Image reconstruction using measurements in volume

Image reconstruction using measurements in volume speckle fields formed by different wavelengths Nikolay V. Petrova *, Mikhail V. Volkovb Andrei A. Gorodetskya , Victor G. Bespalova a St.

Petersburg State Univ. of Information Technology, Mechanics and Optics, 3 Kadetskaya Linia, St. Petersburg, Russia, 199004; b Fock Institute of Physics, St. Petersburg State University, 1 Ulianovskaya Street, Petrodvorets, St. Petersburg, Russia, 198504 ABSTRACT

An iterative wavefront retrieval method based on intensity measurements formed by several wavelengths is investigated in the present contribution. This multiwavelength technique is extended to use the intensity distributions recorded in various planes of the volume speckle field. The ability to retrieve the wavefront using speckle patterns is demonstrated in experiment. Two different experimental techniques have been used. The first proposed method allows one to record three different intensity distributions corresponding to the three CCD RGB channels at single exposure. This gives the advantage in the analysis of fast processes, e.g. phase microscopy of moving biological cell-like objects investigation. The second technique involves using a large number of wavelengths of supercontinuum radiation formed by photonic-crystal fiber. This approach provides faster and more accurate convergence of the proposed method, has simple and rugged recording scheme with fiber optic elements. Keywords: phase retrieval, image reconstruction techniques, speckle, diffraction theory, cell analysis

1. INTRODUCTION The problem of wave field phase information recovery, more usually referred to as the phase problem arises in optical metrology very often. Cell biology is one of the research areas where this problem is rather important as most of the cells are optically transparent and in amplitude microscopy contrast or fluorescent probes are, usually used. However any probes modify cell and affect cell processes. Phase microscopy is a non-invasive investigation method that allows studying cells without any cell modification or cell membrane damage. Nowadays there’re enough prerequisites to introduce a phase microscopy method by phase retrieval. Calculation power growth allows to approach the phase problem solution by iterative algorithms. The first algorithm of this kind was introduced by R. Gerchberg and W. Saxton.1, 2 They used amplitude information in pupil plane and image plane. Important update was reported by J. Fienup3 who introduced limitations for object reconstruction by the amplitude distribution in Fourier plane, namely - nonnegativity of image and its concentration in a certain area. These approaches use a single measurement, and therefore are not sensitive to any external disturbance, so they can be used for fast processes. However despite the fact that successful reconstruction can be achieved using these methods there are certain hurdles concerning usage limits. Another method, that allows phase reconstruction is the variation method of supporting error functional minimization, achieving an extremum at signal reconstruction.4 Such method allows using of additional information obtained from data accumulation for lost wavefront phase compensation, thus ensuring iteration procedure convergence. Wavefront registration at multiple planes is discussed in the previously mentioned paper.4 Afterwards such approach was successfully developed in the works of P. Almoro, G. Pedrini and W. Osten.5–7 Convergence of phase retrieval iterative algorithms using additional speckle field information can be interpreted in terms of D. Gabor theorem8 on degrees of freedom in the optical system. Optical system can be described by N invariant degrees of freedom, i.e. basis set of orthogonal modes in the spatial and frequency domains. Additional intensities registration compensates for the loss of the phase information thus equalizing the number of variables with the number of unknowns. The mentioned facts research led to the appearance of the variety of methods that use additional *E-mail: [email protected]

Biomedical Applications of Light Scattering V, edited by Adam P. Wax, Vadim Backman, Proc. of SPIE Vol. 7907, 790718 · © 2011 SPIE · CCC code: 1605-7422/11/$18 · doi: 10.1117/12.876151

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information. Besides the registration in multiple planes, defocusing,9, 10 rotation transform, or the use of multiple wavelengths11, 12 are used. In this paper we develop an approach comprising the use of multiple wavelengths for phase problem solving. Contemporary laser sources cover the full visible spectrum as well as the registration systems often have several spectral channels, that offers extra benefits in wavefront registration. Advances in fiber femtosecond lasers development allow to implement the rugged setup for wavefront retrieval using the spectral components of supercontinuum. Principles of the reconstruction by supercontinuum and multiple wavelengths are the same: the substitution of the digitally propagated wavefront amplitude to the square root of the intensity of different wavelength, retention the phase information. The whole iteration procedure for multiple wavelengths and distances allows to combine specklegram registration process and allows the registration of 6 to 9 intensity distributions at single exposure. A more detailed discussion on this algorithm can be found in section 2. Experimental and simulation results are presented in section 3. The description of the experiment, comprising supercontinuum light components is discussed in section 4.

2. ITERATIVE ALGORITHM FOR WAVEFRONT RETRIEVAL Let the planar object be illuminated by laser radiation comprising several wavelengths i = 1, ...N . We can measure several diffraction intensity patterns for all N at several distances l = 1, ...M . As a result we get a set of speckle patterns; each being characterized by wavelength λi and distance lj from object to registration plane. 2D Fresnel transform can be taken as an equation for free wave propagation in paraxial approach: exp(ikl) ik 2 2 [(x − x ) + (y − y ) ] Fλ (x , y )dx dy . exp (1) Fλ (x, y, l) = iλl 2l 2 R here Fλ (x, y, l) is a complex function; Fλ (x , y ) = Fλ (x, y, 0) complex object transparency; x, y — coordinates in registration plane; x , y — coordinates in object plane; l — variable distance; k = 2π/λ. For proper understanding let’s describe the simple case of diffraction on planar amplitude object. If we settle all parameters in (1) except λ, we get various diffraction patterns growing in size with wavelength growth. If we change distance l instead we get the same dependency. We get identical diffraction pattern when making distance m times greater and wavelength the same amount smaller at the same time. (Fig. 1). A non-uniform phase object characteristic

Figure 1. Two equivalent intensity distributions formed by different wavelengths.

leads to speckle structures appearance which in turn makes visual estimation harder. The greater wavelength pattern has the smaller phase incursion that can be used to avoid lines that appear due to 2π phase skip in retrieved phase distribution. Another advantage of this approach is a possibility to use wavelength and distance dependent intensities set Iλi ,lj (x, y) for wavefront retrieval. As this set has two varying parameters, speckle patterns can be used for reconstruction in any order. To describe the algorithm in general, without specifying the order of priority let’s take the set of intensities I s , where s = 1, 2, ...N · M , in which λs and ls will finally


describe all the λi and lj combinations. The retrieval process starts with calculating square root of a certain intensity pattern I 1 . Obtained amplitude is multiplied by a constant phase to form a complex field in record plane Fλ11 (x, y, l1 ). This field is then back-propagated to object plane yielding the reconstructed object field F 1 (x , y ) which differs greatly from required at starting iterations. Then the field is again propagated to recording plane with parameters corresponding to the second intensity in the set (λ2 and l2 ). In the resulting complex field Fλ22 (x, y, l2 ) we keep the phase and substitute the amplitude with the square root of the second intensity pattern √ I 2 , forming Fλ22 (x, y, l2 ). Denote it as a step - the resulting complex field is back-propagated to the object plane and then reversed to the plane and parameters corresponding to the next intensity pattern where phase is kept and amplitude is substituted by the square root of the corresponding intensity distribution. Iteration is a set of steps performed for N · M wavelength and distance combinations. For each intensity pattern we compare √ calculated wavefront amplitude Fλss (x, y, ls ) with square root of measured intensity I s to obtain normalized root-mean-square(RMS) error, or misfit: E = s

√ |As − I s |2 √ s2 . | I |

(2)

The whole iteration RMS error, which characterizes the convergence of iterative algorithm defined as: E=

N ·M 1 Es N · M s=1

(3)

3. THREE-WAVELENGTH SINGLE EXPOSURE SYSTEM For fast microscopic processes study, we need a system that is capable to register all the necessary information immediately.

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(c) Figure 2. (а) — Profile/cross-section of BGGR sensor; (b) — Spectral response of Nikon D70/D50 camera CCD ; (c) — Scheme of three-wavelength setup. Each laser beam was scattered on an object put into diaphragm after passing two mirrors m and beam splitters bs. Registration at different distances is performed using beamsplitters.


In CMOS and CCD arrays Bayer filter is used for forming spectral RGB channels in photodetectors lattice. (Fig. 2 (a)). Selecting narrow spectral sources, corresponding to channel sensitivity maxima, one can register up to three independent intensity patterns at one exposure. Use of 2-3 5mm beamsplitters allows to increase the number of patterns up to 6-9. Fig. 2 (b) shows the spectral response of CCD array of Nikon D70/D50 camera.13 Each channel has a spectral region, where its sensitivity is much higher than sensitivity of other channels. It lies between 400 − 450 nm for blue channel, 520 − 560 nm for green, and 620 − 650 for red. Speckle patterns can be formed at these wavelengths, each of them being registered by corresponding spectral channel of the CCD array during one exposure. In the paper7 use of beamsplitters array for simultaneous registration of greater number of speckle patterns is discussed. Proposed registration scheme (Fig. 2 (с)) allows to use less number of beamsplitters. In current work, we present results on experimental and modeling study of three-wavelength speckle recording and subsequent numerical wavefront retrieval.

3.1 Modeling In modeling we used both amplitude object with random phase disturbance (Fig. 3, (a)) and phase-only object (Fig. 3, (c)). For these objects, wavefront intensity patterns were modeled for use in iterative algorithm. Fig. 3 depicts retrieved images for amplitude object with random phase distribution (Fig. 3 (b)) and for phase-only object (Fig. 3 (d)). Pixel size for each CCD array spectral channel was taken 15.6 μm, that corresponds to doubled real pixel size of Nikon D50 camera. This doubling of pixel size comes from pixels position in Bayer filter array. Object size is 256 × 256 pixels, that corresponds to physical size of 2mm × 2mm. The width of amplitude object elements varies from 2 to 70 pixels, i.e. 16-547 μm, the width of phase object elements is 18 pixels = 281 μm. Maximum phase shift at phase object was 2.36 rad. Modeling was performed at wavelengths of 455 nm (Ar laser), 532 nm (2nd harmonic of Nd:YAG laser), and 632.8 nm (He-Ne laser) at distances of 75 mm, 80 mm, and 85 mm for each wavelength. As it was previously mentioned in section 2, identical speckle patterns can be formed with various wavelengths, so, the only distance can be taken and 9 wavelengths instead, namely: 417.2 nm; 498.8 nm; 593.3 nm; 445 nm; 532 nm; 632.8 nm; 472.8 nm; 565,3 nm; 672.4 nm.

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(b)

(c)

(d) (e)

Figure 3. Initial — (а) and reconstructed — (b) amplitudes of test-object with random phase distribution. Initial — (c) and reconstructed — (d) phases of pure phase test-object; (e) — convergence of the method with different number of used intensities distribution

Fig. 3, (e) shows the iterations number dependency of total normalized error for various number of intensity patterns. If only two or three wavelengths are used, amplitude and phase object characteristics are distinguishable, but are modulated with noise, small details are indistinct. Use of beamsplitters in registration scheme crucially increases quality of retrieved wavefront. In this case, the fastest convergence is seen at first ten iterations. The order of intensity patterns use affects the retrieval quality as well. We found that the order of


”three wavelengths at each three distances” (λ1 l1 , λ2 l1 , λ3 l1 , λ1 l2 , ...λ3 l3 ) gives the most fastest and most precise convergence than any other order (e.g. change of distance through wavelength growth λi lj < λp lq ). This could be explained as follows: in the first order, the resulting indexes λi lj of the intensity patterns differ more, this results in greater error at moving from one intensity pattern to another, this error being killed by substitution of amplitude, thus minimizing the resulting residual.

3.2 Experiment Experiment was designed to prove the possibility of three speckle patterns simultaneous recording, that are suitable for wavefront retrieval. We used the setup from Fig. 2(с), with the only modification - beamsplitters at registration were substituted with motorized translation stage from Standa. We used the following laser sources: He-Ne (632.8 nm), Nd:YAG (1064 and 532 nm), and DPSS (473 nm). The logo of St. Petersburg State University of Information Technology, Mechanics and Optics (SPbSU ITMO) on a microfilm was used as amplitude object (Fig. 4 (a)). Letters height at this logo is about 200 μm. Diffraction patterns were registered with Nikon D50 digital SLR camera without a lens (2014 × 3039 pixels, ∼ 7.8μm by ∼ 7.8μm each). The patterns were saved in RAW format, that contains separate information for each spectral channel. The shots were processed by DCRaw converter with the following parameters: -4 -T -D *.NEF. With such parameters DCRaw converter performs processing from camera’s 12-bit ADC to 16-bit TIFF file, without any color interpolation or scaling, usually performed by camera’s processor. Later, data array in BGGR format was split into 3 arrays (Fig. 2 (a)), each corresponding to separate spectral RGB channel, with double pixel size. Thus, in current experiment, during one exposure, two ready-for-processing intensity distributions were obtained: red and green, or red and blue (Fig. 4 (b)). This is simply explained by high sensitivity of camera green channel to 473 nm DPSS blue laser radiation (Fig. 2 (b)). The result of reconstruction from two intensity patterns is shown on fig. 4 (c)).

(a)

(b)

(c)

Figure 4. Experimental results: (a) — Amplitude object picture through microscope; (b) — intensity patterns, uppermost — two-wavelength, lowermost — single wavelength from separate spectral channels; (c) — the result of wavefront retrieval.

4. USE OF SUPERCONTINUUM SPECTRAL COMPONENTS Spectral supercontinuum can be used like a beneficial source of radiation for the formation of a large data set, which can provide faster and more accurate convergence of proposed technique. We performed wavefront retrieval from speckle patterns obtained with spectral supercontinuum radiation. Using the components of fiber optics, it is possible to implement a simple and rugged device with wavelength division within the fiber. We performed a tentative experiment to demonstrate the acceptability of such an approach. Experimental setup scheme is presented at fig. 5 (a). The radiation of 2nd harmonic of Er3+ femtosecond fiber laser LR (λ = 780 nm, pulse duration 130 fs, mean power 47 mW) with microlens OB on fiber translation stage FTS is led into photoniccrystal fiber with quartz core PCF, where spectral supercontinuum radiation is generated. Using diffraction grating DG on a rotation stage RS and spatial filter SF, situated 50cm far, certain spectral components 1-2 nm in width were separated, then the narrow band beam expanded in collimator C, enlights previously described object T, diffraction patterns are registered by Canon EOS 450D digital SLR camera (CMOS sensor, pixel size 5.1 μm) situated 83 mm far from object. The difference in energy of various spectral components was compensated


Spectrum of su percontin U urn radiation 0

08 06 -

04 02 -

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650

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750

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850

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950

1000

Wavelength (nrn)

(b)

(a)

Figure 5. (а) — Scheme of experimental setup using spectral components of supercontinuum radiation, LR - Er 3+ femtosecond fiber laser, OB - microlens, FTS - 3D stage, PCF - quartz core photonic-crystal fiber, DG - diffraction grating, RS - rotation stage, SF - spatial filter, C - collimator, T - test object, CMOS - Canon EOS 450D digital SLR camera; (b) — reconstructed image of object.

by changing exposure time. Just as in previously described experiment, patterns were saved in RAW format (*.CR2 for this camera). The shots were processed by DCRaw converter with the following parameters: -4 -T -D *.cr2. Camera’s red channel data were used for wavefront retrieval. The result is shown in fig. 5 (b).

5. CONCLUSION In our paper, we present the generalization of wavefront retrieval iterative algorithm in case of multizone registration in various speckle field cross sections. The algorithm itself is universal and can be applied both at registration of subjective speckle patterns and if optics is used to register objective speckles. Three-wavelength system for immediate multiple speckle pattern registration is proposed that could be useful for fast processes, e.g. phase microscopy of moving biological cell-like objects investigation. The possibility of wavefront retrieval from spectral supercontinuum components diffraction patterns is demonstrated.

6. ACKNOWLEDGMENTS Authors would like to thank V.S. Shevandin (Vavilov State Optical Institute) for photonic-crystal fibers provision, and Russian National Library reprography center officers for help in making sample microfilm objects. The work was supported by the U.M.N.I.K. project of the The Foundation for Promotion of Small Enterprises in Science and Technology established by the government of Russian Federation.

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