The first method falls under the broad category of Interferometric Phase-Retrieval Techniques and is applicable to cases where the recorded intensity is dominated by the holographic diffraction terms.31 (link)–33 The first step is the digital reconstruction of the hologram, which is achieved by propagating the hologram intensity by a distance of z2 away from the hologram plane yielding the initial wavefront Urec. As a result of this computation, the virtual image of the object is recovered together with its spatially overlapping defocused twin-image. It is important to note that the recorded intensity can also be propagated by a distance of −z2. In this case, the real image of the object can be recovered, while the defocused virtual image leads to the twin-image formation.
Due to the small cell-sensor distance in the incoherent holographic microscopy scheme presented here, the twin-image may carry high intensities, especially for relatively large objects like white blood cells. In such cases, the fine details inside the micro-objects may get suppressed. Similarly, the twin-images of different cells which are close to each other get superposed, leading to an increase in background noise. This issue is especially pronounced for microscopy of dense cell solutions, where the overlapping twin images of many cells lowers the counting accuracy due to reduced SNR.
In order to eliminate the twin-image artifact, an iterative approach using finite support constraints is utilized.33 Basically, this technique relies on the fact that duplicate information for the phase and amplitude of the object exists in two different reconstruction planes at distances +z2 and −z2 from the hologram plane, where the virtual and real images of the object are recovered, respectively. Therefore, a twin-image-free reconstruction in one of the image planes can be obtained, while filtering out the duplicate image in the other plane. Without loss of generality, we have chosen to filter out the real image to obtain a twin-image-free reconstruction in the virtual image plane at −z2. Due to the finite size of the micro-objects, the real image of the object only occupies the region inside its support, while the defocused twin-image image spreads out to a wider region around the object, also overlapping with the real image inside the support. Hence, deleting the information only inside the support ensures that the real image is completely removed from the reconstructed wavefront. Nevertheless, the virtual image information inside the support is also lost, and the iterative technique tries to recover the missing information of the virtual image by going back and forth between the virtual and real image planes, recovering more of the lost information at each iteration. The success of this algorithm is highly dependent on the Fresnel number of the recording geometry, which is given by Nf = n(object size)2/(λz). It is reported that the technique proves successful for Fresnel numbers as high as 10.33 For RBCs of approximately 7µm diameter, the typical recording geometries presented here involve Fresnel numbers of <0.2; hence, the twin-image elimination method yields highly satisfactory results.
The steps of twin-image elimination are detailed below:

Initially the real image, which is the back-projected hologram at a distance of +z2, is used for determining the object support. Object support can be defined by either thresholding the intensity of the reconstructed image, or searching for its local minima.

The region inside the support is deleted and a constant value is assigned to this region as an initial guess for the deleted part of the virtual image inside the support as shown below: Uz2(i)(x,y)={Urec,x,y∉SU¯rec,x,y∈S where Uz(i)(x,y) denotes the field at the real image plane after the ith iteration. S represents the area defined by the object support, and Ūrec is the mean value of Urec within the support.

Then, the field at the real image plane is back propagated by −2z2 to the virtual image plane. Ideally, the reconstruction at this plane should be free from any twin-image distortions. Therefore, the region outside the support can be set to a constant background value to eliminate any remaining out-of-focus real image in the virtual image plane. However, this constraint is applied smoothly as determined by the relaxation parameter β below, rather than sharply setting the image to d.c. level outside the support: U−z2(i)(x,y)={D−D−U−z2(i)β,x,y∉SU−z2    (i)   ,x,y∈S where D is the background in the reconstructed field, which can either be obtained from a measured background image in the absence of the object, or can simply be chosen as the mean value of the field outside the object supports at the virtual image plane. β is a real valued parameter greater than unity, and is typically chosen around 2–3 in this article. Increasing β leads to faster convergence, but compromises the immunity of the iterative estimation accuracy to background noise.

The field at the virtual image plane is forward propagated to the real-image plane, where the region inside the support now has a better estimate of the missing part of the virtual image. The region outside the support can be replaced by Uz2(1)(x,y), the original reconstructed field at the real image plane, as shown below: Uz2(i+1)(x,y)={Uz2(1),x,y∉SUz2(i+1),x,y∈S

Steps c to d can be repeated iteratively until the final image converges. In most cases in this article, convergence is achieved after 10–15 iterations. This iterative computation takes around 4 seconds for an image size of ~5 Mpixels using a regular CPU (central processing unit – e.g., Intel Q8300) and it gets >40× faster using a GPU (graphics processing unit – e.g., NVIDIA GeForce GTX 285) achieving <0.1 sec computation time for the same image size.