# Speckle Noise Reduction in Digital Holography Using a DMD and Multi-Hologram Resampling
## 6 Results and Discussion  
### Recorded Holograms  
We separately evaluated our proposed strategies to decrease the level of speckle noise. The comparison of the results is summarized in Figure 5. For reference, we included reconstructed images from a single hologram in the first row, presenting the unavoidable speckle. In the second row, we show the resulting averaged images with decreased speckle. These images are the result of the average of 20 reconstructions. The improvement is qualitatively evident as shown by these images. More specifically, in Figure 6 we show a quantitative evaluation for comparing some results of the strategy using synthetic holograms, and we expand on the analysis of the results of this strategy in the next section. But regarding the two strategies using only recorded holograms, in the first two columns of Figure 7 we showcase the behavior of their speckle metrics. As is shown by the curves of the speckle  
Figure 4: Digital resampling of the recorded holograms for \(b=16\), \(h=20\). Adding the individual Hadamard multiplications between holograms and binary masks produces a synthetic hologram. A hologram H1 contributes the small patch h1 to the final synthetic hologram.  
contrast, the improvement is also quantitatively evident when more images are averaged. In the final row, the correlation coefficients for all reconstructed images are presented as a lower triangular matrix. On this plot, the indices of the matrix for the plane wave approach also denote an increment of the angle of tilt on the illuminating wave, so the color distribution in this matrix indicates that with more difference in the angles between plane waves, the correlation is smaller; specifically, the pair of images \((s=19,t=0)\) present the smallest value of \(\gamma\), as we increase the value of the index \(t\) keeping the value of \(s\) constant, the two plane waves have gradually more similar tilts, producing an increase in the correlation coefficient. As such, these images will present similar speckle noise distributions. For the checkered random patterns, the correlation behaves randomly for any pair of images, as the computation of the checkered patterns introduces random changes. However given the conditions in our optical setup, the values obtained for \(\gamma\) are not significantly better than the plane wave strategy. The diagonal in all matrices represents the result of computing the correlation of an image with itself, which results in a unitary value.  
The speckle coefficient was evaluated for the square indicated in Figure 5, all plots for \(C\) in Figure 7 demonstrate improvement with the total number of averaged images, with a tendency for saturation indicating that no significant speckle reduction could be obtained with more images. The values for \(C\) are similar between the two optical strategies of plane wave and random checkered phase which use original holograms, however, both are far from the theoretical ideal of \(1/\sqrt{h}\).  
To single out the best one of these two optical strategies, we compared the quality on the averaged images. Since the random checkered pattern includes randomly placed phase jumps, the result after the Fourier filtering is a residual amplitude modulation; consequently, the reconstructed images present some darkening in the places of the phase jumps. As we created the patterns to be randomly distributed, the averaged image \(\langle I\rangle\) tends to hide these effects, but a minimum degree of deterioration is inevitable. In contrast, the plane waves do not present any degree of amplitude modulation and so the perceived quality of the averaged image is better than the one from the checkered phase strategy.