Time Multiplexing Super Resolving Technique for Imaging from a Moving Platform

Summary

A method for overcoming the optical diffraction limit is presented. The method includes a two-step process: optical phase retrieval using iterative Gerchberg-Saxton algorithm, and imaging system shifting followed by repetition of the first step. A synthetically increased lens aperture is generated along the direction of movement, yielding higher imaging resolution.

Abstract

We propose a method for increasing the resolution of an object and overcoming the diffraction limit of an optical system installed on top of a moving imaging system, such as an airborne platform or satellite. The resolution improvement is obtained in a two-step process. First, three low resolution differently defocused images are being captured and the optical phase is retrieved using an improved iterative Gerchberg-Saxton based algorithm. The phase retrieval allows to numerically back propagate the field to the aperture plane. Second, the imaging system is shifted and the first step is repeated. The obtained optical fields at the aperture plane are combined and a synthetically increased lens aperture is generated along the direction of movement, yielding higher imaging resolution. The method resembles a well-known approach from the microwave regime called the Synthetic Aperture Radar (SAR) in which the antenna size is synthetically increased along the platform propagation direction. The proposed method is demonstrated through laboratory experiment.

Introduction

In radar imaging, a narrow angle beam of pulse Radio Frequency (RF) is transmitted using an antenna that is mounted on a platform. The radar signal transmits in a side-looking direction towards the surface^1,2. The reflected signal is backscattered from the surface and is received by the same antenna². The received signals are converted to a radar image. In Real Aperture Radar (RAR) the resolution in the azimuth direction is proportional to wavelength and inversely proportional to the aperture dimension³. Thus, a bigger antenna is required for higher azimuth resolution. However, it is difficult to attach large antenna to a moving platforms such as airplanes and satellites. In 1951 Wiley⁴ suggested a new radar technique called Synthetic Aperture Radar (SAR), which uses the Doppler effect created by the movement of the imaging platform. In SAR, the amplitude as well as the phase of the received signal are recorded⁵. This is possible since the SAR optical frequency is about 1-100 GHz⁶ and the phase is recorded using a reference local resonator installed on top of the platform. In optical imaging, shorter wavelengths are being used, such as the visible and the near infra-red (NIR), which is about 1 μm, i.e. frequency of about 10¹⁴ Hz. The field intensity, rather than the field itself, is being detected since the optic phase changes too fast for detection using standard silicon based detectors.

While imaging an object through an optical system, the aperture of the optics serves as a low-pass filter. Thus, the high-frequency spatial information of the object is lost⁷. In this paper we aim to solve each of the above mentioned issues separately, i.e. the phase lost and the diffraction limit effect.

Gerchberg and Saxton (G-S)⁸ suggested that the optical phase can be retrieved using an iterative process. Misell^9-11 has extended the algorithm for any two input and output planes. These approaches are proven to converge to a phase distribution with a minimal mean square error (MSE)^12,13. Gur and Zalevsky¹⁴ presented a three planes method which improves the Misell algorithm.

We propose and demonstrate experimentally that restoring the phase while shifting the imaging lens, as done with the antenna in SAR application allows us to synthetically increase the effective size of the aperture along the scanning axis and eventually improve the resulted imaging resolution.

The application of SAR in optical imaging using interferometry and holography is well-known^16,17. However, the suggested method is aimed for mimicking a scanning imaging platform, making it suitable for noncoherent imaging (such as side-looking airborne platform). Thus, the concept of holography, which uses a reference beam, is not suitable for such an application. Instead, the revised Gerchberg-Saxton algorithm is used in order to retrieve the phase.

Protocol

1. Setup Alignment

1. Start by roughly aligning the laser, the beam expender, the lens, and the camera on the same axis; this would be the optic axis.
2. Turn on the laser (without the USAT target), and make sure that the light passes through the center of the lens. Use an aperture iris to verify.
3. Turn on the camera, and make sure that the light focuses on the center of the camera.
4. Shift back the camera, using the linear z stage. Since the system is going out of focus, the spot of light will grow. Make sure that the center of the spot remains in the same lateral position. If not, carefully change the position of the imaging system and repeat this step until the spot remains at the same spatial position, up to a pixel level.

2. Imaging at Three Defocus Planes

1. Insert the test target in front of the beam expender. Place the target so that the light that passes through it will pass through the center of the lens.
2. Capture an image. This image will be an anchor point, and its location will be z₀,x₀ (all the other images will be in reference to its location). This image will be I_1,b.
3. Shift back the camera (using the linear z stage) a distance of dz = 5.08 mm (or 0.2 in) and capture an image. This image will be I_2,b.
4. Shift back the camera another distance of dz = 5.08 mm (10.16 mm relative to z₀) and capture an image. This image will be I_3,b.
5. Go back to z₀.

3. Scanning the Aperture

1. Shift the entire imaging system laterally (using the linear x stage) a distance of dx = 2.5 mm and capture an image. This image will be I_1,a.
2. Repeat the process in Protocol 2. Shift back the camera (using the linear z stage) a distance of dz = 5.08 mm, and capture an image (I_2,a). Shift back the camera another distance of dz = 5.08 mm, and capture an image (I_3,a).
3. Now, repeat the procedure for the other side. Shift the imaging system a distance of dx = -2.5 mm and capture a set of three images in three z positions (I_1-3,c).
4. Go back to z₀,x₀.

4. Phase Retrieval (Numerical Calculation)

1. Using the three planes method¹⁴, and images I_1-3,b, retrieve the optical phase of image I_1,b.Using the phase that was retrieved, define q_1,b.
2. Monitor the correlation coefficient between I_1,b and |q_1,b|², in order to verify that the iterative process does converge. To do so, use the 'corr2' function in MATLAB.
3. Repeat the phase retrieval process for I_1-3,a, and I_1-3,c.

5. Super Resolved Image (Numerical Calculation)

1. Using Fresnel free space propagation (FSP) integral¹⁵, back propagate the fields q_1,a-c to the lens plane. These fields will be Ê_lens,a-c⁺.
2. Multiply the resulting fields Ê_lens,a-c⁺ by exp(+πix₀²)/λf), in order to pass back through the lens. These fields will be Ê_lens,a-c^-.
3. In order to place the field Ê_lens,a in its original position, shift it laterally a distance of dx = 2.5 mm.
4. In order to place the field Ê_lens,c in its original position, shift it laterally a distance of dx = -2.5 mm.
5. Sum the three fields Ê_lens,a-c, in order to combine them, and synthetically increase the aperture size.
6. Multiply the resulting field by exp(-πix₀²)/λf), and free space propagate it to image plane.
7. A resolution improvement by a factor of 3 in the scanning direction should be witnessed.

Results

An example for the nine captured images (three defocus images in three lateral positions) is shown in Figure 3.

An example for the G-S convergence is shown in Figure 4. The correlation coefficient for the central image I_1,b is above 0.95, and the correlation coefficient for the side images I_1,a, and I_1,c is above 0.85 (in full numerical simulation they all passed 0.99).

A representative result for th...

Discussion

The optical synthetic aperture RADAR (OSAR) concept that is presented in this paper is a new super resolved approach that uses the G-S algorithm and scanning technique in order to improve the spatial resolution of an object in the direction of the scan. The movement of the imaging platform can be self-generated while using an airborne or satellite platform. Unlike many time multiplexing SR techniques, our method does not require any a priori information of the object, other than the fact that it is stationary du...

Materials

List of materials used in this article

Name	Company	Catalog Number	Comments
Red Laser Module	Thorlabs	LDM635
10X Galilean Beam Expander	Thorlabs	BE10M-A
Negative 1951 USAF Test Target	Thorlabs	R3L3S1N
Filter holder for 2 in Square Filters	Thorlabs	FH2
1 in Linear Translation Stage	Thorlabs	PT1	2x
Lens Mount for Ø1 in Optics	Thorlabs	LMR1
Lens f = 100.0 mm	Thorlabs	AC254-100-A
Graduated Ring-Activated Iris Diaphragm	Thorlabs	SM1D12C
2.5 mm x 2.5 mm Aperture Ø1 in			Indoor production
High Resolution CMOS Camera	Thorlabs	DCC1545M