IEEE 2017 Conference on Computer Vision and Pattern Recognition
Simultaneous Visual Data Completion and Denoising Based on Tensor Rank and Total Variation Minimization and Its Primal-Dual Splitting Algorithm T. Yokota and H. Hontani (Nitech)
Introduction (data completion)
Noise threshold δ VS Lagrange parameter μ
Completion is a procedure to recover missing values by using Available parts of data Structural assumption -- Concept of completion problem -Observed incomplete data
Original data 20% of data was lost
Completed data
δ: can be decided from signal to noise ratio of data, and it does not depend regularizers. μ: is difficult to decide since it depends regularizers. Noise constraint form is appropriate in practice!! But optimization is little bit complicated.
difficult to know
Consider unconstrained form
Indicator functions
Experimental Results
Conv. behavior & times
Various parameter settings
Weight for TV & LR regularizations: Weights in multi-mode low-rank regularizations:
Completion
Convertible, but corresponding values of λ and δ are
Convex Optimization
Variable splitting
Ex.) Vector completion Linear interpolation Polynomial interpolation
PSNRs were compared with
Ex.) Matrix completion
Convex approaches [1] GTV: Generalized total variation regularization (Guo et al., CVPR, 2015.) [2] LNRTC: low-n-rank tensor completion (Gandy et al., Inverse Problem, 2011.) Non-convex approaches [3] SPCQV: smooth PARAFAC tensor completion with quadratic variation regularization (Yokota et al., IEEE-TSP, 2016.) [4] SPCTV: smooth PARAFAC tensor completion with total variation regularization (Yokota et al., IEEE-TSP, 2016.)
Proposed Method
Low-rank matrix completion Bilinear interpolation
Ex.) Tensor completion
MR images [256*256*24]
Original
Missing
We propose a direct solution method for Low-rank and TV regularizations with noise inequality and box constraints. Tensor TV norm
If given incomplete data is with noise, ordinary completion techniques are not so useful (or can not be applied).
N-th order tensor:
Function:
Tensor TV norm
Regularization function (e.g., nuclear-norm, TV-norm, L1-norm etc)
Tensor nuclear norm
Completion &Denoising problem
original
missing (30%)
SPCQV
SPCTV
2nd mode
Noise threshold 1st mode
SPCQV
SPCTV
Missing rate
proposed
GTV
LNRTC
SPCQV
SPCTV
citrus
10%
25.646
25.186
23.852
23.743
23.706
citrus
30%
23.410
22.920
20.948
22.251
22.115
citrus
50%
20.919
20.644
18.112
20.459
20.162
tomato
10%
27.980
27.865
26.231
24.896
24.890
tomato
30%
27.187
26.782
24.516
24.492
24.460
tomato
50%
26.014
25.429
22.785
23.825
23.717
Color movie (4d-tensor: 120*160*3*100)
Definition of mode matrix unfolding 3rd mode
LNRTC
GTV
proposed
LNRTC
Differential with respect to n-th axis
(only) Completion problem Support set projection (missing elements to be zero)
GTV
Dual step (parallelizable)
Simultaneous Tensor Completion and Denoising
proposed
original
Definition of proximal map
Missing rate
proposed
GTV
LNRTC
SPCQV
SPCTV
10%
31.045
30.947
28.820
30.018
30.021
30%
28.942
28.485
26.920
29.642
29.659
50%
26.750
26.101
25.006
28.995
29.996
Conclusions Convex optimization based visual data recovery is proposed. Convex approach is fast & efficient, but non-convex approach is more accurate for highly missing cases.
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Computer Vision and Pattern ... Convex optimization based visual data recovery is proposed. â ... Completion is a procedure to recover missing values by using.
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