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104 4. A Variational approach for the destriping issue
supported by quantitative measurements reported in tables 4.1 and 4.2.
Although the UVM was derived intuitively by combining edge-preserving models and gradient
field integration approaches, similar results can be achieved with slightly different
variational models. These are based on the minimization of the following energy functionals
:
E 1 (u) =TV x (u − f)+λT V (u) (4.110)
∫ ( ) ∂ (u − f) 2
E 2 (u) =
+ λT V (u) (4.111)
Ω ∂x
∫ ( ) ∂ (u − f) 2
E 3 (u) =
+ λT V y (u) (4.112)
∂x
Ω
In the model (4.112), the L 1 -norm on the x-gradient is replaced by the L 2 -norm. This
modification does not cause bluring artifacts as the edges are still preserved by the regularizing
term which includes the L 1 -norm of the y-gradient. Nevertheless, the original
UVDM is more suited to isolate striping due to the L 1 -norm being weaker than the standard
L 2 -norm (L 2 (Ω) ∈ L 1 (Ω)). For u-components holding the same ID index and derived
from (4.112) and the UVDM, the noisy component v extracted with the UVDM contains
more stripe-related texture. This feature comes in handy in the TVN hierarchical decomposition
where the destriped image is obtained by progressively extracting striping from
oversmoothed u estimates. The L 1 -norm in the fidelity term then ensures a minimal number
of iterations compared to the L 2 -norm.
The model (4.110) replaces the regularizing term of the UVM model with the commonly
used TV-norm which provides similar results than the UVM but increases the computational
cost required for the minimization of its energy functional. This drawback, together
with the use of L 2 -norm in the fidelity term are both present in the variational model
(4.111).
From a denoising perspective, the UVDM and models (4.110), (4.111), (4.112) satisfy the
destriping requirements imposed in chapter 3, namely complete stripe removal without
blur/ringing artifacts. In fact, both TVN and/or Osher et al. iterative procedures ensure
that the ID index of the optimally destriped image is close to 1. This emphazises the
importance of a suitable fidelity term which, for all destriping models described above,
takes into account the unidirectional property of striping.