Morphing from the TV-Norm to the l -Norm

Area of Science:

• Image reconstruction
• Optimization techniques
• Applied mathematics

Background:

• Image reconstruction from under-sampled measurements is challenging for piecewise constant objects.
• Sparse solutions are often desired, typically achieved via L1-norm minimization.
• Minimizing the non-differentiable L1-norm presents computational difficulties.

Purpose of the Study:

• To introduce a novel 'meta L1-norm' function for image reconstruction.
• To develop a straightforward gradient descent algorithm for enforcing sparsity.
• To apply the meta L1-norm to few-view tomography.

Main Methods:

• Development of a differentiable meta L1-norm function, bridging L1 and Total Variation (TV) norms.
• Derivation of an explicit subdifferential for the meta L1-norm.
• Implementation of a gradient descent algorithm for iterative image reconstruction.

Main Results:

• The meta L1-norm facilitates a tractable gradient descent algorithm for sparse solutions.
• Computer simulations demonstrate effective reconstruction of piecewise constant objects.
• The meta L1-norm shows promise in few-view tomography applications.

Conclusions:

• The proposed meta L1-norm offers a viable alternative to traditional L1 and TV norms for image reconstruction.
• This method simplifies the enforcement of sparsity in inverse problems.
• The meta L1-norm effectively guides image reconstruction towards piecewise constant solutions.