1 MSU CSE 803 Fall 2014 Vectors [and more on masks] Vector space theory applies directly to several image processing/representation problems

2 MSU CSE 803 Fall 2014 Image as a sum of “basic images” What if every person’s portrait photo could be expressed as a sum of 20 special images?  We would only need 20 numbers to model any photo  sparse rep on our Smart card.

3 MSU CSE 803 Fall 2014 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”

4 MSU CSE 803 Fall 2014 The image as an expansion

5 MSU CSE 803 Fall 2014 Different bases, different properties revealed

6 MSU CSE 803 Fall 2014 Fundamental expansion

7 MSU CSE 803 Fall 2014 Basis gives structural parts

8 MSU CSE 803 Fall 2014 Vector space review, part 1

9 MSU CSE 803 Fall 2014 Vector space review, Part 2 2

10 MSU CSE 803 Fall 2014 A space of images in a vector space M x N image of real intensity values has dimension D = M x N Can concatenate all M rows to interpret an image as a D dimensional 1D vector The vector space properties apply The 2D structure of the image is NOT lost

11 MSU CSE 803 Fall 2014 Orthonormal basis vectors help

12 MSU CSE 803 Fall 2014 Represent S = [10, 15, 20]

13 MSU CSE 803 Fall 2014 Projection of vector U onto V

14 MSU CSE 803 Fall 2014 Normalized dot product Can now think about the angle between two signals, two faces, two text documents, …

15 MSU CSE 803 Fall 2014 Every 2x2 neighborhood has some constant, some edge, and some line component Confirm that basis vectors are orthonormal

16 MSU CSE 803 Fall 2014 Roberts basis cont. If a neighborhood N has large dot product with a basis vector (image), then N is similar to that basis image.

17 MSU CSE 803 Fall 2014 Standard 3x3 image basis Structureless and relatively useless!

18 MSU CSE 803 Fall 2014 Frie-Chen basis Confirm that bases vectors are orthonormal

19 MSU CSE 803 Fall 2014 Structure from Frie-Chen expansion Expand N using Frie- Chen basis

20 MSU CSE 803 Fall 2014 Sinusoids provide a good basis

21 MSU CSE 803 Fall 2014 Sinusoids also model well in images

22 MSU CSE 803 Fall 2014 Operations using the Fourier basis

23 MSU CSE 803 Fall 2014 A few properties of 1D sinusoids They are orthogonal Are they orthonormal?

24 MSU CSE 803 Fall 2014 F(x,y) as a sum of sinusoids

25 MSU CSE 803 Fall 2014 Spatial direction and frequency in 2D

26 MSU CSE 803 Fall 2014 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

27 MSU CSE 803 Fall 2014 Power spectrum from FT

28 MSU CSE 803 Fall 2014 Examples from images Done with HIPS in 1997

29 MSU CSE 803 Fall 2014 Descriptions of former spectra

30 MSU CSE 803 Fall 2014 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.

31 MSU CSE 803 Fall 2014 Bandpass filtering

32 MSU CSE 803 Fall 2014 Convolution of two functions in the spatial domain is equivalent to pointwise multiplication in the frequency domain

33 MSU CSE 803 Fall 2014 LOG or DOG filter Laplacian of Gaussian Approx Difference of Gaussians

34 MSU CSE 803 Fall 2014 LOG filter properties

35 MSU CSE 803 Fall 2014 Mathematical model

36 MSU CSE 803 Fall D model; rotate to create 2D model

37 MSU CSE 803 Fall D Gaussian and 1 st derivative

38 MSU CSE 803 Fall nd derivative; then all 3 curves

39 MSU CSE 803 Fall 2014 Laplacian of Gaussian as 3x3

40 MSU CSE 803 Fall 2014 G(x,y): Mexican hat filter

41 MSU CSE 803 Fall 2014 Convolving LOG with region boundary creates a zero-crossing Mask h(x,y) Input f(x,y)Output f(x,y) * h(x,y)

42 MSU CSE 803 Fall 2014

43 MSU CSE 803 Fall 2014 LOG relates to animal vision

44 MSU CSE 803 Fall D 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]

45 MSU CSE 803 Fall 2014 Experience the Mach band effect

46 MSU CSE 803 Fall 2014 Simple model of a neuron

47 MSU CSE 803 Fall 2014 Output conditioning: threshold versus smoother output signal

48 MSU CSE 803 Fall D situation in the eye Neuron c has + input to neuron A but - input to neuron B. Neuron d has + input to neuron B but – input to neuron A. Neuron b gives no input to neuron B: it is not in the receptive field of B.

49 MSU CSE 803 Fall 2014 Receptive fields

50 MSU CSE 803 Fall 2014 Experiments with cats/monkeys Stabilize/drug animal to stare Place delicate probe in visual network Move step edge across FOV Probe shows response function when the edge images to receptive field Slightly moving the probe produces similar signal when edge is nearby

51 MSU CSE 803 Fall 2014 Canny edge detector uses LOG filter

52 Cornsweet Illusion See, e.g., wikipedia

53 MSU CSE 803 Fall 2014 Summary of LOG filter Convenient filter shape Boundaries detected as 0-crossings Psychophysical evidence that animal visual systems might work this way (your testimony) Physiological evidence that real NNs work as the ANNs

54 MSU CSE 803 Fall 2014 Morphology Operations

55 Morphology Operations

56 MSU CSE 803 Fall 2014 Applications

57 Applications