Wavelets - The College of Engineering at the University of ...cs6640/Wavelets_2017.pdfWavelets •...

Wavelets

• Unlike the Fourier transform, whose basis functions are sinusoids, wavelet transforms are based on small waves of limited duration

• Applications – Image compression – Image denoising

• Background we need – Multiresolution image pyramids – Subband coding – Multiresolution analysis and scaling functions

Multiresolution

•  Statistics of images such as histograms can vary significantly from one part of the image to another

•  Small objects – Analyze at high-resolution

•  Large objects – Analyze at low-resolution

•  Need to analyze images at multiple resolutions

© 1992–2008 R. C. Gonzalez & R. E. Woods

Image pyramids

N=2j j=log2N

• Level 0 is of little value • Normally truncated at level P

• j=J-P,…,J (P+1 levels)


• Approximation filter • Gaussian • Mean • No filtering

• Interpolation filter • Nearest neighbor • Bilinear interpolation

f2↑ n( ) =f (n / 2) if n is even

0 otherwise"#$

f2↓ n( ) = f (2n)

Approximation and residual pyramids

• Approximation pyramid – 512 x 512 (j=9)

to 64 x 64 (j=6)

• Residual pyramid – 64 x 64

approximation image at top of pyramid, rest are residuals

– Higher compressibility • Fewer bits © 1992–2008 R. C. Gonzalez & R. E. Woods

Reconstruction from residual pyramid


Upsample 64 x 64 approximation image to 128 x 128 Apply interpolation filter to predict 128 x 128 image Add 128 x 128 residual image (Now have 128 x 128 approximation image) Upsample 128 x 128 approximation image from previous step to 256 x 256 Apply interpolation filter to predict 256 x 256 image Add 256 x 256 residual image (Now have 256 x 256 approximation image) Upsample 256 x 256 approximation image from previous step to 512 x 512 Apply interpolation filter to predict 512 x 512 image Add 512 x 512 residual image

Exact reconstruction if the approximation/residual images were not quantized

Functionally related impulse responses

Subband Coding

•  Perfect reconstruction filters:

– Biorthogonal filters •  Two prototyped required • Cross modulation

constraint:

•  Biorthogonality condition

Approximation of f(n)

Detail part of f(n)

g0 n( ) = −1( )n h1 n( )g1 n( ) = −1( )n+1 h0 n( )

f̂ (n) = f (n)


hi 2n− k( ),gj k( ) = δ i− j( )δ n( )

Subband Coding

•  Perfect reconstruction filters:

– Orthonormal filters

• Given a single prototype filter g0, remaining 3 can be computed to satisfy the orthonormality constraints

Approximation of f(n)

Detail part of f(n)

g1 n( ) = −1( )n+1 g0 Keven −1− n( )hi n( ) = gi Keven −1− n( )

f̂ (n) = f (n)


gi n( ),gj n+ 2m( ) = δ i− j( )δ m( )

Daubechies filter

g1 n( ) = −1( )n+1 g0 Keven −1− n( )hi n( ) = gi Keven −1− n( )

Two-dimensional subband coding

• Daubechies orthonormal filters – Prototype:

Approximation

Horizontal detail

Vertical detail

Diagonal detail © 1992–2008 R. C. Gonzalez & R. E. Woods

Reconstruction


1.  Upsample columns of all subimages

2.  Filter a and dH along columns with g0

3.  Filter dV and dD along columns with g1

4.  Result of step 2: upsample rows, filter with g0

5.  Result of step 3: upsample rows, filter with g1

6.  Add result of steps 4 and 5

The Haar basis •  Recursively keep

replacing approximation image with its decomposition – Stop at some level

and keep approximation

•  Properties: – Histograms of all

detail images very similar

– Can reconstruct image at various resolutions

h0 = 1 2 1 2!" #$

h1 = 1 2 −1 2!" #$


Orthonormal Haar basis

Series expansions

• Example Fourier series – Expansion set: Sines and cosines – Span: Periodic functions

f (x) = α kϕk (x)k∑Function

Expansion coefficients

Basis functions

V = Spank

ϕk x( ){ }

Expansion set: The set of basis functions

Span of expansion set: The space of functions expressible with the chosen expansion set

Expansion coefficients

•  Orthonormal basis (basis and dual equivalent):

– Computation of expansion coefficients:

α k = !ϕ(x), f (x) = !ϕk

*(x)∫ f (x)dx

Dual of basis function

f (x) = α kϕk (x)k∑

ϕ j (x),ϕk (x) = ϕ j*(x)ϕk (x)∫ dx = δ jk =

0 j ≠ k1 j = k%&'

α k = ϕk (x), f (x)

Inner product Defn. of inner product

Scaling functions

• Width scaling changes shape

• Amplitude scaling makes sure that

ϕ j ,k x( ) = 2 j /2ϕ 2 j x − k( )

Scaling function: Real, square-integrable prototype function

Amplitude scaling

Scaling of width

Position along x-axis


ϕ j ,k ,ϕ j ,k = 1

Example (Haar)::

Example: Haar scaling function

•  Vj is the span achieved by fixing j and varying k

•  As j increases, the size of Vj increases to include functions with finer detail


ϕ(x) =1 0 ≤ x < 10 otherwise#$%

Shifted j=0,k=1

Shifted & Scaled j=1,k=1

Scaled j=1,k=0

Vj = Spank

ϕ j ,k x( ){ }

f (x) = 0.25ϕ1,0 (x) +ϕ1,1(x) − 0.25ϕ1,4 (x)

Nested function spaces

•  All V0 expansion functions are contained in V1

•  All Vj expansion functions are contained in Vj+1

•  Any function in Vj is also in Vj+1


Shifted j=0,k=1

Shifted & Scaled j=1,k=1

Scaled j=1,k=0

ϕ0,k (x) =12ϕ1,2k (x) +

12ϕ1,2k+1(x)

Multiresolution analysis requirements 1.  The scaling function is orthogonal to its integer

translates 2.  The subspaces spanned by the scaling function

at low scales are nested within those spanned at higher scales

3.  The only function common to all Vj is f(x)=0 4.  Any square integrable function can be

represented with arbitrary precision: •  Haar scaling function obeys all these

requirements •  Under these conditions:

V∞ = L2 R( ){ }

ϕ j ,k x( ) = αnϕ j+1,n (x)n∑

V−∞ = 0{ }

ϕ j ,k x( ) = αnϕ j+1,n (x)n∑

Substitute ϕ j+1,n x( ) = 2j+1

2ϕ 2 j+1x − n( )and change αn to hϕ (n), then

ϕ j ,k x( ) = hϕ (n)n∑ 2

j+12ϕ 2 j+1x − n( )

set j = k = 0,also note that ϕ0,0 (x) = ϕ(x)

ϕ x( ) = hϕ (n)n∑ 2ϕ 2x − n( ) Refinement equation:

Expansion functions can be built from double resolution copies of themselves

Scaling function coefficients

Refinement equation ϕ j ,k x( ) = 2 j /2ϕ 2 j x − k( )Recall

Refinement equation example

• Haar function

• Scaling coefficients

• Refinement equation

ϕ(x) =1 0 ≤ x < 10 otherwise#$%

hϕ (0) = hϕ (1) = 12

ϕ x( ) = hϕ (n)n∑ 2ϕ 2x − n( )

ϕ x( ) = ϕ 2x( ) +ϕ 2x −1( )

Wavelet functions

• Wavelet functions span the difference between adjacent scaling subspaces Vj and Vj+1 – Given a scaling function that meets the requirements

discussed, we can design a wavelet function

ψ j ,k x( ) = 2 j /2ψ 2 j x − k( )Wavelet function

Amplitude scaling

Scaling of width

Position along x-axis

Wj = Spank

ψ j ,k x( ){ }Vj+1 = Vj ⊕Wj

Union of function spaces © 1992–2008 R. C. Gonzalez & R. E. Woods

Function representation

L2 R( ) = V0 ⊕W0 ⊕W1⊕ ...L2 R( ) = V1⊕W1⊕W2 ⊕ ...L2 R( ) = Vjo

⊕Wjo⊕Wjo +1

⊕ ...


Generating wavelet functions

•  For the Haar scaling function:

•  Then:

ψ(x) = hψ (n)n∑ 2ϕ 2x − n( )

ϕ j,k (x),ψ j,l (x) = 0

hψ (n) = −1( )n hϕ (1− n) Wavelet function coefficients Modulation, time reversal

hϕ (0) = hϕ (1) = 12

hψ (0) = −1( )0 hϕ (1− 0) = 12

hψ (1) = −1( )1 hϕ (1−1) = − 1 2

For any k, l

Vj+1 = Vj ⊕Wjbecause

Haar wavelet function

ψ (x) = hψ (n)n∑ 2ϕ 2x − n( )

Substitute hψ (0) = 12

and hψ (1) = − 12

ψ (x) = ϕ(2x) −ϕ(2x −1)

ψ (x) =1 0 ≤ x < 0.5−1 0.5 ≤ x < 10 otherwise

&

'(

)(

Haar wavelet function

•  f(x) –  In space V1

•  fa(x) approximation –  In space V0

•  fd(x) difference –  In space W0

ψ (x) =1 0 ≤ x < 0.5−1 0.5 ≤ x < 10 otherwise

$

%&

'&

f (x) = fa (x) + fd (x)

V1 = V0 ⊕W0© 1992–2008 R. C. Gonzalez & R. E. Woods

1D Wavelet Transforms

Wavelet Series Fourier Series

Continuous Wavelet Transform

(Continuous) Fourier Transform

Discrete Wavelet Transform

Discrete Fourier Transform

Wavelet Series

•  The first sum uses scaling functions to provide an approximation of f(x) at a chosen starting scale jO. This sum is over translations (k) only.

•  The second sum is over scales (greater than jO) and over translations (k). It provides ever increasing detail (finer resolution)

f (x) = cjO (k)ϕ jO ,k(x) + dj (k)ψ j ,k (x)

k∑

j= jO

∞

∑k∑

Approximation (scaling) coefficients

Detail (wavelet) coefficients

Wavelet Series Coefficients

•  This assumes an orthonormal basis (or a tight frame) which is often the case.

f (x) = cjO (k)ϕ jO ,k(x) + dj (k)ψ j ,k (x)

k∑

j= jO

∞

∑k∑

cjO (k) = f (x)∫ ϕ jO ,k(x)dx

dj (k) = f (x)ψ j ,k (x)∫ dx

Wavelet Series Example


Discrete Wavelet Transform (DWT) •  Like the DFT, the DWT operates on discrete

functions (length M=2J).

f (n) = 1M

Wϕ jO,k( )ϕ jO ,k(n)

k=0

2 j0−1

∑ +1M

Wψ j,k( )ψ j,k (n)k=0

2 j−1

∑j= jO

J−1

∑

Sampled scaling function

Sampled wavelet function

Wϕ ( jO,k) = 1M

f (n)ϕ jO ,k (n)n=0

M−1

∑

Wψ ( j,k) = 1M

f (n)ψ j,k (n)n=0

M−1

∑ for j ≥ jO

Haar transform

ϕ(x) =1 0 ≤ x < 10 otherwise#$%

ψ (x) =1 0 ≤ x < 0.5−1 0.5 ≤ x < 10 otherwise

#

$(

%( T = HFHT

H2 =121 11 −1"

#$

%

&'

H4 =12

1 1 1 11 1 −1 −1

2 − 2 0 0

0 0 2 − 2

"

#

$$$$$

%

&

'''''

ϕ jk (x) = 2 j /2ϕ(2 j x − k)sample ϕ jk (x) to get ϕ jk (n)

ψ jk (x) = 2 j /2ψ (2 j x − k)sample ψ jk (x) to get ψ jk (n)

Wϕ (0,k) =1M

f (n)ϕ0,k (n)n∑

Wψ ( j,k) =1M

f (n)ψ j ,k (n)n∑

Fast Wavelet Transform (FWT)

•  It can be shown that

•  This is a very useful relationship between the DWT coefficients at adjacent levels.

Wψ ( j,k) = hψ (m − 2k)m∑ Wϕ ( j +1,m)

Wϕ ( j,k) = hϕ (m − 2k)m∑ Wϕ ( j +1,m)

Convolution with hψ(-n)

Convolution with hϕ(-n)


Fast Wavelet Transform


Coefficients at highest scale are samples of the function itself

Another iteration would split VJ-2 into 1/8th frequency bands

Fast Wavelet Transform


Coefficients at highest scale are samples of the function itself

Another iteration would split VJ-2 into 1/8th frequency bands

Wϕ ( jO,k) = 1M

f (n)ϕ jO ,k (n)n=0

M−1

∑

Wψ ( j,k) = 1M

f (n)ψ j,k (n)n=0

M−1

∑ for j ≥ jO

Do this:

Instead of this:

Haar FWT example


hϕ (0) = hϕ (1) = 12

hψ (0) = 12

hψ (1) = − 1 2

Inverse FWT


f (n) = 1M


k=0

2 j0−1

∑ +1M


2 j−1

∑j= jO

J−1

∑

Do this:

Instead of this:

Example: Haar inverse FWT


f (n) = 1M


k=0

2 j0−1

∑ +1M


2 j−1

∑j= jO

J−1

∑

J= 2, j0=0

Do this:

Instead of this:

2D Wavelet transform

• Separable 2D scaling function – ϕ(x,y)=ϕ(x)ϕ(y)

• Directionally sensitive separable 2D wavelet functions – ψH(x,y)=ψ(x)ϕ(y) : horizontal detail – ψV(x,y)=ϕ(x)ψ(y) : vertical detail – ψD(x,y)=ψ(x)ψ(y) : diagonal detail

Scaled and translated basis functions

• We can enumerate the last 3 equations as

ϕ j ,m,n x, y( ) = 2j2ϕ 2 j x − m,2 j y − n( )

ψ j ,m,nH x, y( ) = 2

j2ψ H 2 j x − m,2 j y − n( )

ψ j ,m,nV x, y( ) = 2

j2ψ V 2 j x − m,2 j y − n( )

ψ j ,m,nD x, y( ) = 2

j2ψ D 2 j x − m,2 j y − n( )

ψ j ,m,ni x, y( ) = 2

j2ψ i 2 j x − m,2 j y − n( ), i = H ,V ,D{ }

2D Wavelet Transform

•  Forward transform

•  Inverse transform

f (x, y) = 1MN

Wϕ jO,m,n( )ϕ jO ,m,n(x, y)

n=0

2 j0−1

∑m=0

2 j0−1

∑ +

1MN

Wψ j,m,n( )ψ j ,m,n

i (x, y)n=0

2 j0−1

∑m=0

2 j0−1

∑j= jO

J−1

∑i= H ,V ,D{ }∑

Wϕ jO ,m,n( ) = 1MN

f (x, y)ϕ jO ,m,n (x, y)y=0

N −1

∑x=0

M −1

∑

Wψi j,m,n( ) = 1

MNf (x, y)ψ j ,m,n

i (x, y)y=0

N −1

∑x=0

M −1

∑ , i = H ,V ,D{ }

N = M = 2J

j = jO ,.., J −1m = 0,1,2,...,2 j −1n = 0,1,2,...,2 j −1

2D Wavelet Transform Implementation


Example: DWT with various jO

•  Top left – 128 x 128

image, J=7

•  Top right: – 2D DWT jO=6

• Bottom left: – 2D DWT jO=5

• Bottom right: – 2D DWT jO=4


Inverse Transform

Wavelet image processing

•  Similar to processing in frequency domain •  Approach

1.  Compute 2D DWT of input image 2.  Alter the transform 3.  Compute inverse 2D DWT to get output image

•  First row: edge detection

• Second row: vertical edge detection

Example: thresholding wavelet coefficients


Wavelet image denoising

• Reduce influence of higher scale detail coefficients – Set to 0 all the detail wavelet coefficients that fall

below a threshold level – How many detail levels to threshold? – Should the threshold be the same for all levels? – Soft vs. hard thresholding

Example: Two highest levels thresholded with an interactively chosen global threshold


•  Throwing away the highest scale detail coefficients – Noise reduced – Edges preserved

•  Throwing away 2 highest scale detail coefficients – Some detail and

edges lost

Throwing away detail coefficients Output Image Residual Image


Wavelet Compression

• Wavelets pack the most important visual information into a small number of coefficients. The remaining coefficients can be quantized coarsely or truncated with little loss of visual information

•  JPEG: Quantizes Discrete Cosine Transform on 8x8 blocks. – Need to subdivide into blocks.Creates a blocking

effect •  JPEG-2000: Quantized discrete wavelet

transforms. Provides increased compression ratios. – Wavelets are inherently local basis functions. No

need to subdivide image. No blocking effect

Wavelet selection

Decomposition level selection

•  Fixed global threshold of 25 applied to detail coefficients (not to approximation coefficients)

• Different levels of wavelet decomposition levels can be used – Initial levels provide the most gain


• Detail images have a very peaked probability distribution around zero which allows for a small number of quantization levels


Quantizing wavelet coefficients

• Coefficients need to be quantized • Can use an uniform quantizer • Better results can be achieved by

– Introducing a larger quantization interval around zero – Adapting the size of the quantization intervals from

scale to scale (JPEG-style scaling)

JPEG


JPEG-2000


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