Deep Learning-Based Wrapped Phase Denoising Method for Application in Digital Holographic Speckle Pattern Interferometry  
[PERSON]  
1 Department of Precision Mechanical Engineering, Shanghai University, Shanghai 200072, China; [EMAIL_ADDRESS] (K.  
system providing high-density information and high fringe contrast data has been designed and developed for application in the remote examination of cultural heritage [13]. Methods of recovering data from speckle interferograms formed from another portable system based on digital speckle pattern interferometry were presented by [PERSON] et al. [14]. Since artworks measured with DHSPI are usually complex, multilayered and inhomogeneous structures deteriorating with time and environmental conditions, starting from inside and affecting the precious surface [15; 16; 17; 18; 19], the resulting DHSPI wrapped phase is complicated, carrying complex fringe pattern changes and containing severe speckle noise. In many antique and heavily deteriorated artworks, the resulting interferograms may carry extremely discontinuous or even difficult to resolve fringe patterns. Noise can cause discontinuities in the fringes, making them appear to be interrupted. In this way, to properly process and evaluate the interferograms, tedious long-term fringe skeleton corrections and software for interactive interventions in the fringe pattern reconstruction process necessitating a priori knowledge are required [20; 21]. Therefore, the denoising process is necessary in order to accurately observe and evaluate the state of the defects in artworks. Many methods have been reported to denoise speckle noise from digital holography [22; 23; 24]. Phase-shifting technology provides the wrapped phase distributed in \((-\pi,\pi]\), and a sine/cosine average filter (SCAF) [25] is proposed to denoise the numerator (sine fringes) and denominator (cosine fringes) of arctangent for processing noise and \(2\pi\) discontinuities in the wrapped phase. Windowed Fourier transform (WFT) [26] is an effective method to denoise the wrapped phase, but it requires a long-processing time and adjusts the parameters to process data. The regularized quadratic cost function (RQCF) [27] is proposed by [PERSON] et al. to denoise the wrapped phase, but this method requires the estimation of fringe orientation. There are still many efforts to automatically denoise the wrapped phase map while achieving high accuracy and fast processing. Recently, optical image denoising using deep learning has shown excellent processing speed and accuracy including fringe pattern denoising [28; 29; 30] and wrapped phase denoising [31]. Deep learning has demonstrated effective performance for reducing Gaussian noise in the wrapped phase [31], but the proposed method in [31] does not process the speckle noise of the wrapped phase for application in DHSPI. In this paper, a deep learning method is further proposed for denoising DHSPI wrapped phase to help the structural diagnosis of artworks. In phase-shifting technology, the sine and cosine fringes are calculated by arctangent function to obtain the wrapped phase distributed in \((-\pi,\pi]\). In order to address noise and \(2\pi\) discontinuities of the wrapped phase, we present a deep learning method to denoise the sine and cosine fringes, and the filtered sine and cosine fringes are further calculated by arctangent to give an enhanced wrapped phase. Simulation results show that the proposed method can effectively reduce speckle noise and provide high-quality phase maps. The proposed method is verified by captured data from the portable DHSPI system. By comparing the high-frequency region of the phase map, the proposed method is shown to be superior to the post-processing of portable DHSPI systems and demonstrates a useful processing method for accurately assessing defects in artworks. The rest of this paper is arranged as follows. Section 2 introduces the speckle fringe simulation method and phase retrieval method, which provide a theoretical basis for establishing a database.