1. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.1 Image Measurements and Detection Davi Geiger geiger@cs.nyu.edu

2. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.2 Images are Intensity Surfaces and Edges are Intensity Discontinuities I 1 (x, y) I 2 (x, y)

3. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.3 //  /4  /3  /6  /3 // //  s=3 pixels s=7 pixels Discrete approximation

4. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.4 //  /4  /3  /6  /3 // // , where Removing the center pixel:

5. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.5  /4 s=3 s=5  /3  /6 s=5 s=3 s=5 s=3 s=5 s=7 s=3 for  for  for  In order to obtain more robust measures, for an angle , we can average the value of along  to obtain

6. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.6 Taking the derivative along the angle  i.e., in a direction Analogously, we obtain. We now have a bank of sixteen (16) distinct oriented filters at different scales, namely { ;  and s . will have maximum response (highest value) among the possible orientations, while will have the minimum response.  / 3  

7. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.7 128 64 16 0 -16 -64 -128

8. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.8 255 128 32 0 -32 -128 -255

9. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.9  /3  /6 Corners and Junctions will respond to high values of at distant locations from the center. The maximum responses here among the different  are and  

10. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.10 The Hessian allows for computing the second derivative. In order to compute the (second) derivative along the direction  of a (first) derivative along a direction  we compute the projections For example, for we obtain and for we obtain Measurements: 2 nd Derivatives

11. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.11 In order to compute the (second) derivative along a direction  of a (first) derivative along a direction  we obtain a continuous and a lattice approximation as   x y Measurements: 2 nd Derivatives (cont.)

12. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.12 Features such as edges, corners, junction, and eyes are obtained by making some decision from the image measurements. Decisions are the result of some comparison followed by a choice. Examples: (i) if a measurement is above a threshold, we accept; otherwise, we reject; (ii) if a measurement is the largest compared to others, we select it. Image Features – Decisions!

13. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.13 Edge threshold: Decision! Edge orientation: Decision! A step edge at  The value is (equally) large for both  and  as shown (in red) for the scale s = 3 pixels. It is also large for values  not shown  However, the quantity is significantly larger for  Decisions: Edgels (Edge-pixels and Orientation)

14. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.14 The gray level indicates the angle: the darkest one is 0 degrees. The larger the angle, the lighter its color, up to  Strength of the Edgel Edgel(x, y) if else Nil Decisions: Edgels (cont.)

15. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.15 Decisions: Connecting Edgels (Pseudocode)  max ( x c, y c )

16. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.16 Decisions: Connecting Edgels (Pseudocode)

17. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.17 We have considered at least four parameters: How to estimate them? One technique is Histogram partitioning: plot the histogram and find the parameter that “best partitions it”. Threshold Parameters: Estimation

18. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.18 Angle change y x A contour segment is the contour curvature multiplied by the arc length, where Decisions: Local Angle Change

19. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.19 Curvature of Isocontour

20. January 19, 2006Computer Vision © 2006 Davi GeigerLecture 1.20 Curvature of Isocontour (cont.)