Chapter 7 Wavelets and Multiresolution Processing

Chapter 7 Wavelets and Multiresolution Processing

Background Multiresolution Expansions Wavelet Transforms in One Dimension Wavelet Transforms in Two Dimensions

Image Pyramids Subband Coding The Haar Transform

The total number of elements in a P+ level pyramid for P> is 4 N 4 4 4 3 N + + + + P

Image Pyramids Subband Coding The Haar Transform

The Z-transform a generaliation of the discrete Fourier transform is the ideal tool for studying discrete-time sampled-data systems. The Z-transform of sequemce xn for n is Where is a complex variable. X x n n

Downsampling by a factor of in the time domain corresponds to the simple Z-domain operation [ ] / / x down n xn X down X + X 7.- Upsampling-again by a factor of ---is defined by the transform pair x up x n / n 4... n otherwise X up x 7.-3

If sequence xn is downsampled and subsequently upsampled to yield x n Eqs.7.- and 7.-3 combine to yield X [ X + X ] [ ] where x n Z X is the downsampledupsampled sequence. Its inverse Z-transform is Z [ ] n X x n

We can express the system s output as The output of filter is defined by the transform pair [ ] [ ] X H X H G X H X H G X + + + h n k X H k x k n h n x n h

As with Fourier transform convolution in the time or spatial domain is equivalent to multiplication in the Z-domain. [ ] [ ] X G H G H X G H G H X + + +

For error-free reconstruction of the input and. Thus we impose the following conditions: To get Where denotes the determinant of. n x n x X X + + G H G H G H G H det H H H G G m det H m H m

Letting and taking the inverse Z- transform we get Letting and taking the inverse Z- transform we get det + k m H α α n h n g n h n g n n + α n h n g n h n g n n +

Three general solution: Quadrature mirror filters OMFs Conugate quadrature filters CQFs Orthonormal

Image Pyramids Subband Coding The Haar Transform

The Haar transform can be expressed in matrix form T T HFH Where F is an N*N image matrix H is an N*N transformation matrix T is the resulting N*N transform.

For the Haar transform transformation matrix H contains the Haar basis functions h k.they are defined over the continuous closed interval [ ] for n k N- where N. To generate H we define the integer k such that P k + q q p where p n or for. p for p q

Then the Haar basis functions are and [ ] N h h [ ] < < / /.5 /.5 / / / otherwise q q q q N h h p p p p p p pq k

The ith row of an N*N Haar transformation matrix contains the elements of h i for / N/ N / N... N / N. If N4 for example kq and p assume the values k 3 p q

The 4*4 transformation matrix H 4 is H 4 4

Background Multiresolution Expansions Wavelet Transforms in One Dimension Wavelet Transforms in Two Dimensions

Series Expansion Scaling Functions Wavelet Functions

A signal of function fx can often be better analyed as a linear combination of expansion functions f x α kϕ k x k k is an interger index of the finite or infinite sum; α k are real-valued expansion coefficients; ϕ k x are real-valued expansion functions.

These coefficients are computed by taking ~ the integral inner products of the dual ϕ k x s and function fx. That is α k ~ ϕ k x f x ~ ϕ * k x f x dx

Series Expansion Scaling Functions Wavelet Functions

The set of expansion functions composed of integer translations and binary scaling of the real square-integrable function ϕx ; that is the set { ϕ k x } where / ϕ k x ϕ x k

V { ϕ } Span x k If f x V it can be written k f x α kϕ k x k We will denote the subspace spanned over k for any as V Span k { ϕ } k x

The simple scaling function in the preceding example obeys the four fundamental requirements of multiresolution analysis: MRA Requirement : The scaling function is orthogonal to its integer translates; MRA Requirement : The subspaces spanned by the scaling function at low scales are nested within those spanned at higher scales.

MRA Requirement 3: The only function that is common to all V is fx. MRA Requirement 4: Any function can be represented with arbitrary precision.

Series Expansion Scaling Functions Wavelet Functions

Given a scaling function that meets the MRA requirements of the previous section we can define a wavelet function ψ x that together with its integer translates and binary scaling spans the difference between any two adacent scaling subspaces V and V +. We define the set { ψ k x } of wavelets { / ψ x ψ x } k k

As with scaling functions we write W Span k { ψ } k x And note that if f x W f x α kψ k x k The scaling and wavelet function subspaces are related by V + V W

We can now express the space of all measurable square-integrable functions as or L R V W W... L R V W W...

The Haar wavelet function is ψ x x <.5.5 x < elsewhere

Background Multiresolution Expansions Wavelet Transforms in One Dimension Wavelet Transforms in Two Dimensions

The Wavelet Series Expansions The Discrete Wavelet Transform The Continuous Wavelet Transform

Defining the wavelet series expansion of function f x L R relative to wavelet ψ x and scaling function ϕx. fx can be written as f x c k ϕ + k x d k ψ k k x c k' s : the approximation or scaling coefficients; d k' s : the detail or wavelet coefficients.

If the expansion functions form an orthonormal basis or tight frame the expansion coefficients are calculated as c k f x ϕ x k f x k x dx ϕ and d k f x ψ x k f x ψ k x dx

The Wavelet Series Expansions The Discrete Wavelet Transform The Continuous Wavelet Transform

If the function being expanded is a sequence of numbers like samples of a continuous function fx the resulting coefficients are called the discrete wavelet transformdwt of fx. and x k x x f M k W ϕ ϕ x k x x f M k W ψ ψ + k k k k x k W M x k W M x f ψ ϕ ψ ϕ

Consider the discrete function of four points: f f4 f-3 and f3 Since M4 J and with the summations are performed over x3 and k for or k for.

We find that W ϕ W ψ W ψ W ψ 3 x f x ϕ x [ + 4 3 + ] [ + 4 3 + ] 4 [ + 4 3 + ].5 [ + 4 3 + ].5

[ W ϕ x + W ψ x + W ψ x W ψ ] f x ϕ ψ ψ + ψ x For x3. If x for instance f [ + 4.5.5 ]

The Wavelet Series Expansions The Discrete Wavelet Transform The Continuous Wavelet Transform

The continuous wavelet transform of a continuous square-integrable function fx relative to a real-valued wavelet ψ x is Where W x dx s ψ s τ f x ψ τ ψ s τ x ψ s τ τ s And s and are called scale and translation parameters. x

Given W ψ s τ fx can be obtained using the inverse continuous wavelet transform f Where x C C ψ ψ ψ s τ x s τ dτds s Wψ Ψ u du u And Ψu is the Fourier transform of. ψ x

The Mexican hat wavelet ψ x π 3 / 4 x e x /

Background Multiresolution Expansions Wavelet Transforms in One Dimension Wavelet Transforms in Two Dimensions

In two dimensions a two-dimensional scaling function ϕ x y and three two-dimensional H V V wavelet ψ x y ψ x y and ψ x y are required.

Excluding products that produce onedimensional results like ϕ x ψ x the four remaining products produce the separable scaling function ϕ x y ϕ x ϕ y And separable directionally sensitive wavelets H ψ x y ψ x ϕ y V ψ x y ϕ x ψ y D ψ x y ψ x ψ y

The scaled and translated basis functions: / n y m x y x m n ϕ ϕ { } D V H i n y m x y x n m i / ψ ψ

The discrete wavelet transform of function fxy of sie M*N is then M x N y m n y x y x f MN n m W ϕ ϕ { } M x N y m n i i D V H i y x y x f MN n m W ψ ψ

Given the and fxy is obtained via the inverse discrete wavelet transform ϕ W i W ψ + D V H i m n i n m i m n n m y x n m W MN y x n m W MN y x f ψ ϕ ψ ϕ

Fig. 7.4 Con t