Color encoding of phase: a new step in imaging by structured light and single pixel detection

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1 Color encoding of phase: a new step in imaging by structured light and single pixel detection Edoardo De Tommasi, Luigi Lavanga Institute for Microelectronics and Microsystems, Department of Naples National Research Council Naples, Italy Stuart Watson, Michael Mazilu School of Physics and Astronomy University of St. Andrews North Haugh, St. Andrews, UK Abstract In this paper the imaging of extended targets by means of structured light and single-pixel detection is examined. In particular, different kinds of illumination patterns (discrete Hadamard; continuous sinusoidal; complex Laguerre-Gauss) are compared both numerically and experimentally in terms of obtainable resolution. Furthermore, a new approach for encoding the complex phase which allows to reduce the number of illuminations for a given resolution is presented. Keywords: imaging; modulators; structured light; single-pixel image reconstruction; encoding of phase. I. INTRODUCTION In a conventional imaging system, the light coming from a source (e.g. a lamp or a laser) uniformly invests the object to be imaged. The reflected, transmitted or diffused light is then detected typically by a CCD camera, characterized by m n pixels; the resolution of the final image obviously depends on the total number of pixels. This implies, for example, that in all the microscopy systems based on pixel-by-pixel scanning (fluorescence, confocal, two-photon, Raman and CARS microscopies) the more resolution is required, the more the pixels; the more the pixels, the more time is spent in scanning, which could be a problem, for example, in performing spectral imaging on a living (i.e. moving) sample, like a single cell [1]. Despite all these limitations, the reciprocal nature of optical systems allows to transfer the information relative to spatial resolution to the illumination source and, on the other side, reduce the detecting camera to a "single pixel" device (e.g. a photodiode or a photomultiplier). The light emitted by the source is encoded in more or less complex patterns by means of a spatial light modulator, e.g. a digital-mirror-device (DMD), which can reach refresh rates of tens of khz, allowing to acquire the signal in a very short time interval. The patterns act like probes which allow to obtain higher spatial information of the object the higher is their spatial complexity. The image itself is then retrieved applying to the detected signal algorithms based, for example, on compressive sensing techniques [2]. The main applications of this approach are, as stated above, in spectral microscopy [3,4] and, furthermore, in all the imaging techniques where no efficient CCD are available in a given spectral range. In the present paper three different light patterns (Hadamard, sinusoidal and Laguerre-Gauss probes) are used in a single-pixel detection scheme and compared in terms of resolution performances, by retrieving both the Point Spread Function (PSF) of the system and the images of several extended targets. Furthermore, the ability of this system to encode with colors the complex phase of Laguerre-Gauss beams by means of the so called 2-simplex algorithm is described. Laguerre-Gauss beams thus resulted the most performing probes in terms of obtained resolution, and, in virtue of the 2-simplex encoding, we demonstrated that, for a given resolution, the Laguerre-Gauss probes need the lowest number of illuminations of the target respect to the other tested patterns. A. Orthogonal Patterns II. THEORETICAL BACKGROUND Orthogonal patterns are derived from orthogonal matrices which consist of orthogonal unit vectors both along its columns and down its rows. Light patterns which are mutually orthogonal optimize the efficiency of imaging since guarantee no redundancy or overlap in information. Furthermore, orthogonal patterns represent a complete set for imaging an object, i.e. there is a fixed number of patterns that must be shown in order to image an object with a given resolution. Finally, they are characterized by a high signal-to-noise ratio if compared to other kind of patterns (e.g. raster scan masks, random patterns). Hadamard, sinusoidal and Laguerre-Gauss beams represent typical examples of binary, continuous grey and complexphase orthogonal patterns, respectively, and have been used in present work in order to test encoding techniques described below. B. Encoding "negative intensities" Single matrix elements can have negative values. Of course negative intensities have no physical meaning. A simple way to encode negative intensities is to use, for each mask, two patterns, one of which is inverted respect to the other so that negative values are positive and vice versa. The intensity

2 value obtained for the inverted pattern is subtracted from the value obtained from the positive pattern, so as to simulate a negative intensity. Obviously this procedure implies that, if N patterns are used in a single measurement, the number of actual, corresponding illuminations will be 2N. with at least one of them equal to zero; in such a representation the projection of N patterns correspond to N+1 illuminations instead of 2N. C. Encoding the phase of complex fields In case of complex light patterns, a simple way to encode their phase is to make use of what we called 2-simplex encoding technique. A complex number z is associated to components x and y, referring to its real and imaginary part. Both x and y can assume positive or negative values, so for N complex patterns, 4N illuminations are needed. Actually the same complex number can be represented in a positive, threedimensional space such that at least one of its components will equal 0. The three components of z can thus be associated to the three fundamental colors red, green and blue (r,g,b). In this new base the illuminations needed to project N complex patterns will be 3N instead of 4N. In Fig. 1 the representation of a complex number in complex plane and in "rgb" plane, respectively, is shown. Figure 2: For each pixel P, a set of N orthogonal fields can be represented in a N dimensional space with real components (e 1,...,e N ) or in a N+1 dimensional space with real, positive components (φ 1,...,φ N+1 ) where at least one of them equals zero. Figure 1: A complex number z can be represented in the complex plane with components x and y respect to the real and imaginary axis or in a three-dimensional "rgb space" where the components of z can assume only positive values and at least one of them equals zero ("red" component in figure). C. N-simplex encoding technique N-simplex encoding technique can be viewed as a generalization of 2-simplex when applied to an ensemble of N orthogonal fields. For each pixel P these fields are characterized by N real components which can assume both positive and negative values, so that N patterns in fact correspond to 2N illuminations. The same ensemble can be represented in a N+1 dimensional space, where the components of the fields can assume only real positive values, III. EXPERIMENTAL SET-UP The schematic layout of our experimental setup is shown in Fig.3. A digital light projector (DLP, Light Crafter 4500 Texas Instrument) sequentially illuminates the target to be reconstructed by a set of orthogonal structured patterns or masks of light generated by a computer. The DLP is composed by two principal elements: a digital micro-mirror device (DMD) and a light engine composed by three colored (red, green and blue) light emitting diodes (LEDs). The DMD consists of a matrix of electronically controlled micro-mirrors, acting as a reflective spatial light modulator on the light coming from the light engine [5]. The DLP also provides trigger signals that allow a synchronization between the illumination and the acquisition processes. The backscattered light impinges on a photodiode (Thorlabs Silicon Amplified Detector) which acts like a single-pixel detector. The detected, analogic signal is then digitalized by a National Instruments Multifunction DAQ and sent back to the computer to feed the reconstruction algorithm. Both the DAQ and the DMD were controlled by customized Matlab scripts and the pattern generation, data treatment and reconstruction algorithm were implemented in Matlab environment, too. The reconstruction algorithm is based on the correlation between the projected pattern and the measured backscattered intensity, i.e. each measured value is used as a coefficient to weight the corresponding mask, and the weighted sum of the patterns

3 provide the reconstructed image (see section IV A). Figure 4: a) PSF reconstructed using 810 LG beams; b) reconstruction of two separated Dirac functions using 810 LG beams. Figure 3: Schematic layout of the experimental set-up. A. Numerical simulations IV. RESULTS AND DISCUSSION We made use of a numerical model that simulates the whole process of illumination of the target and retrieval of its image. The correlation between the orthogonal illumination patterns E 1,...,E N and the illuminated target provides a series of coefficients a 1,...,a N such that: I = a! E! i = 1,..., N (1)! where I is the image to be reconstructed. The a 1,...,a N coefficients represent the detected signals corresponding to every illumination. We proposed three different families of illumination patterns: binary masks generated from a Hadamard matrix, continuous greyscale sinusoidal patterns and Laguerre-Gaussian (LG) beams. Fig. 4 a) and b) show the simulated point spread function (PSF) of the system, i.e. the reconstruction of Dirac function placed in the center of the illuminated region, and the reconstruction of two separated Dirac functions, respectively, using a set of 810 LG beams. We simulated a Dirac function by a single white pixel in a black background and reconstructed it using families of patterns of increasing cardinality. This allows us to understand how the number of probes, i.e. of illuminations, affects the resolution capability of the system. Fig. 5 a) shows the full-width at half maximum (FWHM) of the PSF vs the number of probes for four different families of beams: Hadamard masks, sinusoidal patterns and two set of LG beams built ordering the beams in two differen ways. In particular a family of cardinality N of LG beams (LG1) is built choosing the radial index p=n-1,...,0 and the azimuthal index l=-p,...,p. Another family of LG beams (LG2) includes the LG beams corresponding to cardinality n, where n = 2p+ l. Fig. 5 a) shows how the resolution improves increasing the number of illuminations and that the choice of the kind of pattern and also the way in which the masks are ordered affect the resolution. These results are confirmed by Fig. 5 b) in which the minimum distance at which two points are resolved vs the number of illuminations is reported. As criterium to establish whether two reconstructed Dirac funtions are resolved or not we took a more conservative version of the Rayleigh criterium: two points are considered resolved if the minimum values between them is lower than the half of the pick value of the points themselves (see Fig. 6). The presented simulations provides a proof of the feasibility of the proposed approach and the goodness of the encoding stategy, in particular of the 2-simplex one, presented here as a novelty. It also shows the importance of the choice of the type of illumination and of the way in which the beams belonging to a family are selected. Figure 5: a) FWHM of the reconstructed PSF vs number of illuminations; b) minimum distance at which two point are resolved vs number of illuminations.

4 increasing cardinality. Experimental PSF has been retrieved imaging a piece of silver paper over a black screen. The piece of silver paper, in order to work as a point Dirac function, was smaller than the smallest feature of the pattern of highest cardinality which has been used for a given measurement (see Fig. 8). Example of experimental PSFs for different kinds of patterns and for different number of illuminations are showed in Fig. 9. Figura 6: Two points are considered resolved if the minimum value between them (b in figure) is lower than the half of the pick value of the points themselves (a in figure), i.e. if d = a/b > 2. B. Experimental results Fig. 7 provides an example of three different illumination patterns used in our experiment: a binary mask generated from a Hadamard matrix, a continuous greyscale sinusoidal pattern and the superposition of three channels corresponding to the 2- simplex components of a Laguerre-Gaussian mode. Throughout the experiment we used the projector at a frame rate of 30 Hz. Respect to Laguerre-Gauss probes, each complex value of the pattern is encoded by a triad of positive numbers associated to the three color channels red, green and blue. For each pattern we end up with three new patterns (with only positive values). The three mono-colour patterns correspond to three measured intensity values. The three values are converted back to a unique complex value by the inverse 2-simplex transformation, and the latter is used as a weight of the corresponding complex pattern in the reconstruction algorithm. Figure 8: Experimental realization of a point source for the determination of the PSF: the probes are projected onto a black screen where a piece of silver paper, smaller than the smallest feature of the highest cardinality probe, is placed. In Fig. 10 the trend of the experimental PSF Full Width at Half Maximum (FWHM) versus the number of probes is shown for different kinds of patterns. In agreement with the numerical simulations, the best performances are obtained by Laguerre Gauss beams, in the sense that, for a given, local resolution, they need less probes (i.e less illuminations) in order to obtain the same results, in terms of sharpness of the PSF, respect to the other orthogonal patterns; this result is obtained in virtue of the 2-simplex encoding technique. Figure 7: Example of illumination patterns. a) bynary mask generated from Hadamard matrix; b) continuous grayscale sinusoidal pattern; c) 2-simplex coloured encoding of a Laguerre-Gaussian beam. In order to test how the quality of the reconstructed image is affected by the number of illumination masks, for each target and for each type of pattern we performed several set of measurements using a complete set of orthogonal pattern of

5 Figure 9: Experimental PSF retrieved by 3969 sunusoidal probes (a); 4096 Hadamard probes (b); 2340 LG1 probes (c). Figure 9: Experimental PSF Full Width at Half Maximul (FWHM) as a function of the number of probes for different kinds of orthogonal patterns. Fig. 11 and 12 provide examples of reconstructed images of extended targets for each type of patterns mentioned above and for a fixed cardinality N. Figure 11: Examples of reconstruction of target a) using: b) bynary mask generated from Hadamard matrix, N=1024; c) sinusoidal patterns, N=625; d) Laguerre-Gaussian beam, N=816. Figure 12: Retrieved images of extended targets of increasing spatial frequency (a) by 625 sinusoidal probes (b), 1024 Hadamard probes (c), and 816 Laguerre-Gauss beams (d).

6 V. CONCLUSIONS REFERENCES A new technique aimed at the encoding of phase for complex optical fields has been introduced and applied in an imaging system based on structured illumination and single pixel detection. The encoding technique allows, for a given value of local resolution, to use a lower number of probes (thus illuminations) of the object. [1] A. D. Rodrìguez, P. Clemente, E. Irles, E. Tajahuerce, and J. Lancis, Resolution analysis in computational imaging with patterned illumination and bucket detection, Opt. Lett., vol. 39, pp , July [2] M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, R. G. Baraniuk, "Single pixel imaging via compressive sampling," IEEE SIGNAL PROCESSING MAGAZINE, pp , March [3] V. Studer, J. Bobin, M. Chahid, H. S. Mousavi, E. Candes, and M. Dahan, "Compressive fluorescence microscopy for biological and hyperspectral imaging," PNAS, vol. 109, pp. E1679-E1687, June [4] S. Kosmeier, S. Zolotovskaya, A. C. De Luca, A. Riches, C. S. Herrington, K. Dholakia, and M. Mazilu, "Nonredundant Raman imaging using optical eigenmodes," Optica, vol.1, pp , [5] Y. Wang, B. Bhattacharya, E. H. Winer, P. Kosmicki, W. H. El-Ratal, S. Zhang, "Digital micromirror transient response influence on superfast 3D shape measurement," Opt. Laser Eng., vol. 58, pp , 2014.