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Quantifying Intermembrane Distances with Serial Image Dilations
Published on: September 28, 2018
Interpreting translation-invariant wavelet shrinkage as a new image smoothing scale space
1CEREMADE (CNRS UMR 7534), Université de Paris-Dauphine, 75775 Paris CEDEX 16, France. Antonin.Chambolle@ceremade.dauphine.fr
Summary
Iterated translation-invariant wavelet shrinkage offers a novel nonlinear image smoothing scale space. This method, equivalent to gradient descent, provides an alternative to partial differential equations for image denoising.
Area of Science:
- Image Processing
- Wavelet Theory
- Numerical Analysis
Background:
- Wavelet shrinkage is a technique for image noise reduction.
- Translation-invariant wavelet shrinkage was introduced by Coifman and Donoho (1995).
- The original method applies wavelet shrinkage to a 2-D semi-discrete wavelet representation.
Purpose of the Study:
- To develop a mathematical framework for iterated translation-invariant wavelet shrinkage.
- To establish the equivalence of this iterative method to gradient descent.
- To interpret the method as a new nonlinear wavelet-based image smoothing scale space.
Main Methods:
- Developing a mathematical framework for iterated translation-invariant wavelet shrinkage.
- Utilizing a theorem by Kato and Masuda (1978).
- Analyzing the connection to gradient descent in L(2)(I) along the Besov space B(1)(1)(L(1)(I)) with orthogonal wavelets.
Main Results:
- A mathematical framework for iterated translation-invariant wavelet shrinkage is presented.
- The iterative shrinkage method is shown to be equivalent to gradient descent using orthogonal wavelets.
- This equivalence leads to a new nonlinear wavelet-based image smoothing scale space.
Conclusions:
- Iterated translation-invariant wavelet shrinkage provides a novel nonlinear image smoothing scale space.
- This scale space is characterized via gradient descent, not a nonlinear partial differential equation.
- The findings offer a new perspective on wavelet-based image processing and denoising.
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