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Simultaneous multi-color optical sectioning fluorescence microscopy with wavelength-coded volume holographic gratings

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Abstract

Optical sectioning fluorescence microscopy provides high contrast images of volumetric samples and has been widely used for many biological applications. However, simultaneously acquiring multi-color fluorescence images require additional optical elements and devices, which are bulky, wavelength specific, and not cost-effective. In this paper, wavelength-coded volume holographic gratings (WC-VHGs) based optical sectioning fluorescence microscopy is proposed to simultaneously offer multi-color fluorescence images with fine out-of-focus background rejection. Due to wavelength degeneracy, multiplexed WC-VHGs are capable of acquiring multi-wavelength fluorescence images in a single shot, and displaying the laterally separated multi-wavelength images onto CCD. In our system optical sectioning capability is achieved through speckle illumination and HiLo imaging method. To demonstrate imaging characteristics of our system, dual-wavelength fluorescence images of both standard fluorescent microspheres and ex vivo mT/mG mice cardiac tissue are presented. Current results may find important applications in hyperspectral imaging for biomedical research.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Fluorescence imaging is one of the important techniques for biomedical research [1]. Wide-field fluorescence microscopy has been widely used for a variety of biological applications, but it has no optical sectioning capability [2]. Confocal microscopy is a common optical sectioning imaging technique to acquire high contrast fluorescence images of volumetric samples, by removing out of focus background light using a pin-hole [3]. Although the scanning speed for confocal microscopes have been improved in recent years [4], due to point by point scanning mechanism, confocal microscopes have fundamental limitations in terms of image acquisition time. On the other hand, optically sectioned images in the widefield detection can be achieved by structured illumination microscopy [5,6]. Various kinds of structured light patterns have been used to obtain optically sectioned images in the widefield detection. It has been shown that speckle illumination offers advantage for imaging volumetric organ tissues with structured illumination microscopy [713]. Comparison with periodic grid patterns, speckle contrast helps in obtaining large depth of field. HiLo microscopy is one kind of structured illumination techniques, and provides optically sectioned images in high-speed operation [1420]. By choosing appropriate cutoff frequencies, both high and low spatial frequency components at in-focus planes can be extracted using pair-wise uniform and structured illumination images, respectively to reconstruct optically sectioned images. Using speckle illumination based HiLo image processing technique, object thickness can be measured and optical sectioning thickness can be varied [18,21].

Often multi-color fluorescence images are required to study fine structural details of biological samples [1]. Different portions of a sample may be tagged with different fluorescence dyes, and multi-color fluorescence images corresponding to each section can be acquired by simultaneous multi-color excitation. Several imaging techniques for multi-color fluorescence images have been proposed [2224]. In the simplest way, different color fluorescence images are obtained by sequentially switching both excitation wavelength and corresponding optical filter while imaging. The sequential switch imaging scheme may impose limitation in real-time observation. To simultaneously observe multi-color fluorescence images, a beam separation mechanism has been developed to split different wavelengths of light onto specific locations of a CCD or several CCDs [25]. But this method is limited with the requirement of multiple optical components and detectors, which increase system size and complexity. In order to reduce the number of optical components and detectors in the system, the multispectral filters have been used in imaging systems. To achieve wavelength separation multi-color filters wheel or liquid crystal tunable filter can be used. Nevertheless, multispectral filters techniques need high-cost optical components and have limitations in providing real-time multi-color images [26,27]. Spectral imaging techniques have been used to resolve multiple labeled fluorescence images for biological applications [28]. However, spectral imaging needs dispersive gratings or prisms to measure spectrum at each point of an image, which causes long acquisition time. To solve the above described problem, digital spectral imaging has been recently introduced [29], but it still requires scanning and extensive computational process. The other method for wavelength separation is by color separation diffraction gratings (CSGs) [3032]. The surface etched technique is used to form the CSGs which has the microrelief to generate the phase delay period more than (0-2π) values, which can segregate the different color beam into corresponding diffraction orders [30]. However, usually the CSGs methods need complex error functionality, and the etching manufacturing process is complicated and costly. Furthermore, CSGs belong to the thin grating and provide broad angular selectivity, which means the different wavelength diffraction beams may crosstalk to each other to degrade the final image quality [31].

Holographic imaging can be utilized to obtain multiple wavelength images. Recently, it is shown that in-line digital holographic multiplexing can reconstruct different wavelength images of spare cell samples by Fresnel back-propagation and symmetric phase shifting [33,34]. However, in-line holographic multiplexing only works for thin transparent samples, with poor image quality and no optical sectioning. In contrast to digital holographic imaging, volume holography offers many advantages in multi-dimensional imaging [3537], and spatial-spectral filtering [38,39]. Unique diffractive properties, including high angular selectivity, single diffraction order, large multiplexing capacity, and wavelength degeneracy, makes a volume holographic grating (VHG) suitable for optical imaging in broadband operation [40]. The multiplexed VHGs follow the Bragg degeneracy property to provide the high wavelength and wavefront selectivity, which can be used for acquire simultaneous spatial and spectral information for hyperspectral (x,y,z,λ) imaging [4143]. To fabricate such gratings proper design of wavelength-coded volume holographic gratings (WC-VHGs) will be required [4446].

In this work, we present a simultaneous multi-color optical sectioning fluorescence microscope. In our proposed system, angularly multiplexed WC-VHGs are utilized to simultaneously acquire multi-color fluorescent images of tissue samples, while speckle illumination with HiLo imaging process is utilized to significantly improve image contrast, with fine optical sectioning in wide-field fashion. Design principles and fabrication process of angularly multiplexed WC-VHGs are discussed in detail. Rather than using complicated system design or long acquisition time in the previously described conventional multi-color fluorescence imaging techniques, here, we report a single, compact multiplexed holographic element to effectively acquire simultaneous multi-color optical sectioning fluorescence images. Our WCVHG can perform wavelength separation in real time. In addition, the narrow angular selectivity of WCVHG can avoid the cross-talk between the different color images [4446]. Due to their excellent spectral properties VHGS can provide high spectral resolution in broadband range which may not be possible with the other techniques. Performance characteristics of our system are evaluated with experimental imaging measurements through standard fluorescently labeled microspheres, as well as ex vivo mice cardiac tissue samples.

In Section 2, design details and recording geometry of WC-VHGs will be described. Overview and working principle of speckle illumination with HiLo imaging process for optically sectioned images will be presented in Section 3. The experimental configuration of the proposed system and measurements of performance parameters will be discussed in Section 4. Section 5 shows experimental results of simultaneous multi-color optical sectioning fluorescence images of mT/mG mice cardiac tissue samples. Conclusion will be then summarized in Section 6.

2. Wavelength-coded volume holographic gratings (WC-VHGs)

In this study, photopolymer based WC-VHGs are recorded with one wavelength (488 nm), and reconstructed for imaging two different colors (532&633 nm) of interest. Two angularly multiplexed VHGs (diameter∼10 mm) were recorded onto phenanthrenequinone poly (methyl methacrylate) (PQ-PMMA) photosensitive materials using a single wavelength (488 nm). The manufacturing process of the photosensitive materials are following our previous work [39], the optimum weight ratio PQ-PMMA liquid mixture is injected into the mold, which consists of two glass plates and a flexible spacer. After the heat curing process, the PQ-PMMA substrates are demolding and cutting properly to form approximately 1.8 mm thickness high Bragg selectivity recording material. The function of WC-VHGs can be explained using K-Sphere diagram [3638]. Figure 1 shows a K-sphere diagram of two angularly multiplexed grating having grating vectors ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over K} _{G1}}$ and ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over K} _{G2}}$ designed for the wavelength 633 nm and 532 nm respectively [44]. Grating vectors of each grating can be expressed as

$${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over K} _{Gm}} = {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} _{Rm}} - {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} _{Sm}},\;m = 1,2$$
where ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over K} _{Gm}}$ represent the m-th grating vector, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} _{Rm}}$, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} _{Sm}}$ are propagation vectors of the reference and signal beam, respectively. According to wavelength degeneracy property [47], when the reconstruction beam wavelength (λP) and angle (θP) is deviated from Bragg wavelength (λ0) and angle (θ0) with the variation value of Δλ and Δθ, as shown in the following
$${\theta _P} = {\theta _0} + \Delta \theta $$
$${\lambda _P} = {\lambda _0} + \Delta \lambda $$
the detuning parameter ($\vartheta $) for Bragg mismatch condition is given by [47]
$$\vartheta = K\cos [\varphi - {\theta _P}] - \frac{{{K^2}}}{{4\pi n}}{\lambda _P}$$
where the $\varphi $ is the angle between the grating vector (K) and the z axis, and the n is the refraction index. Under Bragg matched condition $\vartheta $ reduces to zero. Equation (4) suggests that volume holographic gratings can be recorded at one wavelength and reconstructed at other wavelengths.

 figure: Fig. 1.

Fig. 1. The K-sphere diagram and recording geometry for the two multiplexed wavelength coded gratings. (a) The K-sphere for the first grating (KG1) that is recorded with blue color (488 nm) and reconstructed with red color (633 nm). (b) The K-sphere for the second grating (KG2) that is recorded with blue color (488 nm) and reconstructed with green color (532 nm). (c) Recording geometry for angularly multiplexed WC-VHGs. The reference beam incident angles are θR1 and θR2 for the KG1 and KG2, respectively. The signal beam incident angles for two gratings are θS1 and θS2. (d) The WC-VHGs functions as beam splitting component, the first and second grating are respectively probed in the red (633 nm) and green (532 nm).

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In our case, the first multiplexed grating was recorded at 488 nm and reconstructed at 633nm, while the second multiplexed grating was recorded at 488 nm and reconstructed at 532 nm. Figure 1(a) shows the K-sphere to record the first grating with an incident angle of θR1 and θS1 for reference and signal beams, respectively. The Bragg-matched reconstructed red beam angle is θP,Red, and diffraction beam angle is θD,Red. In Fig. 1(b), the incident angles of reference and signal beams in blue are θR2 and θS2, and the corresponding angle of the reconstruction and the diffraction beams in green, under Bragg matched condition, are θP,Green and θD,Green. Figure 1(c) illustrates the recording geometry for our WC-VHGs at 488 nm. The wave vectors of reference and the signal beam for first grating and the second grating is (kR1, kS1) and (kR2, kS2) respectively. Following wavelength degeneracy property WC-VHGs reconstruction process can be shown in Fig. 1(d).

3. HiLo imaging principle

HiLo imaging process provides optically sectioned high contrast images for fluorescence microscopy. The principle of the HiLo imaging method requires a pair of images under different illumination conditions to obtain an optically sectioned image. One is uniformly illuminated image (iu), and the other is non-uniformly illuminated image, which can be speckle illuminated image (is) [15]. The uniform-illumination image iu can be written as

$${i_u}(x,y) = {i_{in}}(x,y) + {i_{out}}(x,y)$$
where x, y are spatial coordinates, and iin(x,y) and iout(x,y) are in-focus and out-of-focus components respectively. The in-focus high spatial frequencies content (iHi) can be extracted from iu as
$${i_{Hi}}(x,y) = {F^{ - 1}}({I_u}\textrm{(}{f_x}\textrm{,}{f_y}\textrm{)} \times H{P_{cf}}\textrm{(}{f_x}\textrm{,}{f_y}\textrm{)})$$
where fx and fy are frequency coordinates, Iu is the Fourier transform of the iu, and HPcf is the Gaussian high-pass filter with cut-off frequency cf.

The speckle illuminated image is can then be represented as

$${i_s}(x,y) = {i_{in}}(x,y)s(x,y) + {i_{out}}(x,y)$$
where s(x, y) is the modulation coefficient under speckle illumination. In terms of contrast, the speckle illuminated image at defocus region has poor image contrast, while the image contrast at in-focus plane is high. Hence, low spatial frequency components (iLo) of the in-focus plane can be obtained by
$${i_{Lo}}(x,y) = {F^{ - 1}}(F(C(x,y) \times {i_u}(x,y)) \times L{P_{cf}}({f_x},{f_y}))$$
where LPcf is the Gaussian low-pass filter with the same cutoff frequency cf. C is the local spatial contrast of the is, and can be expressed as
$$C(x,y) = \frac{{ < SD({i_s}){ > _w}}}{{ < M({i_s}){ > _w}}}$$
where < M(is)>w and < SD(is)>w represents the standard deviation and mean value of is, and are calculated within the sampling window (w).

The final optically sectioned HiLo image (iHiLo(x,y)), including the in-focus information of the entire spatial frequency range, can be obtained as

$${i_{HiLo}}(x,y) = {i_{Hi}}(x,y) + \eta {i_{Lo}}(x,y)$$
where η is the scaling factor to balance the intensity between the iHi(x,y) and iLo(x,y). The typical values of η are in the range 0.5∼3, and in our case we choose η=1 for high contrast.

4. Experimental setup and results

A schematic diagram of the proposed WC-VHGs based multi-color fluorescence microscopy is shown in Fig. 2. A blue excitation laser light (Cobolt Calypso 200, 491nm) propagates through an optical diffuser that can generate volumetric speckle pattern. A dichroic mirror is placed between a 4-f relay (R1 & R2), and reflects blue excitation light, which goes through an objective lens to illuminate the sample. Once the fluorescently labeled sample is excited, red and green fluorescent excitation light beams pass through the dichroic mirror along the same incident angle onto multiplexed WC-VHGs, which is located at the conjugate Fourier plane of the objective lens. Under the Bragg matched condition, emitted green and red fluorescent light beams are diffracted by WC-VHGs at different propagation angles to simultaneously form images onto a CCD. It is noted that the single diffraction order of the WC-VHGs provides a solid advantage to simultaneously display laterally separated dual-color images in a single shot.

 figure: Fig. 2.

Fig. 2. Schematic diagram of the proposed WC-VHGs based multi-color fluorescence microscopy. The speckle illumination pattern is generated by the optical diffuser (Lummit, KCN1 25) and projected onto the in-focus plane of the sample. A dichroic mirror (FITC dichroic mirror, MD499, THORLAB) is used to separate excitation and emission wavelengths. Objective lens (NA=0.55, ULWDMSPlan50X, OLYMPUS). Tube lens (NA=0.42, MPlanAPO20X, MITUTOYO). R1, R2 and R3, R4 are relay lens with focal length of f=50 mm. CCD: (C11440, HAMAMATSU).

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The function of WC-VHGs in the microscope is shown by imaging fluorescently labeled 45.0 µm microspheres. The microspheres are evenly distributed in the ∼1mm thick agarose (Invitrogen) that is fixed on the microscope slides. The microspheres excitation wavelength ranges from 350 nm to 530 nm, and the emission wavelength ranges is about 450 nm to 630 nm. Thus, the blue laser light source (λ = 491 nm) is used to excite these microspheres to simultaneously generate the green and red fluorescent light. By utilizing the beam splitting property of the WC-VHGs, the system can laterally separate the two different wavelength images at CCD as shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. Simultaneous multi-color laterally separated images of 45-µm fluorescent microspheres (Polysciences Inc, YG microspheres). (a) Fluorescent image of microspheres in red color, and (b) fluorescent image in green color.

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To quantify the optical sectioning ability of our system with speckle illumination, the fluorescently labeled microspheres are mechanically scanned along axial direction using a piezo stage (OME). Intensity profile along axial direction is shown in Fig. 4. The intensity of each step is computed by selecting a fixed region of interest (ROI) that covers a fluorescent microsphere, and averaging the gray value of each pixel inside the ROI. The intensity variation of the speckled pattern projected onto the microsphere quantifies the optical sectioning ability of the system. The full width at half maximum (FWHM) of the optical sectioning by speckled illumination is ∼50 µm from Gaussian curve fitting of experimental data.

 figure: Fig. 4.

Fig. 4. Normalized intensity distribution of fluorescent microspheres along the axial position. The yellow dots are the measured experimental value, and the axial separation between each step is 1 µm. The size of microspheres is 45 µm. The blue line corresponds to Gaussian curve fitting.

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To demonstrate simultaneous multi-color optical sectioning fluorescence imaging by our system, we acquired images of 90 µm microspheres without and with speckled illumination. Figure 5(a) shows the image captured by the system under uniformly illuminated condition, the presence of out-of-focus light results in poor depth selectivity. Figure 5(b) shows the speckle illuminated images of microspheres. The speckle pattern can be resolved for in-focus image Fig. 5(c), whereas, for defocused images barely show any speckle pattern. Figure 5(d) shows the resultant HiLo optical sectioning image that is processed from the standard uniform illumination image Fig. 5(a) and speckle illumination image Fig. 5(b). Figures 5(e) and 5(f) is the comparison of the intensity cross-section along the selected dash line of uniform illumination (blue dash line) and the HiLo images (green and red dash line), which shows the out of focus signal can be suppressed and the intensity of in-focus region remains the same.

 figure: Fig. 5.

Fig. 5. (a) Uniform illumination images, and (b) speckle illumination images of 90.0 µm fluorescent microspheres. The red and green square respectively indicate the red and green wavelength images, and the in-focus zoom in image in (c) shows the speckle pattern can be clearly observed. (d) Optical sectioning images obtained from (a) and (b) by using HiLo image processing, which eliminates the out of focus light. (e, f) The intensity profile comparison of the (a) and (d) along the cross-section line. The green and red dash lines are the intensity of the HiLo images and the blue lines are the corresponding intensity of the uniform images.

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5. Simultaneous dual color fluorescence optical sectioning of ex-vivo mice cardiac tissue

To demonstrate high contrast fluorescence imaging ability of our system for volumetric tissue samples, we performed simultaneous dual-color fluorescence imaging of the mice cardiac tissue, which is about 100 µm thick. The mT/mG mice sample we used is one kind of the genetically engineered mouse model, which is a two-color fluorescent Cre-reporter allele. Before Cre-mediated excision, the membrane-targeted tandem dimer Tomato (mT) make the tissue emit red fluorescence. After Cre-mediated excision, the mT is excised and the membrane-targeted green fluorescent protein (mG) can be expressed to make the tissue emit green fluorescence [48]. After being excited, the cardiac tissue simultaneously emits fluorescence light in both green and red bands, which express the areas next to each other. Figure 6 shows the dual-wavelength fluorescence images of the mT/mG mice cardiac tissue sample. Figure 6(a) is taken under standard uniform illumination condition, while Fig. 6(b) is HiLo processed optically sectioned images. Figures 6(c) and 6(d) are zoom-in images of the selected rectangular regions. To compare both zoom-in images, we select the same feature along the transverse line in both red and green color bands in Figs. 6(c) and 6(d), respectively. The resultant intensity profiles are shown in Fig. 6(e) and Fig. 6(f), and the out-of-focus noise is significantly eliminated, and high contrast myocardial fiber images can be observed.

 figure: Fig. 6.

Fig. 6. (a) Uniform illumination, and (b) HiLo processed dual wavelength images of ex vivo mT/mG mice cardiac tissue. (c, d) are the zoom in images of the same region of interest in (a) and (b) (white square and blue square region), respectively. (e, f) Intensity profile in the cross section of white and blue dash line, which shows the HiLo image processing technique can improve the depth discrimination ability of the system.

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6. Conclusions

In summary, we demonstrate WC-VHGs provide an efficient and compact way to obtain simultaneous multi-color fluorescence images in a single shot. Speckle illumination in our system offers an advantage for imaging volumetric tissues with high contrast. Dual-wavelength fluorescence optical sectioning images of standard fluorescent microspheres, as well as ex-vivo mT/mG mice cardiac tissue, shows great potentials of our system for biomedical applications. In combination with speckle illumination, optically sectioned images in both green and red emission bands are achieved simultaneously using the HiLo imaging process. The Bragg degeneracy properties of a volume hologram allow to record all multiplexed gratings using a single wavelength, and the gratings can be simultaneously reconstructed at different wavelengths. The design principle of recording and reconstruction of the WC-VHGs is discussed in detail. The number of color channels in our system can be further increased by designing a high-dimensional angular multiplexed WC-VHGs. Our approach is straightforward and does not require any additional components for multi-color imaging. Present method can further be adapted with other microscope and endoscope modalities. We believe present results can find important applications in multi-color fluorescence microscopy and hyperspectral imaging.

Funding

National Taiwan University (08HZT49001, 108L7714, 109L7839); Ministry of Science and Technology, Taiwan (MOST 108-2221-E-002-168-MY4).

Acknowledgments

The authors gratefully acknowledge Dr. Wen-Pin Chen for providing biological samples, and we thank Sunil Vyas and Chou-Min Chia for valuable discussions.

Disclosures

The authors declare no conflicts of interest.

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Figures (6)

Fig. 1.
Fig. 1. The K-sphere diagram and recording geometry for the two multiplexed wavelength coded gratings. (a) The K-sphere for the first grating (KG1) that is recorded with blue color (488 nm) and reconstructed with red color (633 nm). (b) The K-sphere for the second grating (KG2) that is recorded with blue color (488 nm) and reconstructed with green color (532 nm). (c) Recording geometry for angularly multiplexed WC-VHGs. The reference beam incident angles are θR1 and θR2 for the KG1 and KG2, respectively. The signal beam incident angles for two gratings are θS1 and θS2. (d) The WC-VHGs functions as beam splitting component, the first and second grating are respectively probed in the red (633 nm) and green (532 nm).
Fig. 2.
Fig. 2. Schematic diagram of the proposed WC-VHGs based multi-color fluorescence microscopy. The speckle illumination pattern is generated by the optical diffuser (Lummit, KCN1 25) and projected onto the in-focus plane of the sample. A dichroic mirror (FITC dichroic mirror, MD499, THORLAB) is used to separate excitation and emission wavelengths. Objective lens (NA=0.55, ULWDMSPlan50X, OLYMPUS). Tube lens (NA=0.42, MPlanAPO20X, MITUTOYO). R1, R2 and R3, R4 are relay lens with focal length of f=50 mm. CCD: (C11440, HAMAMATSU).
Fig. 3.
Fig. 3. Simultaneous multi-color laterally separated images of 45-µm fluorescent microspheres (Polysciences Inc, YG microspheres). (a) Fluorescent image of microspheres in red color, and (b) fluorescent image in green color.
Fig. 4.
Fig. 4. Normalized intensity distribution of fluorescent microspheres along the axial position. The yellow dots are the measured experimental value, and the axial separation between each step is 1 µm. The size of microspheres is 45 µm. The blue line corresponds to Gaussian curve fitting.
Fig. 5.
Fig. 5. (a) Uniform illumination images, and (b) speckle illumination images of 90.0 µm fluorescent microspheres. The red and green square respectively indicate the red and green wavelength images, and the in-focus zoom in image in (c) shows the speckle pattern can be clearly observed. (d) Optical sectioning images obtained from (a) and (b) by using HiLo image processing, which eliminates the out of focus light. (e, f) The intensity profile comparison of the (a) and (d) along the cross-section line. The green and red dash lines are the intensity of the HiLo images and the blue lines are the corresponding intensity of the uniform images.
Fig. 6.
Fig. 6. (a) Uniform illumination, and (b) HiLo processed dual wavelength images of ex vivo mT/mG mice cardiac tissue. (c, d) are the zoom in images of the same region of interest in (a) and (b) (white square and blue square region), respectively. (e, f) Intensity profile in the cross section of white and blue dash line, which shows the HiLo image processing technique can improve the depth discrimination ability of the system.

Equations (10)

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K G m = k R m k S m , m = 1 , 2
θ P = θ 0 + Δ θ
λ P = λ 0 + Δ λ
ϑ = K cos [ φ θ P ] K 2 4 π n λ P
i u ( x , y ) = i i n ( x , y ) + i o u t ( x , y )
i H i ( x , y ) = F 1 ( I u ( f x , f y ) × H P c f ( f x , f y ) )
i s ( x , y ) = i i n ( x , y ) s ( x , y ) + i o u t ( x , y )
i L o ( x , y ) = F 1 ( F ( C ( x , y ) × i u ( x , y ) ) × L P c f ( f x , f y ) )
C ( x , y ) = < S D ( i s ) > w < M ( i s ) > w
i H i L o ( x , y ) = i H i ( x , y ) + η i L o ( x , y )
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