Computer and Robot Vision I

1. Computer and Robot VisionI Chapter 8 The Facet Model ppt.cc/C8SJx Presented by: 陳毅 b03202042@ntu.edu.tw 指導教授: 傅楸善 博士 Digital Camera and Computer Vision Laboratory Department of Computer Science and Information Engineering National Taiwan University, Taipei, Taiwan, R.O.C.

2. 8.0 Outline • 8.1 Introduction • 8.2Relative Maxima • 8.3 Sloped Facet Parameter and Error Estimation • 8.4 Facet-Based Peak Noise Removal • 8.5 Iterated Facet Model • 8.6 Gradient-Based Facet Edge Detection • 8.7 Bayesian Approach to Gradient Edge Detection

3. 8.0 Outline • 8.8 Zero-Crossing Edge Detector • 8.9 Integrated Directional Derivative Gradient Operator • 8.10 Corner Detection • 8.11 Isotropic Derivative • 8.12 Ridges and Ravines on Digital Images • 8.13 Topographic Primal Sketch

5. 8.1Introduction • The facet model principle Image as continuum or piecewise continuous intensity surface • Observed digital image Noisy discretized sampling of distorted version of this surface

6. 8.1 Introduction Question To actually carry out the processing with the observed digital image Solution Require a model describes what the general form of the surface would be in the neighborhood of any pixel.

7. 8.1 Introduction • General forms • piecewise constant (flat facet model) Ideal region: constant gray level • piecewise linear (sloped facet model) Ideal region: sloped plane gray level • piecewise quadratic Ideal region: bivariate quadratic gray level • piecewise cubic Ideal region: cubic surface gray level

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9. 8.2 Relative Maxima • Example: a simple labeling application How to detect and locate all relative maxima? • Definition: Relative maxima 1. first derivative zero 2. second derivative negative

10. 8.2 Relative Maxima (1-D case) • One-dimensional observation sequence • To find the relative maxima, we can least-squares fit a quadratic function

11. 8.2 Relative Maxima (1-D case) The squared fitting error for the group of can be expressed by

12. 8.2 Relative Maxima • Taking partial derivatives of with respect to the free parameter results in the following slide.

13. 8.2 Relative Maxima

14. 8.2 Relative Maxima (1-D case) • Assume , setting these partial derivatives to zero

15. 8.2 Relative Maxima (1-D case) • Finally, we can get

16. 8.2 Relative Maxima • The quadratic has relative extrema at • The extremum is a relative maxima when • The algorithm then amount to the following • Test whether . If not, then there is no chance of maxima. • If , compute • If , then mark the point as a relative maxima. • reference: relative maxima.pdf, relative maxima.xlsx

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18. 8.3 Sloped Facet Parameter and Error Estimation • Least-squares procedure • To estimate sloped facet parameter • Noise variance

19. 8.3 Sloped Facet Parameter and Error Estimation • Assume the coordinate of the given pixel are in its central neighborhood, and assume each . • Image function is modeled by • Where is a random variable indexed on , which represents noise. • We assume thatis noise having mean 0 and variance and that the noise for any two pixels is independent.

20. 8.3 Sloped Facey Parameter and Error Estimation • The least-squares procedure determine parameters that minimize the sum of the squared differences between the fitted surface and the observed one. • Taking the partial derivatives of and setting them to zero.

21. 8.3 Sloped Facet Parameter and Error Estimation Due to the center is , Hence

22. 8.3 Sloped Facet Parameter and Error Estimation

23. 8.3 Sloped Facet Parameter and Error Estimation • Solving for , we obtain

24. 8.3 Sloped Facet Parameter and Error Estimation • Replacing by :

25. 8.3 Sloped Facet Parameter and Error Estimation • is noise having mean and variance

26. 8.3 Sloped Facet Parameter and Error Estimation review DC & CV Lab. CSIE NTU

28. 8.3 Sloped Facet Parameter and Error Estimation • Examining the squared error residual

29. 8.3 Sloped Facet Parameter and Error Estimation Using the fact that We obtain

30. 8.3 Sloped Facet Parameter and Error Estimation Chi-distribution • Assume , , …… , are independent variable, where • is distributed according to the chi-squared distribution withn degrees of freedom. • It can be write

31. 8.3 Sloped Facet Parameter and Error Estimation Chi-distribution

32. 8.3 Sloped Facet Parameter and Error Estimation is a chi-squared distribution with degrees of freedom. • , , Independent normal distributions is chi-squared distribution with 3 degrees of freedom.

33. 8.3 Sloped Facet Parameter and Error Estimation • distributed as a chi-squared variate with degrees of freedom. Conclusion • can be used as an unbiased estimator for

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37. 8.4 Facet-Based Peak Noise Removal • Peak noise pixel A pixel whose gray level intensity significantly differs from neighborhood pixels.

38. 8.4 Facet-Based Peak Noise Removal • Let N be a set of neighborhood pixels that does not contain the center pixel: for is assumed to be independent additive Gaussian noise having mean 0 and variance

39. 8.4 Facet-Based Peak Noise Removal • We can use sloped facet model • The minimizing ,, are given by

40. 8.4 Facet-Based Peak Noise Removal Hypothesis • is not peak noise • has a Gaussian distribution with mean 0 and variance • Hence has mean 0 and variance 1

41. 8.4 Facet-Based Peak Noise Removal t-distribution (Student's t-distribution) • and are independent • is t-distributed withn degrees of freedom, denoted as . • reference: LinWYHypeTest.doc

42. 8.4 Facet-Based Peak Noise Removal t-distribution (Student's t-distribution)

44. 8.4 Facet-Based Peak Noise Removal Statistical variable • degrees of freedom. Then,

45. 8.4 Facet-Based Peak Noise Removal The center pixel is judged to be a peak noise pixel if a test of the hypothesis rejects the hypothesis. • Let be the number satisfying • is a threshold.

46. 8.4 Facet-Based Peak Noise Removal If The hypothesis of the equality of and is rejected, and the output value for the center pixel is given by . If The hypothesis of the equality of and is not rejected, and the output value for the center pixel is given by .

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48. 8.5 Iterated Facet Model The iterated model for ideal image • The spatial of the image can be partitioned into connected regions called facets. • Each of which satisfies certain gray level and shape constraints.

49. 8.5 Iterated Facet Model Gray level constraint gray levels in each facet must be a polynomial function of row-column coordinates Shape constraint each facet must be sufficiently smooth in shape

50. 8.5 Iterated Facet Model • Each pixel in image is contained in different blocks. • Each block fits a polynomial model. • Set the output gray value to be that gray value fitted by the block with smallest error variance.