DISCRETE WAVELET TRANSFORM (DWT) This method is used in numerous fields of engineering, mathematics, medical and space research. The process involves the transform of a signal (1D) and image (2D) in forms of wavelets, which is rather discrete, compared to the regular sine and cosine waveforms from the fourier transform. The frequency of the signal is reduced to half by applying Nyquist rate. Similarly, the frequency of image is decreased by 4 times of each and every sub-band we get after we apply the DWT transform as the image gets divided into 4 different sub-bands of different frequencies. This method will help us in transferring large volumes of data from one place to another by reducing the extended information of a signal/image and also videos along large distances. By degrading its quality at the transmitter and retrieving its original quality at the receiver end. WAVELET A wavelet is a wave like oscillation which starts from zero, increases and decreases in its amplitude and only exists for a finite duration. There are numerous type of wavelets used for different purposes .
here, we will use Haar wavelet for studying our process of transformation. Ex FIG.1Haar wavelet is used for studying the mathematical behaviour of signal s(n) in 1-D. And of image s(n1,n2) in 2-D .
We will be dealing with haar wavelet & haar scaling methods for decomposing an image into its frequency sub-bands using DWT and retrieving the synthesized image from original image by applying Inverse DWT . a)- Haar Scaling b)- Haar WaveletPROCESS In DWT, we normally take an image of “N*N” resolution and pass it through LPF & HPF which decomposes the original image into two different frequency compositions. Now, further we down-sample the two parts using Haar Scaling .
And again the down-sampled parts are made to pass through LPF and HPF each. Yeilding us with four different sub-bands of different frequencies of LL,LH,HL,HH sub-bands. Reducing the frequencies of each sub-bands by a factor of ‘2’ i.e. the image of frequency ‘2f’ is reduced by ‘f’ of each of the four sub-bands.
FIG.3 (a) Figure 3 (a) and 3 (b) are the circuit and pictorial representations of DWT process. (Notice that after passing the image from LPF at the beginning shows us that it has filtered the higher frequencies, and they are shifted into the part of HPF’ed form of image which contains the darker color or denser frequencies). Among all the 4 sub-bands the LL sub-band contains the maximum information about the image. Whereas the other three bands LH, HL, HH contains horizontal, vertical and diagonal components of the image. The difference b/w color variations present in an image along different directions are represented along these LH, HL, HH sub-bands FIG.4 From clock-wise direction HL, HH, LH sub-bands, respectively representing horizontal, diagonal and vertical distributions of an image. From looking at the original image the differences in varying frequencies can be easily detected and noticed.
For ex. in the HL sub-band the camera stand can be easily noticed and slightly other horizontal details are visible. The dark color patches contain an area of dense frequencies as compared to that with the light color ones. The HPF is responsible for such behavior of these three sub-bands. As the images are normally rich in low frequencies. The LL sub-band contains the maximum frequencies related to the image, which provides us with the maximum information about the image. In this way, we get the decomposed form of image by increasing the level of decomposition slowly removing the denser frequency regions from the previous decomposed level.
The above block diagram represents the 3-level decomposition of an image. NOTICE:- Here only the LL sub-band is decomposed in order to get the maximum information which can be transferred without the loss of any relevant information.