Vectors [and more on masks] Vector space theory applies directly to several image processing/ representation problems

Transcription

1 Vectors [and more on masks] Vector space theory applies directly to several image processing/ representation problems 1

2 Image as a sum of basic images What if every person s portrait photo could be expressed as a sum of 20 special images? è We would only need 20 numbers to model any photo è sparse rep on our Smart card. 2

3 Efaces 100 x 100 images of faces are approximated by a subspace of only x 100 images, the mean image plus a linear combination of the 3 most important eigenimages 3

4 The image as an expansion 4

5 Different bases, different properties revealed 5

6 Fundamental expansion 6

7 Basis gives structural parts 7

8 Vector space review, part 1 8

9 Vector space review, Part 2 2 9

10 A space of images in a vector space n M x N image of real intensity values has dimension D = M x N n Can concatenate all M rows to interpret an image as a D dimensional 1D vector n The vector space properties apply n The 2D structure of the image is NOT lost 10

11 Orthonormal basis vectors help 11

12 Represent S = [10, 15, 20] 12

13 Projection of vector U onto V 13

14 Normalized dot product Can now think about the angle between two signals, two faces, two text documents, 14

15 Every 2x2 neighborhood has some constant, some edge, and some line component Confirm that basis vectors are orthonormal 15

16 Roberts basis cont. If a neighborhood N has large dot product with a basis vector (image), then N is similar to that basis image. 16

17 Standard 3x3 image basis Structureless and relatively useless! 17

18 Frie-Chen basis Confirm that bases vectors are orthonormal 18

19 Structure from Frie-Chen expansion Expand N using Frie- Chen basis 19

20 Sinusoids provide a good basis 20

21 Sinusoids also model well in images 21

22 Operations using the Fourier basis 22

23 A few properties of 1D sinusoids They are orthogonal Are they orthonormal? 23

24 F(x,y) as a sum of sinusoids 24

25 Continuous 2D Fourier Transform To compute F(u,v) we do a dot product of our image f(x,y) with a specific sinusoid with frequencies u and v 26

26 Power spectrum from FT 27

27 Examples from images Done with HIPS in

28 Descriptions of former spectra 29

29 Discrete Fourier Transform Do N x N dot products and determine where the energy is. High energy in parameters u and v means original image has similarity to those sinusoids. 30

30 Bandpass filtering 31

31 Convolution of two functions in the spatial domain is equivalent to pointwise multiplication in the frequency domain 32

32 LOG or DOG filter Laplacian of Gaussian Approx Difference of Gaussians 33

33 LOG filter properties 34

34 Mathematical model 35

35 1D model; rotate to create 2D model 36

36 1D Gaussian and 1 st derivative 37

37 2 nd derivative; then all 3 curves 38

38 Laplacian of Gaussian as 3x3 39

39 G(x,y): Mexican hat filter 40

40 Convolving LOG with region boundary creates a zero-crossing Mask h(x,y) Input f(x,y) Output f(x,y) * h(x,y) 41

41 42

42 LOG relates to animal vision 43

43 1D EX. Artificial Neural Network (ANN) for computing g(x) = f(x) * h(x) level 1 cells feed 3 level 2 cells level 2 cells integrate 3 level 1 input cells using weights [-1,2,-1] 44

44 Experience the Mach band effect 45

45 Simple model of a neuron 46

46 Output conditioning: threshold versus smoother output signal 47

47 Canny edge detector 51

48 Summary of LOG filter n Convenient filter shape n Boundaries detected as 0-crossings n Psychophysical evidence that animal visual systems might work this way (your testimony) n Physiological evidence that real NNs work as the ANNs 53