# Speckle Noise Reduction in Digital Holography Using a DMD and Multi-Hologram Resampling
## 3 Optical Speckle Reduction  
We evaluated two different methods for changing the illumination over the object. The first consists of using the DMD to illuminate the object with plane waves with varying tilt; the second method consists of illuminating the object with a random checkered phase function. Both strategies involve registering a hologram for each pattern on the DMD, producing multiple reconstructed images that are later averaged to reduce the speckle. The pipeline to generate a final intensity reconstruction with reduced speckle is shown in Figure 2. The holograms can be obtained with any of the techniques illustrated in the following sections. Each hologram is propagated with the discrete Fresnel Transform to obtain a reconstructed complex wave. This contains all the information stored in the hologram, the zero order located at the center and the separated twin images to its sides. The separation comes from the off-axis nature of the holographic recording. If the propagation distance \(d\) is properly chosen,  
Figure 1: Digital holographic recording setup with a Digital Micromirror (DMD) for phase modulation. BE: Beam Expander, BS: Beam Splitter, DMD: Digital Micromirror Device, M:Mirror, FF: Fourier Filtering.  
one of the images will be in perfect focus. From this image, we crop a square area containing the object, facilitating the further analysis of the reconstruction quality and its speckle. All the cropped intensity images are averaged and the result is a final reconstruction with reduced speckle noise.  
These former strategies can be supported by characterizing the speckle as a multiplicative noise [23; 24]. Thus, the result of the holographic reconstruction given in Equation (1) can be understood as the object field \(E_{o}\exp(i\phi_{o})\) modulated by a speckle noise \(E_{s}\exp(i\phi_{s})\) as follows [6],  
\[\Gamma=E_{o}\exp(i\phi_{o})E_{s}\exp(i\phi_{s}). \tag{4}\]  
Likewise, the intensity of the reconstruction will be given by  
\[I=\Gamma\cdot\Gamma^{*}=E_{o}^{2}E_{s}^{2}, \tag{5}\]  
where \(\Gamma^{*}\) stands for the complex conjugate of \(\Gamma\). Equation (5) allows us to continue interpreting the speckle \(E_{s}^{2}\) as a multiplicative noise that modulates the distribution of intensities \(E_{o}^{2}\) produced by the object. In this way, it can be observed that the intensity distribution of the object in the reconstruction does not change if only the phase characteristics of the object illumination are modulated between acquisitions. On the other hand, the intensity distribution of the speckle noise does change. This is because the phase modulation of the illuminating beam changes the phases at each point of the surface of the object, which affects the superposition of random phasors that gives rise to the intensity distribution of the speckle noise [16; 25].  
Therefore, the superposition process of several hologram reconstructions with different speckle patterns can be written as  
\[\sum_{i=1}^{h}I_{i}=E_{o}^{2}(E_{s1}^{2}+E_{s2}^{2}+...+E_{sh}^{2}), \tag{6}\]  
where \(h\) is the number of superposed reconstructions. In Equation (6), we have a new speckle given by the superposition of intensities of \(h\) different speckle patterns while preserving the information of the object. The contrast of the resulting speckle pattern is reduced by \(1/\sqrt{h}\) for the superposition of \(h\) totally uncorrelated speckle patterns [25].  
Figure 2: Pipeline for hologram reconstruction and speckle reduction.