Combining the Calculus of Variations and Wavelets for Image Enhancement

Coifman, Ronald R.; Sowa, Artur

doi:10.1006/acha.2000.0299

Cited by 58 publications

(44 citation statements)

References 6 publications

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“…Their method is also close in spirit to approaches by Chan and Zhou [19] who postprocessed images obtained from wavelet shrinkage by a TV-like regularization technique. Coifman and Sowa [23] used functional minimization with wavelet constraints for postprocessing signals that have been degraded by wavelet thresholding or quantization. Candes and Guo [15] also presented related work, in which they combined ridgelets and curvelets with TV minimization strategies.…”

Section: Introductionmentioning

confidence: 99%

On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and SIDEs

Steidl¹,

Weickert²,

Brox³

et al. 2004

SIAM J. Numer. Anal.

218

169

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Soft wavelet shrinkage, total variation (TV) diffusion, total variation regularization, and a dynamical system called SIDEs are four useful techniques for discontinuity preserving denoising of signals and images. In this paper we investigate under which circumstances these methods are equivalent in the 1-D case. First we prove that Haar wavelet shrinkage on a single scale is equivalent to a single step of space-discrete TV diffusion or regularization of two-pixel pairs. In the translationally invariant case we show that applying cycle spinning to Haar wavelet shrinkage on a single scale can be regarded as an absolutely stable explicit discretization of TV diffusion. We prove that space-discrete TV diffusion and TV regularization are identical, and that they are also equivalent to the SIDEs system when a specific force function is chosen. Afterwards we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularization of the Laplacian pyramid of the signal. We analyse possibilities to avoid Gibbs-like artifacts for multiscale Haar wavelet shrinkage by scaling the thesholds. Finally we present experiments where hybrid methods are designed that combine the advantages of wavelets and PDE / variational approaches. These methods are based on iterated shift-invariant wavelet shrinkage at multiple scales with scaled thresholds.

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Section: Introductionmentioning

confidence: 99%

On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and SIDEs

Steidl¹,

Weickert²,

Brox³

et al. 2004

SIAM J. Numer. Anal.

218

169

View full text Add to dashboard Cite

show abstract

“…To improve upon these methods, combinations of such techniques with sparse representations have recently been proposed (e.g. [7,15,28,58,80,89]). A similar combination has been proposed using shearlets in [31].…”

Section: Denoising Using Shearlet-based Total Variation Regularizationmentioning

confidence: 99%

Image Processing Using Shearlets

Easley

Labate

2012

Shearlets

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Since shearlets provide nearly optimally sparse representations for a large class of functions that are useful to model natural images, many image processing methods benefit from their use. In particular, the error rates of data estimation from noise are highly dependent on the sparsity properties of the representation, so that many successful applications of shearlets center around restoration tasks such as denoising and inverse problems. Other imaging problems, where also the application of the shearlet representation turns out to be very beneficial, include image enhancement, image separation, edge detection, and estimation of the geometric features of an object.

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“…Therefore, current researches are increasingly focusing on the combinations of total variation and wavelet-like methods. In [23,25], in order to reduce the Gibbs artifacts, a set of wavelet coefficients was interpolated according to a total variation criterion. This was close to the approach of [9] where PDE techniques were used to reduce edge artifacts generated by wavelet shrinkage.…”

Section: Introductionmentioning

confidence: 99%

Alternating Minimization Method for Total Variation Based Wavelet Shrinkage Model

Zeng

2010

CICP

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Abstract. In this paper, we introduce a novel hybrid variational model which generalizes the classical total variation method and the wavelet shrinkage method. An alternating minimization direction algorithm is then employed. We also prove that it converges strongly to the minimizer of the proposed hybrid model. Finally, some numerical examples illustrate clearly that the new model outperforms the standard total variation method and wavelet shrinkage method as it recovers better image details and avoids the Gibbs oscillations.

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Combining the Calculus of Variations and Wavelets for Image Enhancement

Cited by 58 publications

References 6 publications

On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and SIDEs

On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and SIDEs

Image Processing Using Shearlets

Alternating Minimization Method for Total Variation Based Wavelet Shrinkage Model

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