Estimating the missing tensor observations from the limited uncorrupted tensor observations is very difficult ^{1}. To recover missing observations, repeated patterns present in the image may be used ^{2}. It is providing the solutions to wide range of problems in machine learning, computer vision, appearance acquisition, video coding, scan completion, subspace clustering, and compressed sensing applications ^{3, 4, 5, 6}. The twodimensional data completion difficulties are addressed with the nuclear norm approach; it replaces the rank function ^{7} and it is termed as Low Rank Minimization. Singular value thresholding technique used to provide solutions to numerous LowRank optimization problems. It had poor recovery in the twodimensional data because the allsingular values are minimized, it led to the loss of major structure information. The truncated nuclear norm regularization ^{8} minimizes singular values based on rank by performing the twostep process. The singular value shrinkage process is performed along with the optimization process. Identification of a rank for the twodimensional is simple and computation costs are also low. If the missing ratio is increasing the recovering of the image becomes more difficult and less accurate. An introduction of a sparse regularizer can provide a better approximation. The sparse regularizer is framed with a twostep process^{ }
Kilmer et al. ^{10} introduced the tSVD approach; it is well performing in capturing the spatialshifting correlation of high dimensional data. Let the threedimensional data tensor be decomposed as two orthogonal tensors and one diagonal tensor as similar to twodimensional SVD. The tensor completion algorithms High Accuracy Low Rank Tensor Completion (HaLRTC), Fast Low Rank Tensor Completion (FaLRTC) and Simple Low Rank Tensor Completion (SiLRTC) are introduced ^{6}. Further, to improve the recovery of high dimensional data, the truncated nuclear norm is extended to tensors and introduced with the Discrete Cosine Transformation (DCT)^{ 11. H}ere the inclusion of DCT provides piecewise similarity among the recovered tensors in the image. The unitary transformbased tSVD approach was introduced to improve PSNR in hyperspectral images ^{12}. Here another transformbased tSVD approach is presented to improve the visual quality of the highly corrupted highdimensional data. The secondgeneration wavelet transform ^{13} features: the original data may be changed with its inverse transform without affecting the basic structural information; auxiliary memory is not needed to implement. It is also a reversible integer wavelet transform, easy to implement and understand.
A linked transformbased lowrank tensor representation that fully uses the redundancy in spatial and spectral / temporal dimensions and multiscale spatial nature, resulting in an efficient multi low tensor rank approximation. They have used two spatial dimensions with the twodimensional framelets, and for temporal /spectral dimension one / twodimension Fourier transform is used, and a Karhunen  Loéve transform (by singular value decomposition) to the changed tensors ^{14}. Tensor Train rank reduction is conducted on a created higherorder tensor called group by stacking comparable cubes, which naturally and completely exploits Tensor Train rank's ability to create highorder tensors. To every group of Tensor Train, lowrankness is produced by perturbation analysis ^{15}.
The residual texture information in a spectrum variation image is considered by spatial domain spectral residual total variation (SSRTV). SSRTV calculating a 2DTV for the residual image after evaluating the difference between the pixel values of neighboring bands. Lowrank Tucker decomposition takes advantage of hue, saturation, and intensity (HSI) worldwide, lowrankness and spatial–spectral correlation (LRTD). Furthermore, the l_{2,1} norm was shown to be more effective in dealing the sparse noise, particularly samplespecific noise like stripes ^{16}. In the Tensor Ring (TR) decomposition methodology, the block Hankelization method is used to convert the corrupted tensor data into a 7D higher order tensor. With rank, incremental and multistage methods, the higher order tensor is recovered by employing TR decomposition. Here the rank incremental approach is used to determine TR rankings ^{17}.
The existing and proposed approaches were evaluated using the Full Reference Assessment metrics named PSNR and SSIM, and NoReference Assessment metrics named NIQE ^{18}, and PIQE ^{19}. The values obtained are superior to the existing approaches.
The main contributions of the work are
• The 5/3 wavelet filterbased lifting scheme induced tSVD is proposed to recover the high dimensional data, when the data consists of more than 80 percent of corrupted observations.
• The visual quality of highly corrupted highdimensional data under low tensor rank assumptions is improved.
• The PSNR and SSIM are calculated for the reported high when compared with the existing approaches. The NIQE and PIQE are very small, which indicates extreme visual recovery with local and global information.
The notations used are: Lowercase characters, such as
The following properties and notations are used in the solving of the objective function.
The


._{F} 
Frobinous Norm 
._{2} 


Corrupted high dimensional image 

Unitary tensors obtained from tensor SVD 

Prediction coefficients 

Updation coefficients 

Reconstructed matrix 

Recovered tensor 

Low pass and high pass filters 

Even and Odd tensors obtained with low pass filter 

Even and Odd tensors obtained with high pass filter 
candecomp 
Canonical Decomposition 
parafac 
Parallel Factorization 
The highdimensional data processing requires huge storage, high data speeds for transmission, and accessing is difficult; to simplify the usage of storage and transmission data rates; the data will be compressed or encoded, during these processes, the tensor gets damaged. The corrupted tensors are less; the tensor can be recovered with conventional algorithms easily. If more tensors are corrupted the existing approaches are unable to recover. Here a transformbased optimization model is discussed to recover the tensors.
The lifting wavelet transforms (LWT) play a vital role in image denoising, image compression, and image inpainting. The LWT features are: less computational complexity, fast, allows the one to custom design the filters, polyphase representation and allows inplace implementation. To build sparse approximations of most real time data, the LWT generated correlation structure may be used. The lifting structure begins with the wellknown filters
The polyphase matrix is factorized using the successive division method, which involves utilizing the Euclidean algorithm of the greatest common divisor (GCD) and selecting the proper Laurent polynomials. The factorization process aims to express the polyphase matrix as a series set of lower and upper triangular matrices. The first column of the polyphase matrix is factorized which results in the following matrix decomposition
Complementary polyphase filter g^0can be found where another polyphase matrix
The final polyphase factorization may be written as
k is the nonzero constant and
The analysis polyphase matrix gives the direct implementation of the forward transform of the lifting scheme. When compared to the standard wavelet approach, this reduces computing complexity. The better performance of the lifting scheme is dependent on the choice of the wavelet filter. Wavelet family has wide range of wavelet filters are available. The wavelet filter
The diagonal tensor will be reduced by using the truncation of uncorrelated singular values, it may be used the convex optimization, by utilizing
Keep
Based on the singular value thresholding shrinkage operation eqn. 9. can be solved easily.
By fixing the
The update of
The proposed steps are framed as below
given
Calculate the lifiting coefficnets to tensor
Approximations
Compute
Keep
By fixing
Update
The 5/3 lifting wavelet inclusion in tSVD, which is improving the effective calculation of missing tensors. The analysis filter coefficients are predicted by the odd location tensors by using the even location tensors. The even locations tensors are the approximation information it will have and the odd location tensors will have the details information of tensors. The odd location tensors may not be used in the prediction and updation of the odd location tensors. The missing or corrupted tensors will be updated with this process. The tensor SVD is applied to the retrieved tensors to minimize the rank of a tensor in the augmented lagrangian approach.
The proposed approach is tested with the various classes of images like natural images, and scenes, etc. The system configuration used for the work is Intel i5 8^{th} generation processor with 8GB RAM, MATLAB 2019a. The images used for experimentation are of size 256⨉256 and they are listed in
The color images are degraded by 10% to 90% random pepper noise to make the intensity values to zero. The 5/3 wavelet with decomposition and reconstruction level is 2. The tensor rank ‘r’ is considered to be from 1 to 10. The optimal values are achieved at rank 3. The ε= 10^{−3}, ρ=1.05, l = 50, and µ = 5 × 10^{−4} to balance the efficiency and the accuracy of our approach. The FRIQA metrics PSNR and SSIM are used to evaluate the approaches. The PSNR is evaluated as
where
The SSIM is evaluated with the luminance distortion, contrast distortion and loss of correlation.
Where
Here





24.29 
26.91 
32.53 

17.93 
20.57 
29.54 

19.96 
22.39 
31.38 
In
Similarly, the PSNR achieved as 31.62dB to the same missing ratio and rank. The NIQE is describes the naturalness in the recovered image, while calculating the NIQE score, reference image is not considered. The value is attained as 9.4 to the recovered image at 80% missing ratio with rank 5. Minimum values always indicate the good recovery of the image. Another NRIQA metric is used to evaluate the recovered data i.e., PIQE. The PIQE is also not required to have any reference to evaluate and the value is 27.33. The proposed approach has got the minimum values compared with existing approaches.
The proposed approach is able to recover the high dimensional data from very less observations efficiently. The lifting wavelets features inplace and custom design of filters are made the design simple, fast and accurate with tensor low rank assumptions. The tSVD helps in finding the minimum tensor rank. The proposed approach is evaluated with the FRIQA and NRIQA measures along existing methods. The PSNR and SSIM recorded as high values 31.62dB and 0.8918 respectively to recovered image at 80% corrupted observations. The NRIQA measures NIQE and PIQE values are reported as minimum values (9.4 and 27.33 respectively) at highly corrupted ratios. The missing ratio is increasing, the recovery is consistent in proposed approach compared with the stateoftheart methods.