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Explain discrete wavelet transform in detail. Discuss its relevance in compression
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Solution:

In wavelet decomposition, the signal is separated into slow and fast components using a pair of finite impulse response filters.

The original signal $x[n]$ passes through two complementary filters and emerges as two signals $x_l[n]$ and $x_n[n]$.

This process of filtering and downsampling can be repeated to get a Multi-level decomposition.

As seen from the figure, the low-frequency component from each stage is again fed to similar filters in the next stage so that every stage extracts lower and lower frequency component that is desired.

The low pass and the high pass filter outputs are written as

\begin{aligned} &x_l[n]=x[n] * g[n] \\ &x_n[n]=x[n] * h[n] \end{aligned}

When h[n] and g[n] are the impulse response of the high-pass and low-pass filters. respectively.

$\phi(x, y)=\phi(x) \phi(y) .\\$

where $\phi(x) \rightarrow$ one dimensional function

• Let $\psi(x) \rightarrow$ its companion wavelet.

• Since the scaling and wavelet functions are separable, each convolution breaks down into one-dimensional convolutions on the rows and columns.

Let the size of the original image $f(x, y)$ be NaN

In the first stage, we convolve the rows of the image with $h[n]$ and $g[n]$ and The columns of each of the $\mathrm{N} / 2 \times \mathrm{N}$ data are then convolved with $h[n]$ and g[n] and the alternate rows are discarded.

The result of this entire operation gives us four $\mathrm{N} / 2 \times \mathrm{N} / 2$ images.

This decomposition can be cared out recursively.

This decomposition can be cared out $\rightarrow$ recursively.

The first N/2 matrix is again decomposed into four $\mathrm{N} / 4$ matrices.

The upper leftmost square represents the smooth information (lowest frequency) ie. a blurred version of the image.

The other squares represent detailed information (edges) in different directions and Image processing based on wavelet transform typically proceeds by applying a 2D wavelet transform on the image, manipulating the values of the wavelet transform, and performing the processed image.

DWT relevance in compression:-

The wavelet transform decomposes an image into a set of different resolution sub-images, corresponding to the various frequency bands.

This results in a multi-resolution representation of images with localization in both the spatial and frequency domains.

The main advantages of wavelet-based compression are as given below:

(1) wavelets have hon-uniform frequency spectra which facilitate multi-scale analysis.

(2) The multi-resolution property of the wavelet transform can be wed to exploit the fact that the response of the hum an eye is different to high and low-frequency components of an image.

(3) DWT can be applied to an entire image without imposing a block structure.