Short Time Fourier Transform • Time/Frequency localization depends on window size. • Once you choose a particular window size, it will be the same for all frequencies. • Many signals require a more flexible approach - vary the window size to determine more accurately either time or frequency.

The Wavelet Transform • Overcomes the preset resolution problem of the STFT by using a variable length window: – Narrower windows are more appropriate at high frequencies (better time localization) – Wider windows are more appropriate at low frequencies (better frequency localization)

The Wavelet Transform (cont’d) Wide windows do not provide good localization at high frequencies.

The Wavelet Transform (cont’d) A narrower window is more appropriate at high frequencies.

The Wavelet Transform (cont’d) Narrow windows do not provide good localization at low frequencies.

The Wavelet Transform (cont’d) A wider window is more appropriate at low frequencies.

What is a wavelet? • It is a function that “ waves ” above and below the x-axis; it has (1) varying frequency, (2) limited duration, and (3) an average value of zero. • This is in contrast to sinusoids, used by FT, which have infinite energy. Sinusoid Wavelet

Wavelets • Like sine and cosine functions in FT, wavelets can define basis functions ψ k (t): f t ( ) a ( ) t = ∑ ψ k k k • Span of ψ k (t): vector space S containing all functions f(t) that can be represented by ψ k (t).

Wavelets (cont’d) • There are many different wavelets: Morlet Haar Daubechies

Basis Construction - Mother Wavelet jk t ( ) = ψ

Basis Construction - Mother Wavelet (cont’d) j /2 ( j ) jk t ( ) 2 2 t k ψ = ψ − j scale =1/2 j (1/frequency) k

Discrete Wavelet Transform (DWT) * a f t ( ) ( ) t = ∑ (forward DWT) ψ jk jk t = ∑∑ f t ( ) a ( ) t ψ (inverse DWT) jk jk k j j /2 ( j ) jk t ( ) 2 2 t k ψ = ψ − where

DFT vs DWT • FT expansion: one parameter basis f t ( ) a ( ) t = ∑ ψ or l l l two parameter basis • WT expansion f t ( ) a ( ) t = ∑∑ ψ jk jk k j

Multiresolution Representation using jk t ( ) ψ high resolution f t ( ) (more details) ˆ ( ) f t 1 j ˆ ( ) f t 2 … low resolution ˆ ( ) f t (less details) s f t ( ) a ( ) t = ∑ ∑ ψ jk jk k j

Prediction Residual Pyramid - Revisited • In the absence of quantization errors, the approximation pyramid can be reconstructed from the prediction residual pyramid. • Prediction residual pyramid can be represented more efficiently. (with sub-sampling)

Efficient Representation Using “ Details ” details D 3 details D 2 details D 1 L 0 (without sub-sampling)

Efficient Representation Using Details (cont’d) representation: L 0 D 1 D 2 D 3 in general : L 0 D 1 D 2 D 3 …D J A wavelet representation of a function consists of (1) a coarse overall approximation (2) detail coefficients that influence the function at various scales.

Reconstruction (synthesis) H 3 =H 2 +D 3 H 2 =H 1 +D 2 details D 3 details D 2 H 1 =L 0 +D 1 details D 1 L 0 (without sub-sampling)

Example - Haar Wavelets • Suppose we are given a 1D "image" with a resolution of 4 pixels: [9 7 3 5] • The Haar wavelet transform is the following: L 0 D 1 D 2 D 3 (with sub-sampling)

Example - Haar Wavelets (cont’d) • Start by averaging the pixels together (pairwise) to get a new lower resolution image: [9 7 3 5] • To recover the original four pixels from the two averaged pixels, store some detail coefficients. 1

Example - Haar Wavelets (cont’d) • Repeating this process on the averages (i.e., low resolution image) gives the full decomposition: 1 Harr decomposition:

Example - Haar Wavelets (cont’d) • We can reconstruct the original image by adding or subtracting the detail coefficients from the lower- resolution versions. 1 -1 2

Example - Haar Wavelets (cont’d) Detail coefficients become smaller and smaller as j increases. D j D j-1 How to compute D i ? D 1 L 0

Multiresolution Conditions • If a set of functions can be represented by a weighted sum of ψ (2 j t - k), then a larger set , including the original, can be represented by a weighted sum of ψ (2 j +1 t - k): high resolution j low resolution

Multiresolution Conditions (cont’d) • If a set of functions can be represented by a weighted sum of ψ (2 j t - k), then a larger set, including the original, can be represented by a weighted sum of ψ (2 j +1 t - k): f t ( ) a ( ) t = ∑ ψ V j : span of ψ (2 j t - k): j k jk k f ( ) t b ( ) t = ∑ V j+1 : span of ψ (2 j+1 t - k): ψ j 1 k ( j 1) k + + k V V + ⊆ j j 1

Nested Spaces V j V j : space spanned by ψ (2 j t - k) Basis functions: ψ ( t - k) f t ( ) a ( ) t = ∑ ∑ ψ V 0 f(t) ϵ V j jk jk k j ψ (2 t - k) V 1 … V j ψ (2 j t - k) V V + Multiresolution conditions à nested spanned spaces: ⊂ j j 1 i.e., if f(t) ϵ V j then f(t) ϵ V j+1

How to compute D i ? (cont’d) f t ( ) a ( ) t = ∑ ∑ ψ f(t) ϵ V j jk jk k j IDEA: define a set of basis functions that span the difference between V j+1 and V j

Orthogonal Complement W j • Let W j be the orthogonal complement of V j in V j+1 V j+1 = V j + W j

How to compute D i ? (cont’d) If f(t) ϵ V j+1 , then f(t) can be represented using basis functions φ (t) from V j+1 : j 1 f t ( ) c (2 + t k ) ∑ = ϕ − V j+1 k k Alternatively, f(t) can be represented using two sets of basis functions, φ (t) from V j and ψ (t) from W j : V j+1 = V j + W j j j f t ( ) c (2 t k ) d (2 t k ) ∑ ∑ = ϕ − + ψ − k jk k k

How to compute D i ? (cont’d) Think of W j as a means to represent the parts of a function in V j+1 that cannot be represented in V j j 1 f t ( ) c (2 t k ) + ∑ = ϕ − k k differences j j f t ( ) c (2 t k ) d (2 t k ) ∑ ∑ = ϕ − + ψ − k jk between k k V j and V j+1 V j W j

How to compute D i ? (cont’d) • à using recursion on V j : V j+1 = V j + W j V j+1 = V j-1 +W j-1 +W j = …= V 0 + W 0 + W 1 + W 2 + … + W j if f(t) ϵ V j+1 , then: j f t ( ) c ( t k ) d (2 t k ) ∑ ∑∑ = ϕ − + ψ − k jk k k j V 0 W 0 , W 1 , W 2 , … basis functions basis functions

Summary: wavelet expansion (Section 7.2) • Wavelet decompositions involve a pair of waveforms (mother wavelets): encode low encode details or φ (t) ψ (t) resolution info high resolution info j f t ( ) c ( t k ) d (2 t k ) ∑ ∑∑ = ϕ − + ψ − k jk k k j scaling function wavelet function

1D Haar Wavelets • Haar scaling and wavelet functions: φ (t) ψ (t) computes details computes average

1D Haar Wavelets (cont’d) • V 0 represents the space of one pixel images • Think of a one-pixel image as a function that is constant over [0,1) width: 1 Example: 0 1

1D Haar Wavelets (cont’d) • V 1 represents the space of all two-pixel images • Think of a two-pixel image as a function having 2 1 equal-sized constant pieces over the interval [0, 1). width: 1/2 0 ½ 1 Examples: V V ⊂ • Note that 0 1 e.g., + =

1D Haar Wavelets (cont’d) • V j represents all the 2 j -pixel images • Functions having 2 j equal-sized constant pieces over interval [0,1). Examples: width: 1/2 j V V • Note that − ⊂ j 1 j ϵ V j e.g., ϵ V j

1D Haar Wavelets (cont’d) V 0 , V 1 , ..., V j are nested V V + ⊂ i.e., j j 1 V J fine details … V 1 V 0 coarse details

Define a basis for V j • Mother scaling function: 1 0 1 • Let ’ s define a basis for V j : j ( ) x ( ) x ϕ ≡ ϕ note alternative notation: i ji

Define a basis for V j (cont’d)

Define a basis for W j (cont’d) • Mother wavelet function: 1 -1 0 1/2 1 • Let’s define a basis ψ j i for W j : j ( ) x ( ) x note alternative notation: ψ ≡ ψ i ji

Define a basis for W j Note that the dot product between basis functions in V j and W j is zero!

Define a basis for W j (cont’d) Basis functions ψ j i of W j form a basis in V j+1 Basis functions φ j i of V j

Example - Revisited f(x)= V 2

Example (cont’d) (divide by 2 for normalization) V 1 and W 1 φ 1,0 (x) V 2 =V 1 +W 1 φ 1,1 (x) ψ 1,0 (x) ψ 1,1 (x)

Example (cont’d) (divide by 2 for normalization) V 0 ,W 0 and W 1 φ 0,0 (x) V 2 =V 1 +W 1 =V 0 +W 0 +W 1 ψ 0,0 (x) ψ 1,0 (x) ψ 1,1 (x)

Example

Example (cont’d)

Filter banks • The lower resolution coefficients can be calculated from the higher resolution coefficients by a tree- structured algorithm (i.e., filter bank). Subband encoding (analysis) h 0 (-n) is a lowpass filter and h 1 (-n) is a highpass filter

Example (revisited) [9 7 3 5] low-pass, high-pass, down-sampling down-sampling (9+7)/2 (3+5)/2 (9-7)/2 (3-5)/2 V 1 basis functions

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