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Digital Image Procesing Unitary Transforms Discrete Fourier Trasform (DFT) in Image Processing

Digital Image Procesing Unitary Transforms Discrete Fourier Trasform (DFT) in Image Processing. DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON. 1-D Signal Transforms. Scalar form Matrix form. 1-D Signal Transforms-Remember the 1-D DFT.

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Digital Image Procesing Unitary Transforms Discrete Fourier Trasform (DFT) in Image Processing

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  1. Digital Image ProcesingUnitary TransformsDiscrete Fourier Trasform (DFT) in Image Processing DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSINGIMPERIAL COLLEGE LONDON

  2. 1-D Signal Transforms • Scalar form • Matrix form

  3. 1-D Signal Transforms-Remember the 1-D DFT • General form • DFT

  4. 1-D Inverse Signal Transforms-General Form • Scalar form • Matrix form

  5. 1-D Inverse Signal Transforms-Remember the 1D DFT • General form • Inverse DFT

  6. 1-D Unitary Transforms • Matrix form

  7. Signal Reconstruction

  8. Why do we use Image Transforms? • Often, image processing tasks are best performed in a domain other than the spatial domain. • Key steps: • Transform the image • Carry the task(s) of interest in the transformed domain. • Apply inverse transform to return to the spatial domain.

  9. 2-D (Image) Transforms-General Form

  10. 2-D (Image) Transforms-Special Cases • Separable • Symmetric

  11. 2-D (Image) Transforms-Special Cases (cont.) • Separable and Symmetric • Separable, Symmetric and Unitary

  12. Energy Preservation • 1-D • 2-D

  13. Energy Compaction • Most of the energy of the original data is concentrated in only a few transform coefficients, which are placed close to the origin; remaining coefficients have small values. • This property facilitate compression of the original image.

  14. Let’s talk about DFT in images: Why is it useful? • It is easier for removing undesirable frequencies. • It is faster to perform certain operations in the frequency domain than in the spatial domain. • The transform is independent of the signal.

  15. Example: Removing undesirable frequencies • Example of removing a high frequency using the transform domain • noisy signal • frequencies • Result after removing high frequencies

  16. How do frequencies show up in an image? • Low frequencies correspond to slowly varying information (e.g., continuous surface). • High frequencies correspond to quickly varying information (e.g., edges) Original Image Low-passed version

  17. 2-D Discrete Fourier Transform • It is separable, symmetric and unitary • It results in a sequence of two 1-D DFT operations (prove this)

  18. Visualizing DFT • The dynamic range of F(u,v)is typically very large • Small values are not distinguishable • We apply a logarithmic transformation to enhance small values. original image before transformation after transformation

  19. Amplitude and Log of the Amplitude

  20. Amplitude and Log of the Amplitude

  21. Original Image and Log of the Amplitude

  22. DFT properties: Separability

  23. DFT properties: Separability

  24. DFT properties: Separability

  25. DFT properties: Separability • The DFT and its inverse are periodic.

  26. DFT properties: Conjugate Symmetry

  27. DFT properties: Translation • Translation in spatial domain: • Translation in frequency domain:

  28. DFT properties: Translation • Warning: to show a full period, we need to translate the origin of the transform at

  29. DFT properties: Translation

  30. DFT properties: Translation without translation after translation

  31. DFT properties: Rotation

  32. DFT properties: Addition and Multiplication

  33. DFT properties: Average value of the signal

  34. Original Image Fourier Amplitude Fourier Phase

  35. Magnitude and Phase of DFT • What is more important? • Hint: use inverse DFT to reconstruct the image using magnitude or phase only information magnitude phase

  36. Magnitude and Phase of DFT Reconstructed image using magnitude only (i.e., magnitude determines the contribution of each component!) Reconstructed image using phase only (i.e., phase determines which components are present!)

  37. Magnitude and Phase of DFT

  38. Low pass filtering using DFT

  39. High pass filtering using DFT

  40. Experiment: Verify the importance of phase in images

  41. Reconstruction fromphase of one image and amplitude of the other phase of cameraman phase of grasshopper amplitude of grasshopper amplitude of cameraman

  42. Experiment: Verify the importance of phase in images

  43. Reconstruction fromphase of one image and amplitude of the other phase of buffalo phase of rocks amplitude of rocks amplitude of buffalo

  44. DFT of a single edge • Consider DFT of image with single edge. • For display, DC component is shifted to the centre. • Log of magnitude of Fourier Transform is displayed

  45. DFT of a box

  46. DFT of rotated box

  47. DFT computation: Extended image • DFT computation assumes image repeated horizontally and vertically. • Large intensity transition at edges = vertical and horizontal line in middle of spectrum after shifting.

  48. Windowing • Can multiply image by windowing function before DFT to reduce sharp transitions between borders of repeated images. • Ideally, causes image to drop off towards ends, reducing transitions

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