Amplitude image processing by diffractive optics

Manuel P. Cagigal; Pedro J. Valle; V. F. Canales

doi:10.1364/OE.24.003268

1. Introduction

Image processing has been successfully used for improving image quality from the middle sixties. Since then, the development of processing techniques has been exponential. The first step in most computer recognition algorithms relies on the use of low level techniques like smoothing or derivative masks for removing the noise effect and for extracting image borders, respectively [1]. Low level digital image processing techniques are always carried out over a detected image. It means that the image intensity, the squared modulus of the electromagnetic field, at every pixel of the camera has been saved and low level techniques are applied by convolving the digital image by a particular filter function. For example, a low pass filter is useful for removing noise whilst a high pass filter increases borders signal. The filter size will depend on different factors and the filter parameters will also depend on the type of processing to be applied.

In contrast to digital techniques, we propose optical image processing based on diffractive optics elements (DOE) acting on the optical field instead of on the optical intensity. DOE have been successfully used for designing multifocal lenses [2], extended-focal-depth lenses [3] or variable-focal-length devices [4]; likewise for creating annular vortices [5], calibrating cameras [6] or dispersion management in two-photon microscopy [7]. An interesting former review of DOE design and applications was addressed by M. A. Golub [8]. In particular, we propose the idea of performing the filtering process with an optical setup and a DOE placed at the entrance pupil of the optical system. This DOE will produce a series of amplitude image replicas distributed over the detection plane. Each replica may be weighted with a different coefficient. The superimposition of these replicas is equivalent to the convolution of the amplitude image by a filter and its result depends on the replica separation and the relative weight of the coefficients. Thus, different filters can be carried out by selecting the proper coefficients.

In previous papers [9,10] we presented a simple method for designing DOE using Zernike polynomials. Now, we apply this method for designing an AMplitude ImaGe Optical processing (AMIGO) device. The advantage of the technique is that it acts over the field complex amplitude instead of the intensity as in common image processing techniques, which allows the improvement of the detected image contrast keeping a high sharpness level. Besides, the filters can be designed in a simple and controlled way. In this paper, we show how to design masks for generating weighted image replicas distributed across the transversal plane. The first example corresponds to an averaging filter and shows the difference between amplitude and intensity processing. The second one is a Gaussian filter, which introduces the procedure for modulating the replicas intensity. The third example is the Laplacian filter which is commonly known as a point contrast enhancer. The AMIGO Laplacian filter is checked by numerical simulation as a border enhancer when imaging amplitude objects. It detects borders as efficiently as a digital image processing. Finally, we investigate the capability of the AMIGO Laplacian filter for improving the resolution of ground-based telescopes. The technique can only be applied when D/r₀ is smaller than about 8, where D is the telescope diameter and r₀ is the atmospheric Fried parameter, what implies a telescope with D < 2.5 m in the visible range with Lucky Imaging technique. Numerical simulations show that, under these atmospheric conditions, the AMIGO Laplacian filtering allows the detection of two objects as close as the Rayleigh criterion. This is quite an interesting result since the telescope diffraction limit cutting frequency can be reached using just a passive device, even in the presence of atmospheric turbulence.

The paper is organized as follows: In Section 2 we present the theoretical foundations of the amplitude processing by DOE. In Section 3, we show how to design diffractive masks for different applications: averaging and border enhancement. In Section 4, a comparison between intensity and amplitude processing is performed. Section 5 checks DOE ability for border enhancement with an amplitude (constant phase) object. Section 6 presents the application of this new technique in Astronomy in order to reach diffraction limit resolution images in ground-based telescopes. Finally, Section 7 summarizes the main results of the paper.

2. AMIGO mask theoretical foundations

The improvement of image quality based on optical averaging has been known for many years. Commonly, the averaging of a set of images has been performed in optical signal processing by means of a transmission mask consisting on an array of pinholes [11]. The use of this mask results in a large number of image replicas. When these replicas overlap, at least partially, the resulting image is a smoothed version of the original one. In a previous work, we designed diffractive elements that, when placed at the pupil of an optical system, produced multiple foci [10]. The number and position of foci were directly governed by the DOE design parameters. Recently, this technique has been successfully applied for improving aberrated images [12]. We propose here a new technique for generating weighted image replicas so that the resulting image can be a filtered version of the original one and not only its average. The number, position and distance between replicas can be easily controlled by the mask design parameters.

Figure 1 depicts a way to accomplish AMIGO. A diffractive mask (the DOE in this set up) is placed at the entrance pupil of a lens. This lens performs the Fourier transform of the input field multiplied over the complex transmittance function of the diffractive mask. The resulting transformed field at the camera plane (CCD) can be described as the addition of a series of weighted amplitude image replicas. This image field can also be understood as the convolution between the Fourier transform of the incoming pupil field and the Fourier transform of the DOE function.

Fig. 1 AMIGO setup consisting on a diffractive optical element (DOE) placed at the entrance pupil of lens L1 which forms an image onto the CCD (in some applications after crossing an aberrating medium, here represented by a phase screen, PS).

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The amplitude transmission function of a DOE providing four copies of the image field, two of them displaced along the x axis an amount proportional to ± α and the other two along the y axis an amount proportional to ± α ´, can be obtained using a previously described procedure [9] (z axis is assumed to be the optical axis and x-y the transversal coordinates at the pupil plane),

M (x, y) = \cos (α x) + \cos (α^{'} y) .

Usually, we will take α´ = α. A more general mask transmission function can be obtained using a higher number of replicas distributed along x, y and diagonal axes. These replicas can be easily obtained from the mask profile,

M (x, y) = C_{0} + \sum_{n, m}^{} {\begin{cases} a_{n} \cos (α_{n} x) + b_{n} \cos ({α^{'}}_{n} y) \\ + c_{m} \cos [β_{m} (x + y)] + d_{m} \cos [{β^{'}}_{m} (x - y)] \end{cases}},

with n = 1 to N_α and m = 1 to N_β, where N_α and N_β control the number of replicas in each pair of axes, while the positions are determined by mask parameters α_n, α'_n, β_m and β'_m. The relative replica weights are given by the coefficients a_n, b_n, c_m and d_m and C₀ is a mask offset that forces the filter to take positive values. In this work, we will restrict the analysis to N_α = N_β = 1, so that the pupil mask used for making the replicas, can be described by:

M (x, y) = C_{0} + a \cos (α x) + b \cos (α y) + c \cos [β (x + y)] + d \cos [β (x - y)],

where the coefficients a, b, c and d, used for controlling the relative peak heights, are shown in Fig. 2.

Fig. 2 Image replica positions and the corresponding weighting coefficients according to Eq. (3).

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Since the diffractive mask is placed at the entrance pupil of the system, part of the incoming energy can be lost. In addition, fabrication of continuous transmittance masks may be difficult and expensive. These difficulties can be overcome using phase-only masks. It is possible to obtain a phase only mask from Eq. (2) using the expression:

P M (x, y) = \exp [i M (x, y)] .

Also binary versions of masks, which are more easily implemented, can be obtained applying a threshold. The binary mask BM(x,y) will take zero value in all points except in those where the M(x,y) values are above a threshold where it will take the unity. Then, binary phase masks can also be obtained introducing BM(x,y) (with a π factor) instead of M(x,y) in Eq. (4). Phase and binary phase masks produce similar results than the corresponding amplitude-only ones, as shown in previous works [9].

In all these cases the pupil field is multiplied by the mask M(x,y) so that at the image plane we have the convolution of the Fourier Transform of the mask, FTM(u,v), by the complex field A(u,v) and the detected intensity will be given by (u and v are the transversal coordinates at the image plane):

I_{0} (u, v) = {| A (u, v) * F T M (u, v) |}^{2},

while in digital image processing the intensity is convolved with a filter FTM(u,v) to obtain the processed image [13]:

I_{D} (u, v) = {| A (u, v) |}^{2} * F T M (u, v) .

Once again, it is necessary to note that in digital image processing the convolution is performed on image intensity whilst in AMIGO it is carried out on the image complex amplitude.

3. Masks design

Let us pay attention to the procedure for designing the AMIGO masks according to Eq. (3) in some simple cases of image filtering which are frequently applied, like mean and Gaussian averaging and Laplacian filter.

3.1 Averaging mask

A simple procedure to reduce detection noises in digital imaging is to convolve the detected image by an averaging NxN filter like that shown in Fig. 3(a) for N = 3. In contrast, we propose to obtain the set of displaced amplitude replicas by a DOE built with Eq. (3) and all identical weighting coefficients, like that shown in Fig. 3(b). In particular, when we have a plane incoming wavefront the image given by this mask is shown in Fig. 3(c). It can be seen that the superimposition of the displaced replicas at the detection plane reproduces the effect of convolving the complex amplitude obtained from the unmasked pupil with the 3x3 filter (the replicas are shown separated for picture clearness).

Fig. 3 (a) One-pixel radius averaging mask. (b) Mask used for obtaining image replicas, Eq. (3), and (c) displaced replicas of the image for the case of a plain incoming wavefront (the displacement, α = β = 4π, is exaggerated to be easily appreciated), 2C₀ = a = b = c = d = 0.22.

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3.2 Gaussian mask

Gaussian filter, commonly used for averaging images and for removing detection noise, is just a low pass filter. Figure 4(a) shows the 3x3 Gaussian filter which is convolved with the detected image in digital image processing. In amplitude optical processing the effect of convolving with a Gaussian filter can be reproduced using the pupil mask shown in Fig. 4(b). This DOE is described by

M (x, y) = 1 + \frac{\cos (α x) + \cos (α y)}{\sqrt{2}} + \frac{\cos [β (x + y)] + \cos [β (x - y)]}{2},

and provides the set of the displaced and modulated replicas appearing again in Fig. 4(c). The peak distribution corresponds to the Gaussian coefficients of the digital filter. The mask of Eq. (7) can be considered as an apodizing function and will produce an effect similar to that of the Gaussian apodizing function which is commonly placed at the entrance pupil of an optical system to cancel out small variations of light. An interesting example of its application is in Astronomy where the Airy rings energy of a star is reduced in order to detect faint companions.

Fig. 4 (a) One-pixel radius Gaussian mask. (b) Mask described by Eq. (7) used for obtaining image replicas. (c) Displaced and weighted replicas of the image for the case of a Gaussian mask (α = β = 4π).

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3.3 Laplacian mask

The Laplacian filter is a high pass filter which is applied for detecting isolated points. In digital image processing the Laplacian 3x3 filter is that shown in Fig. 5(a). The pupil mask we used for making the replicas, shown in Fig. 5(b), is:

M (x, y) = \frac{1}{6} {4 - \cos^{} (α x) - \cos^{} (α y) - \cos^{} [β (x + y)] - \cos^{} [β (x - y)]} .

The peak distribution generated at the image plane through this pupil mask consists of a positive central peak surrounded by eight negative peaks whose height is 1/8 of the central one as shown in Fig. 5(c). This field distribution at the image plane resembles the Laplacian filter used in digital image processing Fig. 5(a).

Fig. 5 (a) One-pixel radius Laplacian filter. (b) Mask described by Eq. (8) with α = β = 4π used for obtaining image replicas. (c) Displaced and weighted replicas of the amplitude image for the case of a Laplacian mask. Grey scale range at (c) reproduces the range values at the filter in (a).

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However, here we are not limited to a mask with a reduced number of pixels or a rectangular distribution. Hence we can design radially symmetric masks able to reproduce the behavior of different filters depending on its parameters. For example, it can yield at the image plane a positive central peak surrounded by a ring of negative amplitude. This mask is given by:

M (x, y) = α - \cos^{2} [β (\sqrt{x^{2} + y^{2}})] .

The effect of the mask can be controlled by the parameters α and β. An interesting case is that of α = 1. In that case we have:

M (x, y) = \sin^{2} [β (\sqrt{x^{2} + y^{2}})],

which is an amplitude-only mask and can be easily implemented.

Figure 6 compares a PSF for a clean pupil and the PSF when the mask given by Eq. (10) is applied at the system entrance pupil. We see that the mask produces a slightly narrower central PSF peak whilst the light of the PSF rings is strongly reduced except at the first one.

Fig. 6 (a) PSF of the clear pupil. (b) PSF of the pupil with the mask of Eq. (10) (c) Transversal cut of the PSF for the clear pupil (solid curve) and with the mask (dashed curve).

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4. Comparison between amplitude and intensity processing

Let us compare the result of the average produced by AMIGO with that obtained in digital image processing. Since AMIGO performs the field complex amplitude weighted average, the amplitude at a pixel will be:

A_{0} = \frac{1}{N} \sum_{m n} w_{m n} A_{m n} = \frac{1}{N} \sum_{m n} w_{m n} R_{m n} + i \frac{1}{N} \sum_{m n} w_{m n} I_{m n} \equiv \bar{w R} + i \bar{w I},

where N is the number of copies produced by the DOE, w_mn are the coefficients which modulate the replica relative weights and A_mn are the fields to be averaged, which has been separated in real and imaginary terms, A_mn = R_mn + iI_mn. The corresponding pixel intensity will be:

I_{0} = | \frac{1}{N} \sum_{m n} w_{m n} A_{m n} |^{^{2}} = {\bar{w R}}^{2} + {\bar{w I}}^{2},

On the other hand, when using the filter in digital image processing, the detected intensities are averaged, so that:

I_{D} = \frac{1}{N} \sum_{m n} w_{m n} I_{m n} = \frac{1}{N} \sum_{m n} w_{m n} R_{m n}^{2} + \frac{1}{N} \sum_{m n} w_{m n} I_{m n}^{2} = \bar{w R^{2}} + \bar{w I^{2}},

The difference between the two intensities I₀ and I_D will depend on the value of the coefficients w_mn. For example, according to Eq. (12), for masks that perform an average (or downsampling) like the constant or Gaussian ones, the contribution of a constant background noise will tend to zero as N increases. In contrast, for their digital counter parts, the value of Eq. (13) tends to the mean background intensity. For example, for the averaging filter w_mn = 1 for every (m, n) this difference is:

I_{D} - I_{0} = σ^{2} (R) + σ^{2} (I),

where σ² stands for the variance.

The difference between AMIGO and digital image processing can be more clearly highlighted if the detection noise is taken into account. The detected intensity in a particular pixel including noise when using AMIGO will be:

I_{0} = | \frac{1}{N} \sum_{m n} w_{m n} A_{m n} |^{^{2}} + r_{0} = {\bar{w R}}^{2} + {\bar{w I}}^{2} + r_{0},

where r₀ is the noise affecting that pixel. The intensity for digital image processing will be:

I_{D} = \frac{1}{N} \sum_{m n} w_{m n} (I_{m n} + r_{m n}) = \bar{w R^{2}} + \bar{w I^{2}} + \bar{w r},

where

\bar{w r} = \frac{1}{N} \sum_{m n} w_{m n} r_{m n} .

When the w_mn coefficients correspond to an averaging mask, for example a Gaussian mask, Eq. (15) shows that AMIGO averages efficiently the field components whilst the detection noise remains unchanged. However, the digital image processing [Eq. (16)] performs a noise average whilst the average of the field components is not so effective given that all the components are squared and, hence, positive.

In the case of a Laplacian filter, the effect of applying digital image processing, given by Eq. (16), is the formation of peaks due to the background noise, as Eq. (17) shows. This effect disappears when AMIGO is applied, whilst the capability for detecting points or borders remains unaffected.

5. Amplitude object checking

We are particularly interested in proving the AMIGO performance when the Laplacian filter is applied. To check it we have taken the amplitude-only (constant phase) object shown in Fig. 7(a). Figure 7(b) shows the image obtained when convolving this object with a 3x3 Laplacian filter in a digital image processing. We have also simulated the image provided by our technique under the same conditions [Fig. 7(c)]. It can be seen that our technique highlights the object borders as efficiently as the digital processing. Hence, we can consider that our technique works for detecting borders and isolated point objects.

Fig. 7 (a) Amplitude-only object. (b) Result of convolving this object with a 3x3 Laplacian filter in digital image processing. (c) The result of using the mask given by Eq. (10) in the AMIGO setup.

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6. Application to astronomy

A technique commonly used for detecting faint astronomical objects is to correlate the detected image with a PSF-like function [14,15]. The correlation function takes its highest values in those points where there is an object. The result of applying this digital processing technique can be compared to the effect produced by AMIGO when a mask like that described by Eq. (7) is placed at the telescope entrance pupil, which is equivalent to correlate the amplitude image with a Gaussian function.

Laplacian filters have also been applied to astronomical images for picking out faint small-scale features against the bright background of a comet [16,17].

In previous papers we analyzed the resolution of a ground-based telescope assisted by an Adaptive Optics (AO) system [18,19]. We saw that the efficiency of any kind of superresolving pupil was greatly conditioned by the site seeing and the compensation degree achieved by the AO system. Now, we are interested in checking the AMIGO Laplacian mask for obtaining high resolution images from ground-based telescopes applying the Lucky Imaging selection. For this goal we have considered the incoming aberrated field at the system entrance pupil E(x,y) = A exp[iϕ(x,y)], where A is the constant amplitude and ϕ(x,y), is the wavefront aberration function. In order to simulate an aberrating media we have used the numerical method employed for atmospheric aberration simulations [20]. Atmospheric condition (i. e. amount of aberration) has been determined by the ratio D/r₀. High aberration is assumed when the halo peak intensity contributes more to the Strehl ratio than the coherent peak intensity, that is for D/r₀ >7.8 [21]. Series of 100 aberrated wavefronts were calculated for the case D/r₀ = 7.8. This situation may be reached at ground-based telescopes with a diameter smaller than 2.5 m when detecting on the I-band and the Lucky Imaging technique is applied.

To check the efficiency of the AMIGO mask, we have simulated two objects at a distance apart equal to the radius of the first Airy ring. The intensity of second object was 0.7 that of the main one. A set of 100 simulated wavefronts was Fourier Transformed to obtain a series of amplitude PSF. The amplitude PSF once squared forms what we call the detected PSF. A Gaussian noise was added so that the peak intensity was five times the noise standard deviation and then a Laplacian filter was applied to series. The result of applying the shift-and-add procedure to the digitally processed PSF series is shown in Fig. 8(a).

Fig. 8 (a) Shift-and-add of 100 aberrated and digitally processed PSF with a clear pupil. (b) Shift-and-add of 100 aberrated PSF with a Laplacian AMIGO pupil.

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We have also multiplied the same series of 100 wavefronts by a Laplacian mask as that shown in Eq. (10). The resulting wavefront series was Fourier Transformed and squared. A Gaussian noise with the same variance as before was then added to every frame of the series before applying a shift-and-add procedure. The result obtained from the Laplacian masked pupil is shown for comparison in Fig. 8(b). The result of applying the digital image processing to the PSF series obtained from the clear pupil is an image containing two peaks affected by noise. This noise is due to the point enhancement effect appearing when the Laplacian filter is applied to noisy detected images. On the other hand, when we average the detected AMIGO series two peaks clearly apart appear and the noise is almost canceled as a result of the series average.

Three additional points have to be considered here. The first one is that, as a result of the optical processing the photometry may be lost. The second one is that the predicted results are heavily dependent on the detector performance. Finally, the results achieved by simulation would not be affected by a finite radius of coherence in a real test, because replicas of the same object are used.

7. Conclusions

We have introduced a new technique for designing diffractive optics elements that once placed at the entrance pupil of an optical system are able to produce an optical processing over the image amplitude (we call that an AMplitude ImaGe Optical processing device, AMIGO). Along the paper we have shown how to design masks for applying AMIGO to different cases of interest.

The main difference with the digital image processing, which operates over the image intensity, is that the optical processing only affects the image field amplitude. Hence, since it averages fields instead of intensities, it cancels more efficiently the detection noise.

Besides, we have checked that the effects produced by the intensity processing, such as averaging or sharpening, can also be obtained by amplitude image processing.

We have paid particular attention to the Laplacian mask, which can be considered in a wide sense as a superresolving mask. In order to check it, we have seen that for an amplitude object optical processing performs as well as digital processing. Finally we have applied the AMIGO processing by a Laplacian mask to simulated images of a ground-based telescope. We have confirmed that the optical processing by means of a Laplacian mask provides images less noisy than those obtained from the standard digital image processing.

Acknowledgments

This research was supported by the Ministerio de Economía y Competitividad under project FIS2012-31079.

References and links

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Amplitude image processing by diffractive optics

Abstract

1. Introduction

2. AMIGO mask theoretical foundations

3. Masks design

3.1 Averaging mask

3.2 Gaussian mask

3.3 Laplacian mask

4. Comparison between amplitude and intensity processing

5. Amplitude object checking

6. Application to astronomy

7. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Equations (17)

Optics Express