Here we provide a simple simulation demonstrating how refocusing a hologram is carried out simply as a convolution with a 2D CSF. The scene consists of three playing cards placed at different depths. The coherent spread functions (CSF) for the cards vary in size according to their depths. The CSFs have the shape of a sinusoidal zone plate. Consequently, high spatial frequencies are preserved in the hologram even though the maximum blur diameter is nearly half the image width. In contrast, the incoherent PSFs in conventional imaging act as low-pass filters.

Our prototype camera utilizes a Michelson interferometer setup. A cube beam splitter divides incoming light from the scene into two beams. Two mirrors then reflect the beams back towards an objective lens that images the wavefront onto the sensor. One of the mirrors has a small amount of curvature to shift the focus of the beam relative to the other path. A piezo-actuated mirror creates sub-micron displacements that are used for phase shifting. The mirrors were both 25.4mm diameter and the objective lens was an 50mm focal length lens (Edmunds Optics 59-873), resulting in an aperture setting of approximately F/2. For the sensor, we used a 1/2" monochrome 10 Mpix sensor with 1.67um pixels (Edmunds Optics 86-749). For the color filter, we used a 10nm bandwidth interference filter with center frequency 632.8nm (Thorlabs FL632.8-10).

The figure shows the geometric parameters of an incoherent holographic camera. A Michelson interferometer setup is unfolded and the two paths are shown independently. The key property is that each point source is divided into two point sources with slightly different radii of curvature. The two point sources are then recombined and interfere on the sensor plane, reproducing a Point Spread Function (PSF) very similar to the optical propagator for a coherent optical field (i.e. the CSF).

The incoherent holography PSF consists of a sum of four different PSFs. As a result, the captured image is a sum of four different blurred images. Two of the images are simply blurred by the incoherent PSF from the two point sources in the Michelson interferometry setup. The other two images are blurred by the CSF. Phase shifting is used to remove the undesired blurred images.

We show that The MTF of an incoherent holography system is inversely proportional to the maximum blur size. In this simulation, the scene consists of three playing cards placed at different depths. Gaussian noise is added to captured images with a std-deviation of &sigma=.005. The camera is focused on the front card (Depth 1). The scene depths are chosen so that the maximum blur size is 32 pixels (top row) and 64 pixels (bottom row). (a) The three input phase-shifted images. (b) The recovered holograms. The front card is in focus, and the other two cards are blurred. (c, d) Images refocused on the middle card (Depth 2) and the back card (Depth 3), respectively. Image noise is higher for the larger blur (bottom row).

We derive analytic expressions showing that The MTF of an incoherent holography system is inversely proportional to the maximum blur size. Since the MTF of is flat over the passband, this means that SNR falls off exactly in the same way, producing significantly greater SNR than a full resolution light field camera. To verify this result, we perform simulations with a variety of blur sizes and add Gaussian noise with &sigma=.005. As shown in this plot, the increase in noise is linear, where it is evident that the ratio of RMSE deblurring error is approximately equal to the ratio of blur sizes.

A Playing cards scene. Depth range is approximately 75cm and the maximum blur size is approximately 1200 pixels. The scene was captured at full 10 Mpix resolution. The exposure time was 1/3 sec due to the amount of light blocked by the interference filter. The small pixels have a very small dynamic range, and we averaged 100 frames and used denoising software to increase the SNR to a reasonable level. The right image shows the coherent field recovered after phase shifting, which is focused on the King in the background. The center image shows the results after refocusing on the Jack in the foreground. The Jack is seen to come into crisp focus while the King becomes significantly blurred.

A Diner scene. The depth range is approximately 50cm and the maximum blur size is approximately 600 pixels. The right image shows the coherent field recovered after phase shifting, which is focused on the diner in the background, while the fence is severely blurred. The high frequencies in the fence are still visible even with an extremely large blur size. The center image shows the the fence come into focus while the diner becomes blurred.