1 / 47

MMAT 5390: Mathematical Imaging

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

manuelak
Download Presentation

MMAT 5390: Mathematical Imaging

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

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

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

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

  6. 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:

  7. 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:

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

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

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

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

  12. 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:

  13. 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

  14. 1D and 2D Discrete Fourier Transform: Recap: Definition of DFT For details, please refer to Lecture Note Chapter 2

  15. Elementary images of DFT decomposition

  16. Elementary images of DFT decomposition

  17. 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…

  18. The flower example: Comparison of errors

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

  20. Original: Compressed: Real example

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

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

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

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

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

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

  27. 1D and 2D Even Discrete Cosine Transform: 1D: Recap: Even Discrete Cosine Transform 2D: For details, please refer to Supplementary note 5

  28. Elementary images of EDCT decomposition

  29. Reconstruction w/ EDCT 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…

  30. Odd Discrete Cosine Transform: Other similar transforms 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)

  31. Elementary images of ODCT decomposition

  32. Reconstruction w/ ODCT 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…

  33. Elementary images of EDST decomposition

  34. Reconstruction w/ EDST 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…

  35. Elementary images of ODST decomposition

  36. Reconstruction w/ ODST 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…

  37. The flower example: Comparison of errors

  38. More example on DCT decomposition

  39. Original image DCT More example on DCT decomposition

  40. Original image DCT More example on DCT decomposition

  41. Original image DCT More example on DCT decomposition

  42. More example on DCT decomposition

  43. More example on DCT decomposition

  44. More example on DCT decomposition

  45. More example on DCT decomposition

  46. More example on DCT decomposition

  47. More example on DCT decomposition

More Related