Digital Signal Processing Reference
In-Depth Information
If the data fidelity term in equation (8.24) is in its constrained form and if the
dictionary is formed from orthobases or tight frames, the DR splitting can be used
and the projector on the constraint set has a closed form (see Section 7.4.2). Other
variants are described by Fadili et al. (2009c), and the interested reader may refer
to that paper for further details.
Let us conclude this subsection by noting some differences and similarities be-
tween the two preceding inpainting algorithms:
Target:
In the MCA-based formulation, the targets are the morphological com-
ponents, and component separation is a by-product of the inpainting process,
while in the second algorithm, the goal is to achieve a good inpainting of the
whole image and not necessarily a good separation and inpainting of each com-
ponent.
Parameters:
In the MCA-based inpainting, the user provides the algorithm with
a threshold-lowering schedule and a stopping threshold
λ
min
, while in the second
λ
version, the regularization parameter
is fixed, or a continuation method that
solves a sequence of problems (8.24) for a decreasing value of
λ
can be used; see
Section 7.7.2 and (Fadili et al. 2009c).
Noise:
Both algorithms handle the presence of noise. The second formulation is
able to estimate the noise variance along with inpainting.
Optimization algorithms:
Despite apparent similarities, the two formulations use
different optimization frameworks. MCA is a stagewise algorithm, formed by hy-
bridizing MP with BCR. The second formulation yields an iterative thresholding
algorithm with a rigorous convergence analysis guaranteed for convex penalties,
as explained in Chapter 7.
8.7.3.1 Examples
Barbara
Figure 8.13 shows the “Barbara” image (512
512) and its inpainted results for
three random masks of 20, 50, and 80 percent missing pixels. The unstructured ran-
dom form of the mask makes the recovery task easier, which is predictable using
a compressed sensing argument (see Chapter 11). Again, the dictionary contained
the curvelet and local DCT transforms. The algorithm is not only able to recover
the geometric part (cartoon), but it performs particularly well inside the textured
areas.
×
Lena
We compared the two inpainting algorithms given in Algorithm 30 and Algo-
rithm 33 on “Lena” 512
512. The masked image is depicted in Fig. 8.14 (top
right), where 80 percent of the pixels were missing, with large gaps. The dictionary
contained the curvelet transform. The parameters chosen for each algorithm are
given in Table 8.2. Despite the challenging nature of this example (large gaps), both
inpainting algorithms performed well. They managed to recover the most impor-
tant details of the image, which are hardly distinguishable by eye in the masked
image. The visual quality is confirmed by measures of the PSNR, as reported in
Fig. 8.14.
×
