$L_{0}$ Gradient Projection

Minimizing <inline-formula> <tex-math notation="LaTeX">$L_{0}$ </tex-math></inline-formula> gradient, the number of the non-zero gradients of an image, together with a quadratic data-fidelity to an input image has been recognized as a powerful edge-preserving filtering method. However, the <inline-formula> <tex-math notation="LaTeX">$L_{0}$ </tex-math></inline-formula> gradient minimization has an inherent difficulty: a user-given parameter controlling the degree of flatness does not have a physical meaning since the parameter just balances the relative importance of the <inline-formula> <tex-math notation="LaTeX">$L_{0}$ </tex-math></inline-formula> gradient term to the quadratic data-fidelity term. As a result, the setting of the parameter is a troublesome work in the <inline-formula> <tex-math notation="LaTeX">$L_{0}$ </tex-math></inline-formula> gradient minimization. To circumvent the difficulty, we propose a new edge-preserving filtering method with a novel use of the <inline-formula> <tex-math notation="LaTeX">$L_{0}$ </tex-math></inline-formula> gradient. Our method is formulated as the minimization of the quadratic data-fidelity subject to the hard constraint that the <inline-formula> <tex-math notation="LaTeX">$L_{0}$ </tex-math></inline-formula> gradient is less than a user-given parameter <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>. This strategy is much more intuitive than the <inline-formula> <tex-math notation="LaTeX">$L_{0}$ </tex-math></inline-formula> gradient minimization because the parameter <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> has a clear meaning: the <inline-formula> <tex-math notation="LaTeX">$L_{0}$ </tex-math></inline-formula> gradient value of the output image itself, so that one can directly impose a desired degree of flatness by <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>. We also provide an efficient algorithm based on the so-called alternating direction method of multipliers for computing an approximate solution of the nonconvex problem, where we decompose it into two subproblems and derive closed-form solutions to them. The advantages of our method are demonstrated through extensive experiments.

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