Introduction to Linear Image Processing

Transcription

1 Introduction to Linear Image Processing 1 IPAM - UCLA July 22, 2013 Iasonas Kokkinos Center for Visual Computing Ecole Centrale Paris / INRIA Saclay

2 Image Sciences in a nutshell 2 Image Processing Image to Image da da Imaging Physics to Image Computer Graphics Symbols to Image Computer Vision Image to Symbols

3 Images as functions Continuous Discrete d=1: Gray d=3: Color 3

4 Image Denoising 4

5 Image Denoising 5 Key assumption: clean image is smooth

6 Moving Average in 2D Slide Source: S. Seitz

7 Moving Average in 2D Slide Source: S. Seitz

8 Moving Average in 2D Slide Source: S. Seitz

9 Moving Average in 2D Slide Source: S. Seitz

10 Moving Average in 2D Slide Source: S. Seitz

11 Moving Average in 2D Slide Source: S. Seitz

12 Denoising: input 12

13 Denoising: first application of averaring filter 13

14 Denoising: tenth application of denoising filter 14

15 Denoising: application of larger box filter 15

16 Weighted averaging 16

17 Weighting kernel 17 Gaussian function: Standard deviation, σ: determines spatial support σ = 2 σ = 5 σ = 10

18 Moving average 18

19 Gaussian blur 19

20 Image Processing 20 image filter image

21 Linear Image Processing 21 Linearity Translation Invariance Linear, Translation-Invariant (LTI) system

22 Linear Image Processing 22 image filter image From time-invariance: useful bases.

23 Linear Image Processing 23 image filter image From time-invariance: useful bases.

24 Linear algebra reminder 24 Basis: N linearly independent vectors Expansion on basis: Orthonormal basis: Expansion coefficients: Expansion:

25 Canonical basis 25

26 Canonical basis for 2D signals 26 Kronecker delta

27 Canonical basis for 2D signals 27 Kronecker delta

28 Canonical basis for 2D signals 28 Kronecker delta

29 Canonical basis for signals: expansion 29 Signal expansion: Identify terms: Rewrite: Unit sample function Sifting property:

30 Canonical basis for signals and LTI filters 30 unit sample impulse response Translation-invariance Any signal: By linearity: Convolution sum Output of any LSI filter for any input: convolution of input with filter s impulse response

31 Convolution discrete and continuous 31 2D convolution sum: 2D convolution integral:

32 Linear Image Processing 32 image filter image From time-invariance: useful bases.

33 Associative property & efficiency 33 Associative Property: Separability of Gaussian: Slow Fast

34 Associative property & accuracy 34 Associative Property: Derivative of Gaussian: exact approximate

35 Associative property & multi-scale processing 35 Associative Property: Semi-group property of Gaussian:

36 Denoising: first application of averaging kernel 36

37 Denoising: 10 th application of denoising kernel 37

38 Distributive property & efficiency 38 Distributive property: Steerable fliter: W. Freeman and E. Adelson, The design and use of steerable filters, PAMI, 1991

39 Linear algebra reminder: eigenvectors 39 Eigenvectors: Full-rank, real and symmetric: eigenbasis

40 Eigenvectors and eigenfunctions 40 Eigenvector: Eigenfunction: Input: Output:

41 Eigenfunctions for LTI filters 41 LTI filter: Let s guess: It works: Frequency response:

42 Expansion on harmonic basis 42 From orthonormality: Inner product for complex functions: Discrete-time: Continuous-time:

43 Change of basis 43 Canonical expansion: Eigenbasis expansion: Rotation matrix from eigenbasis: Fourier transform: change of basis Rotation from canonical basis to eigenfunction basis

44 Fourier Analysis 44

45 Fourier synthesis equation 45 Continuous-time: Discrete-time:

46 Convolution theorem of Fourier transform 46 Input expansion: Output: Expansions:

47 Linear Image Processing 47 image filter image From time-invariance: useful bases.

48 Convolution theorem 48 Fourier Analysis Fourier Synthesis

49 Convolution theorem and efficiency 49 Fast Fourier Transform Fast Fourier Transform

50 Gaussian blur Time 50 Frequency

51 Moving average Time 51 Frequency

52 Modulation property and Gabor filters 52 Modulation property: Gaussian: Time Frequency

53 Modulation property and Gabor filters 53 Modulation property: Gaussian: Gabor: Time Frequency

54 2D Gabor filterbank 54 Consider many combinations of and Increasing Increasing Frequency responce isocurves

55 2D Gabor filterbank and texture analysis 55

56 2D Gabor filterbank and texture analysis 56

57 Summary 57 Linear Time-Invariant filters Convolution Fourier Transform (Derivative-of) Gaussian filters Steerable filters Gabor filters Thursday s lecture: Pyramids, Scale-Invariant Blobs/Ridges, SIFT, HOG, Log-polar features, Harmonic analysis on surfaces Further reading: Fast recursive filters: Recursively implementing the Gaussian and its Derivatives - R. Deriche, 1993 Recursive implementation of the Gaussian filter. I. Young, L. Vliet, 1995 Fast IIR Isotropic 2D Complex Gabor Filters with Boundary Initialization, A Bernardino, J. Santos-Victor, TIP, 2006 Wavelets: A Wavelet Tour of Signal Processing, S. Mallat, 2008