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Math 3360: Mathematical Imaging

Math 3360: Mathematical Imaging. Lecture 7: Discrete Fourier Transform. Prof. Ronald Lok Ming Lui Department of Mathematics, The Chinese University of Hong Kong. What is image decomposition:. coefficients. A quick revision: Image decomposition. Elementary images.

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Math 3360: Mathematical Imaging

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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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