Constrained Restoration and the Recovery of Discontinuities
IEEE Transactions on Pattern Analysis and Machine Intelligence
On the Convergence of the Lagged Diffusivity Fixed Point Method in Total Variation Image Restoration
SIAM Journal on Numerical Analysis
Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations (Applied Mathematical Sciences)
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
IEEE Transactions on Image Processing
Joint Exact Histogram Specification and Image Enhancement Through the Wavelet Transform
IEEE Transactions on Image Processing
Exact Histogram Specification for Digital Images Using a Variational Approach
Journal of Mathematical Imaging and Vision
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We consider a class of convex functionals that can be seen as $\mathcal{C}^{1}$ smooth approximations of the ℓ 1-TV model. The minimizers of such functionals were shown to exhibit a qualitatively different behavior compared to the nonsmooth ℓ 1-TV model (Nikolova et al. in Exact histogram specification for digital images using a variational approach, 2012). Here we focus on the way the parameters involved in these functionals determine the features of the minimizers $\hat{u}$ . We give explicit relationships between the minimizers and these parameters.Given an input digital image f, we prove that the error $\|\hat{u}- f\| _{\infty}$ obeys $b-\varepsilon\leq\|\hat{u}-f\|_{\infty}\leq b$ where b is a constant independent of the input image. Further we can set the parameters so that 驴0 is arbitrarily close to zero. More precisely, we exhibit explicit formulae relating the model parameters, the input image f and the values b and 驴. Conversely, we can fix the parameter values so that the error $\|\hat{u}- f\|_{\infty}$ meets some prescribed b,驴. All theoretical results are confirmed using numerical tests on natural digital images of different sizes with disparate content and quality.