DTh3A.7.pdf Digital Holography and 3D Imaging Technical Digest ©2013 OSA

Compensating Systematic Chromatic Errors in Digital Color Holography

Mathieu Leclercq1, Pascal Picart1,2

1-LUNAM Université, Université du Maine, CNRS UMR 6613, LAUM, Avenue Olivier Messiaen, 72085 LE MANS Cedex 9, France 2-École Nationale Supérieure d’Ingénieurs du Mans, rue Aristote, 72085 LE MANS Cedex 9, France

pascal.picart@univ-lemans.fr

Abstract: This paper proposes a robust method to compensate for the chromatic aberrations induced by optical elements in digital color holography. It combines a zero-padding algorithm and a convolution approach with adjustable magnification, using a single recording of a reference rectangular grid. Experimental results confirm and validate the proposed approach. OCIS codes: (090.0090) Holography, (090.1995) Digital holography, (090.1000) Aberration compensation, (220.1010) Aberrations

1. Introduction

Nowadays, important applications rely on the possibilities of using digital color holography to record and reconstruct colored objects with a simple optical setup [1]. However, using multiple wavelengths in the same optical system can lead to severe chromatic aberrations [2]. Different holograms with different wavelengths can be recorded, but the optical components, with presence of chromatic aberration, will image the same object at different planes for each wavelength and, unavoidably, these images will have different sizes. In addition, in a simultaneous color detection scheme [3] the perfect alignment of the reference beams cannot be obtained, and thus a lateral shift of the reconstructed images occurs. Basically, in a Fresnel configuration, no optical imaging system should be placed between the object and the sensor. However, for large objects, the recoding distance must be increased so far, then this increases the full dimensions of the set-up and this can be prohibited for many applications. In 1996, Schnars proposes to use a negative lens so as to reduce the spatial frequency spectrum of the object and then the recording distance [4]. On the other hand, the use of a negative lens introduces also aberrations that modify both the sensor-to-object distance and the virtual object size along each wavelength. The opportunity provided by digital color holography is that the use of an aberration-free lens, having a high cost, is not mandatory, since aberration can be advantageously corrected by a full numerical compensation. So, this paper proposes a robust method to compensate for chromatic aberrations in digital color holography.

2. Digital color holography The basic scheme for a digital color holographic set-up is described in Fig. 1. Three laser wavelengths in the red,

green and blue domain (i.e. 660nm, 532nm, 457nm) [3] are used to illuminate the object. The laser beams can be combined by dichroic plates before the reference beam and the object beam are separated using a polarizing beam splitter. Then, the object wave illuminates the useful area with a unique illumination direction or with dissociated directions [3]. The reference beam, composed of the superposition of the three wavelengths, is extended and spatially filtered to produce an inclined reference plane wave (off axis holography) that is combined with the diffracted object wave, by using a beam splitter. The sensor records simultaneously the three colors, providing real time capabilities to the holographic setup. For a large object, the negative lens is placed just in front of the beam splitter, in the object optical path. Thus, a virtual object is produced in front of the senor at a smaller distance than the initial one, and this enables the Shannon conditions to be fulfilled [4].

Fig. 1. Basic scheme for digital color holography using 3 wavelengths

DTh3A.7.pdf Digital Holography and 3D Imaging Technical Digest ©2013 OSA

The numerical reconstruction of the three color image necessitates compensating for the inherent image size variations due to the discrete Fresnel transform. Alternative efficient solutions such as wavelength zero-padding algorithm [5] or convolution with adjustable magnification [1] are now available. However, the lens produces different image sizes and positions, which make the perfect superposition of the reconstructed images impossible by simply using these approaches.

3. Robust method to compensate for chromatic aberration The proposed method combines a modified version of the wavelength zero-padding algorithm and the adjustable

magnification approach. The object that has to be used is a reference rectangular grid with a good contrast. The first step is to reconstruct the three images corresponding to the three wavelengths and having the same

pixel pitch using the wavelength zero-padding algorithm at the different distances given by the aberration. However, to set the three pixel pitches equal, constraints are imposed and summarized in Eq. (1), for the x-direction (a similar relation holds for y).

=

==∆

j

i

j

i

d

d

d

d

cstpK

d

r

r

x

r

λ

λ

λ

λλ

λλη

0

0

(1)

In Eq. (1) d0λ is the physical image distance due to the lens aberration, dr

λ is the modified numerical reconstruction distance used in the algorithm, px is the pixel pitch and Kλ is the number of pixels of the FFT. The reconstructed images have now the same physical reconstruction horizon, but they are slightly shifted in space and have a different size. The lateral shift can be estimated and will be compensated at the next step with the convolution algorithm. The difference in sizes must be evaluated. To do this, one needs to measure the pitch of the grid along each wavelength. The lines of the grid are automatically detected with a threshold in the image. Using the Hough transform [6], one can obtain equations of any straight line in the image. So it may be possible to get an average value of the distances between each vertical and horizontal lines of the grid, then providing the measurement of the grid period. We consider the blue color to be the reference image to compensate the red and green color. The ratio of the values of the estimated periods for the red and green images versus the blue one determinates the transverse magnification that has to be applied to the green and red image to get the red, green, blue, images having the same size.

The second step is to use the estimated transverse magnification and lateral shift position as inputs in the algorithm with adjustable magnification [1]. The mean spatial frequency along each color is slightly modified to account for the lateral shift, so as the useful spatial bandwidth is localized at the suitable spectral region [1]. The magnification between colors is used to get the same image sizes from the convolution algorithm.

4. Experimental results In order to validate this approach, we used a grating. Fig. 2 shows the three images reconstructed using the zero-

padding algorithm. Their superposition (on the right) indicates clearly that their size depends on the wavelength, and that a lateral shift exists. One can deduce the lateral shift from these reconstructed images.

Fig. 2. Three reconstructed images (from left to right: λ = 457, 532 and 660 nm) (left), Superposition of the three images clearly shows a pixel shift, and a difference in size (right)

DTh3A.7.pdf Digital Holography and 3D Imaging Technical Digest ©2013 OSA

Fig. 3 shows the result given by the Hough transform (in polar coordinates ρ,θ) of the reconstructed image for the green wavelength. One can observe, within the white squares on the center and at the left of the figure, the points corresponding to the 22 lines seen in the reconstructed image of the grid (11 verticals for θ = 90°, 11 horizontals for θ = 0°). So the distance between all lines can be estimated, and the value of the magnification to be applied can be deduced. Parameters are summarized in the right part of Fig. 3.

λ= 457nm 532nm 660nm Average

horizontal period (pixels)

54.7 55.5 56.7

Average vertical period

(pixels) 54.9 55.5 56.8

Magnification 1 54.8/55.5 54.8/56.75

Fig. 3. Hough Transform of the reconstructed green image of the grid (left), and estimation of the transverse magnification (right) Fig. 4 shows the results obtained at the final step of this new approach, after applying the convolution with

adjustable magnification and the lateral shift compensation using the experimental estimations.

Zoom on the center of the image before lateral shifting

Zoom on the edge of the image before

magnification

Zoom on the center of the image after lateral shifting

Zoom on the edge of the image after

magnification

Fig. 4. Three R-G-B images superposed before (left) and after (right) applying the adjustable magnification and the lateral shift Note that the proposed method can be applied without any a priori knowledge or physical knowledge about the optical imaging system or the reference rectangular grid.

5. Conclusion This paper presents a robust, simple and effective method to compensate the chromatic aberrations induced in

every digital color holographic display, in which an optical system is used to compact the dimensions of the set up. The method combines a modified version of the wavelength dependant zero-padding algorithm and the adjustable magnification approach. The object that has to be used is a reference rectangular grid with a good contrast. Experimental results confirm the suitability and robustness of the proposed approach. This research is funded from the French National Agency for Research (ANR) under grant agreement n°ANR 2010 BLAN 0302.

6. References [1] P. Picart & al., “Spatial bandwidth extended reconstruction for digital color Fresnel holograms”, Opt. Exp. 17 9145-9156 (2009). [2] S. de Nicola & al., “Recovering correct phase information in multiwavelength digital holographic microscopy by compensation for chromatic aberrations”, Opt. Lett. 30 2706-2708 (2005). [3] P. Tankam & al., “Real-time three-sensitivity measurements based on three-color digital Fresnel holographic interferometry”, Opt. Lett. 35 2055-2057 (2010). [4] U. Schnars & al., “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum”, Opt. Eng. 35, 977-982 (1996). [5] P. Ferraro & al., “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms”, Opt. Lett. 29, 854-856 (2004). [6] P.V.C. Hough, “Method and means for recognizing complex patterns”, US Patent 3,069,654.