3 Speckle Noise Reduction and Enhancement for OCT Images
53
Similar to 2D approach, 3D DWT suffers from more serious checkerboard artifact.
It is shown in [61] that 3D dual-tree wavelet transforms is a good candidate for
processing medical volume data and video sequences.
3.4.3 Dictionary Learning with Wise Selection of Start
Dictionary
This method is called 2D/3D complex wavelet-based dictionary learning (2D-CWDL
and 3D-CWDL) [16]. The CWT cannot be used in its algebraic form since in dictionary learning approaches an explicit dictionary is needed to be multiplied by the
data. The matrix representation can be calculated for this purpose.
Suppose that the usual 2D separable Discrete Wavelet Transform (DWT) implemented using the filters {h 0 (n), h 1 (n)} can be represented by the square matrix F hh .
If x is a real image, the real and imaginary parts of the oriented complex 2D dual-tree
wavelet transform can be represented by W r 2D and W i2D , respectively [60], where I
is an identity matrix.
W r 2D
1
2
I −I
I I
F hh
F gg
x.
(3.10)
W i2D
1
2
I I
I −I
F gh
F hg
x.
(3.11)
Therefore, the complex coefficients can be calculated by:
F C2D
1
4
⎡
⎢
⎢
⎢
⎣
I −I I I
I I I −I
I I −I I
I −I −I −I
⎤
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎣
F hh
F gg
i.F gh
i.F hg
⎤
⎥
⎥
⎥
⎥
⎦
(3.12)
This dictionary can now be used as start dictionary in dictionary learing of 2DCWDL.
Similarly, one may use 3D dual-tree wavelet transform as start dictionary of 3DCWDL. It can be shown that real and imaginary parts of the oriented complex 3D
dual-tree wavelet transform can be represented by W r 3D and W i3D :
W r 3D
1
4
⎡
⎢
⎢
⎢
⎣
I −I −I −I
I −I I I
I I −I I
I I I −I
⎤
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎣
F hhh
F ggh
F ghg
F hgg
⎤
⎥
⎥
⎥
⎥
⎦
.
(3.13)