We report on a novel algorithm for high-resolution quantitative phase imaging in a new concept of lensless holographic microscope based on single-shot multi-wavelength illumination. This new microscope layout, reported by Noom et al. along the past year and named by us as MISHELF (initials incoming from Multi-Illumination Single-Holographic-Exposure Lensless Fresnel) microscopy, rises from the simultaneous illumination and recording of multiple diffraction patterns in the Fresnel domain. In combination with a novel and fast iterative phase retrieval algorithm, MISHELF microscopy is capable of high-resolution (micron range) phase-retrieved (twin image elimination) biological imaging of dynamic events. In this contribution, MISHELF microscopy is demonstrated through qualitative concept description, algorithm implementation, and experimental validation using both a synthetic object (resolution test target) and a biological sample (swine sperm sample) for the case of three (RGB) illumination wavelengths. The proposed method becomes in an alternative instrument improving the capabilities of existing lensless microscopes.
© 2015 Optical Society of America
Lensless holographic microscopy (LHM) in the digital domain has nowadays becoming in a valuable and versatile tool for different applications ranging from biology [1–3], point-of-care testing , telemedicine  and optofluidic  applications, just to cite a few. LHM supposes a digital version of the Gabor’s concept implemented for electron microscopy in the middle of the XX century  and later adapted to the optical range by Rogers . Classical LHM working scheme uses a point source (typically in the form of a spatial aperture or pinhole [1,3,5,6]) of coherent (laser) light. But successful implementations have been validated using alternative illumination variations such as LEDs , super luminescent diodes  or pulsed laser radiation , and point sources using GRIN lenses  or fiber optics .
Aside of other restrictions (mainly, coherent noise and weak diffraction assumption ), Gabor’s concept of holography is affected by the twin image problem as a direct consequence of the in-line configuration . Essentially, a misfocused version of the twin image disturbs final image quality when imaging at the real image term. Twin image can be eliminated in Gabor-like layouts by using typically two different strategies: phase-shifting [15–18] and phase retrieval [19–28] procedures. Phase shifting is based on the relative phase variation between both interferometric beams and phase retrieval can be conducted from iterative [19–23] as well as from deterministic [24–28] algorithms. In any case [19–28], several holograms incoming from mechanical displacement of the camera [19,20] or by tuning the illumination wavelength [21,22] or by defocusing the image using a SLM  or by solving deterministic equations of the propagation field [24–28] must be recorded.
Among them, in  Waller et al. refer to a clever method to retrieve phase distribution based on the transport of intensity equation (TIE) where the axial defocus needed between images is provided by the intrinsic chromatic aberration of imaging lenses. Using broadband illumination and conventional RGB color camera, the approach provides 3 axially slightly defocused images (one in focus, one under and other over focus) corresponding with the 3 color camera channels. These 3 intensity images are the TIE algorithm inputs. Moreover, the retrieved quantitative phase information is obtained in a single camera snap-shot meaning that it is the camera acquisition rate the limiting factor for imaging dynamic biological processes. Note that phase-shifting [15–18] as well as phase retrieval [24–27] imply the recording in time sequence of multiple (more than two) in-line images before applying the phase recovery cycle, thus preventing (at least in a factor of two) the analysis of fast dynamic events.
As a continuation of its previous work , Noom et al. have recently presented a quantitative phase contrast LHM method first under sequential illumination/recording mode  and after with high-speed capabilities . Their approach is based on a similar concept than in  but applied in the domain of LHM: while the concept presented by Waller et al.  is based on a refractive principle (axial chromatic aberration), the principle proposed by Noom et al.  is based on the wavelength dependence of diffraction. On both cases, 3 simultaneous sample images (slightly defocused intensity images and Fresnel domain in-line holograms for the Waller et al. and Noon et al. cases, respectively) are recorded in a single snap-shot of a color camera. Noon et al. method  records 3 Fresnel diffraction patterns generated at the digital sensor plane by illuminating the sample with 3 simultaneous RGB wavelengths incoming from a fiber coupled source containing 3 diode lasers (636 nm, 519 nm, and 402 nm). After that, digital image processing based on numerical propagation and phase retrieval algorithm performs final phase contrast image of the inspected sample.
In this manuscript, we report on a similar hardware concept of LHM but having improved capabilities concerning quantitative phase imaging, resolution and processing time. As in the Noom et al. method , a trichromatic illumination source is used to record 3 simultaneous in-line holograms in a single CCD recording. Then, the holograms are digitally processed using a novel algorithm yielding in high-resolution noise-reduced quantitative phase imaging due to coherent noise averaging and twin image elimination. Moreover, the proposed algorithm provides a fast convergence to the final result in comparison with the one reported by Noom et al. . We have named this new microscope layout as MISHELF, initials incoming from Multi-Illumination Single-Holographic-Exposure Lensless Fresnel.
The manuscript is organized as follows. Section 2 presents the experimental layout as well as a detailed description of the phase retrieval algorithm. Section 3 includes experimental validation concerning a synthetic object (resolution test target) and a biological sample (swine sperm cells). And Section 4 discusses and concludes the paper.
2. Experimental layout and digital image processing in MISHELF microscopy
2.1 Experimental layout
The proposed experimental setup is outlined at Fig. 1. Three laser illumination wavelengths at 405 nm (violet laser diode module, 50 mW optical power), 532 nm (green-diode-pumped laser module, 50mW optical power) and 632.8 nm (He–Ne laser source, 35mW optical power) are properly combined (dichroic beam splitters) before impinging onto a pinhole (stainless steel aperture, 1 μm diameter). The pinhole provides a divergent spherical point source of RGB coherent illuminations with numerical apertures (NAs) values of 0.63/0.53/0.4 for RGB wavelengths, respectively. These 3 beams illuminate the sample at a short distance (d – z) from the pinhole plane as it corresponds with a LHM configuration for providing a reasonable magnification factor (typically between 5X and 20X). Then, a color CCD camera (AVT 1394 Stingray F-145C, 1388 × 1038 pixels, 6.45x6.45 μm pixel pitch) records the 3 RGB-coded Fresnel diffraction patters in a single CCD capture. The color CCD camera is placed at d and z distances from the pinhole and sample planes, respectively (see Fig. 1). After that, the information contained in the 3 CCD channels is digitally processed to finally obtain a high-resolution image without twin image contribution.
We have conducted experimental validation of our MISHELF microscope concept using 2 different configurations in the experimental layout. The first one is characterized by a low NA value and it is used to present in detail the digital image processing involved in the MISHELF algorithm. We have selected a positive United State Air Force (USAF) resolution test target to show in depth the algorithm steps. And the second implementation provides a medium NA value and it is applied to image a biological sample under quantitative phase imaging mode.
For the first experimental layout (low NA), we have forced the image to be square (1038 x 1038 pixels) just to simplify digital computing and optimize image visualization when presenting the figures. This fact does not suppose any penalty aside of deteriorating a bit the resolution limit in the reduced down image direction. The USAF test is placed at, approximately, 1.2 mm after the pinhole and at 24 mm in front of the CCD. Thus, the layout magnification factor M can be computed as the test’s geometric projection into the recording device in the form of: M = d/(d-z) ≅ 21X. Considering image (1038 pixels) and pixel (6.45 μm) sizes, the NA defined by our MISHELF microscope is NA = 0.14 (this is the previously referred low NA value) and the resolution limit for the shortest illumination wavelength is R = 0.82λ/NA = 2.37 μm.
In the second arrangement (medium NA), the experimental layout has been slightly modified reducing the distances between components in order to improve NA and hence the resolution limit. Moreover, we have considered full frame image (1388x1038 pixels) provided by the CCD. We have set d = 15 mm and z = 14 mm, so that M = 15X, NA = 0.30 (for the largest CCD direction), and R = 1.1 μm (according to that NA value and the blue wavelength). This configuration has been experimentally validated for a real biological sample, thus enabling sample’s visualization using qualitative and quantitative phase imaging modalities such as phase contrast, differential interference contrast (DIC), 3D representation, spiral phase contrast, etc.
2.2 Digital image processing
Our MISHELF microscope involves a novel and fast convergence algorithm for high-resolution quantitative phase imaging. The algorithm working flow diagram is represented through Fig. 2 where we want to stress that all the variables are spatially dependent (for instance: A0 = A0(x, y)) but we have not included the (x, y) coordinates at Fig. 2 to save space. The sample becomes illuminated with the trichromatic RGB source and one RGB hologram is recorded by the color CCD camera. This RGB hologram is then separated into its 3 elementary RGB channels providing 3 RGB color-coded holograms. After that, each hologram is subjected to an image equalization stage for improving them and consisting in 2 corrections: first, the illumination profile of each illumination beam is homogenized across the full frame to compensate the lower intensity at the borders; and second, the crosstalk resulting from the Bayer color filter array are subtracted from each channel to eliminate the contribution incoming from the other two complementary wavelengths. This image equalization stage needs to be done once and it involves basic (subtraction) digital image processing which can be easily implemented by preliminary calibration of the MISHELF microscope without sample.
After that, the phase iteration algorithm starts with the definition of the initial amplitudes per each RGB wavelength () as the square root of the intensity distribution on each equalized RGB hologram (). The 3 initial RGB amplitudes are propagated to its best focus and the resulting propagation distances are stored in the computer’s memory. Since refraction is wavelength dependent (the thickness of the sample’s slide slightly modifies the effective axial position of the sample because of the wavelength dependence in refractive index), the propagation distances per hologram are slightly different; so, MISHELF microscopy plays an equivalent role than a conventional LHM with only one wavelength but axially shifting the camera to record 3 in-line holograms at slightly different z distances. There are different numerical methods for digital reconstruction of the diffraction integral . Among them, we have selected to solve the diffraction Rayleigh-Sommerfeld integral using convolution operation since it allows an effective and economical calculation without any approximation. After numerically propagation to the sample plane, the 3 in-focus images of the sample () are then used to synthesize a single fused image ().
To synthesize this fused image, we implement the following steps. First, we perform digital fast Fourier transform (FFT) of each in-focus images. Second, the global background phase for the 3 in-focus images is forced to be equal just to avoid phase cancellation when mixing their spectra. Third, the 3 spectra are then used to synthesize a new single spectrum incoming from a weighted addition of them. By weighted we mean that, because of the spatial frequency content contained on each image is different due to wavelength dependence of diffraction, the R spectrum contains a lower cutoff frequency than the G one which, in turn, is also lower than the cutoff frequency of the B spectrum. Thus, the information contained in the R spectrum is also included in the G one, and both are included in the B spectrum. So a weighted mask taken into account this fact is applied for not incurring in a low spatial frequency enhancement. And fourth, inverse FFT of the mixed spectrum recovers an improved image of the sample (), improved in the sense of noise averaging due to the mixing of the 3 images into a single one.
Such improved image becomes the input of a phase retrieval algorithm. The algorithm consists in the iterative application of back and forth numerical propagations between hologram and sample planes, but taking into account two peculiarities. On one hand, the improved image is propagated 3 times taken into account the 3 propagation distances previously obtained in the first back propagation. Thus, 3 color coded holograms are generated at the CCD plane from which their phase distributions () are retained while the amplitude ones () are replaced by the square root of the recorded hologram intensity (). Then, the new complex amplitudes () are back propagated again (each one with its corresponding propagation distance) to their in-focus planes. And on the other hand, a new single and mixed spectrum is synthesized every time that one back and forth cycle is performed, that is, a new single image is synthesized at the sample plane for each cycle in the algorithm. After a few number of iterations, a final image containing complex amplitude information of the sample is obtained with improved capabilities since it contains no information about the twin image and exhibits better image quality concerning noise, halos and contrast. Moreover, the single spectrum generation process allows fast convergence in the phase retrieval algorithm since information of the 3 channels is mixed together in each iteration. All together improves quantitative phase imaging of the sample which can be visualized under several imaging modalities such as 3D views, contrast images, plotting some regions of interest for profiling, etc.
For its better understanding, Fig. 3 presents a schematic chart of the algorithm steps when using the USAF test target as input sample. The RGB recorded hologram is separated into its 3 channels allowing direct focusing of the object on each one of them (upper row). These images can be improved by implementing the image previously commented equalization stage (Fig. 3 also includes the RGB illumination profiles without USAF test). Equalized in-focus images (lower row below the central RGB hologram) are then Fourier transformed and the new mixed spectrum is synthesized using the information contained at the RGB spectra considering a weighted mask for not incurring in spatial frequency enhancement. The new synthesized spectrum is Fourier transformed back to recover an improved image of the object (left image at the lower row). This image is the input for the phase retrieval algorithm previously exposed.
3. Experimental validation
Aside of the USAF images presented in previous section, Fig. 4 shows a direct comparison of the USAF reconstructions obtained under different modalities of LHM using the low NA layout (0.14NA): (a) shows the image obtained under conventional LHM using blue illumination (B-LHM), (b) depicts the same image after image equalization stage, (c) presents the image obtained as inverse FFT after the first synthesized spectrum, and (d) shows the final image after phase retrieval with only 2 iterations (additional iterations do not improve the result). Images from (e) to (h) include a magnification of the USAF central part where the resolution limit is produced. Note that all the images included in Fig. 4 are intensity images depicted in Matlab with the same visualization power to allow direct comparison among them.
We can see as the resolution limit at all the images is the same and it corresponds with a value of 2.46 μm (Group 8-Element 5 as last resolved element), value which is in very good agreement with the theoretical one obtained for the B wavelength (see Section 2.1). However, image contrast is clearly enhanced and coherent noise averaged when comparing the MISHELF image with the conventional B- LHM one. Concerning the field of view (FOV), the black solid line in the lower right-corner of Fig. 4(d) has a length of 50 μm. So the total FOV width is around 280 μm. Assuming we are disregarding the borders of the image (black perimeter rectangle shown in last images of Fig. 3) due to border’s diffraction effects when performing numerical propagation, the CCD useful area is around 900 pixels width. This fact means that the FOV (280 μm) is framed in 5.8 mm, approximately, defining an experimental M value of 20.7X, value that also perfectly matches with the theoretical one (M = 21X). Note that the magnification factor is needed to avoid that the CCD sampling in Fresnel diffraction regime may be a limiting factor in the final image resolution. Moreover, because MISHELF technique uses a color CCD, the pixel number in each channel is not the maximum available (as it happens when using monochrome sensors in LHM); so, high magnifications must be pursued.
In addition, Fig. 5 includes the images obtained for a second experiment under the same experimental conditions (0.14NA) but considering now full frame image size (1388 × 1038 pixels). As in Fig. 4, we have compared the final recovered image by MISHELF microscopy [(c) and (d)] with the conventional single-wavelength B-LHM [(a) and (b)]. Although USAF object does not contains phase information because it is a real amplitude object, we have included phase distributions in Figs. 5(b) and 5(d) just to qualitatively show that phase image is also improved as consequence of the proposed method. Once again, images from (e) to (h) include a magnification of the central part of the USAF. Note as, because of the rectangular detection area, the resolution limit is not now the same on both orthogonal directions. Now, the last resolved element (Group 9-Element 2) in the largest CCD direction defines a resolution limit of 1.74 μm. This value is again in good agreement with the theoretical prediction because, now, we have 1388 pixels defining a NA value of 0.185NA and yielding in a resolution limit of 1.79 μm. Obviously, the NA value for the shortest direction is not changed (0.14NA) and the last resolved element is again the same one (Group 8-Element 5) than in the previous experiment.
To clearly show that twin image is eliminated due to the phase retrieval algorithm, we have included two video movies corresponding with conventional B-LHM (Visualization 1) and MISHELF microscopy (Visualization 2). The first frame at both videos corresponds with and in-focus image of the USAF test and then digital propagation to the twin image plane (last frame of the movies) is applied. Figure 6 presents the propagated images to the plane where the twin image is located. One can see as conventional B-LHM contains twin image contribution while MISHELF microscopy eliminates the twin image (no USAF image in the last movie frame).
Now, the medium NA layout (0.3NA) is assembled and MISHELF microscope is tested for a biological sample in order to show quantitative phase imaging capabilities. The biological sample is an unstained swine sperm sample (head width of 6x9 μm, total length of 55 μm, and a tail’s width of 2 μm on the head side and below 1 μm on the end, approximately) which is dried up for fixing the sperm cells in the counting chamber. According to the theoretical values (see Section 2.1), the resolution limit is enough to resolve the sperm tails in the neck side while it is not at their end. Figure 7 show the experimental results of the MISHELF microscope in comparison with the ones obtained when using single-wavelength (blue) conventional LHM. Figures 7(a)-7(b) show, respectively, the RGB and the B-LHM holograms while images in Figs. 7(c)-(d) present the intensity images provided by MISHELF microscope after 2 phase retrieval algorithm iterations and B-LHM, respectively. As in the USAF case, one can see as MISHEL microscopy provides a global contrast enhancement and background noise reduction along the whole image.
Nevertheless, the biosample contains valuable phase information which provides qualitative (improved visualization modes) as well as quantitative (capability of measure cell’s profile) information about the sperm cells. Figure 8 includes a comparison of the images which can be obtained. First row presents the images in positive phase contrast modality while second row show negative phase contrast images. We have included the images obtained from MISHELF microscopy [Figs. 8(a) and 8(e)] as well as the single-wavelength conventional LHM for R-G-B illuminations [Figs. 8(d) and 8(h), 8(c) and 8(g), and 8(b) and 8(f), respectively]. One can visualize the noise reduction concerning twin image elimination, background homogeneity, coherent artifacts, spurious reflections, etc. resulting in a global image quality improvement. Moreover, resolution is not penalized since, as previously for the USAF case, MISHELF microscopy retains the resolution limit provided by the best (the lower) illumination wavelength.
All those qualitative images are supplemented with quantitative information incoming from the 3D representation of the retrieved unwrapped phase distribution. Figure 8 presents a comparison of the biosample’s 3D phase profile when considering the MISHELF microscope and the B-LHM images. We have computed the Standard Deviation (STD) of the background noise in order to compare the phase stability derived from MISHELF microscopy and conventional B-LHM. The STD values are computed at the two regions marked with dotted black line rectangles in the phase profiles included in Fig. 9. Note that no sperm cells are contained at both areas and that phase values have been normalized previous to the STD calculation, thus allowing direct comparison of the phase noise stability. We have obtained 0.028 and 0.062 rads averaged STD values for MISEHLF microscopy and B-LHM, respectively, meaning that a factor a bit higher than 2 is obtained in phase stability improvement as consequence of the proposed algorithm.
In addition, the recovered phase distribution can also be digitally manipulated to generate a wide variety of phase visualization images in a similar way than optical procedures allow. The main advantage is that the imaging synthesis is performed numerically instead of optically, that is, there is no need to change to special lenses or illumination systems in order to provide such images. Figure 10 depicts a set of digitally-processed classical images for phase visualization incoming from the complex amplitude distribution retrieved by MISHELF microscopy. We present: Fig. 10(a) the darkfield image by blocking the DC term at the Fourier plane, Fig. 10(b) and Fig. 10(c) the DIC images by subtracting two displaced images in the horizontal and vertical directions, respectively, and Fig. 10(d) a spiral phase contrast imaging by adding a spiral phase mask at the Fourier plane. One can observe that each modality enhances the visibility and contrast of different features of the biosample.
Finally and aside of improving qualitative and quantitative phase image quality, the proposed new algorithm also improves convergence in the phase retrieval procedure. For instance, in  Zhang et al. report on a digital in-line holographic layout where 3 holograms are acquired by axially shifting the camera instead of using 3 wavelengths. According to , the phase distribution is retrieved after 75 iterations of the phase retrieval algorithm that performs numerical propagation between the 3 recording planes. The number of iterations is reduced until 30  by using 3 wavelengths with a single and static monochrome camera. Note that nothing is stated concerning the phase retrieval algorithm  but that “a few iterations” are needed and that “the same reconstruction procedure” as in  is used. Here, we have reported on superior image quality than the previous RGB multi-wavelength Fresnel diffraction imaging incoming from a higher resolution limit (0.3NA), a significantly improved visualization of phase through multiple phase imaging modes, and a fast algorithm convergence incoming from a reduced number of iterations. The last sentence is supported by the fact that images included along this manuscript only need 2 iterations in the phase retrieval cycle for both USAF and biosample cases. A similar phase retrieval procedure but without RGB spectrum synthesis (only considering a single wavelength) and applying known constraints about the input object (pure phase or amplitude object) needs around 15-30 iterations (depending on the object type) for twin image elimination ; so, the digital processing reported in this manuscript greatly improves algorithm convergence.
4. Discussion and conclusions
In this manuscript, we have provided a step forward on a recently reported LHM method based on wavelength multiplexing digital in-line holography in the Fresnel domain  but with improved capabilities. The technique, named here as MISHELF microscopy, is based on the simultaneous illumination of the sample with a trichromatic (RGB wavelengths) laser source and the recording of a single-shot multiplexed hologram containing information about the 3 wavelength-coded Fresnel patterns diffracted by the sample. Coherent trichromatic illumination is accomplished by combining three independent laser sources over a single illumination pinhole and simultaneous recording of the 3 Fresnel patters is achieved by single snap-shot of a color CCD camera.
Although the use of multiple illumination wavelengths in LHM has been previously reported in the literature [33–36], their use was restricted to pseudocolor image rendering [33,34,36] and/or multispectral analysis [33,35] of the sample. In MISHELF microscopy, RGB illuminations are used for phase retrieval and quantitative phase imaging in single exposure LHM, in addition to the previously commented color imaging capabilities.
Concerning similarities with , MISHELF microscopy differs from and improves quantitative phase imaging in the following. First, it uses a pinhole as illumination source instead coupling the lasers to a single-mode fiber. Single-mode fibers permits low NAs in illumination (typically in the range of 0.1NA), so resolution is highly compromised by this low NA value, even more considering that the real useful NA is usually smaller than its theoretical value due to the intensity decreasing at the borders. However, medium (in the range of 0.4-0.5) NA values are easily accessible by using a pinhole for illumination, as usually is in digital in-line holography [1,3,9,34,35], thus improving the resolution of the final image. In this manuscript, the hardware has been improved allowing the definition of medium NA values (in the range of 0.3), meaning that resolution limits close to 1 micron are accessible.
Second, our novel phase retrieval algorithm converges extremely faster than others because it mixes together information incoming from the 3 RGB channels. The improvement in image quality is not only as consequence of the weighted mask (spatial-frequency enhancement or band-pass filtering must be avoided) but due to the fact of mixing 3 images into a single one. In addition, this fact results in a reduction of the processing time to a few seconds when using standard desktop computers. Note that the algorithm reported in [30,31] is not exposed in depth and converges to a final image after 30 iterations , meaning about 1 min of processing time on a standard desktop computer. MISHELF microscopy algorithm has demonstrated similar (if not superior) image quality retrieval with only 2 iterations.
And third, the final image presents noise reduction and improved image quality incoming from both a qualitative and a quantitative point of views. Qualitatively, MISHELF images present enhanced contrast, halos and coherent noise reduction, twin image elimination, higher background homogeneity, and reduction of coherent artifacts and spurious reflections in comparison with conventional single-wavelength LHM. And quantitatively, all those image quality enhancements derive in phase noise reduction through the definition of lower STD values.
In summary, we have reported on MISHELF microscopy from a double point of view. First, the hardware has been improved allowing the definition of medium NA values (in the range of 0.3), meaning that resolution limits close to 1 micron are accessible. And second, the software has also been improved and optimized through a novel phase retrieval algorithm based on a customized management of the information provided by the 3 illuminations at the Fourier domain. The final images obtained by our MISHELF microscope embodiment provide quantitative phase imaging improvement as well as increase the convergence of similar algorithms through reducing the number of iterations needed to get the final image. Moreover, since the retrieved quantitative phase distribution is obtained in a single camera snap-shot, the limiting factor for imaging dynamic biological events is the camera acquisition rate thus enabling video-rate movies of the inspected samples. Experimental results have been presented validating all previous statements concerning MISHELF microscopy. The synthetic object (USAF resolution test target) case provides useful images for layout calibration and algorithm in-depth presentation. And the biological sample (swine sperm sample) shows different qualitative and quantitative phase imaging modalities.
We thank Prof. Carles Soler and Francisco Blasco from Proiser R + D S.L. for providing the swine sperm sample. Part of this work has been funded by the Spanish Ministerio de Economía y Competitividad and the Fondo Europeo de Desarrollo Regional (FEDER) under the project FIS2013-47548-P.
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