Digital Signal Processing Reference
In-Depth Information
Figure 5.9. (left) Example of a ridgelet obtained by the fast slant stack implementation. (right)
Its FFT superimposed on the DRT frequency tiling.
Figure 5.9 (left) exemplifies a ridgelet in the spatial domain obtained from the
DRT based on FSS implementation. Its Fourier transform is shown in Fig. 5.9 (right)
superimposed on the DRT frequency tiling (Donoho and Flesia 2002). The Fourier
transform of the discrete ridgelet lives in an angular wedge. More precisely, the
Fourier transform of a discrete ridgelet at scale
j
lives within a dyadic square of
size
2
j
.
∼
5.3.5 Local Ridgelet Transforms
The ridgelet transform is optimal for finding global lines of the size of the image. To
detect line segments, a partitioning must be introduced (Cand es and Donoho 1999).
The image can be decomposed into overlapping blocks of side-length
B
pixels in
such a way that the overlap between two vertically adjacent blocks is a rectangular
array of size
B
by
B
/
2; we use overlap to avoid blocking artifacts. For an
N
×
N
image, we count 2
N
B
such blocks in each direction, and thus the redundancy factor
grows by a factor of 4.
The partitioning introduces redundancy because a pixel belongs to four neigh-
boring blocks. We present two competing strategies to perform the analysis and
synthesis:
/
1. The block values are weighted by a spatial window (analysis) in such a way
that the coaddition of all blocks reproduces exactly the original pixel value
(synthesis).
2. The block values are those of the image pixel values (analysis) but are
weighted when the image is reconstructed (synthesis).
Experiments have shown that the second approach leads to better results, espe-
cially for restoration problems; see Starck et al. (2002) for details.
A pixel value,
f
[
k
,
l
], belongs to four blocks, namely,
B
b
1
,
b
2
[
k
1
,
l
1
](with
b
1
=
2
k
/
B
and
b
2
=
2
l
/
B
),
B
b
1
+
1
,
b
2
[
k
2
,
l
1
],
B
b
1
,
b
2
+
1
[
k
1
,
l
2
], and
B
b
1
+
1
,
b
2
+
1
[
k
2
,
l
2
], such that
