Digital Signal Processing Reference
In-Depth Information
Figure 8.11. (left) Detected edges on the original image. (right) Detected edges on
the curvelet reconstruct component.
texture, rather than the natural part. By separating first the two components, we can
recover the true object's edges. Figure 8.11 shows the edges revealed by the Canny
edge detector on both the original image and the curvelet-reconstructed component
(see Fig. 8.9).
Nonlinear Approximation
The efficiency of a sparse decomposition can be assessed through the decay of its
nonlinear approximation (NLA) error as a function of the sparsity level, as ex-
plained in Section 1.1.2. An NLA curve is obtained by reconstructing the image from
its best
m
-term in the decomposition. Throughout the chapters, we discussed decay
rates of the NLA achieved by different transforms for given classes of functions.
For instance, for smooth functions away from a discontinuity across a
C
2
curve, the
NLA error decays as
O
(
m
−
1
/
2
) in a squared
L
2
-norm in a Fourier basis, as
O
(
m
−
1
)
in a wavelet basis (Mallat 2008), and as
O
(
m
−
2
(log
m
)
3
) (see Section 5.4).
Using the separation algorithm described previously, the “Barbara” image was
decomposed in the redundant dictionary local DCT
orthogonal wavelet transform
(OWT). Since the dictionary used is highly redundant, the exact overall representa-
tion of the original image may require a relatively small number of coefficients due
to the promoted sparsity, and this essentially yields a better NLA curve.
Figure 8.12 presents the NLA curves for the image “Barbara” using the OWT
(dotted line), the DCT (solid line), and the algorithm discussed here (dashed line),
based on the local DCT
+
+
OWT dictionary. We see that for low sparsity levels,
typically,
m
15 percent
N
, the combined representation leads to a better NLA
curve than the local DCT or the OWT alone.
<
8.7 INPAINTING
8.7.1 Problem Formulation
Inpainting is to restore missing image information based on the still available
(observed) cues from destroyed, occluded, or deliberately masked subregions of
