The lower bound of nonlocal gradient for non-convex and non-smooth image patches based regularization

Abstract In the inverse problem of image processing, we have witnessed that the non-convex and non-smooth regularizers can produce clearer image edges than convex ones such as total variation (TV). This fact can be explained by the uniform lower bound theory of the local gradient in non-convex and non-smooth regularization. In recent years, although it has been numerically shown that the nonlocal regularizers of various image patches based nonlocal methods can recover image textures well, we still desire a theoretical interpretation. To this end, we propose a non-convex non-smooth and block nonlocal (NNBN) regularization model based on image patches. By integrating the advantages of the non-convex and non-smooth potential function in the regularization term, the uniform lower bound theory of the image patches based nonlocal gradient is given. This approach partially explains why the proposed method can produce clearer image textures and edges. Compared to some classical regularization methods, such as total variation (TV), non-convex and non-smooth (NN) regularization, nonlocal total variation (NLTV) and block nonlocal total variation(BNLTV), our experimental results show that the proposed method improves restoration quality.

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Lower Bound Theory of Nonzero Entries in Solutions of $\ell_2$-$\ell_p$ Minimization

SIAM Journal on Scientific Computing ◽

10.1137/090761471 ◽

2010 ◽

Vol 32 (5) ◽

pp. 2832-2852 ◽

Cited By ~ 154

Author(s):

Xiaojun Chen ◽

Fengmin Xu ◽

Yinyu Ye

Keyword(s):

Lower Bound ◽

Lower Bound Theory ◽

Bound Theory

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Comparison of an improved self-consistent lower bound theory with Lehmann’s method for low-lying eigenvalues

Scientific Reports ◽

10.1038/s41598-021-02473-y ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Miklos Ronto ◽

Eli Pollak ◽

Rocco Martinazzo

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Energy Levels ◽

Upper Bounds ◽

Hamiltonian Matrix ◽

Lanczos Algorithm ◽

Eigenvalue Estimates ◽

Self Consistent ◽

Lower Bound Theory ◽

Bound Theory

AbstractRitz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The well-known Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple’s lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temple’s results. In this paper, we further improve the SCLBT and compare its quality with Lehmann’s theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann’s theory and is essential for the SCLBT method. Using two lattice Hamiltonians, we compared the improved SCLBT (iSCLBT) with its previous implementation as well as with Lehmann’s lower bound theory. The novel iSCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT variants provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and iSCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for low-lying excited states as well. Compared to Lehmann’s theory, one of the advantages of the iSCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used. Especially owing to this property the iSCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower bounds

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Comparison of an Improved Self-consistent Lower Bound Theory with Lehmann’s Method for Low-lying Eigenvalues

10.21203/rs.3.rs-492850/v1 ◽

2021 ◽

Author(s):

Miklos Ronto ◽

Eli Pollak ◽

Rocco Martinazzo

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Energy Levels ◽

Upper Bounds ◽

Hamiltonian Matrix ◽

Lanczos Algorithm ◽

Eigenvalue Estimates ◽

Self Consistent ◽

Lower Bound Theory ◽

Bound Theory

Abstract Ritz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The well-known Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple’s lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temple’s results. In this paper we further improve the SCLBT and compare its quality with Lehmann’s theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann’s theory and is essential for the SCLBT. Using two lattice Hamiltonians, we compared the improved SCLBT with its previous implementation as well as with Lehmann’s lower bound theory. The novel SCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and SCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for low-lying excited states as well. Compared to Lehmann’s theory, one of the advantages of the SCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used. Especially owing to this property the SCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower bounds.

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A BAND SELECTION METHOD FOR HIGH PRECISION REGISTRATION OF HYPERSPECTRAL IMAGE

ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences ◽

10.5194/isprs-archives-xlii-3-2067-2018 ◽

2018 ◽

Vol XLII-3 ◽

pp. 2067-2071

Author(s):

H. Yang ◽

X. Li

Keyword(s):

Lower Bound ◽

Spatial Resolution ◽

Hyperspectral Image ◽

High Spatial Resolution ◽

Selection Method ◽

Registration Accuracy ◽

High Spatial Resolution Image ◽

Lower Bound Theory ◽

Speed And Accuracy ◽

Bound Theory

During the registration of hyperspectral images and high spatial resolution images, too much bands in a hyperspectral image make it difficult to select bands with good registration performance. Terrible bands are possible to reduce matching speed and accuracy. To solve this problem, an algorithm based on Cram’er-Rao lower bound theory is proposed to select good matching bands in this paper. The algorithm applies the Cram’er-Rao lower bound theory to the study of registration accuracy, and selects good matching bands by CRLB parameters. Experiments show that the algorithm in this paper can choose good matching bands and provide better data for the registration of hyperspectral image and high spatial resolution image.

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