7.Wavelets and Multiresolution Processing

7.1 Background

7.1.1 Image Pyramids -An image pyramid is collection of decreasing resolution images arranged in the shape of a pyramid

7.1.2 Subband Coding -In subband coding, An image is decomposed into a set of bandlimited components, called subbands. -The decomposition is performed so that the subbands can be reassembled to reconstruct the original image with out error

7.1.3 The Haar Transform -It is basis function, the oldest and simplest known orthonormal wavelets

7.2 Multiresolution Expansions

7.2.1 Series Expansions -Functions are used to encode the difference in information between adjacent approximations -Expansion set , φ(x), is calle a basis for the class of function that can be so expressed -The expressible functions form a function space that is referred to as the closed span of the expansion set, denoted V= span{ φ(x) } -The a coefficient are computed by integral inner products a = < φ(x) ,f(x) > = ∫ φ(x) ,f(x) dx

7.2.2 Scaling Functions -φ(x) is called a scaling function. -By choosing φ(x) , { φj,k(x)} can be made to span L^2 (R) which is the set of all measurable, square -integrable functions -Using the notation of the previous section, we can define that subspace as V = Span {φj,k(x) }

7.2.2 Wavelet Functions -We can define a wave function Ψj,k(x) that together with its integertranslates an binary scaling, spans the difference between any two adjacent scaling subspaces V -We define the set {Ψj,k(x)} of wavelets, {Ψj,k(x)} = 2^j/2 Ψ (2^j x - k) -Wj = span {Ψj,k(x)} -Vj+1 = Vj ⨁ Wj *Oplus is the union of space

7.3 Wavelet Transform in One Dimension

7.3.1 The Wavelet Series Expansions -It’s counter parts in the Fourier domain are the Fourier series expansion

7.3.2 The Discrete Wavelet Transform -It’s counter parts in the Fourier domain are the discrete Fourier transform

7.3.3 The Continuous Wavelet Transform -It’s counter parts in the Fourier domain are the integral Fourier transform

7.4 The Fast Wavelet Transform

-The fast wavelet transform is a computationally efficient implementation of the discrete wavelet transform that exploits a surprising

7.5 Wavelet Transforms in Two Dimensions

-The results yields the scale j+1 approximation, an the process is repeated until the original image is reconstructed

7.6 Wavelet Packets

-The fast wavelet transform decomposes a function into a sum of scaling and wavelet functions whose bandwidths are logarithmically related.