Computational Models of V1 cells. Gabor and CORF
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1 1 Computational Models of V1 cells Gabor and CORF George Azzopardi Nicolai Petkov
2 2 Primary visual cortex (striate cortex or V1)
3 3 Types of V1 Cells Hubel and Wiesel, Nobel Prize winners Three main types Simple cells Complex Cells Hyper-complex cells (end-stopping) Etc
4 4 References to origins - neurophysiology Hubel and Wiesel named one type of cell "simple" because they shared the following properties: Their receptive fields have distinct excitatory and inhibitory regions. It is possible to predict responses to stimuli given the map of excitatory and inhibitory regions.
5 5
6 6 Neurophysiology D.H. Hubel and T.N. Wiesel: Receptive fields, binocular interaction and functional architecture in the cat's visual cortex, Journal of Physiology (London), vol. 160, pp , D.H. Hubel and T.N. Wiesel: Sequence regularity and geometry of orientation columns in the monkey striate cortex, Journal of Computational Neurology, vol. 158, pp , D.H. Hubel: Exploration of the primary visual cortex, , Nature, vol. 299, pp , 1982.
7 7 Receptive Fields of Simple Cells [Jones and Palmer, 1987]
8 8 Computational Models 2D Gabor functions Combination of Receptive Fields (CORF) Derivative of Gaussians
9 9 Gabor = Gaussian x Cosine Gaussian (sigma) Cosine: lambda and phase Space domain Frequency domain
10 Parameterization according to N. Petkov: Biologically motivated computationally intensive approaches to image pattern recognition, Future Generation Computer Systems, 11 (4-5), 1995, N. Petkov and P. Kruizinga: Computational models of visual neurons specialised in the detection of periodic and aperiodic oriented visual stimuli: bar and grating cells, Biological Cybernetics, 76 (2), 1997, P. Kruizinga and N. Petkov: Non-linear operator for oriented texture, IEEE Trans. on Image Processing, 8 (10), 1999, S.E. Grigorescu, N. Petkov and P. Kruizinga: Comparison of texture features based on Gabor filters, IEEE Trans. on Image Processing, 11 (10), 2002, N. Petkov and M. A. Westenberg: Suppression of contour perception by band-limited noise and its relation to non-classical receptive field inhibition, Biological Cybernetics, 88, 2003, C. Grigorescu, N. Petkov and M. A. Westenberg: Contour detection based on nonclassical receptive field inhibition, IEEE Trans. on Image Processing, 12 (7), 2003,
11 11 Wavelength
12 12 Wavelength Space domain Frequency domain Smallest Wavelength = 2 pixels
13 13 Wavelength Space domain Frequency domain Wavelength = 4 pixels
14 14 Wavelength Space domain Frequency domain Wavelength = 8 pixels
15 15 Wavelength Space domain Frequency domain Wavelength = 16 pixels
16 16 Wavelength Space domain Frequency domain Wavelength = 32 pixels
17 17 Wavelength Space domain Frequency domain Wavelength = 64 pixels
18 18 Orientation
19 19 Orientation Space domain Frequency domain Orientation = 0
20 20 Orientation Space domain Frequency domain Orientation = 45
21 21 Orientation Space domain Frequency domain Orientation = 90
22 22 Symmetry (phase offset)
23 23 Symmetry (phase offset) Space domain Space domain Phase offset = 0 (symmetric function) Phase offset = -90 (anti-symmetric function)
24 24 Spatial aspect ratio
25 25 Spatial aspect ratio Space domain Frequency domain Aspect ratio = 0.5
26 26 Spatial aspect ratio Space domain Frequency domain Aspect ratio = 1
27 27 Spatial aspect ratio Space domain Frequency domain Aspect ratio = 2 (does not occur)
28 28 Sigma/lambda ratio Determines the spatial frequency bandwidth i.e. number of stripe zones
29 29 Preferred spatial frequency and size Space domain Frequency domain Bandwidth = 1 (σ = 0.56 λ) Wavelength = 8/512
30 30 Bandwidth Space domain Frequency domain Bandwidth = 0.5 Wavelength = 8/512
31 31 Bandwidth Space domain Frequency domain Bandwidth = 2 Wavelength = 8/512
32 32 Bandwidth Space domain Frequency domain Bandwidth = 1 (σ = 0.56 λ) Wavelength = 32/512
33 33 Bandwidth Space domain Frequency domain Bandwidth = 0.5 Wavelength = 32/512
34 34 Bandwidth Space domain Frequency domain Bandwidth = 2 Wavelength = 32/512
35 Semi-linear Gabor filter: What is it useful for? 35 Receptive field G(-x,-y) Input Output of Convolution followed by half-wave rectification bandwidth = 2 Ori = 0 Phi = 90 edges Ori = 180 Phi = 90 edges Ori = 0 Phi = 0 lines Ori = 0 Phi = 180 lines
36 Bank of semi-linear Gabor filters: Which orientations to use? 36 frequency domain Receptive field G(-x,-y) Ori = For filters with s.a.r=0.5 and bw=2, good coverage of angles with 6 orientations
37 Ori = For filters with sar=0.5 and bw=2, good coverage of angles with 12 orientations Input Output BICV Summer School, Shenyang, China, Bank of semi-linear Gabor filters: Which orientations to use Filter in frequency domain
38 Bank of semi-linear Gabor filters: Which orientations to use 38 Result of superposition of the outputs of 12 semi-linear anti-symmetric (phi=90) Gabor filters with wavelength = 4, bandwidth = 2, spatial aspect ratio = 0.5 (after thinning and thresholding lt = 0.1, ht = 0.15).
39 39 V1 gets input from Lateral Geniculate Nucleus (LGN)
40 40 V1 gets input from LGN Center-surround receptive fields Detect changes in light intensity Computational model Laplacian of Gaussian(Marr and Hildreth, 1980) or Difference of Gaussians (DoG)
41 41 [Hubel and Wiesel, 1962]
42 42 Combination Of Receptive Fields (CORF) model of a simple cell
43 43 Configuration of a CORF Model
44 44 Configuration of a CORF model
45 45 Response of a CORF Model
46 46 Modulated response to drifting gratings
47 47 Receptive field profile determined by simulated reverse correlation
48 48 Contrast Invariant Orientation Tuning Real simple cell CORF Model GF Model
49 49 CORF model with linear response
50 50 Cross Orientation Suppression CORF model responds as real simple cells do
51 51 CORF Model with linear response
52 52 Tolerance to Rotation
53 53 Tolerance to Rotation
54 54 Tolerance to Rotation
55 55 Contour Detection
56 56 Contour Detection Consider 12 orientations (intervals of 30) Take maximum superposition Thinning Hysteresis thresholding Compare model contour map with ground truth: Count TP, FP, and FN Precision P = TP/(TP + FP) Recall R = TP/(TP + FN) F-measure F = 2PR/(P + R)
57 57 Results
58 58 Results
59 59 Results: RuG Data Set
60 60 Results: CORF vs. Gabor Right-tailed paired t-test RuG: 40 images t = 4.39, p < 0.01 Berkeley: 500 images t = 4.95, p < 0.01
61 61 Robustness to noise
62 62 Take home message 1. Gabor and CORF: computational models of simple cells 2. CORF is nonlinear, Gabor is linear 3. CORF achieves more properties than Gabor 4. CORF is more effective in contour detection 5. Gabor is mathematically more elegant
63 63 Matlab Code
Nicolai Petkov Intelligent Systems group Institute for Mathematics and Computing Science
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V1-inspired orientation selective filters for image processing and computer vision Nicolai Petkov Intelligent Systems group Institute for Mathematics and Computing Science 2 Most of the images in this
Nicolai Petkov Intelligent Systems group Institute for Mathematics and Computing Science
2D Gabor functions and filters for image processing and computer vision Nicolai Petkov Intelligent Systems group Institute for Mathematics and Computing Science 2 Most of the images in this presentation
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