dimensional pattern. In case of an originally one-dimensional time signal, many types of two-dimensional arrangements are conceivable. This additional freedom of design increases the variety of algorithms for optical processing of one-dimensional signals [1-7]. In the following sections we will first discuss a general system suitable for processing non-two-dimensional signals by optical methods. Then we will turn to the special case of one-dimensional input functions. Coherent as well as incoherent illumination will be used. Incoherent systems provide a better signal to noise ratio. On the other hand they are inherently restricted to the processing of positive functions. To overcome this restriction we combined this optical processing type with TV electronics which permitted
us to perform convolutions with a bipolar point spread function.
2. General processing system A general optical system for processing data which are not two-dimensional can be considered to consist of three components (fig. 1): a dimensional transducer, an optical processor and again a dimensional transducer. The optical processor is able to perform linear and space-variant operations such as Fourier transformation, correlation or general spatial filtering. In all cases the input function should be presented in a two-dimensional version in order to utilize completely the available processing capacity or the space-bandwidth product of the system. Therefore, the first dimensional $ e.- . . . . . . . . . . . . . . . . . . . . . .
r
i
1
N~
r
DIMENSIONN. PROCESSOR DIMENSIONAL TRANSDUCER(Un~lr.~=ace.-irM=,io~'I'RANSITd(~R Fig. 1. General architecture of a system S with two dimensional transducers DN/DP and Dp/DM, and a processor P. The input is fN, the output gM. In our case the dimensionality of the processor is P = 2.
87
Volume 42, number 2
OPTICS COMMUNICATIONS
transducer has to convert the input signal into a twodimensional format. In case of one-dimensional time signals as input a 1D/2D conversion would be necessary. This conversion is performed by appropriate scanning in two dimensions of the one-dimensional function. The dimensional transducer at the output end transforms in a similar manner the signal from two dimensions to a single dimension, or to any other dimensionality N, if so desired. The usefulness of such a system with dimensional transducers can be twofold. One type of application is the synthesis of new integral transformations. As an example we would like to mention the synthesis of a Hankel transformation out of a Fourier transformation and two radial scanning operations [6,7]. If we scan a one-dimensional signal u(t) in a radial manner we get as new, two-dimensional pattern u(r):
u(t) ~ u(r).
(1)
Now we apply a two-dimensional Fourier transformation (in polar coordinates): u(r) ~ 7 [u(r)] :
u(p)
: fu(r) Jo(27rrp)rdr, 0
(2)
J0: zero order Bessel function. The result depends only on the radial coordinate p. If we scan again radially, we then obtain the Hankel
II
15 June 1982
transformation of the original signal (fig. 2). With more general scan patterns we are able to perform a large variety of integral transformations. The second application, besides the synthesis of new operations, is the implementation of typical optical operations (Fourier transformation, correlation, convolution, etc.) but now for scanned, one-dimensional signals. In the following section we will explain this application of spatial filtering in more detail.
3. Spatial f'dterning operations of scanned signals It has been shown by several authors that operations like Fourier transformation or correlation can be implemented for one-dimensional signals by simple raster scanning. We will explain shortly the applications of this type of scanning and we will show that either coherent or incoherent light may be used for illumination. Coherent processing permits the implementation of complex •ter functions or impulse responses whereas incoherent methods are restricted to positive point spread functions. On the other hand stability requirements are lower, if incoherent light is used. Furthermore a larger variety of input devices such as TV monitors, printed pages and oscilloscopes are admissable. According to Spann [ 1] and others [2-5] a raster scanned signal gives a suitable input for optical spatial fflterning. To explain
la
Fig. 2. Synthesis of Hankel transformation. (a) Radially scanned function u(t). (b) 2D Fourier transformed signal. 88
Volume 42, number 2
OPTICS COMMUNICATIONS
15 June 1982
//.z/. :
'/
',
L/
'
/7-/'/ • -i~
f'l (x.y)
-~
! 2(x.y)
Fig. 3. Correlation of a raster scanned function f2 with a single line function fl" the principle let us assume a single line function gl(x) of size Ax and a long function g2(x) which will be repeated N times, mutually shifted and cut-off by a window of width 2Ax. These two-dimensional layouts fl(X, y) and/2(x, y) of the one-dimensional functions gl(x) and g2(x ) are shown in fig. 3. We want to convolve the original signals gl (x) and g2(x). We do this by convolving in two dimensions the pattemsfl(x,y ) andf2(x,y ). The width 2Ax o f f 2 ( x , y ) is wide enough that every portion ofg2(x ) appears twice inf2(x,y ). The need for this redundancy can be understood by visualizing the convolution o f f l and f2 as an overlap o f f 1 and f2, with continuous shifts inx and stepwise shifts in y. : Only the portion Ixl< Ax/2 of the convolution f l **f2 yields valid contributions to the wanted convolution gl *8'2 = C(x). This can be deduced also by inspecting the formulae: f l (X, y) = gl (X) 5(y)
f 2(x,y) = rect(x/2Ax) ~ g 2 ( x - A x ) 6(y-na) (n) rect(x / A x ) f 1(x, y )** f 2(x , y) = ~ C ( x - nAx) 8(y - ha). (n)
(3)
Until now we assumed for simplicity only a single line functionf1 . But thismethod works exactly the same way i f f l consists of several lines. The sum of the convolutions with single lines o f f 1 is equal to the convolution with the complete functionfl, which is de1`reed by
f l ( x , y ) = rect(x/Ax)~gl(x -- m a x ) 6(y - ma) . (4)
Fig. 4. Raster scanned 1D object with decreasingand increasing frequency imaged through an optical imaging system (low pass). We now present some examples of spatial filtering with raster scanned signals. The test signal (fig. 4) shows decreasing and increasing spatial frequencies. This test signal was spatially filtered, which corresponds to a convolution with the point spread function of the 1`alter.In fig. 5 examples of band pass t'11tering with a double slit and in fig. 6 of knife edge faltering with coherent illumination are shown. We would like to point out that incoherent light is also applicable for this type of signal processing. We overcome the restriction to positive point spread functions by using optical transfer function synthesis. We make use of the fact that it is possible to synthesize out of two positive functions F 1 and F 2 one bipolar F by subtraction: F =F1 - F2
(F: bipolar, El,/72: positive).
(5)
Such a subtraction can be performed by electronic means. We recorded, therefore, the results of two different filtering operations using a TV-camera and stored the data in an image memory. Then the two results were subtracted digitally and displayed on a 89
Intensity
Position
,
I
Position
Fig. 6. Optical coherent filtering of object in fig. 4 with a knife edge filter.
Fig. 5. Optical coherent filtering of object in fig. 4 with a bandpass filter (double slit).
i I
L
L Fig. 7. Processing of scanned one-dimensional data with spatially incoherent light using optical transfer function synthesis for implementation of a bandpass f'dter. (a) Scanned one-dimensional function imaged through an optical imaging system and displayed on a monitor. (b) Incoherent filtering operation with double slit. (c) Incoherent filtering operation with single slit. (d) Subtraction of result (c) from result (b) in a digital image memory.
Volume 42, number 2
OPTICS COMMUNICATIONS
monitor. Such a hybrid, electronic-optical method combines the advantages of an incoherent processing system with the higher flexibility of TV-electrics. In fig. 7 the synthesis of a bandpass characteristic with incoherent light is shown. The effective OTF was synthesized out of a double slit pupil and a single slit pupil. The subtraction of the results of the filter operations gives the same effect as a coherent bandpass filter.
15 June 1982
of information processing. We also would like to point out that the concept of using dimensional transducers can be applied in a general sense not only for one-dimensional signals, but also for signals which depend on more than two coordinates. Examples are three.dimensional functions, e.g. scenes depending on space x , y and time t [8].
References 4. Condu~on The high processing capacity of optical systems can be applied to signals which are not two-dimensional. To this end we introduced dimensional transducers which change the number of coordinates, e.g. by displaying the input signal as a two-dimensional pattern. Besides an implementation of well known optical operations for these signals, new types of transformations can also be synthesized by using appropriate display patterns. It has been shown that incoherent illumination is applicable also for this type
[1] [2] [3] [4] [5] [6] [7] [8]
R.A. Spann, Proc. IEEE 53 (1965) 2137. C.S. Weaver, J.W. Goodman, Appl. Optics 5 (1966) 1248. C.E. Thomas, Appl. Optics 5 (1966) 1782. C.S. Weaver, S.D. Ramsey, J.W. Goodman and A.M. Rosie, Appl. Optics 9 (1970) 1672. G. Lebreton, E. de Bazelair, Opt. Eng. 19 (1980) 739. H.O. Bartelt, A.W. Lohmann, SPIE Advanced Institute Transformations in Optical Signal Processing, Seattle (USA) 1980. R.A. Athale, H.H. Szu and J.N. Lee, J. Opt. Soc. Am. 71 (1981) 1626. J. Hofer-Alfeis, SPIE Advanced Institute Transformations in Optical Processing, Seattle (USA) 1980.
91
OPTICS COMMUNICATIONS
15 June 1982
OPTICAL PROCESSING OF ONE-DIMENSIONAL SIGNALS H. BARTELTand A.W. LOHMANN Physikalisches Institut der Universitiit, 8520 Erlangen, Fed. Rep. Germany Received 19 February 1982
Long one-dimensional signals can be processed efficiently by optical methods if they are represented in a two-dimensional version. We show that by choosing an appropriate scanning pattern either new types of operations can be synthesized or classical operations like Fourier transformation or correlation can be implemented for this type of signals. Besides coherent illumination, incoherent light can also be used in order to reduce stability requirements, to increase the signal to noise ratio and to accommodate self-luminous inputs.
1. Efficient signal processing by optics One-dimensional time signals such as speach signals are mostly processed by electronic systems. On the other hand, processing by optical systems may become interesting if very long signals or signals with a very high number of samples have to be processed. The number of degrees of freedom in an optical system can easily be 106 or even 108 due to the twodimensional, parallel processing capacity. The input of an optical system has a two-dimensional character of course. For optimal use of the space-bandwidth product the input functions therefore have to be arranged in a two-dimensional pattern. In case of an originally one-dimensional time signal, many types of two-dimensional arrangements are conceivable. This additional freedom of design increases the variety of algorithms for optical processing of one-dimensional signals [1-7]. In the following sections we will first discuss a general system suitable for processing non-two-dimensional signals by optical methods. Then we will turn to the special case of one-dimensional input functions. Coherent as well as incoherent illumination will be used. Incoherent systems provide a better signal to noise ratio. On the other hand they are inherently restricted to the processing of positive functions. To overcome this restriction we combined this optical processing type with TV electronics which permitted
us to perform convolutions with a bipolar point spread function.
2. General processing system A general optical system for processing data which are not two-dimensional can be considered to consist of three components (fig. 1): a dimensional transducer, an optical processor and again a dimensional transducer. The optical processor is able to perform linear and space-variant operations such as Fourier transformation, correlation or general spatial filtering. In all cases the input function should be presented in a two-dimensional version in order to utilize completely the available processing capacity or the space-bandwidth product of the system. Therefore, the first dimensional $ e.- . . . . . . . . . . . . . . . . . . . . . .
r
i
1
N~
r
DIMENSIONN. PROCESSOR DIMENSIONAL TRANSDUCER(Un~lr.~=ace.-irM=,io~'I'RANSITd(~R Fig. 1. General architecture of a system S with two dimensional transducers DN/DP and Dp/DM, and a processor P. The input is fN, the output gM. In our case the dimensionality of the processor is P = 2.
87
Volume 42, number 2
OPTICS COMMUNICATIONS
transducer has to convert the input signal into a twodimensional format. In case of one-dimensional time signals as input a 1D/2D conversion would be necessary. This conversion is performed by appropriate scanning in two dimensions of the one-dimensional function. The dimensional transducer at the output end transforms in a similar manner the signal from two dimensions to a single dimension, or to any other dimensionality N, if so desired. The usefulness of such a system with dimensional transducers can be twofold. One type of application is the synthesis of new integral transformations. As an example we would like to mention the synthesis of a Hankel transformation out of a Fourier transformation and two radial scanning operations [6,7]. If we scan a one-dimensional signal u(t) in a radial manner we get as new, two-dimensional pattern u(r):
u(t) ~ u(r).
(1)
Now we apply a two-dimensional Fourier transformation (in polar coordinates): u(r) ~ 7 [u(r)] :
u(p)
: fu(r) Jo(27rrp)rdr, 0
(2)
J0: zero order Bessel function. The result depends only on the radial coordinate p. If we scan again radially, we then obtain the Hankel
II
15 June 1982
transformation of the original signal (fig. 2). With more general scan patterns we are able to perform a large variety of integral transformations. The second application, besides the synthesis of new operations, is the implementation of typical optical operations (Fourier transformation, correlation, convolution, etc.) but now for scanned, one-dimensional signals. In the following section we will explain this application of spatial filtering in more detail.
3. Spatial f'dterning operations of scanned signals It has been shown by several authors that operations like Fourier transformation or correlation can be implemented for one-dimensional signals by simple raster scanning. We will explain shortly the applications of this type of scanning and we will show that either coherent or incoherent light may be used for illumination. Coherent processing permits the implementation of complex •ter functions or impulse responses whereas incoherent methods are restricted to positive point spread functions. On the other hand stability requirements are lower, if incoherent light is used. Furthermore a larger variety of input devices such as TV monitors, printed pages and oscilloscopes are admissable. According to Spann [ 1] and others [2-5] a raster scanned signal gives a suitable input for optical spatial fflterning. To explain
la
Fig. 2. Synthesis of Hankel transformation. (a) Radially scanned function u(t). (b) 2D Fourier transformed signal. 88
Volume 42, number 2
OPTICS COMMUNICATIONS
15 June 1982
//.z/. :
'/
',
L/
'
/7-/'/ • -i~
f'l (x.y)
-~
! 2(x.y)
Fig. 3. Correlation of a raster scanned function f2 with a single line function fl" the principle let us assume a single line function gl(x) of size Ax and a long function g2(x) which will be repeated N times, mutually shifted and cut-off by a window of width 2Ax. These two-dimensional layouts fl(X, y) and/2(x, y) of the one-dimensional functions gl(x) and g2(x ) are shown in fig. 3. We want to convolve the original signals gl (x) and g2(x). We do this by convolving in two dimensions the pattemsfl(x,y ) andf2(x,y ). The width 2Ax o f f 2 ( x , y ) is wide enough that every portion ofg2(x ) appears twice inf2(x,y ). The need for this redundancy can be understood by visualizing the convolution o f f l and f2 as an overlap o f f 1 and f2, with continuous shifts inx and stepwise shifts in y. : Only the portion Ixl< Ax/2 of the convolution f l **f2 yields valid contributions to the wanted convolution gl *8'2 = C(x). This can be deduced also by inspecting the formulae: f l (X, y) = gl (X) 5(y)
f 2(x,y) = rect(x/2Ax) ~ g 2 ( x - A x ) 6(y-na) (n) rect(x / A x ) f 1(x, y )** f 2(x , y) = ~ C ( x - nAx) 8(y - ha). (n)
(3)
Until now we assumed for simplicity only a single line functionf1 . But thismethod works exactly the same way i f f l consists of several lines. The sum of the convolutions with single lines o f f 1 is equal to the convolution with the complete functionfl, which is de1`reed by
f l ( x , y ) = rect(x/Ax)~gl(x -- m a x ) 6(y - ma) . (4)
Fig. 4. Raster scanned 1D object with decreasingand increasing frequency imaged through an optical imaging system (low pass). We now present some examples of spatial filtering with raster scanned signals. The test signal (fig. 4) shows decreasing and increasing spatial frequencies. This test signal was spatially filtered, which corresponds to a convolution with the point spread function of the 1`alter.In fig. 5 examples of band pass t'11tering with a double slit and in fig. 6 of knife edge faltering with coherent illumination are shown. We would like to point out that incoherent light is also applicable for this type of signal processing. We overcome the restriction to positive point spread functions by using optical transfer function synthesis. We make use of the fact that it is possible to synthesize out of two positive functions F 1 and F 2 one bipolar F by subtraction: F =F1 - F2
(F: bipolar, El,/72: positive).
(5)
Such a subtraction can be performed by electronic means. We recorded, therefore, the results of two different filtering operations using a TV-camera and stored the data in an image memory. Then the two results were subtracted digitally and displayed on a 89
Intensity
Position
,
I
Position
Fig. 6. Optical coherent filtering of object in fig. 4 with a knife edge filter.
Fig. 5. Optical coherent filtering of object in fig. 4 with a bandpass filter (double slit).
i I
L
L Fig. 7. Processing of scanned one-dimensional data with spatially incoherent light using optical transfer function synthesis for implementation of a bandpass f'dter. (a) Scanned one-dimensional function imaged through an optical imaging system and displayed on a monitor. (b) Incoherent filtering operation with double slit. (c) Incoherent filtering operation with single slit. (d) Subtraction of result (c) from result (b) in a digital image memory.
Volume 42, number 2
OPTICS COMMUNICATIONS
monitor. Such a hybrid, electronic-optical method combines the advantages of an incoherent processing system with the higher flexibility of TV-electrics. In fig. 7 the synthesis of a bandpass characteristic with incoherent light is shown. The effective OTF was synthesized out of a double slit pupil and a single slit pupil. The subtraction of the results of the filter operations gives the same effect as a coherent bandpass filter.
15 June 1982
of information processing. We also would like to point out that the concept of using dimensional transducers can be applied in a general sense not only for one-dimensional signals, but also for signals which depend on more than two coordinates. Examples are three.dimensional functions, e.g. scenes depending on space x , y and time t [8].
References 4. Condu~on The high processing capacity of optical systems can be applied to signals which are not two-dimensional. To this end we introduced dimensional transducers which change the number of coordinates, e.g. by displaying the input signal as a two-dimensional pattern. Besides an implementation of well known optical operations for these signals, new types of transformations can also be synthesized by using appropriate display patterns. It has been shown that incoherent illumination is applicable also for this type
[1] [2] [3] [4] [5] [6] [7] [8]
R.A. Spann, Proc. IEEE 53 (1965) 2137. C.S. Weaver, J.W. Goodman, Appl. Optics 5 (1966) 1248. C.E. Thomas, Appl. Optics 5 (1966) 1782. C.S. Weaver, S.D. Ramsey, J.W. Goodman and A.M. Rosie, Appl. Optics 9 (1970) 1672. G. Lebreton, E. de Bazelair, Opt. Eng. 19 (1980) 739. H.O. Bartelt, A.W. Lohmann, SPIE Advanced Institute Transformations in Optical Signal Processing, Seattle (USA) 1980. R.A. Athale, H.H. Szu and J.N. Lee, J. Opt. Soc. Am. 71 (1981) 1626. J. Hofer-Alfeis, SPIE Advanced Institute Transformations in Optical Processing, Seattle (USA) 1980.
91