OPTICAL REVIEW Vol. 16, No. 1 (2009) 15–21
Indexed Optical Composite Filters Oum El Kheir A BRA, Esmail A HOUZI, Ignacio M ORENO1 , and Fakhita R EGRAGUI2 Institut National des Postes et Te´le´communications, Av. Allal al Fassi Madinat al Irfane, Rabat 10000, Maroc 1 Departamento de Ciencia y Tecnologı´a de Materiales, Universidad Miguel Herna´ndez, Elche 03202, Spain 2 UFR ACSYS, Laboratoire LIMIARF, De´partement de Physique, Faculte´ des Sciences, Universite´ Mohammed V, Rabat 10000, Maroc (Received July 8, 2008; Accepted October 12, 2008) A new technique called indexed composite filters (ICFs) for multiple object pattern recognition is proposed. Contrary to the usual so-called composite filters, the proposed ICF offers the possibility to identify which object of the learning class is detected at the output. The major advantage of ICF is that the output gives for every object detected two correlation peaks encoding the position and identity of the detected object. Computer simulation results are given using alphabetic input scenes that illustrates the efficiency of the proposed method. # 2009 The Optical Society of Japan Keywords: composite filters, phase shifting, optical correlator, optical image processing, pattern recognition
1.
correlations to identify which objects are detected from the learning class. In this paper, we propose a new design of composite filters for multiple pattern recognition based on an optical correlator called indexed composite filters (ICFs). The novel features and improved performance of this indexed multiplexed correlator originate from the possibility of localising and identifying the target present in the input scene between the objects of the learning class. Contrary to the usual output correlation plane where one peak indicates that an object is located in the input scene, the ICF output gives for every object detected two correlation peaks encoding the position and identity of the detected object. The major advantages of these filters can be extended to all kinds of composite filters like SDF in order to achieve recognition and location simultaneously of all the objects in an input scene while using one step optical correlation (Fig. 1); it is also convenient for optical implementation in the VDL correlator using spatial light modulators (SLMs). Section 2 gives the mathematical background of the proposed method and introduces the indexation concept in composite filters. In §3, we present computer simulations using a learning database of 26 alphabetic characters to demonstrate the method proposed. In §4, we give conclusions.
Introduction
Optical correlation-based pattern recognition techniques are generally implemented using the Vander Lugt (VDL) correlator or the joint transform correlator.1,2) These correlators are shift invariants, allowing us to locate the target in an input scene by locating the correlation peak. One of the advantages of using optical correlators in pattern recognition is their capability to process a large amount of data simultaneously and rapidly.3) This is a consequence of the parallelism inherent in optical systems and the two-dimensional (2D) Fourier transforming property of a lens. The key element in the VDL architecture is the matched filter.1) Hence, a variety of filters and techniques have been developed and experimentally demonstrated. The most multiplexing architectures proposed in the literature are based on one of the following techniques: the linear composite filter, introduced by Caulfield and Maloney,4) the generalised synthetic discrimination functions (SDF),3–5) the multiplexed correlation filter of Cottrell et al.,6) the multi-channel correlator7) and multi-object filters.7–9) Reference 10 summarises the main existing multiplexing techniques. The SDF which has received much attention is designed from a weighted sum of the Fourier transform of the references. If any object from the learning class, which includes the training images in the filter design, is presented in the input plane, the same peak appears in the centre of the image. Villarreal et al. proposed in ref. 9 a method to generate filters for multiple object recognition by means of the multiplexing of phase-only filters in a phase modulator. They segmented the correlation plane into eight zones and allocated for each object one zone of the correlation plane. The number of correlations that can be performed simultaneously was limited to eight because of the space available in the filter. In the multi object recognition process task, it is very interesting to identify which object is detected in the input scene. All the above multiplexing techniques require many
2.
Indexed Composite Filter Design
Pattern recognition based optical correlation in the case of images is a processing technique that compares a scene image to a predefined image called a reference. The correlator with which we are concerned is based on the conventional ‘‘4f’’ configuration of the frequency plane coherent optical correlator1) shown in Fig. 2. Let sðx; yÞ and rðx; yÞ denote the input scene and the reference, respectively. The reference information rðx; yÞ is Fourier transformed and its conjugate is introduced in the Fourier plane (P2) whereas the scene image is presented in the input plane (P1). In the Fourier plane behind the filter plane, the correlation function is expressed as
E-mail address:
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One step correlation Identification with the indexed
Location and
composite filter
identification of an object in the training
Fig. 1. Location and identification process using indexed correlation filter.
class Object unknown ...
Input plane P1 Collimating lens
...
Fourier plane P2 Lens
Output plane P3 Lens
Coherent light
Fig. 2. Optical processor.
Cðu; vÞ ¼ Sðu; vÞHðu; vÞ;
ð1Þ
where u and v are the spatial frequencies. Hðu; vÞ ¼ R ðu; vÞ is the matched filter to the reference rðx; yÞ. Sðu; vÞ and Rðu; vÞ are the Fourier transform of the input scene and the reference, respectively and the superscript denotes the complex conjugate. The correlation function is obtained in plane (P3) by cðx; yÞ ¼ sðx; yÞ hðx; yÞ;
ð2Þ
where hðx; yÞ is the impulse response of the filter matched to the reference rðx; yÞ and denotes the correlation operator. The correlation plane will exhibit a peak of correlation when the reference is located in the scene image. In a multi object detection task, one matched filter will be needed for each appearance of an object. To address this challenge, composite filters4) and the synthetic discriminant functions5) were introduced. A common weakness in all these optical composite correlation filters is that the output correlation peaks lead to the conclusion that one of the objects of the training class is present in the input scene but cannot be identified. In the following, the goal is to design a composite optical filter for multi-class recognition that allows both location and identification of an object in an input scene. Let us suppose a mono-object input scene sðx; yÞ containing the reference rðx; yÞ located at the origin, the scene is described as: sðx; yÞ ¼ rðx; yÞ;
ð3Þ
Let us suppose in the optical correlation process that we use a filter matched to the reference described as: hðx; yÞ ¼ rðx 0 ; y 0 Þ þ rðx 1 ; y 1 Þ;
ð4Þ
In the Fourier domain, we obtain a function that denotes the reference hðx; yÞ function as HICMF ðu; vÞ ¼ jRðu; vÞj expf j½’r ðu; vÞ þ ð0 u þ 0 vÞg þ jRðu; vÞj expf j½’r ðu; vÞ þ ð1 u þ 1 vÞg;
ð5Þ
’r ðu; vÞ and jRðu; vÞj are the phase and the amplitude information of the Fourier transform of the object rðx; yÞ, respectively. ð0 u þ 0 vÞ and ð1 u þ 1 vÞ are the linear phases due to the shifts of the reference in the input plane. We define HICMF ðu; vÞ as the indexed classical matched filter corresponding to the object rðx; yÞ. When performing correlation with an input scene containing a mono-object rðx; yÞ centred at the origin given by eq. (3) and the filter described by eq. (4), we will observe in the correlation plane two correlation peaks distant by a distance d equal to p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 0 Þ2 þ ð1 0 Þ2 . We propose to use this distance as a criterion for indexing objects of the learning class of the filter. We will assign to each object of this class an index to identify the objects at the output. In a similar way, we define HIPOF ðu; vÞ as the indexed phase only filter (IPOF) corresponding to an object rðx; yÞ by HIPOF ðu; vÞ ¼ expf j½’r ðu; vÞ þ ð0 u þ 0 vÞg þ expf j½’r ðu; vÞ þ ð1 u þ 1 vÞg;
ð6Þ
Horner and Gianino11) stated that the phase information is more important than the intensity information in images. Consequently, the use of phase only filters improves correlation features12,13) like optical efficiency, correlation peak and discrimination. Moreover, correlation peaks obtained using the POF are sharp and present no secondary lobes.12–14) In the present work, we opted to use only phase information of objects for the synthesis of the proposed filter because of the advantages mentioned above. We propose to synthesise an indexed composite filter starting from the
OPTICAL REVIEW Vol. 16, No. 1 (2009) Scene
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IFFT
image Analysis of the correlation plane
Decision
Indexed Composite Filter
Fig. 3. Block diagram of the correlation process using an indexed composite filter. FFT, fast Fourier transform; IFFT, inverse Fourier transform.
(rn, index n) Indexed composite filter design (r1, index1)
Learning class
linear combination of indexed POFs matched to the objects of the training set of the filter. The proposed composite filter differs from the phase only synthetic discriminant filters (POSDFs) because the proposed filter is able to localise and to identify the object present in the input scene by means of the parameter d, the index assigned to every object in the learning class. According to our approach and for the multiclass pattern recognition process, we propose to design an indexed composite POF (ICPOF) using a set of n objects fri ðx; yÞgi¼1:n (Fig. 3) considered as the learning class of the filter. For each object ri ðx; yÞ of the training set, we define an indexed phase only filter function Ri ðu; vÞ defined as follows: Ri ðu; vÞ ¼ expfj½’ri ðu; vÞ þ ði0 u þ i0 vÞg ð7Þ þ expfj½’ri ðu; vÞ þ ði1 u þ i1 vÞg; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi di ¼ ði1 i0 Þ2 þ ði1 i0 Þ2 is the index corresponding to the object ri ðx; yÞ from the training set. The indexed composite filter is defined as the sum of the indexed POFs of all references belonging to the learning database of the filter. n X Ri ðu; vÞ; ð8Þ Hðu; vÞ ¼ i¼1
Let us suppose that a mono-object input scene sðx; yÞ containing an object rk ðx xk ; y yk Þ from the training class is correlated (Fig. 3) with the filter given by eq. (8). Assuming that ðxk ; yk Þ are the position coordinates of the object rk in the input scene, in the Fourier plane the correlation is formulated as: ð9Þ Cðu; vÞ ¼ Rk ðu; vÞHðu; vÞ; n X Ri ðu; vÞ; ð10Þ Cðu; vÞ ¼ jRk ðu; vÞje j½’r ðu;vÞþðxk uþyk vÞ k
i¼1
jRk ðu; vÞj and ’rk ðu; vÞ are the amplitude and the phase of the input scene Fourier transform, respectively. Cðu; vÞ ¼ jRk ðu; vÞje j½’r ðu;vÞþðxk uþyk vÞ n X ðe j½’r ðu;vÞþði uþi vÞ i
0
þe
0
Þuþð yk k0 Þv
þ jRk ðu; vÞje j½ðxk k þ jRk ðu; vÞj
n X
1
Þuþð yk k1 Þv
ðe j½’r ðu;vÞ’r ðu;vÞþði uþi vÞ i
0
k
0
i6¼k
þe
j½’ri ðu;vÞ’rk ðu;vÞþði1 uþi1 vÞ
Þ;
ð12Þ
The first term on the right-hand side of eq. (12) is the autocorrelation distribution that will give a peak at the position ðxk k0 ; yk k0 Þ and the second term is the autocorrelation distribution which gives a second peak at the position ðxk k1 ; yk k1 Þ. The others terms are the cross correlations of the object k with the rest of the objects of the learning class. The optical correlation plane [Figs. 4(a) and 4(b)] presents two visible sharp peaks separated by a distance pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ ðk1 k0 Þ2 þ ðk1 k0 Þ2 even though the object is centred or is shifted. By computing the distance between the two peaks, we are able to identify which object was presented in the input scene since the index of each object is known. Hence, we proved that the method described above allows us to identify an object in an input scene by the indexation procedure. Obviously, the indexed composite is shift invariant. Let us demonstrate that the location is still conserved by the proposed method. If the object is shifted in the input scene, the correlation peaks will also be shifted [Fig. 4(b)], but the distance d between the two peaks remains constant. Next, we will show that the position of the object in the input scene can be deduced from the position coordinates of the two expected peaks in the correlation plane. Let us consider that ðxp1 ; yp1 Þ and ðxp2 ; yp2 Þ are the position coordinates of the two correlation peaks in the correlation plane: xp1 ¼ xk k0 ; xp2 ¼ xk k1 ;
k
i¼1 j½’ri ðu;vÞþði1 uþi1 vÞ
Cðu; vÞ ¼ jRk ðu; vÞje j½ðxk k
yp1 ¼ yk k0 ; yp2 ¼ yk k1 :
ð13Þ
0
Þ;
ð11Þ
As ðk0 ; k0 Þ and ðk1 ; k1 Þ are known, ðxp1 ; yp1 Þ and ðxp2 ; yp2 Þ are obtained by analysing the correlation plane, hence ðxk ; yk Þ are computed as below:
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α 0k
peaks
β 0k
.............
.....
Fig. 5. Samples of input characters used in simulation. β 1k
d Object at
α 1k
the origin
(a)
Object shifted xk – α 0k
yk –β 0k
Correlation peaks
(a)
yk –β1k
x k –α 1k
(b) Fig. 4. Correlation using indexed composite filter. (a) Correlation plane in the case of a centred object in the input scene. (b) Correlation plane in the case of a shifted object in the input scene.
xp1 þ xp2 k0 k1 ; 2 yp1 þ yp2 k0 k1 ; yk ¼ 2
xk ¼
ð14Þ
In the particular case, where k0 ¼ k1 and k0 ¼ k1 xp1 þ xp2 ; 2 yp1 þ yp2 yk ¼ ; 2
(b) Fig. 6. Correlation example: (a) input scene centred and (b) intensity distribution using ICPOF. The distance between the two peaks is the index corresponding to P.
xk ¼
ð15Þ
From eq. (15), the object position in the input scene is then easily deduced. 3.
Computer Simulations
Computer simulations are performed to illustrate the behaviour of the proposed indexed composite filter in the correlation plane. To show the effectiveness of the ICF, let us take the example of an indexed composite filter involving 26 references of alphabetic characters in the learning class. The input characters used are all upper case, Roman fonts, letters; related examples of these are shown in Fig. 5. Similar results to those presented have been achieved with other fonts and with lower case characters.
The input patterns were inserted into low dimension arrays (32 32 pixels). Then, an ICPOF of (256 256 pixels) was calculated for the twenty six input patterns ri ðx; yÞ according to eq. (6). Each object of the learning class was indexed using the method described above. The matching process was performed by cross correlating an input image with the ICF and analysing the resulting correlation plane. Figure 6(a) presents one of the patterns which consist of the alphabetic character P that is centred in a (256 256 pixels) input scene; in the correlation plane, as depicted in Fig. 6(b) we note two visible peaks, displaced from the centre by an amount equal to the index assigned to the letter P at the design stage. Figures 7(a) and 7(b) show the same performance obtained with the pattern G. We have performed the simulations with all patterns, and they were unambiguously identified and located, although some patterns are very similar.
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(a) (a)
(b) Fig. 8. Correlation example: (a) input scene shifted and (b) intensity distribution using ICPOF. The distance between the two peaks is the index corresponding to P.
(b) Fig. 7. Correlation example: (a) input scene centred and (b) intensity distribution using ICPOF. The distance between the two peaks is the index corresponding to G.
We investigate also when the input scene is off-centre as shown in Figs. 8(a) and 9(a), the expected correlation peaks are displaced but the distance does not change [Figs. 8(b) and 9(b)], and the position of the input in the scene is deduced from the coordinates of the correlation peaks as shown above. To avoid the overflow of the peaks in the correlation plane due to the DFT implementation of the Fourier transform, both the input scene and the filter are inserted in a larger array (512 512) using the technique of zero padding. Finally, we test the robustness of the proposed filter in the presence of additive input white noise. The filter performance is investigated using two well-known metrics, the signal-to-noise ratio (SNR) and the peak-to-correlation energy ratio (PCE). The SNR is defined as the ratio of the expected value squared of the correlation peak amplitude to the variance of the correlation peak amplitude; the PCE metric is defined as the ratio of the expected value squared of the correlation peak to the average expected value of the output-signal energy. Here we present some simulation results to illustrate the noise tolerance of the proposed filter. The SNR and PCE are measured. The input noise on the input scene was simulated
(a)
(b) Fig. 9. Correlation example: (a) input scene shifted and (b) intensity distribution using ICPOF. The distance between the two peaks is the index corresponding to P.
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(d) (b) Fig. 10. (a) Original image. (b) Image corrupted by additive zero-mean white noise with a standard deviation of 0.7. (c) Output correlation obtained with the original image. (d) Output correlation intensity for the noisy image.
for several values of standard deviation. Performance parameters were computed for 50 different sample realisations of input signals corrupted by noise. The input characters were corrupted by Gaussian white noise with standard deviation varying from 0.1 to 1. Figure 10 shows the original image [Fig. 10(a)] and the noisy image with white noise of a standard deviation of 0.7 [Fig. 10(b)] and the corresponding correlation planes [Figs. 10(c) and 10(d)]. Figures 11(a) and 11(b) illustrate the SNR and the PCE, respectively, as functions of different standard deviations of the additive Gaussian white noise. The performances of the proposed filter show similarities to those of phase only filters in terms of sensitivity to an additional Gaussian white noise. 4.
Conclusion A method for designing composite filters for detection
and identification tasks is proposed. Called indexed optical composite filters, the method is used for detection and identification of a target among a set of objects belonging to the learning class of the filter in a Vander Lugt correlator (VLC). The proposed indexed composite filter yields for each object two correlation peaks in the correlation plane, encoding the position and the identity of the detected object. This method is very efficient for locating and identifying targets. The indexed composite filters were obtained by the sum of the IPOFs matched to the objects of the learning database of the filter; nevertheless, major advantages of these filters can be extended to all kinds of composite filters like SDF in order to achieve recognition and location simultaneously of all objects in an input scene while using one step optical correlation. The ICF is also valued because of its conven-
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Fig. 11. Simulation results of the proposed filter in the presence of additive white noise with different standard deviations. (a) Output PCE. (b) Output SNR.
ience for optical implementation in the VLC using spatial light modulators. The simulation results obtained in the case of alphabetic characters show good performance by the proposed method. We also evaluated the performance of the filter in terms of input noise and found it very suitable for segmented input scenes. References 1) Vander Lugt: IEEE Trans. Inf. Theory 10 (1964) 139. 2) S. Weaver and J. W. Goodman: Appl. Opt. 5 (1966) 1248. 3) J. W. Goodman: Introduction to Fourier Optics (McGrawHill, New York, 1996) 2nd ed., p. 256. 4) H. J. Caulfield and W. T. Maloney: Appl. Opt. 8 (1969) 2354. 5) C. F. Hester and D. Casasent: Appl. Opt. 19 (1980) 1758. 6) D. M. Cottrell, J. A. Davis, M. P. Schamschula, and R. A.
Lilly: Appl. Opt. 26 (1987) 934. 7) A. Vargas, J. Campos, M. J. Yzuel, C. Iemmi, and S. Ledesma: Appl. Opt. 37 (1998) 2063. 8) J. Campos, A. Ma´rquez, M. J. Yzuel, J. A. Davis, D. M. Cottrell, and I. Moreno: Appl. Opt. 39 (2000) 5965. 9) M. Villarreal, C. Iemmi, and J. Campos: Opt. Eng. 42 (2003) 2354. 10) B. V. Kumar: Appl. Opt. 31 (1992) 4773. 11) J. L. Horner and P. D. Gianino: Appl. Opt. 23 (1984) 812. 12) C. Iemmi, S. Ledesma, J. Campos, and M. Villarreal: Appl. Opt. 39 (2000) 1233. 13) E. Ahouzi, J. Campos, and M. J. Yzuel: Opt. Lett. 19 (1994) 1340. 14) E. Ahouzi, J. Campos, and M. J. Yzuel: Opt. Eng. 37 (1998) 2351.
