How to prove a function of image lies in sobolev space?
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To prove that a function of an image lies in the Sobolev space, one approach is to use the reconstruction operator based on the values of the function at a given number of nodes. The error in this reconstruction operator is shown to coincide with the corresponding orthogonal width . Another method involves using the Sobolev gradient method in conjunction with the steady-state solution of the Navier-Stokes equation to fill in missing pieces in a digital image. The Sobolev gradient flow obtained from this method is proven to be globally existent . Additionally, a representation of a linear functional in the weighted Sobolev space is obtained without the use of pseudodifferential operators. This representation is deduced via a boundary element of the test space, and the uniqueness of the boundary element is proven using Clarkson's inequalities .
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| Papers (5) | Insight |
|---|---|
| The paper proves that there exists at least one solution of the system of equations that lies in the Sobolev space. | |
| The paper does not provide specific information on how to prove that a function of image lies in a Sobolev space. | |
| The paper discusses the reconstruction operator for functions from the Sobolev space based on their values at a given number of nodes. | |
| The paper provides a method to represent a linear functional in the weighted Sobolev space without using pseudodifferential operators. | |
8 Citations | The paper presents an image inpainting algorithm based on the Sobolev gradient method, which demonstrates the effectiveness of the Sobolev gradient flow in filling in missing pieces in a digital image. |
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