Spectral-domain denoising

Precise estimation of phase structure of THz vortex beams is a key moment, partic-ularly in topological charge measurement which can be hindered by the presence of noise. In this way the spectral domain algorithm CCF is very useful because of its efficiency in complex-valued hyperspectral data denoising.

Fig. 4.22 and 4.23 illustrate the experimental and simulation results of

spatio-Figure 4.19 ToF representation of THz vortex wavefront denoising in experiment. (a-c) correspond to the case of noisy data ‘before VBM3D’ E˜(x,y,t =t_i,z_r); (d-f) correspond to the case

‘after VBM3D’Eˆ(x,y,t=t_i,z_r)^.

spectral CCF denoising for amplitude and phase. Note, that first-step VBM3D de-noising with lowt h_tparameter was also applied. Simulation figures are also supple-mented by the noise-free data. Note, that the phase distribution contains lateral area with undefined phase due to the zero amplitude. This regions are not assumed to be denoised and they are dashed for clarity of visualization (Fig. 4.23-(e-f)).

The wavefield structure for individual frequency provides a valuable information for both object reconstruction [47, 57, 65] and for investigation of beam spatio-spectral dynamics[40, 44]. The denoising at individual frequencies is demonstrated in Fig. 4.24 and Fig. 4.25 for experiment and simulation correspondingly.

In these figures the denoising is demonstrated particularly for the low SNR data.

It allows to reconstruct the information hidden in the noisy data and to recover the fine details. Thus, CCF (combined with first-step VBM3D) demonstrates to expansion of spectral interval of amplitude/phase data reconstruction.

Note, that phase unwrapping is not performed in order to keep the phase delay in the range[−π;π). It allow to visualize vortex distribution with topological charges

>1. For example, spectral components 1.03 THz and 1.49 THz correspond to the

Figure 4.20 ToF representation of THz vortex wavefront denoising in simulation. (a-c) correspond to the case of noisy data ‘before VBM3D’E˜(x,y,t=t_i,z_r); (d-f) correspond to the case ‘after VBM3D’Eˆ(x,y,t=t_i,z_r); (g-i) correspond to the simulated noise-free data.

topological charges 2 and 3. However, the topological charges can be still hardly dis-cerned. For quantitative estimation of topological charges the dependency of phase versus angle of azimutal bypass for various frequencies was calculated (Fig.4.26-a).

These graphs are presented for frequencies corresponding to the integer charge. If plotting charges versus frequency we get the peculiar stair-like function (Fig.4.26-b) with integer charge areas (around 0.5 THz, 1 THz, 1.6 THz, 2.3 THz and 2.7 THz) and non-integer areas (climbing areas between integer charges). Thus, it proves in simulation, that frequency 1.03 THz corresponds to charge 2 and 1.49 THz is charge 3. Unfortunately, noise in the experiment do not allow to visually distinguish the charges even after denoising and in this aspect we should rely on the simulation

re-Figure 4.21 Differential signal between noisy and denoised data in spatio-temporal cross-section

E_{d i f f}(x,y =y₀,t,z_r)(a) and in ToFE_{d i f f}(x,y,t = t_i,z_r)(b). (c) corresponds to the standard deviation of differential signal in each ToF slice.

Figure 4.22 Experimental results of spatio-spectralC C Fdenoising for amplitude (top row) and phase (low row). (a,c) - before denoising; (b,d) - after denoising.

sults.

As a quantified criteria NRMSE between denoised and initial noise-free data is

Figure 4.23 Simulation results of spatio-spectralC C F denoising for amplitude (top row) and phase (low row). (a,d) - before denoising; (b,e) - after denoising; (c,f) - noise-free data.

Figure 4.24 Experimental results of denoising at individual frequencies for amplitude (1st and 2nd rows) and for phase (3rd and 4th rows). Results before filtration are in (1,3 rows), after filtration (2,4 rows).

Figure 4.25 Simulation results of denoising at individual frequencies for amplitude (1,2,3 rows) and for phase (4,5,6 rows). Results before filtration are in (1,4 rows), after filtration (2,5 rows), noise-free data (3,6 rows).

Figure 4.26 (a) - Phase versus azimutal angle for frequencies corresponding to integer topological charge. (b) - topological charge versus frequency.

also calculated. This efficiency criteria is also compared with other algorithms such as BM3D (see[16]) and CDBM3D in previous section. The main advantage of CCF is that it filters hyperspectral data simultaneously in comparison with CDBM3D which provides denoising in each frequency separately. Fig. 4.27 illustrates lowest NRMSE of CCF. Thus, its efficiency for THz hyperspectral data denoising is quan-titatively proved.

Figure 4.27 NRMSE between between denoised and initial noise-free data by BM3D (black line), CDBM3D (red line) and CCF (blue line). Each method uses their optimal denoising set-tings.

Hyperspectral THz holography is a powerful method of pulsed THz wavefront investigation in spatial, temporal and spectral domain. It allows both object recon-struction and wavefield evolution analyzing. Low SNR, however, is a valuable lim-itation in terms of image reconstruction and wavefront characterization. For this case, the chapter 4 is devoted to hyperspectral data denoising methods based on block-matching algorithm BM3D[16]. Its modifications for spatio-temporal (VBM3D) and spatio-spectral (CDBM3D and CCF) data were considered.

Spatio-temporal wavefront analyzing is actual for various beams such as vortex beams, Gauss-Bessel beams or any others. For this objective VBM3D algorithm with hard settings is efficient what makes the fine details in THz wavefront to be resolvable. However, hard settings in VBM3D can lead to the oversmoothing in the corresponding spectral images.

In this case of spectral data the more efficient CDBM3D and CCF could be used.

They process directly a spectral domain complex-valued data with amplitude and phase joint processing. Combination of spectral domain algorithms with VBM3D provides more efficient phase images reconstruction. It was show that the soft first-step VBM3D filtration increases the further efficiency of CDBM3D and CCF and thus allowing to increase the THz spectral interval of quality amplitude-phase recon-struction. Further, the advantage of CCF over CDBM3D was also demonstrated.

Sections 4.1 and 4.2 differ by the peculiarities of detection systems. In section 4.1 the specificity of photoconductive antenna with Gaussian noise distribution is con-sidered. Contrary, in section 4.2 the balance detector system is investigated. This BD system is characterized by Skellam noise distribution originating from the dif-ferential signal from two detectors. Note, that in this case the noise variance is not in ratio to the THz signal.

The results were proved in both simulation and experimental research. Image visual quality as well as numerical efficiency criteria RMSE were presented.

5 PECULIARITIES OF CORRELATION MEASUREMENTS OF PULSED

HYPERSPECTRAL OPTICAL FIELDS

The proposed THz hyperspectral holography approach provides huge possibility not only to THz wavefront analysis. Its basis technique could be transferred to other frequency range. For example, the femtosecond (fs) laser pulses are also classified as pulses with few field oscillations and they contain also broadband temporal spec-trum. The measurements of such fs pulses are usually associated with interferomet-ric detection which can be sensitive to the wavefront pertrubations. However, the peculiarities of fs wavefront with broad spatial spectrum and phase fluctuations are not sufficiently studied for now. Further we consider the case of interferometric system where this features could be actual.