# Image Completion with Hybrid Interpolation in Tensor Representation
## 4 Results
### Setup  
The incomplete images were obtained by removing some entries from tensors representing the original images. The following test cases were analyzed:  
* **A**: 90% uniformly distributed random missing tensor fibers in its third mode (color), which corresponds to 90% missing pixels (\"dead pixels\"),
* **B**: 95% uniformly distributed random missing tensor entries (\"disturbed pixels\"),
* **C**: 200 uniformly distributed random missing circles--created in the same way as in the first case, but the disturbances are circles with a random radius not exceeding 10 pixels,
* **D**: resolution up-scaling--an original image was down-sampled twice by removing the pixels according to a regular grid mask with edges equal to 1 pixel. The aim was to recover the missing pixels on the edges. We compared the proposed methods with the following: FAN (filtering by adaptive normalization) and EFAN (efficient filtering by adaptive normalization) [60], SmPC-QV (smooth PARAFAC tensor completion with quadratic variation) [38], LRTV (low-rank total-variation) [61], TMAC-inc (low-rank tensor completion by parallel matrix factorization with the rank-increasing) [62], C-SALSA (constrained split augmented Lagrangian shrinkage algorithm) [63], fALS (filtered alternating least-squares) [64], and KA-TT (ket augmentation tensor train) [65]. FAN and EFAN are based on adaptive Gaussian low-pass filtration. SmPC-QV performs low-rank tensor completion with smoothness-penalized CP decomposition and gradually increasing CP rank. LRTV accomplishes low-rank matrix completion using total variation regularization. TMAC-inc also belongs to a family of low-rank tensor completion, and in this approach, an incomplete tensor is unfolded with respect to all modes, and the resulting matrices are completed by applying low-rank matrix factorizations together with the adaptive rank-adjusting strategy. C-SALSA performs image completion using the variable splitting approach to solve an LS image reconstruction problem with a strong nonsmooth regularizer. fALS and KA-TT combine low-pass filtration with standard Tucker decomposition and tensor train models, respectively. The proposed algorithm is referred to as Tensorial Interpolation for Image Completion (TI-IC), and it is presented in Algorithm 1. It combines two strategies: RBF-interpolation with an exponential function, and multivariate polynomial regression. To emphasize the importance of both terms in the model (16), we also present the results obtained separately for each of them, using the same partitioning strategy in each case. The TI-IC algorithm with only the exponential term is referred to as TI-IC(Exp). When only the polynomial regression is used, TI-IC will be denoted as TI-IC(Poly). TI-IC is flexible with respect to the choice of the distance function \(d^{(n)}(\cdot\cdot\cdot)\), degrees of the interpolation polynomials, partitioning, and overlapping rates. Since \(\{i_{n},j_{n}\}\) lie on a line, \(d^{(n)}(i_{n},j_{n})=|i_{n}-j_{n}|\) seems to be the best choice. Factor matrices \(\{\mathbf{P}^{(n)}\}\) were determined by quadratic polynomials, hence \(\forall n:R_{n}=3\). Higher-order polynomials result in ill-conditioning of the system  
Figure 1: Original images: [PERSON], Lena, Peppers, and Monarch (from left to right). matrix in (19) and do not noticeably improve the performance. The partitioning and overlapping rates were set experimentally to \([S_{1},S_{2},S_{3}]=[32,\ 33,\ 1]\) and \([\theta_{1},\theta_{2},\theta_{3}]=[33.33,\ 33.33,\ 0]\). As the resolution of \(\mathcal{M}\) is \(512\times 512\), the overlapping amounts to 5 pixels across the first and second mode for each subtensor \(\hat{\mathcal{Y}}\in\mathbb{R}^{16\times 16\times 3}\). For larger subtensors, computational time increased considerably, and we did not observe a noticeable improvement in the quality of recovered images. For smaller subtensors, the performance decreased.