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

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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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