Time vs. frequency resolution Each coefficient of a Fourier spectra - - PowerPoint PPT Presentation

Image Analysis

Wavelets and Multiresolution Processing Niclas Börlin niclas.borlin@cs.umu.se

Department of Computing Science Umeå University

February 20, 2009

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 1 / 24

Time vs. frequency resolution

Each coefficient of a Fourier spectra of a signal (image) provides information about the signal contents of that frequency. However, the Fourier coefficients contain no spatial knowledge. On the other hand, each spatial coefficient, i.e. sample, contains no frequency information. In image analysis, different frequencies correspond to objects of different sizes. Low frequencies correspond to slow changes over a large area. High frequencies correspond to fast changes over a small area. Wavelets allow us to analyse a combination of spatial and frequency information.

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 2 / 24

The Fourier vs. Wavelet transformation

The Fourier transform uses sinusoids of infinite duration as basis functions. Wavelet transforms uses small waves (wavelets) of limited duration as basis functions. The Fourier transform gives information about images frequency decomposition. The Wavelet transforms have resolution in the frequency domain as well as in the spatial domain (what frequencies are in the image — and where). Wavelets are conceptually similar to musical scores:

◮ Which tones, and ◮ when to play them? Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 3 / 24

The Image Pyramid

A closely related concept is the Image pyramid. Images of different spatial resolutions, form levels of the pyramid. The original image with the highest resolution is at level J, the lowest pyramid level. Each higher level contain a lower resolution image, usually half. The image at the top Level 0 contain only one pixel. The total number of pixels in a P + 1 level pyramid is N2

• 1 +

1 (4)1 + 1 (4)2 + · · · + 1 (4)P

• ≤ 4

3N2.

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 4 / 24

The Approximation and Prediction Residual Pyramids

Level j of the Approximation Pyramid is formed by smoothing and downsampling the image at level j + 1. Level j of the Prediction Residual Pyramid is formed as the difference between the upsampled and interpolated level j − 1 image, and the level j approximation image.

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 5 / 24

The Approximation and Prediction Residual Pyramids

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 6 / 24

Subband coding and filter banks

The concept of subband coding is to split a signal into two (or more) components that completely describes the input. This is performed in conjunction with two filter banks. The analysis filter bank uses filters h0(n) and h1(n) to split the input sequence f(n) into two half-length sequences (subbands) flp(n) and fhp(n). The synthesis filter bank g0(n) and g1(n) combines flp(n) and fhp(n) to form the reconstructed signal ˆ f(n).

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 7 / 24

Perfect reconstruction

The lowpass filter h0(n) produce the approximation subband flp(n). The highpass filter h1(n) produce the detail subband fhp(n). If the filter banks are chosen properly, the signal can be perfectly reconstructed from its subbands flp(n) and fhp(n). If the analysis filter bank is recursively applied to the approximation subband, we obtain an approximation pyramid.

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 8 / 24

2D subbands

Subband coding for a 2D signal is separable and may be applied sequentially along the rows and columns. Furthermore, the downsampling can be applied once after each analysis stage, significantly improving performance. The result is four subbands: The approximation a(m, n), the vertical detail dV(m, n), the horizontal detail dH(m, n), and the diagonal detail dD(m, n).

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 9 / 24

2D subband example

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 10 / 24

Multiresolution analysis

Multiresolution analysis (MRA) uses a scaling function ϕ(x) to create a series of approximations of a signal or image, each differing by a factor of 2 from its nearest neighboring approximation. Wavelet functions ψ(x) are then used to encode the difference (detail) in information between adjacent approximations. MRA defines a set of requirements for the scaling functions. Given a scaling function that meets these requirements we define a wavelet function to use.

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 11 / 24

The Haar Transform

The Haar transform basis functions are the oldest and simplest known orthonormal wavelets. The Haar transform is separable and expressible in matrix form T = HFHT, where F is an N × N image, H is an N × N transformation matrix that contains the Haar basis functions, and T is the resulting N × N transform. The basis functions are scaled and translated versions of a mother wavelet.

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 12 / 24

The Haar Transform

Assume N = 2n, 0 ≤ p ≤ n − 1, k = 2p + q − 1, and q =

• 0 or 1,

p = 0 1 ≤ q ≤ 2p, p = 0 . Then the Haar basis functions hk(z) are defined as h0(z) = h00(z) = 1 √ N , z ∈ [0, 1], and hk(z) = hpq(z) = 1 √ N    2p/2 (q − 1)/2p ≤ z < (q − 0.5)/2p, −2p/2 (q − 0.5)/2p ≤<< q/2p,

• therwise, z ∈ [0, 1].

We get H2 = 1 √ 2 1 1 1 −1

• and H4 =

1 √ 4     1 1 1 1 1 1 −1 −1 √ 2 − √ 2 √ 2 − √ 2    

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 13 / 24

The Haar scaling and wavelet functions

The Haar transform correspond to the Haar scaling function ϕ(x) =

• 1

0 ≤ x < 1

• therwise

and the Haar wavelet function ψ(x) =    1 0 ≤ x < 0.5 −1 0.5 ≤ x < 1

• therwise

. The shifted and scaled versions of the Haar functions are defined as ϕj,k(x) = 2j/2ϕ(2jx − k), ψj,k(x) = 2j/2ψ(2jx − k), where k determines the position of the functions and j determines the width. The value of j corresponds to the layer in the image pyramid. The wavelet function ψ(x) corresponds to the difference between layer j and j + 1.

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 14 / 24

The Haar scaling and wavelet functions

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 15 / 24

The Discrete 1D wavelet transform

Wϕ(j0, k) = 1 √ M

• x

f(x)ϕj0,k(x) Wψ(j, k) = 1 √ M

• x

f(x)ψj,k(x) f(x) = 1 √ M

• k

Wϕ(j0, k)ϕj0,k(x) + 1 √ M

∞

• j=j0
• k

Wψ(j, k)ψj,k(x), where j0 = 0, M = 2J, x = 0, 1, . . . , M − 1, j = 0, 1, . . . , J − 1, k = 0, 1, . . . , 2j − 1. The functions ϕ(x) and ψ(x) are assumed to form an

• rthonormal base (or tight frame).

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 16 / 24

Time-frequency resolution

Each tile represent an equal portion of the time-frequency plane. There is a trade-off between the resolution in time and frequency. Higher resolution in frequency (low frequencies) ⇔ lower resolution in time. Higher resolution in time (high frequencies) ⇔ lower resolution in frequency.

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 17 / 24

The Discrete 2D wavelet transform

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 18 / 24

The Fast Wavelet Transform (FWT)

The Fast Wavelet Transform (FWT) is essentially a recursive interleaving application of the smoothing (ϕ(x)) and differencing (ψ(x)) filters and the downsampling. The first application corresponds to the highest frequencies (narrowest wavelets). Each successive downsampling corresponds to widening of the wavelet by a factor of two.

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 19 / 24

2D discrete wavelet transform using Haar basis

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 20 / 24

Multiscale DWT

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 21 / 24

Wavelet edge detection

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 22 / 24

Wavelet noise removal

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 23 / 24

Other wavelets

The optimal wavelet function is called Daubechies and have self-similar, i.e. fractal, characteristics.

Niclas Börlin (CS, UmU) Wavelets and Multiresolution Processing February 20, 2009 24 / 24