# Improvement of Fresnel Diffraction Convolution Algorithm
###### Abstract  
With the development of digital holography, the accuracy requirements for the reconstruction phase are becoming increasingly high. The transfer function of the double fast transform (D-FFT) algorithm is distorted when the diffraction distance is larger than the criterion distance \(\mathrm{d}_{\mathrm{r}}\), which reduces the accuracy of solving the phase. In this paper, the Fresnel diffraction integration algorithm is improved by using the low-pass Tukey window to obtain more accurate reconstructed phases. The improved algorithm is called the D-FFT (Tukey) algorithm. The D-FFT (Tukey) algorithm adjusts the degree of edge smoothing of the Tukey window, using the peak signal-to-noise ratio (PSNR) and the structural similarity (SSIM) to remove the ringing effect and obtain a more accurate reconstructed phase. In a simulation of USAF1951, the longitudinal resolution of the reconstructed phase obtained by D-FFT (Tukey) reached 1.5 \(\mathrm{\SIUnitSymbolMicro m}\), which was lower than the 3 \(\mathrm{\SIUnitSymbolMicro m}\) obtained by the T-FFT algorithm. The results of Fresnel holography experiments on lung cancer cell slices also demonstrated that the phase quality obtained by the D-FFT (Tukey) algorithm was superior to that of the T-FFT algorithm. D-FFT (Tukey) algorithm has potential applications in phase correction, structured illumination digital holographic microscopy, and microscopic digital holography.  
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correction, structured illumination digital holographic microscopy, and microscopic digital holography, the two existing algorithms cannot meet the demand, and a more accurate calculation method is urgently needed.  
To pursue more accurate calculated results, many studies have been carried out. For example, digital hologram apodization [17], zero-padding, and then extending the hologram eliminates any ringing noise due to aperture diffraction [18], optimizing the sampling resources of the S-FFT algorithm to remove noise from high-frequency component aliasing [19], analyzing the applicable conditions of each existing diffraction integration algorithm to obtain the best reconstruction results [20], and improving existed diffraction convolution algorithms [21; 22; 23; 24]. In terms of the improvement of diffraction convolution algorithms, there are many methods such as limiting the bandwidth of the transfer function using an ideal low-pass filter [21], extending the unmixed bandwidth of the transfer function by rearranging the sampling points in the spatial frequency domain [22], carrying out polynomial decomposition of the transfer function [23], and replacing the traditional Fourier transform in the reconstruction process with the fractional Fourier transform to utilize more high-frequency components [24]. Among these, for the improvement of the transfer function, two common methods are filtering the transfer function with a rectangular window (D-FFT (rect)) [21] and resampling the distorted part of the transfer function [22]. However, filtering the transfer function with a rectangular window will cause a ringing effect because of the sharp edge, and using a non-uniform FFT to resample the aliasing part of the transfer function will cause a similar amplitude and phase error to the T-FFT algorithm.  
In this paper, the T-FFT algorithm and D-FFT algorithm are briefly introduced at first. Then, the results of the T-FFT algorithm and D-FFT algorithm are compared with examples to illustrate the problems of the D-FFT algorithm. Based on the above analysis, the D-FFT algorithm is improved, and finally, the simulation results before and after the improvement and comparison examples of lung cancer cells' experimental measurements are given. The experimental results show that the improved Fresnel diffraction convolution algorithm eliminates the under-sampling effect of the transfer function without causing the ringing effect, and a higher resolution of the phase distribution is obtained.