By Dr. Rajeev Srivastava CSE, IIT(BHU) Wavelet Transform By Dr. Rajeev Srivastava CSE, IIT(BHU) Dr. Rajeev Srivastava

Its Understanding Wavelet means ‘small wave’. So wavelet analysis is about analyzing signal with short duration finite energy functions. They transform the signal under investigation in to another representation which presents the signal in more useful form. This transformation of the signal is called Wavelet Transform Dr. Rajeev Srivastava

Wavelet Analysis and Synthesis Wavelets : The wavelets are a family of functions generated from a single function by translation and dilation. Have a zero mean. Used for analyzing and representing signals or other functions. Dr. Rajeev Srivastava

A windowing technique with variable-sized regions. Wavelet analysis: A windowing technique with variable-sized regions. Allows the use of long time intervals where we want more precise low-frequency information, and shorter regions where we want high-frequency information. Dr. Rajeev Srivastava

Wavelets……… Translation Dilations(scaling) Time –Frequency plane of Discrete Wavelet Transform Fourier Transform Dr. Rajeev Srivastava

Wavelets…….. In the time domain we have full time resolution, but no frequency localization or separation. In the Fourier domain we have full frequency resolution but no time separation. In the wavelet domain we have some time localization and some frequency localization. Dr. Rajeev Srivastava

Wavelets……. A set of dilations and translations ψ τ,s (t) of a chosen mother wavelet ψ (t) is used for analysis of a signal. The general form of wavelets :   Where s is the scaling (dilations) factor and τ is the translation (location) factor. Manipulating wavelets by translation ( change the central position of the wavelet along the time axis) and scaling ( change the locations or levels). The forward wavelet transform (Analysis Part), calculates the contribution (wavelet coefficients, denoted as C τ,s )of each dilated and translated version of the mother wavelet in the original data set.Wavelet transform is defined as Dr. Rajeev Srivastava

Wavelets…… Inverse wavelet transform (Synthesis Part) uses the computed wavelet coefficients and superimposes them to calculate the original data set. Discrete Wavelet Transform(DWT) The scale and translate parameters are chosen such that the resulting wavelet set forms an orthogonal set. Dilation factors are chosen to be powers of 2. A common choice for τ and s is τ =2m , s=n.2 m where n, m εZ i.e. Where m is the scaling factor and n is the translation factor . Dr. Rajeev Srivastava

Advantages of Using Wavelets: Wavelet Transform and its inverse Discrete Wavelet Transform (DWT) Forward (DWT) and inverse transforms (IDWT) are then calculated using the following equations: f(t)=original signal, 𝐶 𝑚,𝑛 =Wavelet coefficient, =mother wavelet 𝜑 𝑚,𝑛 . Advantages of Using Wavelets: Good de-correlating behavior Easily detect local features in a signal Based on multi-resolution analysis. Fast and stable algorithms are available to calculate the Discrete Wavelet Transform and its inverse Dr. Rajeev Srivastava

A two-band filter bank for 1D sub-band coding and decoding Wavelets and its relation to Sub-band coding y0(n) h0(n) LPF h1(n) HPF 2↓ 2↑ g0(n) g1(n) + x’(t) x(t) y1(n) A two-band filter bank for 1D sub-band coding and decoding |H0(ω)| |H1(ω)| LOW BAND HIGH BAND 0 π/2 π ω Spectrum splitting properties of sub-band coding and decoding NOTE: y0(n) is approximation part of x(n) and y1(n) is detail part of x(n) Dr. Rajeev Srivastava

Splitting the signal spectrum with an iterated filter bank. Dr. Rajeev Srivastava

Wavelets….. The Z-Transform of sequence x(n) for n=0,1,2,3,…. is Where z is a complex variable .If , above equation becomes DFT. Basic advantage of using Z-Transform is that it easily handles the sampling rate changes . Down Sampling by a factor of 2 in the time domain corresponds to the simple Z-domain operation: Up Sampling by a factor of 2 is defined as: for n=0,2,4,….. Otherwise Dr. Rajeev Srivastava

Perfect Reconstruction Filter Bank The following figure illustrates the decomposition and reconstruction process. The filter bank is said to be a perfect reconstruction filter bank when a2 = a0 . If, additionally, h1 = h2 and g1 = g2, the filters are called conjugate mirror filters Dr. Rajeev Srivastava

A 2D ,four –band filter bank for sub-band Image Coding (Analysis Part) h0(m) h1 (m) 2↓ h0(n) h1 (n) a(m,n) Columns (along n) Rows (along m) dV(m,n) x(m,n) dH(m,n) dD (m,n) Dr. Rajeev Srivastava

Representation of Spatial and Frequency Hierarchies Spatial Hierarchy for 2D Image Dr. Rajeev Srivastava

Frequency hierarchy for a two level 2D DWT decomposition Representation of frequency hierarchies Frequency hierarchy for a two level 2D DWT decomposition Frequency hierarchy for a two level full 2D WPT decomposition Dr. Rajeev Srivastava

Example of an 128x128 image at different levels of decompositions by 2D DWT Dr. Rajeev Srivastava

Dr. Rajeev Srivastava Applications Image Compression Image De-noising Image Enahancement Image Segmentation Etc. Dr. Rajeev Srivastava

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