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) .

A two-dimensional scaling function,

, and three two-dimensional wavelet

,

and

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):

Where

Arbitrary starting scale

Approximation

coefficients for

at scale

Horizontal,

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.

BANOTH, B. (2007). ROTATION AND SCALE INVARIANT TEXTURE CLASSIFICATION

USING LOG-POLAR WAVELET ENERGY SIGNATURES, National Institute of Technology

Rourkela.

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.