Wavelets Transform & Multiresolution Analysis

Why transform?

Image representation

Noise in Fourier spectrum

Fourier Analysis Breaks down a signal into constituent sinusoids of different frequencies In other words: Transform the view of the signal from time-base to frequency-base.

What’s wrong with Fourier? By using Fourier Transform , we loose the time information : WHEN did a particular event take place ? FT can not locate drift, trends, abrupt changes, beginning and ends of events, etc. Calculating use complex numbers.

Time and Space definition Time – for one dimension waves we start point shifting from source to end in time scale . Space – for image point shifting is two dimensional . Here they are synonyms .

Short Time Fourier Analysis In order to analyze small section of a signal, Denis Gabor (1946), developed a technique, based on the FT and using windowing : STFT

STFT (or: Gabor Transform) A compromise between time-based and frequency-based views of a signal. both time and frequency are represented in limited precision. The precision is determined by the size of the window. Once you choose a particular size for the time window - it will be the same for all frequencies.

What’s wrong with Gabor? Many signals require a more flexible approach - so we can vary the window size to determine more accurately either time or frequency.

What is Wavelet Analysis ? And…what is a wavelet…? A wavelet is a waveform of effectively limited duration that has an average value of zero.

Wavelet's properties Short time localized waves with zero integral value. Possibility of time shifting. Flexibility.

The Continuous Wavelet Transform (CWT) A mathematical representation of the Fourier transform: Meaning: the sum over all time of the signal f(t) multiplied by a complex exponential, and the result is the Fourier coefficients F() .

Wavelet Transform (Cont’d) Those coefficients, when multiplied by a sinusoid of appropriate frequency , yield the constituent sinusoidal component of the original signal:

Wavelet Transform And the result of the CWT are Wavelet coefficients . Multiplying each coefficient by the appropriately scaled and shifted wavelet yields the constituent wavelet of the original signal:

Scaling Wavelet analysis produces a time-scale view of the signal. Scaling means stretching or compressing of the signal. scale factor (a) for sine waves:

Scaling (Cont’d) Scale factor works exactly the same with wavelets:

CWT Reminder: The CWT Is the sum over all time of the signal, multiplied by scaled and shifted versions of the wavelet function Step 1: Take a Wavelet and compare it to a section at the start of the original signal

CWT Step 2: Calculate a number, C, that represents how closely correlated the wavelet is with this section of the signal. The higher C is, the more the similarity.

CWT Step 3: Shift the wavelet to the right and repeat steps 1-2 until you’ve covered the whole signal

CWT Step 4: Scale (stretch) the wavelet and repeat steps 1-3

STFT - revisited Time - Frequency localization depends on window size. Wide window  good frequency localization, poor time localization. Narrow window  good time localization, poor frequency localization.

Wavelet Transform Uses a variable length window, e.g.: Narrower windows are more appropriate at high frequencies Wider windows are more appropriate at low frequencies

What is a wavelet? A function that “waves” above and below the x-axis with the following properties: Varying frequency Limited duration Zero average value This is in contrast to sinusoids, used by FT, which have infinite duration and constant frequency. Sinusoid Wavelet

Types of Wavelets There are many different wavelets, for example: Morlet Daubechies Haar

Basis Functions Using Wavelets Like sin( ) and cos( ) functions in the Fourier Transform, wavelets can define a set of basis functions ψk(t): Span of ψk(t): vector space S containing all functions f(t) that can be represented by ψk(t).

Basis Construction – “Mother” Wavelet The basis can be constructed by applying translations and scalings (stretch/compress) on the “mother” wavelet ψ(t): Example: scale ψ(t) translate

Basis Construction - Mother Wavelet (dyadic/octave grid) scale =1/2j (1/frequency) j k

Continuous Wavelet Transform (CWT) scale parameter (measure of frequency) scale =1/2j (1/frequency) translation parameter (measure of time) Forward CWT: mother wavelet (i.e., window function) normalization constant

Illustrating CWT Take a wavelet and compare it to a section at the start of the original signal. Calculate a number, C, that represents how closely correlated the wavelet is with this section of the signal. The higher C is, the more the similarity.

Illustrating CWT (cont’d) 3. Shift the wavelet to the right and repeat step 2 until you've covered the whole signal.

Illustrating CWT (cont’d) 4. Scale the wavelet and go to step 1. 5. Repeat steps 1 through 4 for all scales.

Visualize CTW Transform Wavelet analysis produces a time-scale view of the input signal or image.

Continuous Wavelet Transform (cont’d) Forward CWT: Inverse CWT: Note the double integral!

Fourier Transform vs Wavelet Transform weighted by F(u)

Fourier Transform vs Wavelet Transform weighted by C(τ,s)

Properties of Wavelets Simultaneous localization in time and scale - The location of the wavelet allows to explicitly represent the location of events in time. - The shape of the wavelet allows to represent different detail or resolution.

Properties of Wavelets (cont’d) Sparsity: for functions typically found in practice, many of the coefficients in a wavelet representation are either zero or very small.

Properties of Wavelets (cont’d) Adaptability: Can represent functions with discontinuities or corners more efficiently. Linear-time complexity: many wavelet transformations can be accomplished in O(N) time.

Discrete Wavelet Transform (DWT) (forward DWT) (inverse DWT) where

DFT vs DWT DFT expansion: DWT expansion one parameter basis or two parameter basis

Multiresolution Representation Using Wavelets fine details wider, large translations j coarse details

Multiresolution Representation Using Wavelets fine details j coarse details

Multiresolution Representation Using Wavelets fine details narrower, small translations j coarse details

Multiresolution Representation Using Wavelets high resolution (more details) j … low resolution (less details)

Pyramidal Coding - Revisited Approximation Pyramid (with sub-sampling)

Pyramidal Coding - Revisited (details) Prediction Residual Pyramid (details) (with sub-sampling) reconstruct Approximation Pyramid

Efficient Representation Using “Details” details D3 details D2 details D1 L0 (without sub-sampling)

Efficient Representation Using Details (cont’d) representation: L0 D1 D2 D3 in general: L0 D1 D2 D3…DJ A wavelet representation of a function consists of a coarse overall approximation detail coefficients that influence the function at various scales

Reconstruction (synthesis) H3=H2 & D3 details D3 H2=H1 & D2 details D2 H1=L0 & D1 details D1 L0 (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: (with sub-sampling) L0 D1 D2 D3

Example - Haar Wavelets (cont’d) Start by averaging and subsampling the pixels together (pairwise) to get a new lower resolution image: To recover the original four pixels from the two averaged pixels, store some detail coefficients. [9 7 3 5] 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) The original image can be reconstructed by adding or subtracting the detail coefficients from the lower-resolution representations. L0 D1 D2 D3 2 1 -1 [6]

Example - Haar Wavelets (cont’d) Detail coefficients become smaller and smaller scale decreases. Dj How should we compute the detail coefficients Dj ? Dj-1 D1 L0

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

Multiresolution Conditions (cont’d) Vj+1: span of ψ(2j+1t - k): Vj: span of ψ(2jt - k):

Multiresolution Conditions (cont’d) Nested Spaces ψ(t - k) ψ(2t - k) ψ(2jt - k) … V0 V1 Vj j=0 j=1 j if f(t) ϵ V j then f(t) ϵ V j+1

How to compute Dj ? IDEA: Define a set of basis functions that span the difference between Vj+1 and Vj

How to compute Dj ? (cont’d) Let Wj be the orthogonal complement of Vj in Vj+1 Vj+1 = Vj + Wj

How to compute Dj ? (cont’d) If f(t) ϵ Vj+1, then f(t) can be represented using basis functions φ(t) from Vj+1: Vj+1 Alternatively, f(t) can be represented using two sets of basis functions, φ(t) from Vj and ψ(t) from Wj: Vj+1 = Vj + Wj

How to compute Dj ? (cont’d) Think of Wj as a means to represent the parts of a function in Vj+1 that cannot be represented in Vj Vj+1 differences between Vj and Vj+1 Vj Wj

How to compute Dj ? (cont’d)  using recursion on Vj: Vj+1 = Vj + Wj Vj+1 = Vj-1+Wj-1+Wj = …= V0 + W0 + W1 + W2 + … + Wj if f(t) ϵ Vj+1 , then: V0 W0, W1, W2, … basis functions basis functions

Summary: wavelet expansion (Section 7.2) Wavelet decompositions involve a pair of waveforms: encodes low resolution info encodes details or high resolution info φ(t) ψ(t) Terminology: scaling function wavelet function

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

1D Haar Wavelets (cont’d) Let’s consider the spaces corresponding to different resolution 1D images: . .. …. 1-pixel 2-pixel 4-pixel (j=0) (j=1) (j=2) V0 V1 V2 etc.

1D Haar Wavelets (cont’d) j=0 V0 represents the space of 1-pixel (20-pixel) images Think of a 1-pixel image as a function that is constant over [0,1) Example: width: 1 0 1

1D Haar Wavelets (cont’d) j=1 V1 represents the space of all 2-pixel (21-pixel) images Think of a 2-pixel image as a function having 21 equal-sized constant pieces over the interval [0, 1). Example: 0 ½ 1 width: 1/2 Note that: e.g., + =

1D Haar Wavelets (cont’d) V j represents all the 2j-pixel images Functions having 2j equal-sized constant pieces over interval [0,1). Vj-1 Example: width: 1/2j Note that: ϵ Vj V1 ϵ Vj width: 1/2j-1 width: 1/2

1D Haar Wavelets (cont’d) V0, V1, ..., Vj are nested i.e., Vj … V1 V0 fine details coarse info

Define a basis for Vj Scaling function: Let’s define a basis for V j : Note new notation:

Define a basis for Vj (cont’d) width: 1/20 width: 1/2 width: 1/22 width: 1/23

Define a basis for Wj Wavelet function: Let’s define a basis ψ ji for Wj : Note new notation:

Define basis for Wj (cont’d) Note that the dot product between basis functions in Vj and Wj is zero!

Basis for Vj+1 Basis functions ψ ji of W j Basis functions φ ji of V j form a basis in V j+1

Define a basis for Wj (cont’d) V3 = V2 + W2

Define a basis for Wj (cont’d) V2 = V1 + W1

Define a basis for Wj (cont’d) V1 = V0 + W0

Example - Revisited f(x)= V2

Example (cont’d) f(x)= φ2,0(x) φ2,1(x) φ2,2(x) φ2,3(x) V2

Example (cont’d) φ1,0(x) φ1,1(x) ψ1,0(x) ψ1,1(x) V1 and W1 V2=V1+W1 (divide by 2 for normalization) V1 and W1 φ1,0(x) φ1,1(x) ψ1,0(x) ψ1,1(x) V2=V1+W1

Example (cont’d)

Example (cont’d) φ0,0(x) ψ0,0(x) ψ1,0 (x) ψ1,1(x) V0 ,W0 and W1 (divide by 2 for normalization) V0 ,W0 and W1 φ0,0(x) ψ0,0(x) ψ1,0 (x) ψ1,1(x) V2=V1+W1=V0+W0+W1

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) h0(-n) is a lowpass filter and h1(-n) is a highpass filter

Example (revisited) [9 7 3 5] (9+7)/2 (3+5)/2 (9-7)/2 (3-5)/2 LP HP low-pass, down-sampling high-pass, down-sampling (9+7)/2 (3+5)/2 (9-7)/2 (3-5)/2 LP HP

Filter banks (cont’d) Next level: h0(-n) is a lowpass filter and h1(-n) is a highpass filter

Example (revisited) [9 7 3 5] LP HP (8+4)/2 (8-4)/2 low-pass, down-sampling high-pass, down-sampling LP HP (8+4)/2 (8-4)/2

Convention for illustrating 1D Haar wavelet decomposition x x x x x x … x x average (LP) … detail (HP) … re-arrange: … … LP HP … re-arrange:

Examples of lowpass/highpass (analysis) filters Haar h1 h0 Daubechies h1

Filter banks (cont’d) The higher resolution coefficients can be calculated from the lower resolution coefficients using a similar structure. Subband encoding (synthesis) g0(n) is a lowpass filter and g1(n) is a highpass filter

Filter banks (cont’d) Next level: g0(n) is a lowpass filter and g1(n) is a highpass filter

Examples of lowpass/highpass (synthesis) filters Haar (same as for analysis): g1 g0 Daubechies: g1 +

2D Haar Wavelet Transform The 2D Haar wavelet decomposition can be computed using 1D Haar wavelet decompositions. i.e., 2D Haar wavelet basis is separable We’ll discuss two different decompositions (i.e., correspond to different basis functions): Standard decomposition Non-standard decomposition

Standard Haar wavelet decomposition Steps: (1) Compute 1D Haar wavelet decomposition of each row of the original pixel values. (2) Compute 1D Haar wavelet decomposition of each column of the row-transformed pixels.

Standard Haar wavelet decomposition (cont’d) average detail (1) row-wise Haar decomposition: re-arrange terms … … x x x … x … … . x x x ... x … … … . … … . …

Standard Haar wavelet decomposition (cont’d) average (1) row-wise Haar decomposition: detail row-transformed result … … … … … … . … … . …

Standard Haar wavelet decomposition (cont’d) average detail (2) column-wise Haar decomposition: row-transformed result column-transformed result … … … … … … … . … … . … …

Example … … … . row-transformed result … … … . re-arrange terms

Example (cont’d) column-transformed result … … … … . …

Standard Haar wavelet decomposition (cont’d)

What is the 2D Haar basis for the standard decomposition? To construct the standard 2D Haar wavelet basis, consider all possible outer products of the1D basis functions. φ0,0(x) ψ0,0(x) ψ1,0(x) ψ1,1(x) Example: V2=V0+W0+W1

What is the 2D Haar basis for the standard decomposition? To construct the standard 2D Haar wavelet basis, consider all possible outer products of the1D basis functions. φ00(x), φ00(x) ψ00(x), φ00(x) ψ10(x), φ00(x) Notation:

What is the 2D Haar basis for the standard decomposition? Notation: V2

Non-standard Haar wavelet decomposition Alternates between operations on rows and columns. (1) Perform one level decomposition in each row (i.e., one step of horizontal pairwise averaging and differencing). (2) Perform one level decomposition in each column from step 1 (i.e., one step of vertical pairwise averaging and differencing). (3) Rearrange terms and repeat the process on the quadrant containing the averages only.

Non-standard Haar wavelet decomposition (cont’d) one level, horizontal Haar decomposition: one level, vertical Haar decomposition: … … x x x … x … … . x x x . . . x … … … … . … … . … …

Non-standard Haar wavelet decomposition (cont’d) re-arrange terms … … one level, vertical Haar decomposition on “green” quadrant one level, horizontal Haar decomposition on “green” quadrant … … . … … … … … … … . …

Example re-arrange terms … … … … . … …

Example (cont’d) … … … … . …

Non-standard Haar wavelet decomposition (cont’d)

What is the 2D Haar basis for the non-standard decomposition? Define 2D scaling and wavelet functions: Apply translations and scaling:

What is the 2D Haar basis for the non-standard decomposition? Notation: V2

2D Subband Coding Three sets of detail coefficients (i.e., subband coding) LL: average LL Detail coefficients LH: intensity variations along columns (horizontal edges) HL: intensity variations along rows (vertical edges) HH: intensity variations along diagonal

2D Subband Coding LL LH HL HH

Wavelets Applications Noise filtering Image compression Special case: fingerprint compression Image fusion Recognition G. Bebis, A. Gyaourova, S. Singh, and I. Pavlidis, "Face Recognition by Fusing Thermal Infrared and Visible Imagery", Image and Vision Computing, vol. 24, no. 7, pp. 727-742, 2006. Image matching and retrieval Charles E. Jacobs Adam Finkelstein David H. Salesin, "Fast Multiresolution Image Querying", SIGRAPH, 1995.