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A method of recognition of blurred-edge objects
Volume 39, number 4
OPTICS COMMUNICATIONS
15 October 1981
A METHOD OF RECOGNITION OF BLURRED-EDGE OBJECTS A.M. BEKKER, N.I. BUKHTOYAROVA and B.G. TURUKHANO
Leningrad NuclearPhysicsInstitute Acad. Sci. USSR, Gatchina,LeningradDistrict, 188350 USSR Received 9 February 1981 Revised manuscript received 2 June 1981
The problem of recognition of blurred-edge objects is discussed. An optimal algorithm for the recognition of such objects is proposed. The holographic filter realizing this algorithm can be formed with a spatially modulated reference wave by using the non-linear regime of the filter recording. The choice of the optimal filter recording regime was made by computer simulation of the holographic process.
1. Introduction In analyzing citological preparations, bacterial colonies and similar problems, one deals with approximately circular objects. It is possible to define an object as approximately circular if its rim lies within a certain ring of average radius R 0 and of width 2d (fig. 1). Assume that such objects belong to the class ARod. Thus the problem o f determining the dimensions of a blurrededge object reduces to dividing the input object set into nonintersecting classes ARoidi, it being assumed that some of input objects belong to none of the classes ARoidi. As will be shown, the optical filter for a frequency plane correlator (FPC) is the filter with impulse response ~0(r) given in polar coordinates by 8(r - ( g 0 - - a)) ~ ( r , O ) c~
R o - d
Fig. 1. An example of the blurred-edge object. plates has been extensively analysed and discussed in a number of papers [ 1 - 4 ] . In this work we use the nonlinearity of the t A - E curve to synthesize the optimal transfer function of a FPC.
a(r - (g 0 + a)) -
R 0 +a
(1)
where 6 (r) is the Dirac delta function. In the conventional method of matched filter recording, one must obtain a transparency having amplitude transmittance defined by the expression (1), Such a complex transparency is difficult to achieve since the phase of the function ~ ) changes. A different method of optimal filter realization is proposed in this paper. The method involves the use of a spatially modulated reference wave and a non-linear regime for filter recording. The nonlinearity of the t A - E curve for film and holographic 0 030-4018/81/0000-0000/$ 02.75 © 1981 North-Holland
2. Theory We will define the system by the functional L(q) determined on the set of input objects. This functional describes the value of the output signal versus the amplitude transmittance function of the input transparency. Let us require that L(q) should have the following properties:
1. L(q)=C=~O
forqEARoid i
(2a)
2. L(q) = 0
for q EARo]d] f o r / ~ i
(2b) 231
Volume 39, number 4
3. L ( q ) < C
OPTICS COMMUNICATIONS
for q E Ano]d j for any ].
(2c)
It will be shown that to meet the conditions (2) it is necessary and sufficient to assure that
~oi(r,O) =t0i(r) C (5(r - (Roi - di) )
-G--<',
:
15 October 1981
condition (3) is necessary for fulfilment of eqs. (2) we consider the subset of the input objects to be the set of the circles of various radii. Let us note that the circles belong to the class ARo d if their radii lie within the interval [R 0 - d, R 0 + d ] . The response of the system to the circle qR(r) of the radius R can be calculated by
6(r - (Roi + di))~ -
(3)
+4
Supposing that the output signal is measured at the point of the output plane corresponding to the center of the ring including the rim of the object, we obtain the system response V ~ q ) to the input object q(r, 0):
R
x,/L(qR ) = 2re f
~o(R) r dr.
(6)
0 On differentiation of both sides of the eq. (6) with respect to R we obtain 1
d
(7)
~0(R) = 27rR dR (X/~(qR))" 2?r
f f
0 x
0
o)
6(r - (Roi - di) )
Conditions (2) for the objects of the class in question can be written as:
-
L(qR)OCLI(R)
6(r - (Roi + di))~ r
j drdO
R E [Roi - di, Roi + di]
1, =
Then
(8)
O, 2rr
f
R E [Roi -
/'5(. --
~oi(r) = C 2 ~ 2~T
q(Roi + di,O ) dO.
(4)
As follows from eq. (4), condition (3) guarantees that condition (2) will be met. In fact if the object belongs to the class ARoid i we can write by definition:
q(Roi - di) = e l ,
q(Roi + di) = C2,
and then assuming for the sake of simplicity that C 1 = 1 and C 2 = 0:
x/-L(q) o: 27rlq(Roi _ di ) _ q(Ro i + di)] = 2n.
(5)
If object q(r , O) belongs to the class ARoll d.), (l" =/=i) there is a position o f the center of the ring such that the rim of the object lies either within the circle of radius Roi d i or outside the circle of radius Roi + di, otherwise classes ARoid i and ARo.d] would intersect In either 7 case, as follows from eq. (4) L(q) = O. If object q(r,O) belongs to none of classes ARo]d], then either the first integral in eq. (4) is less than 1 or the second one is greater than 0 and condition (2c) is met. To prove that 232
Roi + di]
Substitution of eq. (8) in eq. (7) yields:
q(%, - d,, o) dO
0
- f 0
d i,
(Roi-
...... ROi _ di
eli))
6(r.~_(Roi+_._d3))~ RO i + di
]. (9)
Thus condition (3) is necessary for fulfilment of the eq. (2) for the subset of the input objects and hence for the whole set. To choose the means of realization of a filter whose impulse response is close to that of the ideal filter defined by eq. (3), consider the complex transmittance function of the ideal filter ~I(P). From eq. (3) it follows that ¢Pi(p) o¢J0((R 0 - d)p) - J o ( ( R o + d)p)
(10)
where here and elswhere Jk(r) denotes the Bessel function of order k. A plot of the function ~I(P) for R 0 = 1 and d = 0.1 is shown in fig. 2. The function ~bi(p) is seen to be characterized by points Rsi where the function SGN ~i(,o) deviates from a quasiperiodic law of alternation of values 1 and - 1 . Zero points of the function J1 (P)IP and the function ¢PI(P) are close together and the values of these functions agree in sign within the interval [0,Rsl ] and are of opposite sign within [Rsl ,Rs2 ] . This fact allows us to choose the circle of
Volume 39, number 4
OPTICS COMMUNICATIONS
15 October 1981
where TA(E ) is the amplitude transmittance-exposure function of the photoemulsion, which was measured and introduced point by point into the computer. By computer simulation for any parameters Q0 and T O of the filter recording, the output signal of the system Foo ' To(R) versus radius of the input circle can be determined. Then we determine the parameters minimizing the functional
I '"'"'""'"".............
A(Q0 , r 0)
=f
IFO.oTo(R)/FQoTo(RO)-LI(R)I 2 dR
(13)
0
r a d i u s ffm l'ztqumaclr 1)laue
Fig. 2.
(p)
(solid line): complex transmittance function of
the ideal filter; . . . . (dashed line): plot of the function 2J 160)/0;
..... (dotted line): plot of the function 2Jl (0.122 0)/0.122 p. radius I as the object for filter recording. To reverse the phase of the function 4(,o) at the points Rsi we can use a reference wave whose phase changes by rr at these points. In practice, it is convenient to use a reference wave that is the Fourier transformation of the circle of radius 1.22 d/R 0 . To obtain the required ratio of the object-reference amplitudes, non-linear properties of the recording material are employed. A choice of the optimal recording parameters was made by computer simulation of the holographic process (see appendix 2 to ref. [4] ). According to this method the complex transmittance function of the hologram may be expressed as:
where LI(R ) is determined by eq. (8). In calculation the integral in eq. (13) was replaced by a partial sum within the interval [R 0 - 5d, R 0 + 5d]. The Nelder-Midd method of deformed polyhendrons [5] was used, since it does not call for a great accuracy of the calculation of the function values and converges rather quickly. Calculations have shown that the variation of only 1.0
0.5
0.7
0.9
radius
I
1.1
I.)
of inputcix~le (R) a
¢60) = exp(i arg f(p)/g(p)
X D(Q0 [f(P)/g(P)I, TOIg(o)12),
(11)
where f(p) is the amplitude of the subject wave, g(p) is the amplitude of the reference wave, T O is the exposure time, Q0 defines the ratio of the amplitude of the subject wave to the amplitude of the reference wave in the hologram recording plane and
Ir
o(Q,r)=l f rA(r(1 +Q2+2o cos,)) cos, d,, 0
(12)
b
c
d
Fig. 3. Experimental results: a) system response as a function of the radius of the input circle; (solid line): experimental curve; . . . . (dashed line): system response of the ideal filter;
..... (dotted line): calculated curve; b) input transparency; c) impulse response of the filter; d) response of the FPC to the input transparency. 233
Volume 39, number 4
OPTICS COMMUNICATIONS
15 October 1981 LI
1
L2
E~ ~J
4~>~... m
'
0.5
,
F
4~
Fig. 5. Sketch of the set-up for filter recording. L1 - collimating lens; L2 - Fourier-transforming lens; R - object hole; r - reference hole; H - filter recording plane.
,-4
•
.
0.4-
.
i
0.8
,
i
1.2
Ig(energy)
°
i
,
1.6
J
,
1.8
(lg E)
Fig. 4. The tA-E curve for Agfa-Gevaert 8E75 emulsion produced by D-76 developer. two parameters Qo and T O results in too great a value of the functional for optimal parameters Q0 and T O . Hence such a m e t h o d o f filter recording does not allow us to reliably recognize the objects belonging to the class ARod. A suitable value o f the functional was found b y varying three parameters: Q0 - the ratio of amplitude of the subject wave to the amplitude of the reference wave at the point 19 = 0; T 1 - exposure time for the central filter zone of radius Rsl corresponding to the zero order o f the reference beam; T 2 - the exposure time for outside of this zone. The values of the optimal parameters are Qo = 9.8; T1 = 1209; T 2 = 10303. The calculated curve of the system response is shown in fig. 3a. In all calculation we made use of the amplitude transmittance function o f Agfa-Gevaert 8E75 emulsion produced b y D-76 developer (fig. 4).
3. Experimental results
The experimental set-up for the filter recording is shown schematically b y fig. 5. There are two holes of 3.9 and 0 A 8 m m radii in front focal plane of the lens L 2 . The former forms the subject wave front and the latter, that of the reference wave. To obtain the required ratio of amplitudes of the subject wave and reference, an absorbing filter was used. Various exposure times were realized b y a masking a portion of the filter. The opened portion o f the filter was exposed for a period of time T = T 2 - T 1 and the whole of the filter was exposed for T 1 . To study the properties of the filter 234
recorded we measured the o u t p u t signal intensity as a function of the radius of the input circle. This curve is shown in fig. 3a and one can see a good agreement between the experimental and calculated data. The input transparency composed b y b o t h objects belonging to the class ARod for R 0 = 3.9 and d = 0.39 (left column) and those not belonging to this class (right column) is shown in fig. 3b. The response of the FPC with the filter under discussion (the photograph of its impulse response can be seen in fig. 3c) to this transparency is shown in fig. 3d. Distributions of intensity in the right and in the left of this photograph correspond to the convolution and correlation of the input signal with the impulse response o f the filter. In our case these operations are equivalent because the impulse response of the filter is centro-symmetric. The middle area corresponds to zero diffraction order. One can see that the high intensity o f the correlation signal corresponds to the centers of the objects belonging to the class ARod and correlation signal is rather weak for objects not belonging to this class. A high selectivity o f the filter is illustrated in this figure. This method o f using a spatially modulated reference wave and a non-linear t A - E curve for filter recording can also be applied for recognition of objects whose rim is a more complicated shape.
References
[I ] A. Kozma, J. Opt. Soc. Am. 56 (1966) 428. [2] J.W. Goodman and G.R. Knight, J. Opt. Soc. Am. 58 (1968) 1276. [3] O. Bryngdahl and A. Lohmann, J. Opt. Soc. Am. 58 (1968) 1325. [4] A.M. Bekker, N.I. Bukhtoyarova and B.G. Turukhano, Optics Comm. 31 (1979) 290. [5 ] D.M. HimmeIblau, Applied nonlinear programming (McGrawHill, 1972).
OPTICS COMMUNICATIONS
15 October 1981
A METHOD OF RECOGNITION OF BLURRED-EDGE OBJECTS A.M. BEKKER, N.I. BUKHTOYAROVA and B.G. TURUKHANO
Leningrad NuclearPhysicsInstitute Acad. Sci. USSR, Gatchina,LeningradDistrict, 188350 USSR Received 9 February 1981 Revised manuscript received 2 June 1981
The problem of recognition of blurred-edge objects is discussed. An optimal algorithm for the recognition of such objects is proposed. The holographic filter realizing this algorithm can be formed with a spatially modulated reference wave by using the non-linear regime of the filter recording. The choice of the optimal filter recording regime was made by computer simulation of the holographic process.
1. Introduction In analyzing citological preparations, bacterial colonies and similar problems, one deals with approximately circular objects. It is possible to define an object as approximately circular if its rim lies within a certain ring of average radius R 0 and of width 2d (fig. 1). Assume that such objects belong to the class ARod. Thus the problem o f determining the dimensions of a blurrededge object reduces to dividing the input object set into nonintersecting classes ARoidi, it being assumed that some of input objects belong to none of the classes ARoidi. As will be shown, the optical filter for a frequency plane correlator (FPC) is the filter with impulse response ~0(r) given in polar coordinates by 8(r - ( g 0 - - a)) ~ ( r , O ) c~
R o - d
Fig. 1. An example of the blurred-edge object. plates has been extensively analysed and discussed in a number of papers [ 1 - 4 ] . In this work we use the nonlinearity of the t A - E curve to synthesize the optimal transfer function of a FPC.
a(r - (g 0 + a)) -
R 0 +a
(1)
where 6 (r) is the Dirac delta function. In the conventional method of matched filter recording, one must obtain a transparency having amplitude transmittance defined by the expression (1), Such a complex transparency is difficult to achieve since the phase of the function ~ ) changes. A different method of optimal filter realization is proposed in this paper. The method involves the use of a spatially modulated reference wave and a non-linear regime for filter recording. The nonlinearity of the t A - E curve for film and holographic 0 030-4018/81/0000-0000/$ 02.75 © 1981 North-Holland
2. Theory We will define the system by the functional L(q) determined on the set of input objects. This functional describes the value of the output signal versus the amplitude transmittance function of the input transparency. Let us require that L(q) should have the following properties:
1. L(q)=C=~O
forqEARoid i
(2a)
2. L(q) = 0
for q EARo]d] f o r / ~ i
(2b) 231
Volume 39, number 4
3. L ( q ) < C
OPTICS COMMUNICATIONS
for q E Ano]d j for any ].
(2c)
It will be shown that to meet the conditions (2) it is necessary and sufficient to assure that
~oi(r,O) =t0i(r) C (5(r - (Roi - di) )
-G--<',
:
15 October 1981
condition (3) is necessary for fulfilment of eqs. (2) we consider the subset of the input objects to be the set of the circles of various radii. Let us note that the circles belong to the class ARo d if their radii lie within the interval [R 0 - d, R 0 + d ] . The response of the system to the circle qR(r) of the radius R can be calculated by
6(r - (Roi + di))~ -
(3)
+4
Supposing that the output signal is measured at the point of the output plane corresponding to the center of the ring including the rim of the object, we obtain the system response V ~ q ) to the input object q(r, 0):
R
x,/L(qR ) = 2re f
~o(R) r dr.
(6)
0 On differentiation of both sides of the eq. (6) with respect to R we obtain 1
d
(7)
~0(R) = 27rR dR (X/~(qR))" 2?r
f f
0 x
0
o)
6(r - (Roi - di) )
Conditions (2) for the objects of the class in question can be written as:
-
L(qR)OCLI(R)
6(r - (Roi + di))~ r
j drdO
R E [Roi - di, Roi + di]
1, =
Then
(8)
O, 2rr
f
R E [Roi -
/'5(. --
~oi(r) = C 2 ~ 2~T
q(Roi + di,O ) dO.
(4)
As follows from eq. (4), condition (3) guarantees that condition (2) will be met. In fact if the object belongs to the class ARoid i we can write by definition:
q(Roi - di) = e l ,
q(Roi + di) = C2,
and then assuming for the sake of simplicity that C 1 = 1 and C 2 = 0:
x/-L(q) o: 27rlq(Roi _ di ) _ q(Ro i + di)] = 2n.
(5)
If object q(r , O) belongs to the class ARoll d.), (l" =/=i) there is a position o f the center of the ring such that the rim of the object lies either within the circle of radius Roi d i or outside the circle of radius Roi + di, otherwise classes ARoid i and ARo.d] would intersect In either 7 case, as follows from eq. (4) L(q) = O. If object q(r,O) belongs to none of classes ARo]d], then either the first integral in eq. (4) is less than 1 or the second one is greater than 0 and condition (2c) is met. To prove that 232
Roi + di]
Substitution of eq. (8) in eq. (7) yields:
q(%, - d,, o) dO
0
- f 0
d i,
(Roi-
...... ROi _ di
eli))
6(r.~_(Roi+_._d3))~ RO i + di
]. (9)
Thus condition (3) is necessary for fulfilment of the eq. (2) for the subset of the input objects and hence for the whole set. To choose the means of realization of a filter whose impulse response is close to that of the ideal filter defined by eq. (3), consider the complex transmittance function of the ideal filter ~I(P). From eq. (3) it follows that ¢Pi(p) o¢J0((R 0 - d)p) - J o ( ( R o + d)p)
(10)
where here and elswhere Jk(r) denotes the Bessel function of order k. A plot of the function ~I(P) for R 0 = 1 and d = 0.1 is shown in fig. 2. The function ~bi(p) is seen to be characterized by points Rsi where the function SGN ~i(,o) deviates from a quasiperiodic law of alternation of values 1 and - 1 . Zero points of the function J1 (P)IP and the function ¢PI(P) are close together and the values of these functions agree in sign within the interval [0,Rsl ] and are of opposite sign within [Rsl ,Rs2 ] . This fact allows us to choose the circle of
Volume 39, number 4
OPTICS COMMUNICATIONS
15 October 1981
where TA(E ) is the amplitude transmittance-exposure function of the photoemulsion, which was measured and introduced point by point into the computer. By computer simulation for any parameters Q0 and T O of the filter recording, the output signal of the system Foo ' To(R) versus radius of the input circle can be determined. Then we determine the parameters minimizing the functional
I '"'"'""'"".............
A(Q0 , r 0)
=f
IFO.oTo(R)/FQoTo(RO)-LI(R)I 2 dR
(13)
0
r a d i u s ffm l'ztqumaclr 1)laue
Fig. 2.
(p)
(solid line): complex transmittance function of
the ideal filter; . . . . (dashed line): plot of the function 2J 160)/0;
..... (dotted line): plot of the function 2Jl (0.122 0)/0.122 p. radius I as the object for filter recording. To reverse the phase of the function 4(,o) at the points Rsi we can use a reference wave whose phase changes by rr at these points. In practice, it is convenient to use a reference wave that is the Fourier transformation of the circle of radius 1.22 d/R 0 . To obtain the required ratio of the object-reference amplitudes, non-linear properties of the recording material are employed. A choice of the optimal recording parameters was made by computer simulation of the holographic process (see appendix 2 to ref. [4] ). According to this method the complex transmittance function of the hologram may be expressed as:
where LI(R ) is determined by eq. (8). In calculation the integral in eq. (13) was replaced by a partial sum within the interval [R 0 - 5d, R 0 + 5d]. The Nelder-Midd method of deformed polyhendrons [5] was used, since it does not call for a great accuracy of the calculation of the function values and converges rather quickly. Calculations have shown that the variation of only 1.0
0.5
0.7
0.9
radius
I
1.1
I.)
of inputcix~le (R) a
¢60) = exp(i arg f(p)/g(p)
X D(Q0 [f(P)/g(P)I, TOIg(o)12),
(11)
where f(p) is the amplitude of the subject wave, g(p) is the amplitude of the reference wave, T O is the exposure time, Q0 defines the ratio of the amplitude of the subject wave to the amplitude of the reference wave in the hologram recording plane and
Ir
o(Q,r)=l f rA(r(1 +Q2+2o cos,)) cos, d,, 0
(12)
b
c
d
Fig. 3. Experimental results: a) system response as a function of the radius of the input circle; (solid line): experimental curve; . . . . (dashed line): system response of the ideal filter;
..... (dotted line): calculated curve; b) input transparency; c) impulse response of the filter; d) response of the FPC to the input transparency. 233
Volume 39, number 4
OPTICS COMMUNICATIONS
15 October 1981 LI
1
L2
E~ ~J
4~>~... m
'
0.5
,
F
4~
Fig. 5. Sketch of the set-up for filter recording. L1 - collimating lens; L2 - Fourier-transforming lens; R - object hole; r - reference hole; H - filter recording plane.
,-4
•
.
0.4-
.
i
0.8
,
i
1.2
Ig(energy)
°
i
,
1.6
J
,
1.8
(lg E)
Fig. 4. The tA-E curve for Agfa-Gevaert 8E75 emulsion produced by D-76 developer. two parameters Qo and T O results in too great a value of the functional for optimal parameters Q0 and T O . Hence such a m e t h o d o f filter recording does not allow us to reliably recognize the objects belonging to the class ARod. A suitable value o f the functional was found b y varying three parameters: Q0 - the ratio of amplitude of the subject wave to the amplitude of the reference wave at the point 19 = 0; T 1 - exposure time for the central filter zone of radius Rsl corresponding to the zero order o f the reference beam; T 2 - the exposure time for outside of this zone. The values of the optimal parameters are Qo = 9.8; T1 = 1209; T 2 = 10303. The calculated curve of the system response is shown in fig. 3a. In all calculation we made use of the amplitude transmittance function o f Agfa-Gevaert 8E75 emulsion produced b y D-76 developer (fig. 4).
3. Experimental results
The experimental set-up for the filter recording is shown schematically b y fig. 5. There are two holes of 3.9 and 0 A 8 m m radii in front focal plane of the lens L 2 . The former forms the subject wave front and the latter, that of the reference wave. To obtain the required ratio of amplitudes of the subject wave and reference, an absorbing filter was used. Various exposure times were realized b y a masking a portion of the filter. The opened portion o f the filter was exposed for a period of time T = T 2 - T 1 and the whole of the filter was exposed for T 1 . To study the properties of the filter 234
recorded we measured the o u t p u t signal intensity as a function of the radius of the input circle. This curve is shown in fig. 3a and one can see a good agreement between the experimental and calculated data. The input transparency composed b y b o t h objects belonging to the class ARod for R 0 = 3.9 and d = 0.39 (left column) and those not belonging to this class (right column) is shown in fig. 3b. The response of the FPC with the filter under discussion (the photograph of its impulse response can be seen in fig. 3c) to this transparency is shown in fig. 3d. Distributions of intensity in the right and in the left of this photograph correspond to the convolution and correlation of the input signal with the impulse response o f the filter. In our case these operations are equivalent because the impulse response of the filter is centro-symmetric. The middle area corresponds to zero diffraction order. One can see that the high intensity o f the correlation signal corresponds to the centers of the objects belonging to the class ARod and correlation signal is rather weak for objects not belonging to this class. A high selectivity o f the filter is illustrated in this figure. This method o f using a spatially modulated reference wave and a non-linear t A - E curve for filter recording can also be applied for recognition of objects whose rim is a more complicated shape.
References
[I ] A. Kozma, J. Opt. Soc. Am. 56 (1966) 428. [2] J.W. Goodman and G.R. Knight, J. Opt. Soc. Am. 58 (1968) 1276. [3] O. Bryngdahl and A. Lohmann, J. Opt. Soc. Am. 58 (1968) 1325. [4] A.M. Bekker, N.I. Bukhtoyarova and B.G. Turukhano, Optics Comm. 31 (1979) 290. [5 ] D.M. HimmeIblau, Applied nonlinear programming (McGrawHill, 1972).