III the two-dimensional DWT can be implemented

 III The Discrete
Wavelet Transform

The Wavelet Series is just a sampled
version of Continuous Wavelet Transform (CWT) and its computation may consume a
significant amount of time and resources,
depending on the resolution required. The Discrete Wavelet Transform (DWT),
which is based on sub-band coding is found to yield a fast computation of
Wavelet Transform. It is easy to implement and reduces the computation time and
resources required (BANOTH 2007) .

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A two-dimensional scaling function,

, and three two-dimensional wavelet



are critical elements for wavelet transforms in two
dimensions. Given separable 2-D scaling and wavelet functions, 2-D DWT can be
defined. First, we define the scaled and translated or shifted basis functions
are defined as follows (Gonzalez and Woods 2008):






Where i = directional wavelet index.
Therefore, 2-D DWT of  an image

 of size

is given by (Gonzalez and Woods 2008):






Arbitrary starting scale

coefficients for

at scale

vertical and diagonal details coefficients at scales




Then the two-dimensional DWT can be
implemented using digital filters and downsamplers

. The block diagram in Fig. 2 shows the process of taking the
one-dimensional FWT of the rows of

and the subsequent one-dimensional FWT of the resulting
columns. Three sets of detail coefficients including the horizontal, vertical,
and diagonal details are produced.



The proposed work has examined, Haar discrete wavelet transform
based, 7th  level
decomposition of the breast cancer histopathology images . The discrete wavelet
transform named Haar, have originally been designed by (Haar 1911). At first
level of decomposition, breast cancer histopathology images  are

being divided into four equal size sub-images, namely LL1 (approximation
coefficients), LH1 (horizontal coefficients), HL1 (vertical coefficient) and
HH1 (diagonal coefficient). Subsequently at the second level of decomposition
LL1 (approximation coefficient) sub-image is further decomposed into four equal
size sub-images LL2, LH2, HL2 and HH2. Continuously until we reach the seventh level of decomposition. In
this manner 28 sub-images have been formed from the every channel (red, green
& blue) thus, we get 28 x 3 sub-images have been established from the original image.
Then we calculated nine of traditional statistical features (Mean, Standard deviation,
Skewness, kurtosis, Entropy, Energy, Root mean square, Mean
Absolute Deviation, Median Absolute Deviation). Overall, nine statistical
features have been acquired from each sub-images; and 756 features for each of
the breast cancer histopathology image samples.  
























Gonzalez, R. and R. Woods
(2008). “Digital image processing: Pearson prentice hall.” Upper
Saddle River, NJ.


Haar, A. (1911). “Zur
theorie der orthogonalen funktionensysteme.” Mathematische Annalen 71(1): 38-53.




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