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Robust Low Transformed Multi-Rank Tensor Methods for Image Alignment

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Abstract

Aligning a group of linearly correlated images is an important task in computer vision. In this paper, we propose a combination of transformed tensor nuclear norm and tensor \(\ell _1\) norm to deal with this image alignment problem, where the observed images, stacked into a third-order tensor, are deformed by unknown domain transformations and corrupted by sparse noise like impulse noise, partial occlusions, and illumination variation. The key advantage of the proposed method is that both spatial correlation and images variation can be captured by the use of transformed tensor nuclear norm. We show that when the underlying of correlated images is a low multi-rank tensor, an upper error bound of the estimator of the proposed model can be established and this bound can be better than the previous result. Besides the proposed convex transformed tensor model, the method can be further studied by incorporating nonconvex functions in the transformed tensor nuclear norm and the sparsity norm. Both the proposed convex and nonconvex optimization models are solved by generalized Gauss–Newton algorithms. Also the global convergence of the numerical methods for solving the subproblems of convex and nonconvex optimization models can be provided under very mild conditions. Extensive numerical experiments on real images with misalignment and sparse corruptions demonstrate the performance of our proposed methods is better than that of several state-of-the-art methods in terms of accuracy and efficiency.

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Notes

  1. http://people.eecs.berkeley.edu/~yima/matrix-rank/rasl.html

  2. http://sites.google.com/site/hejunzz/t-grasta

  3. http://www2.ece.ohio-state.edu/~aleix/ARdatabase.html

  4. http://yann.lecun.com/exdb/mnist/

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Acknowledgements

We would like to thank Professor Xiaoqin Zhang and Dr. Xiai Chen for sending us the codes of [69] and [12], respectively. We would also like to thank Professor Di Wang for helpful discussions on the domain transformations and sharing the code of [70]. Thanks also go to the anonymous referees for their constructive suggestions to improve the presentation of the paper.

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Authors and Affiliations

  1. School of Mathematics, Hunan University, Changsha, 410082, China

    Duo Qiu & Minru Bai

  2. Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong

    Michael K. Ng

  3. School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China

    Xiongjun Zhang

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  1. Duo Qiu
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Correspondence to Minru Bai.

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M. Bai: The research of this author was partially supported by the National Natural Science Foundation of China under grant 11971159. M. K. Ng: The research of this author was partially supported by the HKRGC GRF 12306616, 12200317, 12300218, 12300519 and 17201020, and HKU Grant 104005583. X. Zhang: The research of this author was partially supported by the National Natural Science Foundation of China under grants 11801206, 11871025, Hubei Provincial Natural Science Foundation of China under grant 2018CFB105, and Fundamental Research Funds for the Central Universities under grant CCNU19ZN017.

Appendix

Appendix

We recall the definitions of subdifferential of a nonconvex function and the Kurdyka–Łojasiewicz (KL) property. The most fundamental works on KL property are due to Łojasiewicz [39] and Kurdyka [31].

Consider a function \(f:{\mathbb {R}}^n\rightarrow (-\infty , +\infty ]\) and a point \({\mathbf {x}}\) with \(f({\mathbf {x}})\) finite. The subdifferential of f at \({\mathbf {x}}\), denoted by \(\partial f({\mathbf {x}})\), is defined as [53, Definition 8.3]

$$\begin{aligned} \partial f({\mathbf {x}}):=\left\{ {\mathbf {v}}\in {\mathbb {R}}^n:\ \exists \ {\mathbf {x}}^k\rightarrow {\mathbf {x}}, f({\mathbf {x}}^k)\rightarrow f({\mathbf {x}}), {\mathbf {v}}^k\rightarrow {\mathbf {v}} \ \ \text {with} \ \ {\mathbf {v}}^k\in {\widehat{\partial }} f({\mathbf {x}}^k) \ \ \text {as} \ \ k\rightarrow +\infty \right\} , \end{aligned}$$

where \({\widehat{\partial }} f({\mathbf {x}})\) denotes the Fréchet subdifferential of f at \({\mathbf {x}}\in \text {dom}(f)\), and is the set of all \({\mathbf {v}}\) satisfying

$$\begin{aligned} \liminf _{\begin{array}{c} {\mathbf {z}}\rightarrow {\mathbf {x}},\\ {\mathbf {z}}\ne {\mathbf {x}} \end{array}} \frac{f({\mathbf {z}})-f({\mathbf {x}})-\langle {\mathbf {v}},{\mathbf {z}}-{\mathbf {x}} \rangle }{\Vert {\mathbf {z}}-{\mathbf {x}}\Vert }\ge 0. \end{aligned}$$

Here the effective domain of a function f is defined as \(\text{ dom }(f):=\{x:f(x)<+\infty \}\). The proximal mapping of f at \({\mathbf {y}}\) is defined as

$$\begin{aligned} \text {Prox}_{tf}({\mathbf {y}}):=\arg \min _{\mathbf {x}}\left\{ tf({\mathbf {x}})+\frac{1}{2}\Vert {\mathbf {x}}-{\mathbf {y}}\Vert ^2\right\} , \end{aligned}$$

where \(t>0\) is a given parameter.

Let \({\mathfrak {L}}\) be a Euclidean space and \({\mathfrak {D}}\subset {\mathfrak {L}}\). For any \({\mathcal {X}}\in {\mathfrak {L}}\), the distance from \({\mathcal {X}}\) to \({\mathfrak {D}}\) is defined and denoted by \(\text{ dist }({\mathcal {X}},{\mathfrak {U}}):=\inf _{{\mathcal {Y}}\in {\mathfrak {D}}}\{\Vert {\mathcal {X}}-{\mathcal {Y}}\Vert _F\}\). Next, we recall the KL property [3, Definition 3.1], see also [4, Definition 2.4]. Let \(f:{\mathbb {R}}^n\rightarrow (-\infty ,+\infty ]\) be a proper lower semicontinuous function. For \(-\infty<\xi _1< \xi _2\le +\infty \), we define \([\xi _1<f({\mathbf {x}})<\xi _2] := \{{\mathbf {x}}\in {\mathbb {R}}^n: \xi _1< f({\mathbf {x}})<\xi _2\}\).

Definition 6

[3, Definition 3.1] The function f is said to have the Kurdyka-Łojasiewicz (KL) property at \({\mathbf {x}}\in \text{ dom }(\partial f)\) if there exist \(\eta \in (0,+\infty ]\), a neighborhood \({\mathfrak {U}}\) of \({\mathbf {x}}\) and a continuous concave function \(\varphi :[0,\eta )\rightarrow [0,+\infty )\) such that: (i) \(\varphi (0)=0\); (ii) \(\varphi \) is continuous and differentiable on \((0,\eta )\); (iii) \(\varphi {'}(s)>0\) for all \(s\in (0,\eta )\); (iv) for all \({\mathbf {z}}\in {\mathfrak {U}}\cap [f({\mathbf {x}})<f({\mathbf {z}})<f({\mathbf {x}})+\eta ]\), the KL inequality holds: \( \varphi {'}(f({\mathbf {z}})-f({\mathbf {x}}))\text{ dist }(0,\partial f({\mathbf {z}}))\ge 1.\)

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Qiu, D., Bai, M., Ng, M.K. et al. Robust Low Transformed Multi-Rank Tensor Methods for Image Alignment. J Sci Comput 87, 24 (2021). https://doi.org/10.1007/s10915-021-01437-8

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