Deep Plug-and-play Prior for Low-rank Tensor Completion

TL;DR

This paper introduces a novel tensor completion model combining low-rank tensor nuclear norm and deep denoising priors, effectively recovering multi-dimensional images with missing data, outperforming existing methods.

Contribution

It proposes a new regularized tensor completion approach integrating low-rank tensor nuclear norm with deep denoising priors within a PnP framework.

Findings

Achieves superior image recovery quality compared to competing methods.

Effectively preserves global structure and fine details in multi-dimensional images.

Demonstrates robustness across color images, videos, and multi-spectral images.

Abstract

Multi-dimensional images, such as color images and multi-spectral images, are highly correlated and contain abundant spatial and spectral information. However, real-world multi-dimensional images are usually corrupted by missing entries. By integrating deterministic low-rankness prior to the data-driven deep prior, we suggest a novel regularized tensor completion model for multi-dimensional image completion. In the objective function, we adopt the newly emerged tensor nuclear norm (TNN) to characterize the global low-rankness prior of the multi-dimensional images. We also formulate an implicit regularizer by plugging into a denoising neural network (termed as deep denoiser), which is convinced to express the deep image prior learned from a large number of natural images. The resulting model can be solved by the alternating directional method of multipliers algorithm under the plug-and-play (PnP) framework. Experimental results on color images, videos, and multi-spectral images demonstrate that the proposed method can recover both the global structure and fine details very well and achieve superior performance over competing methods in terms of quality metrics and visual effects.

Tables

Table 1. Table 1: Quantitative comparison of the results by SNN [ 18 ] , TNN [ 28 ] , TNN-3DTV [ 37 ] , and the proposed method on color images. The best and second best values are highlighted in bold and underlined, respectively.

Image	SR	PSNR					SSIM					Time (s)
		SNN	TNN	TNN-3DTV	DP3LRTC		SNN	TNN	TNN-3DTV	DP3LRTC		SNN	TNN	TNN-3DTV	DP3LRTC
Starfish $321\times 481\times 3$	10%	18.48	19.47	22.59	27.43		0.3617	0.3007	0.6345	0.8270		6	8	38	17
	20%	22.27	22.55	24.85	31.25		0.5476	0.5052	0.7519	0.9143		6	8	38	17
	30%	24.66	25.71	26.52	34.38		0.6883	0.6685	0.8217	0.9533		5	9	40	15
Airplane $512\times 512\times 3$	10%	20.91	21.94	23.46	28.48		0.6706	0.6623	0.8414	0.9471		10	17	71	54
	20%	24.81	25.17	25.83	30.57		0.8279	0.8242	0.9220	0.9709		12	19	76	33
	30%	27.20	27.90	28.45	31.36		0.9060	0.9014	0.9531	0.9806		10	18	75	30
Baboon $512\times 512\times 3$	10%	17.43	17.83	19.56	21.68		0.4077	0.3959	0.5835	0.7798		11	17	71	30
	20%	19.34	20.07	20.67	23.44		0.5974	0.6005	0.7271	0.8433		9	18	77	34
	30%	21.17	21.66	21.93	24.06		0.7308	0.7346	0.8083	0.9118		8	19	80	60
Fruits $512\times 512\times 3$	10%	20.73	20.72	24.81	31.47		0.6046	0.5646	0.8363	0.9362		10	17	72	54
	20%	24.23	24.23	27.31	34.90		0.7777	0.7510	0.9122	0.9621		10	18	75	57
	30%	26.90	26.99	29.22	36.48		0.8689	0.8541	0.9434	0.9723		9	19	77	44
Lena $512\times 512\times 3$	10%	21.43	21.89	25.96	30.93		0.6415	0.6177	0.8396	0.9241		10	17	71	54
	20%	24.98	25.68	28.41	32.92		0.8034	0.7888	0.9069	0.9423		10	19	77	34
	30%	27.71	28.06	30.07	33.46		0.8844	0.8719	0.9380	0.9575		10	18	75	54
Watch $768\times 1024\times 3$	10%	22.47	23.01	26.27	33.96		0.7128	0.7490	0.8863	0.9825		39	58	250	143
	20%	25.64	26.61	28.46	38.18		0.8641	0.8923	0.9466	0.9888		38	62	255	141
	30%	28.37	29.70	30.31	40.89		0.9332	0.9502	0.9709	0.9971		36	62	262	140
Opera $586\times 695\times 3$	10%	24.23	25.05	25.56	30.51		0.7499	0.7486	0.8075	0.9300		17	28	121	87
	20%	27.55	28.24	29.13	33.57		0.8649	0.8734	0.9132	0.9628		15	29	124	89
	30%	29.80	30.89	31.14	35.90		0.9188	0.9323	0.9447	0.9812		15	34	124	48
Water $768\times 1024\times 3$	10%	20.20	21.00	22.57	26.17		0.5860	0.6126	0.7756	0.9223		38	64	254	180
	20%	22.75	23.37	24.29	28.74		0.7790	0.6685	0.8822	0.9472		34	60	256	103
	30%	24.67	25.76	25.88	30.65		0.8767	0.8922	0.9302	0.9806		31	61	262	142
Average	10%	20.74	21.36	23.85	28.83		0.5919	0.5814	0.7756	0.9061		18	28	119	81
	20%	23.95	24.49	26.12	31.70		0.7578	0.7380	0.8703	0.9415		17	29	122	68
	30%	26.31	27.08	27.94	33.40		0.8509	0.8507	0.9138	0.9668		16	30	124	72

Table 2. Table 2: Quantitative comparison of the results by SNN [ 18 ] , TNN [ 28 ] , TNN-3DTV [ 37 ] , and the proposed method on videos. The best and second best values are highlighted in bold and underlined, respectively.

Video	SR	PSNR					SSIM					Time (s)
		SNN	TNN	TNN-3DTV	DP3LRTC		SNN	TNN	TNN-3DTV	DP3LRTC		SNN	TNN	TNN-3DTV	DP3LRTC
Akiyo $144\times 176\times 30$	5%	20.18	28.33	28.76	30.59		0.5976	0.8630	0.8949	0.9294		5	20	56	54
	10%	23.64	31.16	31.97	33.33		0.7340	0.9272	0.9452	0.9623		3	20	56	51
	20%	27.54	34.82	36.06	36.96		0.8642	0.9674	0.9777	0.9816		3	20	57	52
Suzie $144\times 176\times 30$	5%	20.84	26.37	27.39	29.37		0.5944	0.7203	0.7989	0.8373		4	19	56	51
	10%	24.40	28.41	29.27	31.80		0.7046	0.7976	0.8513	0.8899		3	18	56	51
	20%	28.12	31.14	31.95	34.75		0.8189	0.8739	0.9123	0.9358		2	18	57	52
Container $144\times 176\times 30$	5%	19.81	26.27	26.63	26.87		0.6446	0.8305	0.8611	0.8695		4	19	56	51
	10%	22.26	29.53	30.05	30.44		0.7413	0.9023	0.9272	0.9230		3	19	56	51
	20%	25.56	33.65	34.69	34.82		0.8495	0.9553	0.9677	0.9654		2	19	57	52
News $144\times 176\times 30$	5%	18.34	26.65	27.20	28.29		0.5610	0.8182	0.8738	0.9023		4	20	56	51
	10%	21.47	30.31	30.56	31.96		0.6955	0.9106	0.9285	0.9473		3	20	57	51
	20%	25.00	33.82	34.21	35.55		0.8256	0.9554	0.9665	0.9740		3	21	57	52
Bus $256\times 256\times 30$	5%	15.79	18.30	18.36	20.78		0.3300	0.3331	0.3973	0.5932		11	52	185	145
	10%	17.70	19.61	19.66	22.83		0.4174	0.4421	0.5083	0.7222		9	50	185	146
	20%	19.86	21.74	21.91	25.48		0.5560	0.5993	0.6778	0.8359		5	50	187	148
Average	5%	18.99	25.18	25.67	27.18		0.5455	0.7130	0.7652	0.8263		6	26	82	70
	10%	21.89	27.80	28.30	30.07		0.6586	0.7960	0.8321	0.8889		4	25	82	70
	20%	25.22	31.03	31.76	33.51		0.7828	0.8703	0.9004	0.9385		3	26	83	71

Table 3. Table 3: Quantitative comparison of the results by SNN [ 18 ] , TNN [ 28 ] , TNN-3DTV [ 37 ] , and the proposed method on MSIs. The best and second best values are highlighted in bold and underlined, respectively.

MSI	SR	PSNR					SSIM					Time (s)
		SNN	TNN	TNN-3DTV	DP3LRTC		SNN	TNN	TNN-3DTV	DP3LRTC		SNN	TNN	TNN-3DTV	DP3LRTC
Balloons $256\times 256\times 31$	5%	24.29	31.35	32.66	39.04		0.8232	0.8670	0.9381	0.9843		8	58	191	144
	10%	31.27	35.63	37.66	42.98		0.9242	0.9380	0.9735	0.9937		7	61	200	155
	20%	37.34	41.11	42.77	47.54		0.9737	0.9803	0.9904	0.9975		7	59	192	147
Beads $256\times 256\times 31$	5%	16.50	19.86	21.27	22.53		0.3259	0.4569	0.6506	0.6912		9	60	199	147
	10%	17.99	22.92	23.95	26.08		0.4191	0.6512	0.7907	0.8369		7	61	198	153
	20%	21.34	27.61	27.78	30.20		0.6404	0.8420	0.9024	0.9319		7	58	197	152
Cd $256\times 256\times 31$	5%	22.19	26.51	29.35	30.22		0.8497	0.8029	0.9251	0.9597		10	62	191	144
	10%	25.98	29.54	31.92	35.30		0.9018	0.8871	0.9550	0.9790		10	69	196	157
	20%	30.85	33.22	36.04	39.35		0.9498	0.9447	0.9791	0.9890		8	64	193	150
Clay $256\times 256\times 31$	5%	28.83	33.62	34.25	39.89		0.9224	0.9052	0.9487	0.9680		10	59	194	145
	10%	35.78	38.22	39.66	43.80		0.9726	0.9657	0.9819	0.9907		9	62	193	144
	20%	41.58	43.69	44.69	48.02		0.9895	0.9813	0.9890	0.9952		9	62	191	147
Face $256\times 256\times 31$	5%	25.64	32.32	32.70	36.82		0.8346	0.8981	0.9381	0.9663		9	61	198	146
	10%	30.51	36.50	37.52	39.66		0.9125	0.9548	0.9736	0.9829		8	60	192	145
	20%	36.20	41.33	42.00	42.96		0.9664	0.9842	0.9778	0.9916		8	61	192	147
Feathers $256\times 256\times 31$	5%	20.45	27.55	28.23	31.14		0.6565	0.7694	0.8712	0.9375		7	58	196	151
	10%	24.58	31.19	31.83	34.59		0.7925	0.8717	0.9308	0.9656		8	58	193	147
	20%	29.30	35.86	36.73	38.68		0.8995	0.9481	0.9726	0.9838		8	60	193	150
Flowers $256\times 256\times 31$	5%	20.47	26.75	27.90	30.34		0.6293	0.7153	0.8288	0.8894		10	60	196	148
	10%	24.88	30.36	31.37	33.70		0.7605	0.8357	0.9025	0.9419		7	58	192	147
	20%	29.49	35.33	36.25	37.66		0.8804	0.9342	0.9621	0.9745		7	60	194	149
Average	5%	22.62	28.28	29.48	32.85		0.7202	0.7735	0.8715	0.9138		9	60	195	146
	10%	27.28	32.05	33.42	36.59		0.8119	0.8720	0.9297	0.9558		8	61	195	150
	20%	32.30	36.88	38.04	40.63		0.9000	0.9450	0.9676	0.9805		8	60	193	149

Table 4. Table 4: Quantitative comparison of the results by different methods on MSIs. The best are highlighted in bold.

MSI	SR	PSNR			SSIM
		TNN	DPR	DP3LRTC	TNN	DPR	DP3LRTC
Average	5%	28.28	24.80	32.85	0.7735	0.8631	0.9138
	10%	32.05	30.64	36.59	0.8720	0.9111	0.9558
	20%	36.88	36.24	40.63	0.9443	0.9516	0.9805

Table 5. Table 5: Quantitative comparison of the results by different methods on color images. The best are highlighted in bold.

	SR	PSNR		SSIM
		DP3LRMC	DP3LRTC	DP3LRMC	DP3LRTC
Color images (average)	10%	23.89	28.83	0.7630	0.9061
	20%	26.76	31.70	0.8719	0.9415
	30%	29.07	33.40	0.9308	0.9668
MSIs (average)	5%	25.76	32.85	0.6948	0.9138
	10%	28.75	36.59	0.7863	0.9558
	20%	32.27	40.63	0.8706	0.9805

Table 6. Table 6: Quantitative comparison of the recovered results by different methods on color images. The best are highlighted in bold.

Color image	SR	PSNR			SSIM
		SNN-DPR	TNN-BM3D	DP3LRTC	SNN-DPR	TNN-BM3D	DP3LRTC
Average	10%	28.55	27.80	28.83	0.9013	0.8754	0.9061 \bigstrut[t]
	20%	31.24	30.91	31.70	0.9461	0.9382	0.9415
	30%	32.59	33.25	33.40	0.9597	0.9641	0.9668

Table 7. Table 7: Quantitative comparison of the results by different methods on color video. The best are highlighted in bold.

Color Video	SR	PSNR			SSIM
		SNN	WSTNN	DP3LRTC	SNN	WSTNN	DP3LRTC
Suzie	1%	7.44	23.26	24.96	0.0372	0.6967	0.7371
	5%	17.97	28.24	30.73	0.5148	0.8384	0.8891
	10%	21.22	30.94	33.47	0.6139	0.8943	0.9285

Equations

$$\arg \underset{\mathbf{X}}{\min }\text{rank}(\mathbf{X})\text{s.t.}{\mathcal{P}}_{\Omega }(\mathbf{X})={\mathcal{P}}_{\Omega }(\mathbf{O}),$$

$$\arg \underset{\mathcal{X}}{\min }\text{rank}(\mathcal{X})\text{s.t.}{\mathcal{P}}_{\Omega }(\mathcal{X})={\mathcal{P}}_{\Omega }(\mathcal{O}),$$

$$\arg \underset{\mathcal{X}}{\min }{\Vert \mathcal{X}\Vert }_{\text{TNN}}\text{s.t.}{\mathcal{P}}_{\Omega }(\mathcal{X})={\mathcal{P}}_{\Omega }(\mathcal{O}),$$

$$\arg \underset{\mathcal{X}}{\min }{\Vert \mathcal{X}\Vert }_{\text{TNN}}+\lambda \Phi (\mathcal{X})\text{s.t.}{\mathcal{P}}_{\Omega }(\mathcal{X})={\mathcal{P}}_{\Omega }(\mathcal{O}),$$

$$\mathcal{C}(i,j,:)=\sum _{k=1}^{n_2}\mathcal{B}(i,k,:)\star \mathcal{C}(k,j,:),$$

$$\mathcal{X}=\mathcal{U}\ast \mathcal{S}\ast {\mathcal{V}}^{\mathrm{H}},$$

$${\Vert \mathcal{X}\Vert }_{\text{TNN}}:=\sum _{i=1}^{n_3}\Vert {\bar{\mathbf{X}}}^{(i)}{\Vert }_{\ast }.$$

$${\text{prox}}_{\Phi }(y)=\text{arg}{\min }_x\left\{\Phi (x)+\frac{\rho }{2}{\Vert x-y\Vert }^2\right\},$$

$$\arg \underset{\mathcal{X}}{\min }{\Vert \mathcal{X}\Vert }_{\text{TNN}}+\lambda \Phi (\mathcal{X})\text{s.t.}{\mathcal{P}}_{\Omega }(\mathcal{X})={\mathcal{P}}_{\Omega }(\mathcal{O}).$$

$$1_{\mathbb{S}}(\mathcal{X})=\left\{\begin{array}[]{ll}{0,}&{\text{ if }\mathcal{X}\in\mathbb{S}},\\ {\infty,}&{\text{ otherwise, }}\end{array}\right.$$

$$\arg \underset{\mathcal{X},\mathcal{Y},\mathcal{Z}}{\min }$$

$$\mathcal{Y}=\mathcal{X},\mathcal{Z}=\mathcal{X}.$$

$$L={\Vert \mathcal{Y}\Vert }_{\text{TNN}}+\lambda \Phi (\mathcal{Z})+1_{\mathbb{S}}(\mathcal{X})+⟨\mathcal{X}-\mathcal{Y},{\Lambda }_1⟩+\frac{\beta }{2}{\Vert \mathcal{X}-\mathcal{Y}\Vert }_F^2+⟨\mathcal{X}-\mathcal{Z},{\Lambda }_2⟩+\frac{\beta }{2}{\Vert \mathcal{X}-\mathcal{Z}\Vert }_F^2,$$

$$\arg \underset{\mathcal{Y}}{\min }{\Vert \mathcal{Y}\Vert }_{\text{TNN}}+\frac{\beta }{2}{\Vert {\mathcal{X}}^l-\mathcal{Y}+{\Lambda }_1^l/\beta \Vert }_F^2.$$

$${\mathcal{Y}}^{l+1}=\mathcal{U}\ast \mathcal{D}\ast {\mathcal{V}}^{\mathrm{H}},$$

$$\arg \underset{\mathcal{Z}}{\min }\lambda \Phi (\mathcal{Z})+\frac{\beta }{2}{\Vert {\mathcal{X}}^l-\mathcal{Z}+{\Lambda }_2^l/\beta \Vert }_F^2.$$

$${\text{prox}}_{\Phi }(\mathcal{Z})=\arg \underset{\mathcal{Z}}{\min }\Phi (\mathcal{Z})+\frac{1}{2{\sigma }^2}{\Vert \mathcal{Z}-{\mathcal{X}}^l-{\Lambda }_2^l/\beta \Vert }_F^2.$$

$${\mathcal{Z}}^{l+1}=\mathtt{FFDNet}({\mathcal{X}}^l+{\Lambda }_2^l/\beta ,\sigma ).$$

$$\arg \underset{\mathcal{X}}{\min }{1}_{\mathbb{S}}(\mathcal{X})+\frac{\beta }{2}{\Vert \mathcal{X}-{\mathcal{Y}}^{l+1}+{\Lambda }_1^l/\beta \Vert }_F^2+\frac{\beta }{2}{\Vert \mathcal{X}-{\mathcal{Z}}^{l+1}+{\Lambda }_2^l/\beta \Vert }_F^2.$$

$$\{\begin{array}{l}\begin{array}{rl}& {\mathcal{P}}_{\Omega }({\mathcal{X}}^{l+1})={\mathcal{P}}_{\Omega }(\mathcal{O}),\\ & {\mathcal{P}}_{{\Omega }^c}({\mathcal{X}}^{l+1})={\mathcal{P}}_{{\Omega }^c}(\frac{\beta {\mathcal{Y}}^{l+1}+\beta {\mathcal{Z}}^{l+1}-{\Lambda }_1^l-{\Lambda }_2^l}{2\beta }),\end{array}\end{array}$$

$$\{\begin{array}{l}\begin{array}{rl}{\Lambda }_1^{l+1}=& {\Lambda }_1^l+\beta ({\mathcal{X}}^{l+1}-{\mathcal{Y}}^{l+1}),\\ {\Lambda }_2^{l+1}=& {\Lambda }_2^l+\beta ({\mathcal{X}}^{l+1}-{\mathcal{Z}}^{l+1}).\end{array}\end{array}$$

$$\text{RelCha}=\frac{{\Vert {\mathcal{X}}^{l+1}-{\mathcal{X}}^l\Vert }_F}{{\Vert {\mathcal{X}}^l\Vert }_F}.$$

Full text

††journal: Neurocomputing

1 Introduction

The image completion problem aims to estimate missing entries from the partially observed entries, which is a fundamental problem in low-level computer vision and computational imaging [1, 2, 3, 4, 5, 6, 7, 8].

By exploiting the redundancy of natural images, the image completion problem can be formulated as the following low-rank matrix completion (LRMC) model:

[TABLE]

where ${\mathbf{X}}$ is the underlying matrix, $\mathcal{\mathbf{O}}$ is the observed incomplete matrix, $\Omega$ is the index set corresponding to the observed entries, $\mathcal{P}_{\Omega}(\cdot)$ is the projection function that keeps the entries in $\Omega$ while setting others be zeros, and the elements of the observation $\mathbf{O}$ in the set $\Omega$ are given while the remaining elements are missing. For gray images, the low-rankness is characterized by the matrix rank or its convex envelope, i.e., the nuclear norm. For multi-dimensional images, the image elements are usually reordered into a matrix, which is known as matricization or unfolding.

However, matricization will inevitably destroy the intrinsic structure of multi-dimensional images. As the higher-order extension of the matrix, the tensor can provide a more natural and compact representation for multi-dimensional images. Thus we can formulate the image completion problem as the following low-rank tensor completion (LRTC) model:

[TABLE]

where $\mathcal{X}$ is the underlying tensor and $\mathcal{O}$ is the observed incomplete tensor (as shown in the top-left of Fig. 1). Here, we generally consider the element-wise sampling in the model (2), which can be extended to the structured sampling, e.g., inpainting [1] and demosaicing [9].

Different from the matrix case, there is not an exact (or unique) definition for tensor rank. Thus, how to define the tensor rank becomes an important task. Many research efforts have been devoted to this topic [10, 11, 12, 13], such as the CANDECOMP/PARAFAC (CP) rank and the Tucker rank. The CP rank is defined based on the CP decomposition, where an $n$-th order tensor is decomposed as the sum of rank-one tensors [14, 15], i.e., the outer product of $n$ vectors. The CP rank is defined as the minimal number of the rank-one tensors required to express the target tensor [16]. However, its related optimization is difficult, and even determining the CP rank of a given tensor is NP-hard [17]. The Tucker rank is based on the Tucker decomposition which decomposes a tensor into a core tensor multiplied by a matrix along each mode [18, 19, 20, 21]. The Tucker rank is defined as the vector consisting of the ranks of unfolding matrices. The Tucker rank has been considered in the LRTC problem by minimizing its convex surrogate, i.e., the sum of the nuclear norm (SNN) [18], or the non-convex surrogates[22, 23]. The top-right of Fig. 1 exhibits the recovered result by the SNN-based model. However, unfolding the tensor along each mode will also destroy the intrinsic structures of the tensor.

The tensor singular value decomposition (t-SVD), based on the tensor-tensor product (t-product), has been emerged as a powerful tool for preserving the intrinsic structures of the tensor [24, 25]. Although the t-SVD is originally suggested for third-order tensors, it has been extended to $n$-th order tensors ($n\geq 3$) [26, 27]. Based on the t-SVD, the corresponding multi-rank and tubal-rank have received considerable attention. The tensor nuclear norm (TNN) [28, 29, 30, 31, 32] is suggested as a convex surrogate of the multi-rank. The TNN-based LRTC model is given as follows:

[TABLE]

where $\mathcal{X}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$, $\left\|\mathcal{X}\right\|_{\text{TNN}}=\sum_{i=1}^{n_{3}}\left\|\bar{\mathbf{X}}^{(i)}\right\|_{\ast}$, $\bar{\mathbf{X}}^{(i)}$ is the $i$-th frontal slice of $\mathcal{\bar{X}}$, and $\bar{\mathcal{X}}$ is the tensor generated by performing discrete Fourier transformation (DFT) along the mode-3 fibers (tubes) of $\mathcal{X}$, i.e., $\mathcal{\bar{X}}={\tt{fft}}(\mathcal{X},[],3)$. The TNN regularizer has the advantage of preserving the global structure of multi-dimensional images. We can observe from Fig. 1 that the recovered result by the TNN-based method (bottom-left) is slightly better than that by the SNN-based method (top-right).

Although promoting the low-rankness of the underlying data has shown its effectiveness for tensor completion, these methods suffer from two drawbacks. First, many real-world multi-dimensional images contain abundant fine details. This makes these multi-dimensional images are not strictly low-rank in mathematics. Second, when the sampling rate is extremely low or structurally sampled, the rank of these observed entries is already low. The low-rank regularization is insufficient for a fancy recovery of the underlying tensor. This phenomena can be observed in Fig.  1, where results by the SNN-based model and TNN-based model are low-quality when the sampling rate is 5%.

As a compensation, many LRTC methods introduce additional regularizers, which express other types of image priors, for a better solution of this ill-posed inverse problem. For example, the total variation (TV) regularizer, framelet regularizer, and nonlocal regularizer have received much attention [33, 34, 35, 36]. In particular, the TV regularizer was incorporated into the TNN-based LRTC model by exploiting the local smoothness [37]. Recently, deep learning-based methods have been developed to learn data-driven priors from a large number of natural images [38, 39, 40] and have shown promising performance on extensive application. Therefore, we consider to leverage the power of deep learning-based methods.

However, from the literatures, convolutional neural networks (CNNs) are successfully applied for the recovery of 2-D gray level images or the color images. There is not a general CNN structure suitable for various types of multi-dimensional imaging data. Fortunately, the multi-dimensional imaging data can be viewed as consisting of the typical 2-D gray level images. For example, each frame of a video is indeed a 2-D image and each band of a multi-spectral image looks also like a 2-D image. Thus, we believe that the CNNs designed for 2-D images can be utilized in LRTC.

In this work, we attempt to exploit the respective strengths of the deterministic low-rankness prior and the data-driven deep image prior. By integrating the low-rankness prior with deep image prior expressed by an implicit regularizer, we suggest the novel LRTC model as follows:

[TABLE]

where $\Phi(\mathcal{X})$ is an implicit regularizer to plug in the denoising CNNs (termed as deep denoisers). In (4), the two regularizers are organically combined and benefit from each other. On the one hand, the implicit regularizer can bring in deep image priors to each 2-D spatial slice of the multi-dimensional imaging data, well characterizing the fine details which can hardly be captured by the TNN regularizer. On the other hand, TNN can enforce the inner global correlations of the multi-dimensional imaging data, avoiding the lackness of lack inherent relations that the slices, which are separately handled by the CNN, . $\left\|\mathcal{X}\right\|_{\text{TNN}}$ and $\Phi(\mathcal{X})$ respectively regularize the coarse structure and the fine details, being complementary to each other. To efficiently solve the proposed model, we develop the alternating direction method of multipliers (ADMM) [41] under the highly flexible Plug-and-Play (PnP) framework [42, 43, 44], which allows us to plug in the stat-of-the-art deep denoisers. We term our method as deep plug-and-play prior for low-rank tensor completion (DP3LRTC). From the bottom-right of Fig. 1, we can observe that the deterministic low-rankness prior and data-driven deep image prior contribute to the superior performance of the proposed DP3LRTC.

The rest of this paper is organized as follows. Section 2 presents minimal preliminaries necessary for the subsequent discussion. Section 3 gives the corresponding solving algorithm to tackle the proposed model. Section 4 gives the experimental results and discusses details about the DP3LRTC. Section 5 concludes this paper.

2 Preliminaries

In this section, we introduce minimal and necessary preliminaries of the modular parts, i.e., TNN and PnP, for the subsequent discussion.

2.1 Notations

In this subsection, we give the basic notations and briefly introduce some definitions. We denote vectors as bold lowercase letters (e.g., $\mathbf{x}$), matrices as uppercase letters (e.g., $\mathbf{X}$), and tensors as calligraphic letters (e.g., $\mathcal{X}$). For a third-order tensor $\mathcal{X}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$, with the MATLAB notation, we denote its $(i,j,k)$-th element as $\mathcal{X}(i,j,k)$ or $\mathcal{X}_{i,j,k}$. Its $(i,j)$-th mode-1, mode-2, and mode-3 fibers are denoted as $\mathcal{X}(:,i,j)$, $\mathcal{X}(i,:,j)$, and $\mathcal{X}(i,j,:)$, respectively. We use $\mathcal{X}(i,:,:)$, $\mathcal{X}(:,i,:)$, and $\mathcal{X}(:,:,i)$ to denote the $i$-th horizontal, lateral, and frontal slices of $\mathcal{X}$, respectively. More compactly, we also use $\mathbf{X}^{(i)}$ to represent the $i$-th frontal slices $\mathcal{X}(:,:,i)$. The Frobenius norm of $\mathcal{X}$ is defined as $\|\mathcal{X}\|_{F}:=\sqrt{\sum_{i,j,k}|\mathcal{X}(i,j,k)|^{2}}$.

2.2 T-SVD and TNN

Recently, Kilmer have introduced the novel tensor-tensor product as follows:

Definition 1** (t-product [25])**

Let $\mathcal{A}$ be $n_{1}\times n_{2}\times n_{3}$ and $\mathcal{B}$ be $n_{2}\times n_{4}\times n_{3}$. Then the tensor-tensor product (t-product) is the $n_{1}\times n_{4}\times n_{3}$ tensor $\mathcal{C}=\mathcal{A}*\mathcal{B}$ whose $(i,j)$th tube is given by

[TABLE]

where “$\star$" denotes circular convolution of two vectors.**

Definition 2** (special tensors [25])**

The conjugate transpose of a third-order tensor $\mathcal{X}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$, denote as $\mathcal{X}^{\mathrm{H}}$, is the tensor obtained by conjugate transposing each of the frontal slices and then reversing the order of transposed frontal slices 2 through $n_{3}$.* The identity tensor $\mathcal{I}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$ is the tensor whose first frontal slice is the identity matrix, and other frontal slices are all zeros. A third-order tensor $\mathcal{Q}$ is orthogonal if $\mathcal{Q}\ast\mathcal{Q}^{\mathrm{H}}=\mathcal{Q}^{\mathrm{H}}\ast\mathcal{Q}=\mathcal{I}.$ A third-order tensor $\mathcal{S}$ is f-diagonal if each of its frontal slices is a diagonal matrix.*

Based on t-product, the tensor singular value decomposition (t-SVD) is suggested.

Theorem 1** (t-SVD [25])**

Let $\mathcal{X}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$ be a third-order tensor, then it can be decomposed as

[TABLE]

where $\mathcal{U}\in\mathbb{R}^{n_{1}\times n_{1}\times n_{3}}$ and $\mathcal{V}\in\mathbb{R}^{n_{2}\times n_{2}\times n_{3}}$ are the orthogonal tensors, and $\mathcal{S}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$ is a f-diagonal tensor whose frontal slices are diagonal matrices.**

The t-SVD can be efficiently obtained by computing a series of matrix SVDs in the Fourier domain; see Algorithm 1 for more details.

Based on t-SVD, we have the corresponding definitions of tensor multi-rank and tubal-rank.

Definition 3** (tensor tubal-rank and multi-rank [28])**

Let $\mathcal{X}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$ be a third-order tensor, the tensor multi-rank, denoted as $\text{rank}_{\text{m}}(\mathcal{X})\in\mathbb{R}^{n_{3}}$, is a vector whose $i$-th element is the rank of the $i$-th frontal slice of $\bar{\mathcal{X}}$, where $\bar{\mathcal{X}}=\mathrm{\tt fft}(\mathcal{X},[],3)$. The tubal-rank of $\mathcal{X}$, denote as $\text{rank}_{\text{t}}(\mathcal{X})$, is defined as the number of non-zero tubes of $\mathcal{S}$, where $\mathcal{X}=\mathcal{U}\ast\mathcal{S}\ast\mathcal{V}^{\mathrm{H}}$.**

The relationship between the tubal-rank and multi-rank is \text{rank}_{\text{t}}(\mathcal{X})=\max\big{(}\text{rank}_{\text{m}}(\mathcal{X})\big{)}. The convex surrogate of the multi-rank is suggested as follows:

Definition 4** (TNN [28])**

The tensor nuclear norm of a tensor $\mathcal{X}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$, denoted as $\|\mathcal{X}\|_{\text{TNN}}$, is defined as the sum of singular values of all the frontal slices of $\bar{\mathcal{X}}$, i.e.,**

[TABLE]

where $\bar{\mathbf{X}}^{(i)}$ is the $i$-th frontal slice of $\mathcal{\bar{X}}$, and $\bar{\mathcal{X}}=\mathrm{\tt fft}(\mathcal{X},[],3)$.**

2.3 Plug-and-Play (PnP) Framework

PnP is a highly flexible framework [42, 43, 44] that leverages the power of the state-of-the-art denoisers in the ADMM or other proximal algorithms. After variable splitting technique, the optimization problem is decoupled into easier subproblems, one of which is the proximal operator of regularization. The proximal operator of regularization $\textrm{prox}_{\Phi}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is defined as

[TABLE]

which maps the input $y$ to the minimizer of (8). Under the PnP framework, the proximal operator of regularization is replaced by the denoising algorithm (termed as denoiser), which maps the noisy image to the clean image. Here the denoiser serves as an implicit regularizer to express the deep image prior. Regularization by Denoising (RED) [45, 46], which uses the denoising engine in defining the regularization of the inverse problem, is also suggested as an alternative framework.

The state-of-the-art denoisers can be flexibly plugged in as a modular part under the PnP framework. In the past decades, the hand-crafted denoisers, which exploit the deterministic priors of the nature images, have been dominantly used, e.g., TV denoiser [47], framelet denoiser [48], BM3D denoiser [49], WNNM denoiser [50], and ITS denoiser [51]. Recently, deep learning-based denoisers have been rapidly developed to learn data-driven image priors from a large number of natural images [38, 39, 40] and have shown promising performance. The PnP framework, which integrates modern denoiser, has shown great empirical success on diverse applications including denoising [52, 1], restoration[53], tomography, super-resolution, and fusion.

However, even the most basic question of convergence is still a challenging problem. The convergence of the PnP framework has been considered from the fixed-point viewpoint in [44]. Very recently, the convergence of the PnP framework with properly trained denoisers has been considered in [54].

3 The Proposed Model and Algorithm

As previously discussed, given an incomplete observation $\mathcal{O}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$ with its support $\Omega$ (namely, the index set corresponding to the observed entries), our DP3LRTC model is formulated as:

[TABLE]

In our model, $\Phi(\mathcal{X})$ is an implicit reuglarizer, whose related subproblem will be solved by a CNN within the PnP framework. We make some remarks for our model. When using CNNs, a straightforward idea is to design a network structure and conduct end-to-end training. However, this may lacks flexibility and generalization ability for different kinds of data and different kinds of degradation, which is corresponding to different sampling situations in the tensor completion problem. Also, training a 3-D CNN also costs much more time than training a 2-D CNN. Therefore, in our model, we directly plug in a 2-D denoising CNN, which is trained on the natural image dataset, to solve the $\Phi(\mathcal{X})$ related subproblem. Specifically, the denoising CNN hereof acts on the spatial slices of the multi-dimensional data, for example bands of MSIs and frames of videos, to preserve the spatial details. Meanwhile, the whole structure of the data is captured by the low-rank term $\left\|\mathcal{X}\right\|_{\text{TNN}}$, which would dominate the correlation along the temporal or spectral direction. Thus, our model can deal with different types of multi-dimensional data and various sampling situations in a flexible and economic manner.

Then, we develop the ADMM [41] to tackle the minimization problem (11) under the PnP framework. First, we denote the indicator function as

[TABLE]

where $\mathbb{S}:=\left\{\mathcal{X}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}},\mathcal{P}_{\Omega}(\mathcal{X})=\mathcal{P}_{\Omega}(\mathcal{O})\right\}$ and introduce two auxiliary variables $\mathcal{Y}$ and $\mathcal{Z}$. Then, we reformulate (4) as a constrained optimization problem, i.e.,

[TABLE]

The augmented Lagrangian function of (11) is

[TABLE]

where $\Lambda_{1}$ and $\Lambda_{2}$ are the Lagrangian multipliers, and $\beta$ is a nonnegative penalty parameter.

According to the ADMM framework, the solution of (4) can be found by solving a sequence of subproblems.

In Step 1, we need to solve the $[\mathcal{Y},\mathcal{Z}]$-subproblem. Since the variables $\mathcal{Y}$ and $\mathcal{Z}$ are decoupled, their optimal solutions can be calculated separately.

1. The $\mathcal{Y}$-subproblem is

[TABLE]

Let the t-SVD of $(\mathcal{X}^{l}+\Lambda_{1}^{l}/\beta)$ be $\mathcal{U}\ast\mathcal{S}\ast\mathcal{V}^{\mathrm{H}}$, the closed-form solution of the $\mathcal{Y}$-subproblem can be exactly calculated via singular value thresholding (SVT) [55, 56] as

[TABLE]

where $\mathcal{D}$ is an $f$-diagonal tensor whose each frontal slice in the Fourier domain is $\bar{\mathcal{D}}(i,i,k)=\max\{\bar{\mathcal{S}}(i,i,k)-\frac{1}{\beta},0\}$. The complexity of computing $\mathcal{Y}$ is ${O}(n_{3}\operatorname{min}({n_{1}}^{2}n_{2},n_{1}{n_{2}}^{2})+n_{1}n_{2}n_{3}\operatorname{log}(n_{3}))$.

2. The $\mathcal{Z}$-subproblem is

[TABLE]

Let $\sigma=\sqrt{\lambda/\beta}$, then, (15) can be rewritten as

[TABLE]

Under the PnP framework, the proximal operator of regularization $\textrm{prox}_{\Phi}:\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\rightarrow\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$ is replaced by the deep learning-based denoiser, which maps the noisy image to the clean image. Here we consider the fast and flexible denoising convolutional neural network, namely FFDNet [39], as the deep denoiser. Feeding $\mathcal{X}^{l}+\Lambda_{2}^{l}/\beta$ into the denoiser FFDNet, we obtain the solution of the $\mathcal{Z}$-subproblem as

[TABLE]

In FFDNet the parameter $\sigma$ is related to the noise level, but here $\sigma$ is related to the error level between the estimation and ground truth. Here we consider the pre-trained FFDNet denoiser which belongs to the CNN denoiser family. We feed the color images into the the FFDNet denoiser trained for color images and the spatial slices of gray videos and MSIs into the the FFDNet denoiser trained for gray images. The complexity of computing $\mathcal{Z}$ is ${O}(n_{1}n_{2}n_{3}n_{l}n_{f}n_{k})$, where $n_{l}$ is the number of layers, $n_{f}$ is the number of features, and $n_{k}$ is the number of kernel pixels.

In Step 2, we need to solve the $\mathcal{X}$-subproblem:

[TABLE]

By minimizing the $\mathcal{X}$-subproblem, we have $1_{\mathbb{S}}(\mathcal{X})=0$, i.e., $\mathcal{X}\in\mathbb{S}$. Thus, the closed-form solution of $\mathcal{X}$-subproblem is given as follows:

[TABLE]

where $\Omega^{c}$ denotes the complementary set of $\Omega$. The complexity of computing $\mathcal{X}$ is ${O}(n_{1}n_{2}n_{3})$.

In Step 3, we update the multipliers $\Lambda_{1}$ and $\Lambda_{2}$ as follows:

[TABLE]

The complexity of updating $\Lambda_{1}$ and $\Lambda_{2}$ is ${O}(n_{1}n_{2}n_{3})$.

Finally, the algorithm is summarized in Algorithm 2. The computational complexity is ${O}(n_{3}\operatorname{min}({n_{1}}^{2}n_{2},n_{1}{n_{2}}^{2})+n_{1}n_{2}n_{3}\operatorname{log}(n_{3})+n_{1}n_{2}n_{3}n_{l}n_{f}n_{k})$. The main part of the ADMM algorithm is implemented on CPU while the FFDNet is evaluated on GPU.

4 Numerical Experiments

In this section, the performance of DP3LRTC will be comprehensively evaluated by experiments on color images, videos, and multi-spectral images (MSIs). Although the FFDNet denoiser is trained for color images, the DP3LRTC can be well generalized for gray videos and MSIs. The DP3LRTC is compared with the baseline methods: SNN [18], TNN [28], and TNN-3DTV [37].

The peak signal to noise ratio (PSNR) and the structural similarity index (SSIM) [57] are chosen as the quality metrics. We report the mean PSNR and mean SSIM of all bands (or frames). The relative change (RelCha) is adopted as the stopping criterion of all methods, which is defined as

[TABLE]

In all experiments, when RelCha is smaller than the tolerance $10^{-4}$, we stop the iterations. All parameters involved in different methods are manually selected from a candidate set $\{10^{-4},10^{-3},10^{-2},10^{-1},1,10,10^{2}\}$ to obtain the highest PSNR value. In this paper, we mainly consider the element-wisely sampled, i.e., entries are randomly sampled for each band (or frame). For color images, we consider the element-wisely sampling, the tubal sampling, i.e., entries are sampled along all RGB channels for the spatial pixel, and the Bayer sampling, i.e., each $2\times 2\times 3$ cell contains two green pixels, one blue pixel, and one red pixel. All the experiments are implemented on Windows 10 and Matlab (R2018a) with an Intel(R) Core(TM) i5-4590 CPU at 3.30 GHz, 16 GB RAM, and NVIDIA GeForce GTX 1060 6GB.

4.1 Color Image Completion

In this subsection, 8 color images222Available at http://sipi.usc.edu/database/database.php. are selected for the color images completion experiments. Each image is rescaled to $[0,1]$, and the observed elements are element-wisely sampled with different sampling rates (SR). For color images, we use the FFDNet network trained for color images as the PnP denoiser.

Tab. 1 presents the PSNR and SSIM values of recovered results by different methods. From Tab. 6, it can be observed that the TNN-based method obtains better performance than SNN. Since TNN-3DTV introduces additional prior knowledge, it outperforms the TNN-based method and achieves the second-best PSNR and SSIM values. The proposed DP3LRTC obtains the highest PSNR and SSIM values. Meanwhile, it is noteworthy that the average margins between the results by DP3LRTC and TNN-3DTV are more than 3.9 dB in PSNR and 0.05 in SSIM.

For visualization, we exhibit the recovered images with the sampling rate $30\%$ in Fig. 2. From the images illustrated in Fig. 2, we can find that the SNN-based method and TNN-based method are able to recover the general structure of the color images, while the details are unclear and results remain artifacts, which can be seen in the enlarged red boxes. The results by TNN-3DTV are relatively better. However, we can observe the over-smoothed phenomena in the enlarged areas. The recovered results by the proposed method are of high quality, being highly clear and very similar to the original images.

Now, we consider the structured samplings, i.e., inpainting and demosaicing, which are more difficult than the element-wise sampling. In image inpainting and demosaicing, we consider the tubal sampling (i.e., entries are sampled along all channels for a spatial location) and the Bayer pattern sampling (i.e., each two-by-two cell contains two green, one blue, and one red), respectively. In inpainting, there are three different kinds of masks: letters, graffiti, and grid, respectively. Fig. 3 shows the inpainting and demosaicing results by different methods on color images: Fruits, Lena, Sailboat, and Baboon. It is easy to observe that the proposed method is superior to competing methods in inpainting and demosaicing experiments.

4.2 Video Completion

In this subsection, 5 grayscale videos333Available at http://trace.eas.asu.edu/yuv/. with different sizes are selected. For gray videos, we feed the spatial slices into the FFDNet denoiser trained for gray images as the PnP denoiser. Tab. 2 lists the PSNR and SSIM values of the recovered video results by different methods with different sampling rates. It is easy to observe that DP3LRTC obtains the results with the highest performance evaluation indices. For visualization, we show the 20-th frame of the recovered videos with the sampling rate $20\%$ in Fig. 4. We can see that the results recovered by DP3LRTC are of higher quality than the results obtained by compared methods.

4.3 MSI Completion

In this subsection, we test 7 MSIs from the CAVE dataset444Available at http://www.cs.columbia.edu/CAVE/databases/multispectral/. which are of size $256\times 256\times 31$ with the wavelengths in the range of $400\sim 700$ nm at an interval of 10 nm. For MSIs, we feed the spatial slices into the FFDNet denoiser trained for gray images as the PnP denoiser.

Tab. 4 exhibits the PSNR and SSIM values of all the results by different methods. The DP3LRTC achieves the highest PSNR and SSIM values, while the TNN-3DTV gets the second best. Fig. 5 displays the recovered MSIs (pseudo-color images composed of the 1st, 2nd, and 31-st bands) by different methods with the sampling rate $10\%$. From Fig. 5, it can be observed that the results recovered by DP3LRTC are of the highest quality. In Fig. 6, we display one selected tube of results recovered by SNN, TNN, TNN-3DTV, and DP3LRTC on MSI Balloons with the sampling rate 10%. We can observe that the selected tube of the MSI recovered by DP3LRTC is closer to the ground truth as compared with those by SNN, TNN, and TNN-3DTV.

4.4 Discussions

4.4.1 Convergence Behaviour

The numerical experiments have shown the great empirical success of DP3LRTC. However, it is still an open question whether the ADMM algorithm under the PnP framework has a good convergence behavior. In Fig. 7, we displayed the relative change curves of the ADMM algorithm (in the logarithmic scale) with respect to the iteration on different videos. We can evidently observe the numerical convergence of the ADMM algorithm.

4.4.2 Contributions of Different Terms

To evaluate the distinct contributions of the low-rankness prior and deep image prior, we test the performance of 3 methods: TNN (only the TNN regularizer), DPR (only the implicit deep regularizer), and DP3LRTC (both the TNN and implicit deep regularizers). In Fig. 8, we exhibit the recovered results by TNN, DPR, and DP3LRTC on color image Airplane and MSI Flowers, together with PSNR values. In Fig. 8, the results by DPR are clearer than those by TNN. This shows the remarkable ability of CNN denoiser to capture the spatial structures. Meanwhile, we can observe that the performance of DPR is comparable to DP3LRTC on color images, since that color images only consists of 3 color channels. However, from the enlarged areas, we can see obvious artifact in the result by DPR. When reconstructing the MSI, which consists of 31 spectral bands, TNN and DP3LRTC outperform DPR. This shows that the TNN regularizer is good at utilizing the spectral redundancy. Therefore, we can confirm that two regularizers are indispensable to DP3LRTC and benefit from each other. Together, they contribute to the superior performance of the proposed DP3LRTC.

To take a little more in-depth look at the mechanism of each term, we conduct experiments on all 8 MSIs in Sec. 4.3 with different sampling rates. The averaged PSNR and SSIM values are reported in Tab. 4. Generally, the metric SSIM evaluates the local structure similarity [57] and the metric PSNR reflects the overall recovery quality. The PSNR values of results by TNN are higher than those by DPR, while DPR achieves higher SSIM values. This phenomenon also reveals that the TNN regularizer is more suitable for characterising the global information and the CNN denoiser is appropriate for recovering the fine structures. As the sampling rate increases, the SSIM margin between TNN and DP3LRTC decreases from 0.140 to 0.036 while the PSNR margin is always around 4dB. This shows that in the low sampling cases, recovering the local structure becomes more difficult than capturing the global information.

Moreover, we study the influence of parameters $\beta$ and $\sigma$. In Fig. 9, we show the PSNR and SSIM values of the results recovered by DP3LRTC on color image Starfish with respect to $\beta$ and $\sigma$. We can see that DP3LRTC delivers the best performance when $\beta\in\{10^{-1},1\}$ and $\sigma\in\{10^{-2},10^{-1},1\}$.

4.4.3 Low-matrix-rank and Low-tensor-rank

To illustrate the superiority of the tensor-rank-based model, we reported the quantitative comparison of the recovered results by our DP3LRTC and DP3LRMC, in which the TNN is replaced by the matrix nuclear norm of the matricization of multi-dimensional images (where the rows represent the vectorized spatial images and the columns the spectral channels), on color images and multi-spectral images (MSIs) in Tabs. 5. The setup of experiments on color images and the multispectral images are the same as Sec. 4.1 and Sec. 4.3, respectively. Similarly, we only report the average indexes of the results. For better visual inspection, we further displayed the recovered results by DP3LRMC and DP3LRTC in Fig. 10. From Tab. 5 and Fig. 10, we confirmed that the tensor-rank-based model DP3LRTC is evidently better than the matrix-rank-based model DP3LRMC.

4.4.4 Beyond TNN, FFDNet, and Third-order Tensors

Actually, in this work, we suggested a user-friendly LRTC framework which integrates low-rankness prior and the denoising prior. Tab. 6 shows the quantitative comparison of the results obtained by different combinations of low-rankness metrics and denoisers. The setup is the same as the color image completion with random missing pixels in Sec. 4.1. For saving space, we only report the average indexes of the results of 8 color images. In Tab. 6, “SNN-DPR” denotes the combination of the sum of the nuclear norm (SNN) and the FFDNet denoiser, while “TNN-BM3D” indicates that the tensor nuclear norm (TNN) is combined with the BM3D denoiser [49]. From Tab. 6, we can observe that the combination of the tensor nuclear norm and the FFDNet denoiser is best among all combinations. Thus, we choose the the combination of the tensor nuclear norm and the FFDNet denoiser as the representative example of DP3LRTC framework.

Next, we extend our DP3LRTC framework to high-order tensors ($n\geq 3$) by replacing the TNN regularizer with its high-order extension, i.e., the WSTNN in [27]. We conduct the experiment on a fourth-order tensor, the color video “Suzie” 555Available at http://trace.eas.asu.edu/yuv/ of size $144\times 176\times 3\times 100$. For color videos, we use the FFDNet network trained on color images as the PnP denoiser. When solving the $\mathcal{Z}$ subproblem in (16), we feed each frame of size $144\times 176\times 3$ into the CNN denoiser in sequence. The quantitative comparison of the recovered results by SNN, WSTNN, and DP3LRTC on color video Suzie is reported in Tab. 7. For better visual inspection, we further displayed the recovered results by SNN, WSTNN, and DP3LRTC in Fig. 11. We can observe that DP3LRTC can be extended to handle high-order tensors with the help of WSTNN. Also, we can generalize our method for high-order tensor via other low-rankness metrics, such as the “SNN-DPR”, and obtain remarkable performances.

5 Conclusions

In this paper, we proposed a hybrid tensor completion model, in which the TNN regularizer is utilized to catch the global information and a data-driven implicit regularizer is used to express the local information. The proposed model simultaneously combines the model-based optimization method with the CNN-based method, in consideration of the global structure and fine detail preservation. An efficient ADMM is developed to solve the proposed model. Numerical experiments on different types of multi-dimensional images illustrate the superiority of the proposed method on the tensor completion problem.