MMAT 5390 Mathematical Imaging Lecture 5 More about

MMAT 5390: Mathematical Imaging Lecture 5: More about Haar and Walsh Transform, DFT, Image Enhancement Prof. Ronald Lok Ming Lui Department of Mathematics, The Chinese University of Hong Kong

Recap: Haar transform For details, please refer to Lecture note Chapter 2 Definition of Haar functions:

Recap: Haar transform For details, please refer to Lecture Note Chapter 2 Definition of Haar transforms: (N = power of 2)

Recap: Walsh transform For details, please refer to Lecture Note Chapter 2 Definition of Walsh functions: (N = power of 2)

Recap: Walsh transform For details, please refer to Lecture Note Chapter 2 Definition of Walsh transforms: (N = power of 2��)

Haar transform elementary images Haar transform basis image. White = positive; Black = negative; Grey = 0 The i-th row j-th column elementary image is given by:

Walsh transform elementary images Walsh transform basis image. White = positive; Black = negative; Grey = 0 The i-th row j-th column elementary image is given by:

More about Haar transform basis image. White = positive; Black = negative; Grey = 0

More about Haar Transform What are the coefficients associated to different elementary images representing?

Elementary images of Walsh transform elementary images. White = positive; Black = negative; Grey = 0

Compared with Haar transform basis image. White = positive; Black = negative; Grey = 0

A quick revision: Image decomposition What is image decomposition: coefficients Elementary images (By truncating the terms with small coefficients, we can compress the image while preserving the important details) Main technique for image decomposition:

A quick revision: Image decomposition Main goal: HOW TO CHOOSE U and V? WHAT IS THE REQUIREMENT of g? So far we have learnt: 1. SVD Diagonal unitary 2. Haar transform Sparse (hopefully) Haar transform matrix 3. Walsh transform Sparse (hopefully) Walsh transform matrix

Recap: Definition of DFT 1 D and 2 D Discrete Fourier Transform: For details, please refer to Lecture Note Chapter 2

Elementary images of DFT decomposition

Reconstruction w/ DFT decomposition (a) = using 1 x 1 elementary images (first 1 row and first 1 column elementary images; (b) = using 2 x 2 elementary images (first 2 rows and first 2 column elementary images…and so on…

Comparison of errors The flower example:

Real example Original: Compressed: 13. 6: 1 Truncating the small coefficients

Real example Original: Compressed:

Real example Truncate smallest X% Fourier coefficients Truncate 70% of Fourier coefficients Truncate 50% of Fourier coefficients Truncate 30% of Fourier coefficients Truncate 10% of Fourier coefficients

Real example Truncate smallest X% Fourier coefficients Difference Image: Original minus Compressed image Original Truncate ~63% of Fourier coefficients

Real example Truncate smallest X% Fourier coefficients Difference Image: Original minus Compressed image Truncate 10% of Fourier coefficients

Real example Truncate smallest X% Fourier coefficients Truncate 30% of Fourier coefficients

Real example Truncate smallest X% Fourier coefficients Truncate 50% of Fourier coefficients

Real example Truncate smallest X% Fourier coefficients Truncate 70% of Fourier coefficients

Recap: Even Discrete Cosine Transform 1 D and 2 D Even Discrete Cosine Transform: 1 D: 2 D: For details, please refer to Supplementary note 5

Elementary images of EDCT decomposition

Reconstruction w/ EDCT decomposition (a) = using 1 x 1 elementary images (first 1 row and first 1 column elementary images; (b) = using 2 x 2 elementary images (first 2 rows and first 2 column elementary images…and so on…

Other similar transforms Odd Discrete Cosine Transform: Even Discrete Sine Transform: Odd Discrete Sine Transform: All of them have explicit formula for their inverses. (For details, please refer to Supplementary note 5)

Elementary images of ODCT decomposition

Reconstruction w/ ODCT decomposition (a) = using 1 x 1 elementary images (first 1 row and first 1 column elementary images; (b) = using 2 x 2 elementary images (first 2 rows and first 2 column elementary images…and so on…

Elementary images of EDST decomposition

Reconstruction w/ EDST decomposition (a) = using 1 x 1 elementary images (first 1 row and first 1 column elementary images; (b) = using 2 x 2 elementary images (first 2 rows and first 2 column elementary images…and so on…

Elementary images of ODST decomposition

Reconstruction w/ ODST decomposition (a) = using 1 x 1 elementary images (first 1 row and first 1 column elementary images; (b) = using 2 x 2 elementary images (first 2 rows and first 2 column elementary images…and so on…

Comparison of errors The flower example:

More example on DCT decomposition

More example on DCT decomposition Original image DCT

More example on DCT decomposition