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Image Decomposition Techniques: Discrete Fourier Transform Overview

Understand image decomposition using DFT, its applications, choosing decomposition factors, and compressing images while preserving details. Explore SVD, Haar, and Walsh transforms, DFT definition, reconstruction, and error comparisons. Dive into real examples.

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Image Decomposition Techniques: Discrete Fourier Transform Overview

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

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

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

  4. 1D and 2D Discrete Fourier Transform: Recap: Definition of DFT For details, please refer to Lecture Note Chapter 2

  5. Elementary images of DFT decomposition

  6. Elementary images of DFT decomposition

  7. 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…

  8. The flower example: Comparison of errors

  9. Original: Compressed: 13.6:1 Real example Truncating the small coefficients

  10. Original: Compressed: Real example

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