Basis Images and The Wavelet Transform · 2020-05-13 · •Wavelet functions (wavelets) are then used to encode the differences between adjacent approximations. •The discrete wavelet

Basis Images andThe Wavelet Transform

Image Processing

CSE 166

Lecture 13

Announcements

• Assignment 4 will be released today

– Due May 20, 11:59 PM

• Reading

– Chapter 6: Wavelet and Other Image Transforms

• Sections 6.5 and 6.10

• (Sections 6.6-6.9 for details of specific transforms)

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Matrix-based transforms

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Forward transform

Inverse transform

where

Matrix-based transforms using orthonormal basis vectors

• In matrix form

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Matrix-based transform

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Example: 8-point DFT of f(x) = sin(2πx)

real part + imaginary part

Matrix-based transforms

• Discrete Fourier transform (DFT)• Discrete Hartley transform (DHT)• Discrete cosine transform (DCT)• Discrete sine transform (DST)• Walsh-Hadamard (WHT)• Slant (SLT)• Haar (HAAR)• Daubechies (DB4)• Biorthogonal B-spline (BIOR3.1)

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Basis vectors of matrix-based 1D transforms

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N = 16

real part

imaginary part

Standard basis(for reference)

Basis vectors of matrix-based 1D transforms

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N = 16

basis dual Standard basis(for reference)

Matrix-based transformsin two dimensions

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Forward transform

Inverse transform

where

Matrix-based transforms in two dimensions using basis images

• Inverse transform

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where

Each Su,v is a basis image

Basis images of matrix-based 2D transforms

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Standard basis images (for reference)

8-by-8 array of8-by-8

basis images

Basis images of matrix-based 2D transforms

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Discrete Fourier transform (DFT) basis images

real part imaginary part

Basis images of matrix-based 2D transforms

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Discrete Hartley transform (DHT) basis images

Basis images of matrix-based 2D transforms

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Discrete cosine transform (DCT) basis images

Basis images of matrix-based 2D transforms

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Discrete sine transform (DST) basis images

Basis images of matrix-based 2D transforms

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Walsh-Hadamard transform (WHT) basis images

Basis images of matrix-based 2D transforms

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Slant transform (SLT) basis images

Basis images of matrix-based 2D transforms

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Haar transform (HAAR) basis images

Wavelet transforms

• A scaling function is used to create a series of approximations of a function or image, each differing by a factor of 2 in resolution from its nearest neighboring approximations.

• Wavelet functions (wavelets) are then used to encode the differences between adjacent approximations.

• The discrete wavelet transform (DWT) uses those wavelets, together with a single scaling function, to represent a function or image as a linear combination of the wavelets and scaling function.

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Scaling functions and set of basis vectors

• Father scaling function

• Set of basis functions

– Integer translation k

– Binary scaling j

• Basis of the function space spanned by

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Scaling function, multiresolution analysis

1. The scaling function is orthogonal to its integer translates

2. The function spaces spanned by the scaling function at low scales are nested within those spanned at higher scales𝑉−∞⊂⋯⊂ 𝑉−1⊂ 𝑉0⊂ 𝑉1⊂⋯⊂ 𝑉∞

3. The only function representable at every scale (all 𝑉𝑗) is f(x) = 0

4. All measureable, square-integrable functions can be represented as 𝑗 → ∞

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• Given father scaling function , there exists a mother wavelet function whose integer translations and binary scalings

span the difference between any two adjacent scaling spaces

• The orthogonal compliment of

Wavelet functions

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Relationship between scaling and wavelet function spaces

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Union

V is basis of the function space spanned by scaling function

Wj is orthogonal complement of Vj in Vj+1

Scaling function coefficients and wavelet function coefficients

• Refinement (or dilation) equation

where are scaling function coefficients

• And

where are wavelet function coefficients

• Relationship

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1D discrete wavelet transform

• Forward

• Inverse

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Approximation

Details

for real signals

2D discrete wavelet transform

• Forward

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Approximation

Details

where

Directionalwavelets

for real signals

2D discrete wavelet transform

• Inverse

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for real signals

2D discrete wavelet transform

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Decomposition

Horizontal detailsApproximation

Vertical details

Diagonal details

2D discrete wavelet transform

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3-levelwavelet

decomposition

Wavelets in image processing

1. Wavelet transform

2. Alter transform

3. Inverse wavelet transform

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Wavelet-based edge detection

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Zero horizontal

details

Zero lowest scale

approximation

Vertical edges

Edges

Wavelet-based noise removal

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Noisy imageThreshold

details

Zero highest

resolution details

Zero details for all levels

Next Lecture

• Image compression

• Reading

– Chapter 8: Image Compression and Watermarking

• Sections 8.1, 8.9, 8.10, and 8.12

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