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Total Variation regularization (TV) is a seminal approach for image recovery.
TV involves the norm of the image's gradient, aggregated over all pixel
locations. Therefore, TV leads to piece-wise constant solutions, resulting in
what is known as the "staircase effect." To mitigate this effect, the Hessian
Schatten norm regularization (HSN) employs second-order derivatives,
represented by the pth norm of eigenvalues in the image hessian, summed across
all pixels. HSN demonstrates superior structure-preserving properties compared
to TV. However, HSN solutions tend to be overly smoothed. To address this, we
introduce a non-convex shrinkage penalty applied to the Hessian's eigenvalues,
deviating from the convex lp norm. It is important to note that the shrinkage
penalty is not defined directly in closed form, but specified indirectly
through its proximal operation. This makes constructing a provably convergent
algorithm difficult as the singular values are also defined through a
non-linear operation. However, we were able to derive a provably convergent
algorithm using proximal operations. We prove the convergence by establishing
that the proposed regularization adheres to restricted proximal regularity. The
images recovered by this regularization were sharper than the convex
counterparts.
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