Abstract

We present a system for shape tolerant three-dimensional (3D) recognition of biological microorganisms using holographic microscopy. The system recognizes 3D microorganisms by analyzing complex images of the 3D microorganism restored from single-exposure on-line (SEOL) digital hologram. In this technique the SEOL hologram is recorded by a Mach-Zehnder interferometer, and then the original complex images are reconstructed numerically at different depths by inverse Fresnel transformation. For recognition, a number of sampling segment features are arbitrarily extracted from the restored 3D image. These samples are processed using a number of cost functions and the sampling distributions for the difference of the parameters (location, dispersion) between the sample segment features of the reference and input 3D image are calculated using a statistical sampling method. Then, a hypothesis testing for the equality of the parameters between reference and input 3D image is performed for a statistical decision about populations on the basis of sampling distribution information. Student’s t distribution and Fisher’s F distribution are used to statistically analyze the difference of means and the ratio of variances of two populations, respectively. The proposed system is designed to be tolerant to recognizing various, plain microorganisms with analogous shape such as bacteria and algae. Preliminary experimental results are presented to illustrate the robustness of the proposed recognition system using statistical inference.

©2005 Optical Society of America

1. Introduction

There have been substantial research and development activities in 2D object recognition [1–6]. Recently, optical systems for recognition of three-dimensional (3D) objects have been investigated [7–9]. 3D microscopy by digital holography has been studied because of its broad applications [10]. One of the advantages of digital holography (DH) is that it automatically produces focused volume images of 3D micro-object from a single digital hologram without any mechanical scanning as is needed in conventional microscopy. It has been applied to the visualization and recognition of 3D objects and 3D microorganisms [11–16]. In phase-shifting digital holography, the optical system should have stable environmental vibration and fluctuation during multiple exposures in order to obtain precise phase-shift digital holograms and the phase-shift in the reference beam should be precise. Consequently these issues make it difficult to recognize or visualize dynamic events such as moving 3D objects or evolving 3D biological microorganisms. Single-exposure on-line (SEOL) digital holography is an attractive technique for 3D recognition and identification of objects [17]. In this technique the optical system is less sensitive to environmental vibration and requires only a single-exposure, so it can provide real-time 3D recognition of moving, growing and reproducing microorganisms.

Automated identifying system of bacteria and micro-organisms has the potential applications. In particular, biological microorganisms such as bacteria and algae are very minute and simple in their thickness and shape, respectively. They are usually unicellular though they often grow and branch in colonies large enough to see. Therefore, the 3D recognition and classification of microorganisms by sufficient morphological similarities or characteristics between them is essential for biology, security and medicine applications. Recently, we proposed 3D microorganisms recognition based on holographic microscopy [18]. The holographic microscope was based on single-exposure on-line (SEOL) digital holography. The algorithm used the shape of biological microorganism for classification.

This paper presents a new approach to provide 3D recognition of biological microorganisms that vary in their thickness and shape, move, grow and reproduce. A SEOL digital holographic microscope is used to sense the microorganism. However, recognition algorithms are developed to be independent of the shape and profile of the microorganisms. Sample segments are extracted from the 3D reconstructed image of the biological cell. These samples are processed using a number of cost functions including mean-square-distance, mean-absolute-distance, and statistical inference algorithms for the equality of locations and equality of dispersions between the sampling segments of the reference and input 3D image. Student’s t distribution and Fisher’s F distribution are used to analyze the difference of means and the ratio of variances of two populations, respectively.

The following sections describe various stages of the proposed approach to perform shape tolerant recognition of 3D microorganisms by use of SEOL digital holography. In Section 2, we present a brief review of SEOL digital holography and its advantages for 3D biological microorganism recognition. In Section 3, the design procedure for shape tolerant 3D microorganism recognition is described and the cost functions to evaluate the recognition system and multiple decision rules for the 3D microorganism recognition are calculated. The algorithms based on statistical inference are described. Experimental results are presented in Section 4 to test the proposed method. The conclusions follow in Section 5.

2. Review of Single Exposure On-Line (SEOL) Digital Holography

In the following, we briefly describe the SEOL digital holography technique. The SEOL digital hologram of a 3D microorganism in the Fresnel diffraction domain is recorded by the CCD (Charge-Coupled Device) array as shown in Fig 1. Coherent light from an Argon laser (center wavelength of 514.5 nm) is used as a source of illumination. A spatial filter and a collimating lens provide the spatial coherence. A beam splitter divides the plane-parallel wave into object and reference wave. The object wave illuminates the specimen magnified by the microscope objective. The digital hologram of 3D microorganism is the interference intensity pattern generated by the plane-parallel reference wave and the diffracted wave-fronts of the specimen.

 

Fig. 1. Experimental setup for recording SEOL digital hologram of a 3D biological microorganism; Ar: Argon laser, BS1, BS2: beam splitter; M1, M2: mirror; MO: microscope objective; CCD: charge coupled device array.

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A brief overview of SEOL is presented in Appendix A. Our holographic microscope system requires only a single exposure recorded for obtaining Fresnel diffracted pattern of 3D microorganism. Therefore, SEOL digital holography technique can be suitable for recognizing a moving 3D object and robust to external noise factors such as fluctuation and vibration.

 

Fig. 2. Frameworks for shape tolerant 3D biological microorganism recognition based on the single-exposure on-line (SEOL) digital holography.

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3. Design Procedure for Shape Tolerant 3D Microorganism Recognition

In the following, we describe the design procedure for shape tolerant 3D microorganism recognition using SEOL digital holography. First, we restore 3D microorganism as a volume image from SEOL digital hologram corresponding to a reference microorganism. Then, we randomly extract N pixels m times in the reconstructed 3D image. Each sample segment consists of N complex values. We denote each pixel value in the trial sampling segment as RNm , where m is the m th reference sampling segment of the p th page [see Fig. 3]. This idea may be suitable for recognizing 3D microorganisms such as bacteria and biological cells that do not have well defined shapes or profiles. For example, they may be simple, unicellular and branched in their morphological traits. It could also be applied to cells that vary in shape and profile rapidly. This process can be automatically performed by a segmentation and edge detection algorithm on the computer [17]. To test our 3D microorganism recognition system, we record the SEOL digital hologram of an unknown input microorganism and then restore the original input image. Next, arbitrary sampling segments are extracted in the restored 3D image. Each sampling segment consists of complex values. We denote the field distribution in the patch as SNn , where n is the n th unknown input sampling segment.

 

Fig. 3. The design procedure for shape tolerant 3D biological microorganism recognition. The windows of sample segment are extracted in the restored 3D image from SEOL digital hologram.

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We use a number of statistical cost functions to recognize the input microorganism. The measurements are made from the reconstructed 3D image of the microorganism. We refer to each reconstruction plane of the 3D volume as “page”. The cost functions measure the statistical distances between a training sample of microorganism (that is, reference images), and an input microorganism under observation obtained by digital holography. We first use metrics of mean-square-distance (MSD) and mean-absolute-distance (MAD) to quantitatively estimate the performance of our 3D microorganism recognition system. The MSD and MAD are defined as:

MSD=i=0N1{Ri(x,y)E[S(x,y)]}2i=0N1{Ri(x,y)}2MAD=i=0N1Ri(x,y)E[S(x,y)]i=0N1Ri(x,y),

where RNm and SNn represent the complex field distribution in the m th reference’s sampling segment at the p th page and in the nth input’s sampling segment, respectively, N are the numbers of nodes (or pixels) within the sampling segment and E[·] is the expectation of RNm or SNn .

 

Fig. 4. The statistical inference method to implement the proposed 3D biological microorganism recognition system.

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From the histograms analysis of the real and imaginary parts of the complex images after a segmentation and Sobel edge detection algorithms [17], we assume that the random variables (real or imaginary parts of the segmented and edge-detected 3D image) in the sampling segment nearly follow Gaussian distribution.

 

Fig. 5. The histogram of (a) real part, (b) imaginary part of the preprocessed (segmentation and edge detection) 3D image.

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For comparing location parameter the sampling distribution of the difference between two sample location parameters from two actual normal populations can be calculated. For comparing dispersion parameters we calculate the sampling distribution of the ratio between two sample variances. We assume that random variables RNm and SNn which are elements inside the reference and input sampling segment are statistically independent with identical Normal distribution N(μ R, σR2) and N(μ s, σS2). Also let the RNm be independent of the SNn . It is noted that the random variables RNm or RNm are elements of the real part or the imaginary part of the reconstructed image, so we perform two separate univariate hypothesis testing. For comparing the location parameters between two samples, we assume that all three statistical parameters are unknown and θ = (μ R, μ s, σ2 = σR2 = σS2). Then, we know that {N R V[R] + N s V[S]}/σ2 has a Chi-square distribution with N R + N s -2 degrees of freedom, where V[·] is sample variance. Therefore, the variable of the difference of the means between two independent normal populations can be represented as follows [19]:

T=NR+NS2NRV[R]+NSV[S]×E[R]E[S]{(NR)1+(NS)1}1/2,

where T has Student’s t distribution with N R + N s-2 degrees of freedom.

To perform the hypothesis testing, we set a null hypothesis H 0 and alterative hypothesis H 1 as follows:

H0:μR=μS,H1:μRμS

where the null hypothesis means that there is no difference between population means and variances. Then for hypothesis H 0, Eq. (2) can be given as:

T=1V¯pE[R]E[S]{(NR)1+(NS)1}1/2,

where V¯P is the pooled estimator of σ2.

On the basis of a two-tailed test at a level of significance α 1, we have the following decision rule:

  1. Accept H 0 if the variable T is placed inside the interval -t NR+Ns-2,1-α1/2 to t NR+Ns-2,1-α1/2,
  2. Reject H 0 otherwise.

We denote the upper 100×(α 1/2)% point of the t NR+Ns-2 distribution as t NR+Ns-2,1-α1/2. Thus the following probability can be claimed [19]:

P{(E[R]E[S])tNR+NS2,1α1/2V¯p[(NR1+NS1)1/2]
<μRμS<(E[R]E[S])+tNR+NS2,1α1/2V¯P[(NR1+NS1)1/2]}=1α1.

This decision rule implies that H 0 is true if Student’s t distribution occurs between percentile value -t 1-α1/2 and t 1-α1/2 given the probability density function of Student’s t distribution. α 1 can be adjusted so that the probability of correct detection is 100× (1-α 1/2)%. This is the area under Student’s t distribution between -t 1-α1/2 and t 1-α1/2.

For comparing the dispersion parameters between two samples, we assume that all four statistical parameters are unknown and θ = (μ R, μ S, σR2, σS2). The ratio of the dispersions of two independent normal populations can be represented as follows [19]:

F(NR1),(NS1)={NR/(NR1)}V[R]/σR2{NS/(NS1)}V[S]/σS2,

where the F distribution has N R -1, N S -1 degrees of freedom. Finally, the null hypothesis H 0 and the alterative hypothesis H 1 are defined as follows:

H0:μR2=μS2,H1:μR2μS2,

where the null hypothesis means that there is no difference between two population variances. Then for hypothesis H 0, the Eq. (34) can be given as:

F(NR1),(NS1)={NR/(NR1)}V[R]{NS/(NS1)}V[S]=V̂[R]V̂[S]

On the basis of a two-tailed test at a level of significance α 2, we have the following decision rule:

  1. Accept H 0 if the statistics of [R]/[S]is placed inside the interval F (NR-1)(NS-1)α2/2 to F (NR-1)(NS-1)α2/2,
  2. Reject H 0 otherwise.

We have denoted the upper 100 × (α 2/2)% point of the F (NR-1), (Ns-1) distribution as F (NR-1)(NS-1)α2/2. This decision rule implies that H 0 is true if the F distribution occurs between percentile value F α 1/2 and F 1-α1/2 given the probability density function of the F distribution. α 2 can be adjusted so that the probability of correct detection is 100×(1 -α 2/2)%. This is the area under Fisher’s F distribution between F α1/2 and F 1-α1/2. Thus the following probability can be claimed [19]:

P{F(NR1),(NS1),α2/21[V̂[R]V̂[S]]<σR2σS2<F(NR1),(NS1),1α2/21[V̂[R]V̂[S]]}=1α2,

4. Experiments Result

We show experimental results of the image formation of two filamentous microorganisms (sphacelaria alga and polysiphonia alga) using SEOL digital holography and identify them by their morphological characteristics with mean-square-distance (MSD), mean-absolute-distance (MAD), and hypothesis testing is performed for the equality of the parameters between two populations for a statistical decision.

6.1 3D microorganism’s image from SEOL digital hologram

In the following subsection, we present the sensing of 3D microorganisms (sphacelaria alga and polysiphonia alga) and visualization using SEOL digital holography. In the experiments the 3D microorganisms are around 40 ~ 100 μm in size. They are recorded using a SEOL digital hologram with a CCD array of 2048 × 2048 pixels and a pixel size of 9 μm × 9 μm , where the specimen was sandwiched between two transparent cover slips.

 

Fig. 6. The magnified algae’s images by use of a 10 × microscope objective: (a) sphacelaria’s 2D image and (b) polysiphonia’s 2D image.

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Fig. 7. sphacelaria’s phase contrast image after applying segmentation and edge detection algorithm at distance d =180 mm as the reference by use of a 10 × microscope objective.

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Fig. 8. Experimental results for input algae by use of a 10 × microscope objective: (a) sphacelaria’s intensity image at distance d = 180 mm and (b) polysiphonia’s intensity image at distance d =180 mm.

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Figure 6(a) and (b) show 2D sphacelaria’s image and polysiphonia’s image, respectively, which were used for testing our recognition system. For the recognition, the reference’s 3D image was reconstructed at the distances of 180 mm by a SEOL digital hologram. Fig. 7 show the reference alga’s phase contrast image restored at the distance of 180 mm. the segmentation by the histogram analysis of reconstructed 3D images and Sobel edge detection algorithm were used for a fast and efficient identification. Figure 8(a), (b) show the input intensity images reconstructed at the distances of 180 mm from SEOL digital hologram, respectively, where we used a different sample for input from the reference to test the robustness of our recognition system.

6.2 Shape tolerant 3D microorganism recognition by mean-square-distance (MSD), mean-absolute-distance (MAD), and hypothesis testing as multiple decision rules

In the following subsection, we evaluate the performance of our shape tolerant 3D microorganism recognition system using SEOL digital holography. First, 100 trial sampling segments were produced randomly selecting the pixel values in the segmented sphacelaria alga 3D image as the reference, where we changed the size of each trial sampling segment from 2 to 500 and applied a Sobel edge-detection method to the segmented 3D image for a fast and efficient recognition.

 

Fig. 9. The average (a) MSD, (b) MAD calculated by the complex amplitude between the reference segments and the input segments versus the sample size of sampling segments.

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Fig. 10. (a) T-test for the equality of the location parameter between two sampling segments versus a sample size, (b) F-test for the equality of the dispersion parameter between two sampling segments versus a trial number with a sample size 500.

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Similarly, a number of sampling segments are randomly selected in the sphacelaria alga 3D image as the true-class inputs and in the polysiphonia alga image as the false-class inputs. We produced 100 true-class and 100 false-class input sampling segments to test the performance of the shape independent 3D recognition, respectively, where we changed the size of each input sampling segment from 2 to 500. The reference and input images are reconstructed at distance d=180 mm. Fig. 9(a), (b) show experimental results of the average MSD and MAD calculated by the complex values between each reference sampling segment and each input sampling segment. As shown in Fig. 9(a), (b), it is noted that the average MSD and MAD for the true-class input patch calculated are around 0.01525 and 0.10355, respectively and for the false-class input patch more than 0.01535 and 0.10425. Fig. 10(a) shows the results of the hypothesis testing for the difference of location parameters between the reference and input sampling segments, where we reject the null hypothesis H 0(μ X = μ Y) if the statistic T defined in Eq. (4) are outside the range -t 0.975 and t 0.975 on the basis of a two-tailed test at level of significance 0.05. It is noted that the percentages of the correct matched segments by the decision rule (-t 0.975Tt 0.975) for the true-class input segments were around 80%, while for the false-class input patches, the percentages rapidly decreased as a sample size increases. We also conducted the hypothesis testing for the ratio of dispersions of the reference and input microorganism, where we reject the null hypothesis H 0(σX2 = σY2) if the statistic F defined in Eq. (8) is outside the range F 0.05 and F 0.95 on the basis of a two-tailed test at level of significance 0.10. Fig. 10(b) shows the ratio of the dispersions of sampling segments between the reference and the number of the correct matched segments with a sample size 500 by the following decision rules (F 0.05FF 0.95). It is noted that the number of the correct matched segments for the true-class inputs was 87, while for the false-class inputs the number was 2.

5. Conclusion

In conclusion, we have presented a shape tolerant 3D biological microorganism recognition system using SEOL digital holography and a statistical inference. We have used the cost functions by calculating the mean-square-distance (MSD) and mean-absolute-distance (MAD) and performed the hypothesis testing for the difference of means and the ratio of variances as a decision tool. We optically measured the complex values of 3D microorganisms and then digitally extracted arbitrary sample segments in the restored 3D image by the proposed design procedure. For recognition, MSD, MAD, and the sampling probability distribution of the difference of locations and the ratio of the dispersions were calculated between the reference and input sample segment varying the sample size of sample segment and then we tested the proposed system by a multiple decision rules. It is shown in experiments that the complex field distribution in the restored 3D microorganism’s image from the single SEOL hologram contains important information for recognition and classification and microorganisms can be identified using a statistical estimation.

SEOL digital holography allows the section images to be obtained at different planes along the longitudinal direction without any mechanical scanning or special optical components. The identification capability for 3D microorganism recognition can be improved using the 3D volume image restored from single hologram. This concept is more suitable for recognizing and classifying much smaller microorganism efficiently, because the higher magnification of the microscope objective causes the lower depth of focus. The sampling segments were extracted in the restored section images from SEOL hologram to recognize 3D microorganisms. The shapes of some bacteria and algae are filamentous, spherical, and branched. They may look similar in terms of shape. This approach allows the proposed system to be tolerant of shape in recognizing 3D microorganisms like bacteria or algae and enables much faster and more efficient recognition.

Appendix A

In this Appendix, we review SEOL digital holography as shown in Fig. 1 [18]. The field distribution of a 3D microorganism at the CCD plane or Fresnel diffraction domain can be represented as follows:

OH(x,y)=d0δ2d0+δ2exp[j2πz/λ]jλzexp[jπλz+(x2+y2)]×
O(ξ,η;z)exp[jπλz(ξ2+η2)]exp[j2πλz(+)]dzdξdη,

where d 0 is the distance between the center of 3D microorganism and CCD plane and δ is microorganism’s depth along z-axis. The plane parallel reference wave at the CCD plane is given as:

R(x,y)=AR(x,y)exp[jφR(x,y)],

The interference pattern recorded at the CCD plane or hologram plane is represented as follows:

I(x,y)=OH(x,y)+R(x,y)2
=AH(x,y)2+AR2+2AH(x,y)ARcos[ΦH(x,y)φR].

The SEOL digital hologram of 3D microorganism at the CCD plane can be represented as follows:

H(x,y)=I(x,y)O(x,y)2R(x,y)2,

where the reference beam’s intensity |R(x,y)|2 is obtained by only a one time measurement on the experiment and the object beam’s intensity |O(x,y)|2 can be approximately obtained by use of the following local window’s averaging technique over Lx by Ly pixels. Namely, for the local hologram we average the values of the L x × Ly adjacent pixels.

O(x,y)2nx=0Nx1ny=0Ny1{1LxLxlx=1Lxly=1Ly[H(nx+lx,ny+ly)R(nx+lx,ny+ly)2]},

where Nx Ny is the whole hologram size in the x, y direction, respectively. The reconstruction of the original 3D microorganism is performed digitally on a computer. The field distribution of the restored 3D microorganism from SEOL digital hologram can be calculated numerically by the following inverse Fresenl transformation with two Fourier transforms, which cancels the scale factor between the input and output:

O′(ξ,η)=IFrT{H(x,y)}=IFrT(FrT{H(x,y)}×exp{jπλdo[u2(ΔxNx)2+v2(ΔyNy)2]}),

where u and v denote transverse discrete spatial frequencies, and (Δx, Δy) are resolution at the hologram plane. The restored 3D microorganism’s image from the SEOL digital hologram originally contains a conjugate image. This undesired component degrades the quality of the reconstructed 3D image, but the intrinsic defocused conjugate image also contains information of the 3D microorganism. As an additional merit, SEOL digital holography allows us to obtain a dynamic time-varying scene restored digitally on the computer for monitoring and recognizing moving and growing microorganisms.

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Figures (10)

Fig. 1.
Fig. 1. Experimental setup for recording SEOL digital hologram of a 3D biological microorganism; Ar: Argon laser, BS1, BS2: beam splitter; M1, M2: mirror; MO: microscope objective; CCD: charge coupled device array.
Fig. 2.
Fig. 2. Frameworks for shape tolerant 3D biological microorganism recognition based on the single-exposure on-line (SEOL) digital holography.
Fig. 3.
Fig. 3. The design procedure for shape tolerant 3D biological microorganism recognition. The windows of sample segment are extracted in the restored 3D image from SEOL digital hologram.
Fig. 4.
Fig. 4. The statistical inference method to implement the proposed 3D biological microorganism recognition system.
Fig. 5.
Fig. 5. The histogram of (a) real part, (b) imaginary part of the preprocessed (segmentation and edge detection) 3D image.
Fig. 6.
Fig. 6. The magnified algae’s images by use of a 10 × microscope objective: (a) sphacelaria’s 2D image and (b) polysiphonia’s 2D image.
Fig. 7.
Fig. 7. sphacelaria’s phase contrast image after applying segmentation and edge detection algorithm at distance d =180 mm as the reference by use of a 10 × microscope objective.
Fig. 8.
Fig. 8. Experimental results for input algae by use of a 10 × microscope objective: (a) sphacelaria’s intensity image at distance d = 180 mm and (b) polysiphonia’s intensity image at distance d =180 mm.
Fig. 9.
Fig. 9. The average (a) MSD, (b) MAD calculated by the complex amplitude between the reference segments and the input segments versus the sample size of sampling segments.
Fig. 10.
Fig. 10. (a) T-test for the equality of the location parameter between two sampling segments versus a sample size, (b) F-test for the equality of the dispersion parameter between two sampling segments versus a trial number with a sample size 500.

Equations (18)

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MSD = i = 0 N 1 { R i ( x , y ) E [ S ( x , y ) ] } 2 i = 0 N 1 { R i ( x , y ) } 2 MAD = i = 0 N 1 R i ( x , y ) E [ S ( x , y ) ] i = 0 N 1 R i ( x , y ) ,
T = N R + N S 2 N R V [ R ] + N S V [ S ] × E [ R ] E [ S ] { ( N R ) 1 + ( N S ) 1 } 1 / 2 ,
H 0 : μ R = μ S , H 1 : μ R μ S
T = 1 V ¯ p E [ R ] E [ S ] { ( N R ) 1 + ( N S ) 1 } 1 / 2 ,
P { ( E [ R ] E [ S ] ) t N R + N S 2 , 1 α 1 / 2 V ¯ p [ ( N R 1 + N S 1 ) 1 / 2 ]
< μ R μ S < ( E [ R ] E [ S ] ) + t N R + N S 2 , 1 α 1 / 2 V ¯ P [ ( N R 1 + N S 1 ) 1 / 2 ] } = 1 α 1 .
F ( N R 1 ) , ( N S 1 ) = { N R / ( N R 1 ) } V [ R ] / σ R 2 { N S / ( N S 1 ) } V [ S ] / σ S 2 ,
H 0 : μ R 2 = μ S 2 , H 1 : μ R 2 μ S 2 ,
F ( N R 1 ) , ( N S 1 ) = { N R / ( N R 1 ) } V [ R ] { N S / ( N S 1 ) } V [ S ] = V ̂ [ R ] V ̂ [ S ]
P { F ( N R 1 ) , ( N S 1 ) , α 2 / 2 1 [ V ̂ [ R ] V ̂ [ S ] ] < σ R 2 σ S 2 < F ( N R 1 ) , ( N S 1 ) , 1 α 2 / 2 1 [ V ̂ [ R ] V ̂ [ S ] ] } = 1 α 2 ,
O H ( x , y ) = d 0 δ 2 d 0 + δ 2 exp [ j 2 πz / λ ] jλz exp [ j π λz + ( x 2 + y 2 ) ] ×
O ( ξ , η ; z ) exp [ j π λz ( ξ 2 + η 2 ) ] exp [ j 2 π λz ( + ) ] dzdξdη ,
R ( x , y ) = A R ( x , y ) exp [ j φ R ( x , y ) ] ,
I ( x , y ) = O H ( x , y ) + R ( x , y ) 2
= A H ( x , y ) 2 + A R 2 + 2 A H ( x , y ) A R cos [ Φ H ( x , y ) φ R ] .
H ( x , y ) = I ( x , y ) O ( x , y ) 2 R ( x , y ) 2 ,
O ( x , y ) 2 n x = 0 N x 1 n y = 0 N y 1 { 1 L x L x l x = 1 L x l y = 1 L y [ H ( n x + l x , n y + l y ) R ( n x + l x , n y + l y ) 2 ] } ,
O′ ( ξ , η ) = IFrT { H ( x , y ) } = IFrT ( FrT { H ( x , y ) } × exp { jπλ d o [ u 2 ( Δ x N x ) 2 + v 2 ( Δ y N y ) 2 ] } ) ,

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