A Nonlinear Structure Tensor with the Diffusivity Matrix Composed of the Image Gradient

Hahn, Jooyoung; Lee, Chang-Ock

doi:10.1007/s10851-009-0138-1

Cited by 18 publications

(20 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The structure tensor J ρ is positive semi-definite and has two orthonormal eigenvectors v 1 || ∇u σ (in the direction of gradient) and v 2 || ∇u σ (in the direction of the isolevel lines). The corresponding eigenvalues μ 1 and μ 2 can be calculated from 6) where j 11 , j 12 and j 22 are the components of J ρ . They are given as…”

Section: Tensor Estimationmentioning

confidence: 99%

“…Anisotropic diffusion tensor can be used to describe the local geometry at an image pixel, thus making it appealing for various image processing tasks [6][7][8][9][10][11][12][13]. Variational methods allow easy integration of constraints and use of powerful modern optimisation techniques such as primal-dual [14][15][16], fast iterative shrinkagethresholding algorithm [17,18], and alternating direction method of multipliers [2][3][4][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Introducing diffusion tensor to high order variational model for image reconstruction

Duan

Ward

Sibbett

et al. 2017

Digital Signal Processing

View full text Add to dashboard Cite

The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.uk JID:YDSPR AID:2153 /FLA [m5G; v1.219; Prn:17/07/2017; 11:02] Second order total variation (SOTV) models have advantages for image restoration over their first order counterparts including their ability to remove the staircase artefact in the restored image. However, such models tend to blur the reconstructed image when discretised for numerical solution [1][2][3][4][5]. To overcome this drawback, we introduce a new tensor weighted second order (TWSO) model for image restoration. Specifically, we develop a novel regulariser for the SOTV model that uses the Frobenius norm of the product of the isotropic SOTV Hessian matrix and an anisotropic tensor. We then adapt the alternating direction method of multipliers (ADMM) to solve the proposed model by breaking down the original problem into several subproblems. All the subproblems have closed-forms and can be solved efficiently. The proposed method is compared with state-of-the-art approaches such as tensor-based anisotropic diffusion, total generalised variation, and Euler's elastica. We validate the proposed TWSO model using extensive experimental results on a large number of images from the Berkeley BSDS500. We also demonstrate that our method effectively reduces both the staircase and blurring effects and outperforms existing approaches for image inpainting and denoising applications.

show abstract

Section: Tensor Estimationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Introducing diffusion tensor to high order variational model for image reconstruction

Duan

Ward

Sibbett

et al. 2017

Digital Signal Processing

View full text Add to dashboard Cite

show abstract

“…Note that it is not straightforward to use the Perona-Malik (PM) model [4] or Rudin-Osher-Fatemi (ROF) model [7] directly for regularizing derivative information of an image [9]. One of reason for regularizing the tangential vector field is that the incompressibility condition, ∇ · t = 0, is numerically computed using the Chorin projection type method which is well developed in the fluid dynamics; see details in Section 3.…”

Section: Review Of Tv-stokes Denoising Algorithmmentioning

confidence: 99%

“…Since qualities of denoised images are seriously dependent on estimated derivative information, it has been a crucial topic to regularize derivatives of an image [9], that is, an orientational information [1,[10][11][12]. Inspired by [1][2][3], we also use a regularization of the tangent vector field of an image with the zero divergence condition.…”

Section: Introductionmentioning

confidence: 99%

Image Denoising Using TV-Stokes Equation with an Orientation-Matching Minimization

Tai

Borok

Hahn

2009

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper, we propose an orientation-matching minimization for denoising digital images with an additive noise. Inspired [1][2][3] by the two-step algorithm in the TV-Stokes denoising process, the regularized tangential vector field with the zero divergence condition is used in the first step. The present work suggests a different approach in order to reconstruct a denoised image in the second step. Namely, instead of finding an image that fits the regularized normal direction from the first step, we minimize an orientation between the image gradient and the regularized normal direction. It gives a nonlinear partial differential equation (PDE) for reconstructing denoised images, which has the diffusivity depending on an orientation of a regularized normal vector field and the weighted self-adaptive force term depending on the direction between the gradient of an image and the vector field. This allows to obtain a denoised image which has sharp edges and smooth regions, even though an original image has smoothly changing pixel values near sharp edges. The additive operator splitting scheme is used for discretizing Euler-Lagrange equations. We show improved qualities of results from various numerical experiments.

show abstract

“…For example, Nagel and Gehrke [34] and Nath and Palaniappan [35] use adaptive Gaussians instead of a Gaussian convolution; Köthe [20] uses a hourglassshaped kernel instead of the Gaussian; van de Weijer and van den Boomgaard [47] use robust statistics to choose one of the ambiguous orientations at every pixel; Brox et al [4] and Hahn and Lee [13] propose non-linear diffusion processes in order to aggregate contributions of the neighbors.…”

Section: Introductionmentioning

confidence: 99%

Adaptation of Tensor Voting to Image Structure Estimation

Moreno

Pizarro

Burgeth

et al. 2012

Mathematics and Visualization

View full text Add to dashboard Cite

Tensor voting is a well-known robust technique for extracting perceptual information from clouds of points. This chapter proposes a general methodology to adapt tensor voting to different types of images in the specific context of image structure estimation. This methodology is based on the structural relationships between tensor voting and the so-called structure tensor, which is the most popu- lar technique for image structure estimation. The problematic Gaussian convolution used by the structure tensor is replaced by tensor voting. Afterwards, the results are appropriately rescaled. This methodology is adapted to gray-valued, color, vectorand tensor-valued images. Results show that tensor voting can estimate image structure more appropriately than the structure tensor and also more robustly.

show abstract

A Nonlinear Structure Tensor with the Diffusivity Matrix Composed of the Image Gradient

Cited by 18 publications

References 23 publications

Introducing diffusion tensor to high order variational model for image reconstruction

Introducing diffusion tensor to high order variational model for image reconstruction

Image Denoising Using TV-Stokes Equation with an Orientation-Matching Minimization

Adaptation of Tensor Voting to Image Structure Estimation

Contact Info

Product

Resources

About