Discrete Fourier Transform The Chinese University of Hong Kong

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Presentation transcript:

Discrete Fourier Transform The Chinese University of Hong Kong Math 3360: Mathematical Imaging Lecture 7: Discrete Fourier Transform Prof. Ronald Lok Ming Lui Department of Mathematics, The Chinese University of Hong Kong

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:

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

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

Elementary images of DFT decomposition

Elementary images of DFT decomposition

Reconstruction w/ DFT decomposition = using 1x1 elementary images (first 1 row and first 1 column elementary images; = using 2x2 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: