Wavelets and Multiresolution Processing Thinh Nguyen - PowerPoint PPT Presentation

Wavelets and Multiresolution Processing Thinh Nguyen

Multiresolution Analysis (MRA) Analysis (MRA) Multiresolution � A scaling function scaling function is used to create a series of approximations of a function or image, each differing by a factor of 2 from its neighboring approximations. � Additional functions called wavelets wavelets are then used to encode the difference in information between adjacent approximations.

Series Expansions Series Expansions f x ( ) � Express a signal as = ∑ α ϕ f x ( ) ( ) x k k k expansion functions expansion coefficients ϕ ( ) k x � If the expansion is unique, the are called basis functions basis functions, and the expansion set { } ϕ is called a basis basis k x ( )

Series Expansions Series Expansions � All the functions expressible with this basis form a function space function space which is referred to as the closed span closed span of the expansion set { } = ϕ V Span ( ) x k k ∈ f x ( ) V f x ( ) � If , then is in the closed span { } ϕ k x ( ) of and can be expressed as = ∑ α ϕ ( ) ( ) f x x k k k

Orthonormal Basis Basis Orthonormal � The expansion functions form an orthonormal basis for V ≠ ⎧ 0 j k ϕ ϕ = δ = ⎨ ( ), x ( ) x j k jk = ⎩ 1 j k � The basis and its dual are equivalent, i.e., ϕ = % ϕ ( ) x ( ) x and k k = ∫ ϕ ∗ α = ϕ ( ), x f x ( ) ( ) ( ) x f x dx k k k

Scaling Functions Scaling Functions � Consider the set of expansion functions composed of integer translations and binary scalings of the ϕ ( ) x real square-integrable function defined by } { } { ϕ = ϕ − j / 2 j j k x , ( ) 2 (2 x k ) ϕ ∈ j k ∈ ฀ 2 ฀ ( ) x L ( ) , for all and ϕ ( ) x � By choosing the scaling function wisely, { } 2 ( ) L ฀ ϕ can be made to span , ( ) j k x

} { } { ϕ = ϕ − j / 2 j j k x , ( ) 2 (2 x k ) ϕ j k x , ( ) � Index k determines the position of along the x -axis, index j determines its width; 2 j / 2 controls its height or amplitude. = j j � By restricting j to a specific value the { } o ϕ j k x , ( ) resulting expansion set is a subset of { } o ϕ j k x , ( ) { } = ϕ V Span , ( ) x � One can write j j k o o k

Example: The Haar Haar Scaling Function Scaling Function Example: The ≤ < ⎧ 1 0 1 x ϕ = ⎨ ( ) x ⎩ 0 otherwise ⊂ V V 0 1 = ϕ + ϕ − ϕ f x ( ) 0.5 ( ) x ( ) x 0.25 ( ) x 1,0 1,1 1,4 1 1 ϕ = ϕ + ϕ ( ) x ( ) x ( ) x + 0, k 1,2 k 1,2 k 1 2 2

MRA Requirements MRA Requirements The scaling function is orthogonal to its 1. integer translates The subspaces spanned by the scaling 2. function at low scales are nested within those spanned at higher scales: ⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂ L L V V V V V V −∞ − ∞ 1 0 1 2

Wavelet Functions Wavelet Functions � Given a scaling function which satisfies the MRA requirements, one can define a wavelet wavelet ψ ( ) x function which, together with its integer function translates and binary scalings, spans the difference between any two adjacent scaling V + V subspaces and j j 1

Wavelet Functions Wavelet Functions � Define the wavelet set } { } { ψ = ψ − j / 2 j j k x , ( ) 2 (2 x k ) k ∈ ฀ W for all that spans the spaces j � We write { } = ψ W Span , ( ) x and, if j j k k ∈ f x ( ) W j = ∑ α ψ f x ( ) ( ) x , k j k k

+ = ⊕ + = ⊕ V V W V V W Orthogonality: : Orthogonality 1 j j j 1 j j j � This implies that ϕ ψ = ( ), x ( ) x 0 , , j k j l j k l ∈ ฀ for all appropriate , , � We can write = ⊕ ⊕ ⊕ 2 ฀ L L ( ) V W W 0 0 1 and also = ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2 ฀ L L L ( ) W W W W W − − 2 1 0 1 2 (no need for scaling functions, only wavelets!)

Example: Haar Haar Wavelet Functions in Wavelet Functions in Example: and W W 0 W 1 W 0 and 1 = + f x ( ) f ( ) x f ( ) x a d low frequencies high frequencies

Wavelet Series Expansions Wavelet Series Expansions ∈ 2 ฀ f x ( ) L ( ) � A function can be expressed as ∞ ∑ ∑ ∑ = ϕ + ψ f x ( ) c ( k ) ( ) x d ( ) k ( ) x j j , k j j k , 0 0 = k j j k 0 approximation or scaling coefficients approximation or scaling coefficients = f x ϕ detail or wavelet coefficients detail or wavelet coefficients c ( ) k ( ), k x ( ) j j , = ψ 0 0 d ( ) k f ( ), x ( ) x j j k , = ⊕ ⊕ ⊕ 2 ฀ L L ( ) V W W + j j j 1 0 0 0

Example: The Haar Haar Wavelet Series Wavelet Series Example: The 2 x 2 Expansion of y = x y = Expansion of ⎧ ≤ < 2 x 0 x 1 = ⎨ y � Consider ⎩ 0 otherwise j = � If , the expansion coefficients are 0 0 1 1 1 1 ∫ ∫ = ϕ = = ψ = − 2 2 c (0) x ( x ) dx d (0) x ( ) x dx 0 0 0 , 0 0,0 3 4 0 0 1 1 2 3 2 ∫ ∫ = ψ = − = ψ = − 2 2 d (0) x ( x ) dx d (1) x ( x d ) x 1 1,0 1 1,1 32 32 0 0 ⎡ ⎤ ⎡ ⎤ 1 1 2 3 2 = ϕ + − ψ + − ψ − ψ + ⎢ ⎥ L , ( ) ( ) ( ) ( ) y x x x x ⎢ ⎥ 0 0 0,0 1, 0 1 ,1 ⎣ ⎦ 3 4 ⎣ 3 2 32 ⎦ 1 4 24 3 14 4 244 3 14444 4 244444 3 V W 1444 4 24444 3 0 W 0 1 = ⊕ V V W 1444444444 4 24444444444 3 1 0 0 = ⊕ = ⊕ ⊕ V V W V W W 2 1 1 0 0 1

Example: The Haar Haar Wavelet Series Wavelet Series Example: The 2 x 2 Expansion of y = x y = Expansion of

The Discrete Wavelet Transform (DWT) The Discrete Wavelet Transform (DWT) = − K f x ( ), x 0,1, , M 1 � Let denote a discrete function � Its DWT is defined as 1 ∑ = ϕ approximation approximation W ( j , ) k f x ( ) ( x ) ϕ 0 j , k 0 M coefficients coefficients x 1 ∑ detail detail = ψ ≥ W ( , ) j k f x ( ) ( ) x j j ψ j , k coefficients 0 coefficients M x ∞ 1 1 ∑ ∑ ∑ = ϕ + ψ f x ( ) W ( j , ) k ( ) x W ( , ) j k ( ) x ϕ ψ 0 j k , j k , M 0 M = k j j k 0 = − ⎧ K x 0,1, , M 1, ⎪ j = M = 2 J = − 0 � Let and so that ⎨ K j 0,1, , J 1, o ⎪ = − j K ⎩ k 0,1, ,2 1

Example: Computing the DWT Example: Computing the DWT � Consider the discrete function = = = − = f (0) 1, f (1) 4, f (2) 3, f (3) 0 = = ⎯⎯ → = 2 M 4 2 J 2 � It is � The summations are performed over x = k = j = 0,1,2,3 and for and 0 0 k = j = for 0,1 1 � Use the Haar scaling and wavelet functions

Example: Computing the DWT Example: Computing the DWT 3 1 1 ∑ [ ] = ϕ = ⋅ + ⋅ − ⋅ + ⋅ = (0,0) ( ) ( ) 1 1 4 1 3 1 0 1 1 W f x x ϕ 0, 0 2 2 = x 0 3 1 1 ∑ [ ] = ψ = ⋅ + ⋅ − ⋅ − + ⋅ − = W (0,0) f x ( ) ( ) x 1 1 4 1 3 ( 1 ) 0 ( 1 ) 4 ψ 0,0 2 2 = x 0 3 1 1 ∑ ⎡ ⎤ = ψ = ⋅ + ⋅ − − ⋅ + ⋅ = − W (1,0) f x ( ) ( ) x 1 2 4 ( 2 ) 3 0 0 0 1.5 2 ⎣ ⎦ ψ 1 , 0 2 2 = 0 x 3 1 1 ∑ ⎡ ⎤ = − = ψ = ⋅ + ⋅ − ⋅ + ⋅ − W (1,1) f x ( ) ( ) x 1 0 4 0 3 2 0 ( 2 ) 1 .5 2 ψ ⎣ ⎦ 1, 1 2 2 = x 0

Example: Computing the DWT Example: Computing the DWT � The DWT of the 4-sample function relative to the Haar wavelet and scaling functions thus is { } − − 1,4, 1.5 2, 1.5 2 � The original function can be reconstructed as 1 ⎡ = ϕ + ψ + f x ( ) W (0,0) ( x ) W (0,0) ( ) x ⎣ ϕ ψ 0, 0 0,0 2 ⎤ ψ + ψ W (1 ,0) ( x ) W (1,1 ) ( x ) for ⎦ ψ ψ 1,0 1 ,1 x = 0,1,2,3

Wavelet Transform in 2- - D D Wavelet Transform in 2 � In 2-D, one needs one scaling function ϕ = ϕ ϕ ( , ) x y ( ) ( ) x y and three wavelets ⎧ ψ = ψ ϕ H ( , ) x y ( ) ( ) x y •detects horizontal details ⎪ ψ = ϕ ψ ⎨ V ( , ) x y ( ) ( ) x y •detects vertical details ⎪ ψ = ψ ψ D •detects diagonal details ( , ) x y ( ) ( ) x y ⎩ ϕ (.) is a 1-D scaling function and � ψ (.) is its corresponding wavelet

2- - D DWT: Definition D DWT: Definition 2 � Define the scaled and translated basis functions ϕ = ϕ − − j / 2 j j ( x y , ) 2 (2 x m ,2 y n ) j m n , , { } ψ = ψ − − = i j / 2 i j j ( x , y ) 2 (2 x m ,2 y n ), i H V D , , j m n , , � Then − − M 1 N 1 1 ∑ ∑ = ϕ W ( j , m n , ) f x y ( , ) ( , x y ) ϕ 0 j m , , n 0 MN = = x 0 y 0 − − M 1 N 1 1 ∑ ∑ { } = ψ = i i W ( , j m n , ) f ( x y , ) ( , x y ) , i H V D , , ψ j , m , n MN = = x 0 y 0 1 ∑∑ = ϕ f x y ( , ) W ( j , m n , ) ( , ) x y ϕ 0 j m , , n MN 0 m n ∞ 1 ∑ ∑ ∑∑ + ψ i i W ( , j m n , ) ( x , ) y ψ , , j m n M N = = i H V D j , , j m n 0