Numerical wave propagation in ImageJ - OSA Publishing

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Engineering and Laboratory Note

Vol. 54, No. 21 / July 20 2015 / Applied Optics

Numerical wave propagation in ImageJ PABLO PIEDRAHITA-QUINTERO, RAUL CASTAÑEDA,

AND

JORGE GARCIA-SUCERQUIA*

School of Physics, Universidad Nacional de Colombia—Sede Medellin, A.A: 3840, Medellin 050034, Colombia *Corresponding author: [email protected] Received 16 April 2015; revised 15 June 2015; accepted 16 June 2015; posted 17 June 2015 (Doc. ID 238292); published 15 July 2015

An ImageJ plugin for numerical wave propagation is presented. The plugin provides ImageJ, the well-known software for image processing, with the capability of computing numerical wave propagation by the use of angular spectrum, Fresnel, and Fresnel–Bluestein algorithms. The plugin enables numerical wave propagation within the robust environment provided by the complete set of built-in tools for image processing available in ImageJ. The plugin can be used for teaching and research purposes. We illustrate its use to numerically recreate Poisson’s spot and Babinet’s principle, and in the numerical reconstruction of digitally recorded holograms from millimetersized and pure phase microscopic objects. © 2015 Optical Society of America OCIS codes: (100.2000) Digital image processing; (100.6890) Three-dimensional image processing; (170.3010) Image reconstruction techniques; (090.1995) Digital holography; (090.1970) Diffractive optics; (260.3160) Interference. http://dx.doi.org/10.1364/AO.54.006410

1. INTRODUCTION ImageJ is software that invites the creation of customized tools in the realm of digital processing of images [1,2]. The main reason to accept this invitation is the possibility of using a complete set of built-in tools common to all the fields that process information in the form of digital images. The field of digital optics, where optical information can be produced by means of computing numerical wave propagations of a digital image, is therefore an area with great interest in accepting that invitation. For numerical propagation, the digital image is placed at the input plane; the complex result of the wave propagation is obtained at the output plane. Specifically, computer generation of holograms [3] and numerical reconstruction of digitally recorded holograms [4,5] are two fields in which the numerical evaluation of wave propagation is mandatory. For instance, in the numerical reconstruction of digitally recorded holograms, the recorded hologram is placed at the input plane and the reconstructed hologram, through the numerical wave propagation, is obtained at the output plane. The needed numerical wave propagation can be computed via the scalar wave equation [6]. It can be solved in different ways to arrive to diverse forms of the scalar diffraction formula, one of which is the Fresnel–Kirchhoff diffraction formula [6]: ZZ i expikjr − r 0 j 0 1 cos χdr: Aout r − Ainp r 2λ Screen jr − r 0 j (1) In Eq. (1), the output complex field Aout r 0 is computed by propagating the input complex field Ainp r. The latter can be composed by complex multiplication of a digital image times 1559-128X/15/216410-06$15/0$15.00 © 2015 Optical Society of America

the complex illuminating wave field [6]. In the same equation, k 2π∕λ is the wavenumber with λ the illuminating wavelength; the vectors r X ; Y ; z and r 0 x; y; z 0 denote positions at the input and output planes, respectively. 1 cos χ is the inclination factor with χ being the angle between the pffiffiffiffiffi normal at the input plane and the vector r − r 0 ; and i −1. The numerical implementation of Eq. (1) can be done via different approaches according to the experimental conditions that should be fulfilled [7–9]. For the reader’s quick reference, we have briefly recalled the three numerical methods of wave propagation implemented in the plugin; a deeper study about them can be read in the corresponding references. On the following expressions, for conciseness, only its one-dimensional forms are considered. On the equations, Δx∕Δξ stand for the input/output sampling pitches, M for the number of data points, z for the propagation distance, and Ap z∕An 0 for the input/output complex fields. The first implemented method is the angular spectrum method [9,10]. It provides a nonapproximate calculation of the diffraction process and regularly is suitable for distances within the range of μm to cm. Its fast Fourier transform (FFT) implementation is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ap z IFFT FFTfAn 0g exp iz k2 − 4π 2 Δf x 2 n2 ; (2) where Δf x 1∕M Δx. This FFT implementation of the angular spectrum method requires the input and output pitches to be equal-sized. Second, we have implemented the Fresnel diffraction approach [11]. It can be obtained from Eq. (1) using the paraxial



approximation; once the resulting equation is transformed into its FFT form, it is written as expikz ik exp pΔξ2 Ap z Δx iλz 2z ik nΔx2 ; (3) × FFT An 0 exp 2z

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Fig. 1. ImageJ main menu. See the OD menu, under which is located the numerical wave propagation plugin (NP).

which constrains the size of the pixels at the output plane to be Δξ MλzΔx . The third implemented method allows the removal of the restriction imposed by the Fresnel diffraction approach in regard to the size of pixels of the output plane. To do this, the product 2np present in the kernel of the Fourier transform needs to be rewritten as 2np n2 p2 − p − n2 . Performing this substitution allows the Fresnel diffraction method to be expressed as a circular convolution, given by expikz ik 2 exp − ΔξΔx − Δξp Ap z iλz 2z × IFFTfF 1n F 2n g; ik with F 1n FFTfAn 0 exp2z ik ΔxΔξn2 g. This FFTfexp2z

(4) Δξn2 g

ΔxΔx − and F 2n method is known as the Fresnel–Bluestein transform [12]. Diverse, ready-to-use tools are available to compute numerical wave propagation. Libraries for programming languages like C and C++ [13]; MATLAB and Mathematica notebooks, functions, and routines [14]; and commercial [15] and noncommercial software [16] are among the multiple choices available. However, most users are inclined to develop their own numerical wave propagation tools to cope with the specific needs of their application or because the post-propagation analysis demands very specialized image-processing tools. In this paper, we present a plugin for the numerical wave propagation of complex wave fields in the environment of the image-processing software ImageJ. The main advantage of performing the numerical wave propagation within ImageJ is the possibility of using the complete and wide set of built-in tools for image processing available in this software, which eases the pre- and post-processing of the input- and output-data images. In the plugin, Eq. (1) is numerically implemented in its angular spectrum, Fresnel, and Fresnel– Bluestein forms to guarantee the covering of the full range of propagation from the very aperture to infinity. Examples of application of the plugin in both teaching and research are shown. 2. PLUGIN The developed plugin for numerical wave propagation has been added to the ImageJ main menu under the OD menu (see Fig. 1). We have included this plugin below a separate menu from the Plugins menu of ImageJ because there are more tools under development by our Opto-Digital group that will be located within the same OD menu very soon. Once the NP label is clicked, the new window shown in Fig. 2 is created. In this window, the user can enter the needed information to perform the numerical wave propagation and can read the history of the recent use of the plugin in a log

Fig. 2. Main window of the developed plugin for numerical wave propagation.

frame. The very details of each of the components and their use can be read on the documentation of the plugin, which is available online along with the plugin itself and the installation instructions [17]. Here, we present a brief summary of the operation of the plugin for quick reference. Four different Methods of numerical propagation are available in the plugin: angular spectrum [9,10], Fresnel [11], Fresnel–Bluestein [12], and automatic; the latter chooses automatically between angular spectrum and Fresnel according to the limit of use as shown in [7,18]. The plugin uses as input any digital images that are open in ImageJ. Two Input combo boxes allow the user to select the real and imaginary parts of the input complex field Ainp r; both can be chosen among the multiple images that could be simultaneously opened in ImageJ, with the only condition being that the two images must be of identical sizes along the x and y axes, respectively. Once the images are selected, the user enters in the input boxes the wavelength, propagation distance, and physical dimensions of the input images. If the Fresnel–Bluestein method is chosen, the size of the output images must also be entered; notice that it is possible to keep the aspect ratio of the output image by activating the lock next to the size of the output. As seen in Fig. 2, the real and imaginary parts of the propagated complex wave field can be shown. In addition, the phase, amplitude, and intensity of the numerically propagated complex wave can be computed and presented as well. All of the above options can be selected simultaneously. For this case, each output is shown in a separate new window. The button Settings allows the configuration of the numerical propagation plugin. The units of the physical quantities and the type scaling of the output data can be set up in the settings

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frame. For the propagation, the spatial filtering can be enabled or disabled, the input wave field can be illuminated by a plane or spherical wave front, and batch operation can be configured. To perform the propagation, the button Propagate is clicked. If the filtering option is enabled, a new window is opened where the spectrum of the Fourier transform of the input image is shown; an emerging message that invites the user to perform any filtering over the Fourier spectrum is also opened. The spectrum of the Fourier transform is shown according with the settings previously selected by the user in the scaling settings. Within that Fourier spectrum, the user can choose the desired set of spatial frequencies by using any of the selection tools provided by ImageJ. Once selected, the plugin propagates the chosen set of spatial frequencies; from the resulting complex propagated wave, the chosen output representations are calculated and shown in individual windows. The user can therefore use the complete set of built-in tools of ImageJ to manipulate the output images and manage the histogram, brightness, contrast of the image, specimen counting, and three-dimensional plotting, among the other very useful opportunities that are immediately available for the user. 3. EXAMPLES OF USE The use of the developed plugin for numerical wave propagation in ImageJ is now illustrated with examples in the teaching and research realms. A. Teaching

The fascinating Poisson’s spot [19] can be recreated in the developed plugin. For this purpose, a black circle was painted within a white background [see Fig. 3(a)]. The whole image was made of 512 × 512 pixels and the circle was drawn with a radius of 65 pixels. The physical dimensions for the whole image were 5 mm × 5 mm, which gives an opaque circle with radius 0.63 mm. We chose an illuminating wavelength of 633 nm and propagated 0.15 m from the aperture to the recording screen. The propagated wavefield, in which the bright spot placed exactly behind the opaque obstacle as was predicted by Poisson, is presented in Fig. 3(b). One challenging concept to teach in optics is Babinet’s principle [20]. According to this principle, the amplitude fields produced by complementary apertures are directly related to each other. For complementary apertures, the portions that are clear

Fig. 3. Recreation of Poisson’s spot. A black obstacle in a clear screen is shown in panel (a). The propagated wavefield with a Poisson’s spot in the very center is shown in panel (b).

Engineering and Laboratory Note in one are opaque in the other and vice versa. Let us consider A1 r 0 , the wavefield diffracted through an opening, and A2 r 0 , the wavefield diffracted through its complementary screen. From the diffraction formula in Eq. (1), it can be seen that if A0 r 0 is the wavefield propagated through the free space, then A0 r 0 A1 r 0 A2 r 0 .

(5)

From Eq. (5), it follows that if for a particular point A0 r 0 0, then A2 r 0 expiπA1 r 0 , namely, the wave fields are identical with a π phase shift. To test Babinet´s principle as for wavefield in the free space A0 r 0 , we have considered that produced by a circular opening at a distance that four Fresnel zones are subtended [20]. With this choice, the very center of the wavefield A0 r 0 vanishes and the π phase shift predicted by Babinet can also be tested. An image of the logo of our University is used as diffracting aperture; by using the Invert tool of ImageJ, we have created its complementary image. Each of these images was then attached to the circular opening to be illuminated with a plane wave [see Fig. 4(a) for the aperture and Fig. 4(b) for the complementary aperture]. In the main window of the numerical propagation plugin, we have set the width and height of the images to 4 mm. For an illuminating wavelength of 633 nm, we have propagated both images at a distance of 0.43 m; the amplitude of the wavefields for the aperture A1 r 0 and the complementary aperture A2 r 0 are shown in panels (c) and (d), respectively. As an initial test of Babinet’s principle, we measured the phase values at the very center of the images in Figs. 4(c) and 4(d). There was a phase difference of 3.07 rad between these two spots, which is just 2.28% off from the expected π phase shift. This offset should be due to numerical errors in the processing of the images. The real parts of A1 r 0 and A2 r 0 were added with the Image Calculator tool of ImageJ; separately, the same summation process was performed for the imaginary parts. The amplitude pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 2 of the sum A1 r A2 r Re sum Im2 sum is

Fig. 4. Illustration of Babinet’s principle in the developed plugin.



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For the latter, a spherical wavefront, with a chosen radius of curvature measured from the center of the digital image, illuminates the recorded hologram. Two different configurations of digital holography are illustrated: macro digital holography or simple digital holography and digital holographic microscopy. 1. Digital Holography

Fig. 5. Verification of Babinet’s principle. The sum of the propagated wavefields for the aperture and its complement is shown in (a). The amplitude produced by the circular opening is shown in (b). The histogram of the subtraction of both amplitudes in (a) and (b) is shown in (c).

computed using the Utilities tool of the numerical propagation plugin [see Fig. 5(a)], and then contrasted with the wavefield produced by the circular opening A0 r 0 [see Fig. 5(b)]. A direct comparison of panels (a) and (b) of Fig. 5 shows that effectively both amplitudes are equal. We have also subtracted both images with the Image Calculator tool of ImageJ. The histogram of the subtraction, Fig. 5(c), shows that the values of that image are distributed in the vicinity of 0 with a mean value of 1.892, indicating a good agreement with the expected value. This comparison and the result of the phase shift at the very center of the wavefield for the complementary screens indicate that effectively the developed plugin can be used for illustrating Babinet’s principle within the robust image processing environment of ImageJ. B. Research

The use of the developed plugin in research is illustrated with the numerical reconstruction of digital recorded holograms, namely digital holography [5,21], in different configurations. The reconstruction process can be understood as the diffraction that an illuminating wave undergoes when it illuminates the recorded digital hologram [6]. Within this concept, the numerical reconstruction can be read from Eq. (1) when Ainp r Aillu rH r;

For the case of digital holography, we have utilized a setup as illustrated in Fig. 6. In this setup, to fulfill the sampling requirements with the commercially available cameras [4,5,22], the distance between the object and the digital camera is of the order of one meter. This figure means that among the options available in the plugin for the numerical wave propagation, Fresnel, Fresnel–Bluestein, or Automatic can be utilized. In either choice, the plugin allows the user to apply a given spatial filtering [23] at its will. For illustrating the use of the plugin, we have used a digital hologram recorded by a CMOS camera with 768 × 576 pixels. The width/height of the digital recording is 8.45 mm∕ 6.34 mm; the illuminating wavelength is 633 nm. As shown in Fig. 7(a), the plugin with the built-in function of ImageJ for free-hand selection allows the user to pick the desired set of spatial frequencies; in Fig. 7(b) is shown the reconstructed amplitude image. For this reconstruction, the utilized option of the plugin was Fresnel. As mentioned above, the distinctive feature of the Fresnel– Bluestein method is the possibility of choosing the size of the plane where the propagated wave is computed [12]. Its use in the plugin can be illustrated by further propagation of the set of spatial frequencies of Fig. 7(a). In the main window, the step back or forward propagation is available through the − or buttons, in that order. As either of these buttons is pressed,

Fig. 6. Digital holography setup. BS, beam splitter; MO, microscope objective; L, lens; M, mirror; O, object wave; R, reference wave; CMOS, digital camera.

(6)

with Aillu r representing the complex illuminating wave and H r the digitally recorded hologram. As mentioned above, the developed plugin allows the user to select between plane and spherical illumination. For the former, a homogenous plane wave impinges perpendicularly onto the digital hologram.

Fig. 7. Reconstruction of a digital hologram with Fresnel method. (a) Free-hand spatial filtering and (b) amplitude reconstruction.

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Fig. 9. Transmission digital holographic microscope. L, lens; BS, beam splitter; M, mirror; S, sample; MO, microscope objective; TL, tube lens; O, object wave; R, reference wave; CMOS, digital camera.

Fig. 8. Reconstruction of digital hologram with the Fresnel– Bluestein method. The reconstruction window was selected to be 40 mm × 20 mm.

the last selected set of spatial frequencies is propagated to a distance equal to the last utilized − or the amount written in the text box between the back and forward buttons. The image shown in Fig. 8 is the reconstructed amplitude obtained as the selected set of spatial frequencies is propagated forward 0.01 m with respect to that propagation in Fig. 7(b). The size of the reconstruction plane was chosen to be 40 mm × 30 mm to provide an enlarged view of the reconstructed amplitude image via the Fresnel–Bluestein method. For the reconstructed images in Figs. 7(b) and 8, any of the available operations for image processing in ImageJ can be utilized. A further option provided by the developed plugin is the presentation of the reconstructed image in units corresponding to the physical parameter represented, easing any measuring process performed with the ImageJ tools. This feature is illustrated by automatically adding the scale bars at those images by means of the Scale Bar tool of ImageJ. The same feature will be also illustrated in the next section by measuring phase values from digital holographic microscopy.

Fig. 10. Reconstruction of a hologram from digital holographic microscopy by angular spectrum method of a pure phase object.

2. Digital Holographic Microscopy

The use of the plugin in digital holographic microscopy is illustrated by reconstructing a digital hologram recorded in a typical transmission microscope built in Mach–Zehnder architecture (see Fig. 9). The microscope was set up to operate at diffraction limit in an off-axis telecentric mode [24]. A CMOS camera placed 10.8 mm away from the image plane of the imaging system recorded a digital hologram of a pure phase test target. This configuration means that the numerical propagation method valid to reconstruct the recorded hologram is angular spectrum [9,10]. In the method combo box, the user should select angular spectrum or automatic to perform this reconstruction. In Fig. 10, the reconstructed phase of the recorded hologram is shown. To reduce the zero-order influence on the reconstructed image, the average value of the hologram has been subtracted from the hologram using the Process > M ath > Subtract… tool of ImageJ. For pseudo-coloring this

Fig. 11. 3D profile of the area enclosed in the red square in Fig. 10. The color bar (in radians) on the right-hand side was added for its correct visualization.

Engineering and Laboratory Note image, we have used the Lookup Tables tool of ImageJ. This option allows the user of the developed plugin to choose among the multiple coloring palettes to enhance the representation of the result. Because the data is represented in phase units (radians) rather than in gray levels, from the image in Fig. 10, the user can make a direct 2D and/or 3D profile in the physical dimensions of the phase for this case. In Fig. 11, a 3D profile of the phase reconstruction is presented. This profile was generated using the Pl ugins > 3D > I nteractive 3D Surface Plot tool included in ImageJ. Additionally, the image is automatically scaled to the physical dimensions in metric scale. These two features simplify the quantitative analysis of the resulting reconstructed holograms. 4. CONCLUSIONS An ImageJ plugin to compute numerical wave propagation has been presented. The plugin includes the angular spectrum, Fresnel, and Fresnel–Bluestein algorithms. The combination of these three propagation methods guarantees the covering of the complete range of propagation distances, from very close to the diffracting aperture (angular spectrum) to the far field (Fresnel and Fresnel–Bluestein). Additionally, for the far field, the Fresnel–Bluestein algorithm provides the option of computing the wave propagation within planes with sizes chosen at will by the user; namely, allowing different magnification levels. The possibility of computing numerical wave propagations within the robust environment of the complete set of built-in tools for image processing provided by ImageJ eases the analysis, enhancing, packing, and sorting (among other tasks) of the retrieved information. Furthermore, the plugin has been equipped with the needed options to do multiprocessing using the stacking option of ImageJ. The use of the plugin has been illustrated for teaching and research. In the former, Poisson’s spot and Babinet’s principle have been recreated with the aid of the plugin and the built-in tools of ImageJ. For research, we have utilized the plugin in the numerical reconstruction of digitally recorded holograms from digital holography and digital holographic microscopy. The feature of the plugin of creating output images with physical dimensions and parameters—for instance, micrometers and phase rather than pixels and gray levels—eases greatly the post-manipulation of the output images using the built-in set of tools provided by ImageJ. Funding. Colciencias - Universidad Nacional de Colombia (Hermes 28751); Universidad Nacional de Colombia (Hermes 19834). REFERENCES 1. M. D. Abràmoff, P. J. Magalhães, and S. J. Ram, “Image processing with ImageJ,” Biophoton. Int. 11, 36–41 (2004).


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